diff --git a/404.html b/404.html
index c65f52b3..14b5da7e 100644
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@@ -6,7 +6,7 @@
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data-v-b05232f3>Problems & Solutions</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/02-20" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 02-20</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-2</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_chemistry_notes_12-5" data-v-c5936a1e><div><h1 id="_12-5-reaction-mechanism" tabindex="-1">12-5: Reaction Mechanism <a class="header-anchor" href="#_12-5-reaction-mechanism" aria-hidden="true">#</a></h1><h2 id="_12-5-1-learning-objectives" tabindex="-1">12-5-1: Learning Objectives <a class="header-anchor" href="#_12-5-1-learning-objectives" aria-hidden="true">#</a></h2><div class="tip custom-block"><p class="custom-block-title">Learning Objectives</p><p>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.</p></div><p>In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><msub><mi mathvariant="normal">C</mi><mn>8</mn></msub><msub><mi mathvariant="normal">H</mi><mn>18</mn></msub><mo stretchy="false">(</mo><mi mathvariant="normal">l</mi><mo stretchy="false">)</mo><mo>+</mo><mn>25</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><mtext> </mtext><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">)</mo><mo>⟶</mo><mn>16</mn><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><mtext> </mtext><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">)</mo><mo>+</mo><mn>18</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi mathvariant="normal">g</mi><mo stretchy="false">)</mo></mrow><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}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">18</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathrm">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">25</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm" style="margin-right:0.01389em;">g</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">16</span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm" style="margin-right:0.01389em;">g</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">18</span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathrm">O</span><span class="mopen">(</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>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.</p><p>The overall sequence of elementary reactions is the mechanism of the reaction.</p><h2 id="_12-5-2-molecularity-and-the-rate-determining-step" tabindex="-1">12-5-2: Molecularity and the Rate-Determining Step <a class="header-anchor" href="#_12-5-2-molecularity-and-the-rate-determining-step" aria-hidden="true">#</a></h2><p>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.</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">N</mi><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo>→</mo><mn>2</mn><mi mathvariant="normal">C</mi><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{2CO + 2NO_2 \rightarrow 2CO_2 + N_2O_4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">2CO</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">2N</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">2C</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with a molecule of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> 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:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>The fact that the reaction is second order in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[\mathrm{NO}_{2}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> and independent of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{CO}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span></span></span></span> tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_{2}\right][\mathrm{CO}] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><p>The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:</p><h3 id="_15-2-1-two-step-mechanism" tabindex="-1">15-2-1: Two-Step Mechanism <a class="header-anchor" href="#_15-2-1-two-step-mechanism" aria-hidden="true">#</a></h3><table><thead><tr><th>Steps</th><th>Reaction</th><th>Reaction Type</th></tr></thead><tbody><tr><td>step 1</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo><mover><mo><mo>⟶</mo></mo><mtext> slow </mtext></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}+\mathrm{NO}_{2} \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}_{3}+\mathrm{NO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.4471em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2971em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">⟶</span></span></span><span style="top:-3.711em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight"> slow </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span></span></span></span></td><td>elementary reaction</td></tr><tr><td>step 2</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo>→</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0333em;vertical-align:-0.35em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.69em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span></span></td><td>elementary reaction</td></tr><tr><td>sum</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo>→</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{NO}+\mathrm{CO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td>overall reaction</td></tr></tbody></table><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO_3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> 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.</p><p>The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.</p><h2 id="_12-5-3-using-molecularity-to-describe-a-rate-law" tabindex="-1">12-5-3: Using Molecularity to Describe a Rate Law <a class="header-anchor" href="#_12-5-3-using-molecularity-to-describe-a-rate-law" aria-hidden="true">#</a></h2><p>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!)</p><div class="warning custom-block"><p class="custom-block-title">About the image</p><p>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)</p></div><p>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 ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\mathrm{A} \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span> products) is</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k[A] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B} \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span> products, the general rate law is</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k[A][B] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><table><thead><tr><th>Elementary Reaction</th><th>Molecularity</th><th>Rate</th><th>Reaction Order</th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Unimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span></span></span></span></td><td>first</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">2\mathrm{~A} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">2</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Bimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td>second</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Bimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}][\mathrm{B}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">B</span><span class="mclose">]</span></span></span></span></td><td>second</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">2 \mathrm{~A}+\mathrm{B} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Termolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">B</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]^2[\mathrm{~B}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">B</span></span><span class="mclose">]</span></span></span></span></td><td>third</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>+</mo><mi mathvariant="normal">C</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Termolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">B</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">C</span><span class="mclose">]</span></span></span></span></td><td>third</td></tr></tbody></table><blockquote><p>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).</p></blockquote><h2 id="_12-5-4-identifying-the-rate-determining-step" tabindex="-1">12-5-4: Identifying the Rate-Determining Step <a class="header-anchor" href="#_12-5-4-identifying-the-rate-determining-step" aria-hidden="true">#</a></h2><p>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.</p><p>Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.</p><p>Rate laws for each elementary reaction in our example as well as for the overall reaction</p><table><thead><tr><th>Steps</th><th>Reaction</th><th>Rate</th></tr></thead><tbody><tr><td>step 1</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo><mover><mo><mo>→</mo></mo><msub><mi mathvariant="normal">k</mi><mn>1</mn></msub></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{NO}_2+\mathrm{NO}_2 \stackrel{\mathrm{k}_1}{\rightarrow} \mathrm{NO}_3+\mathrm{NO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3141em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.578em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mtext> predicted) </mtext></mrow><annotation encoding="application/x-tex">\text { rate }=k_1\left[\mathrm{NO}_2\right]^2(\text { predicted) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord"> predicted) </span></span></span></span></span></td></tr><tr><td>step 2</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo><mover><mo><mo>→</mo></mo><msub><mi>k</mi><mn>2</mn></msub></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\mathrm{NO}_3+\mathrm{CO} \stackrel{k_2}{\rightarrow} \mathrm{NO}_2+\mathrm{CO}_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5141em;vertical-align:-0.35em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-2.8541em;"><span class="pstrut" style="height:3.1641em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.1641em;"><span class="pstrut" style="height:3.1641em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.578em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mtext> predicted </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k_2\left[\mathrm{NO}_3\right][\mathrm{CO}](\text { predicted })</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span><span class="mopen">(</span><span class="mord text"><span class="mord"> predicted </span></span><span class="mclose">)</span></span></span></span></td></tr><tr><td>step 3</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo><mover><mo><mo>→</mo></mo><mi>k</mi></mover></mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_2+\mathrm{CO} \stackrel{k}{\rightarrow} \mathrm{NO}+\mathrm{CO}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mtext> observed) </mtext></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_2\right]^2(\text { observed) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord"> observed) </span></span></span></span></span></td></tr></tbody></table><p>The experimentally determined rate law for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">C O</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">CO</span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> for the overall reaction must equal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">k_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. That is, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.</p><p>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.</p><h3 id="_12-5-4-1-example-a-reaction-with-an-intermediate" tabindex="-1">12-5-4-1: Example-A Reaction with an Intermediate <a class="header-anchor" href="#_12-5-4-1-example-a-reaction-with-an-intermediate" aria-hidden="true">#</a></h3><p>In an alternative mechanism for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> appearing as an intermediate.</p><p>alternative mechanism for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> appearing as an intermediate.</p><p>Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mrow><mi mathvariant="normal">k</mi><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mi mathvariant="normal">t</mi></mrow></mrow><annotation encoding="application/x-tex">=\mathrm{k[{NO}_{2}]^{2}t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">k</span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathrm">t</span></span></span></span></span>)</p><p>Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law</p><h4 id="strategy" tabindex="-1">Strategy <a class="header-anchor" href="#strategy" aria-hidden="true">#</a></h4><ul><li><p>Determine the rate law for each elementary reaction in the reaction.</p></li><li><p>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.</p></li></ul><h4 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h4><details class="details custom-block"><summary>View solution</summary><p>A The rate law for step 1 is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">=k_{1}\left[\mathrm{NO}_{2}\right]^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>; for step 2 , it is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo fence="true">[</mo><msub><mrow><mtext> </mtext><mi mathvariant="normal">N</mi></mrow><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">=k_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{4}\right][\mathrm{CO}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">N</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span></span></span></span>.</p><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">k_{1}\left[\mathrm{NO}_{2}\right]^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>. This is the same as the experimentally determined rate law. Hence this mechanism, with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as an intermediate, and the one described previously, with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, directly.</p></details></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/notes/12-5.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><!----></div><div class="pager" data-v-e033cd21><a class="pager-link next" href="/academic/chemistry/problems/02-20" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Problem: 02-20</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>Problems & Solutions</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/02-20" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 02-20</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_chemistry_notes_12-5" data-v-c5936a1e><div><h1 id="_12-5-reaction-mechanism" tabindex="-1">12-5: Reaction Mechanism <a class="header-anchor" href="#_12-5-reaction-mechanism" aria-hidden="true">#</a></h1><h2 id="_12-5-1-learning-objectives" tabindex="-1">12-5-1: Learning Objectives <a class="header-anchor" href="#_12-5-1-learning-objectives" aria-hidden="true">#</a></h2><div class="tip custom-block"><p class="custom-block-title">Learning Objectives</p><p>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.</p></div><p>In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and water:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><msub><mi mathvariant="normal">C</mi><mn>8</mn></msub><msub><mi mathvariant="normal">H</mi><mn>18</mn></msub><mo stretchy="false">(</mo><mi mathvariant="normal">l</mi><mo stretchy="false">)</mo><mo>+</mo><mn>25</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><mtext> </mtext><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">)</mo><mo>⟶</mo><mn>16</mn><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><mtext> </mtext><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">)</mo><mo>+</mo><mn>18</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi mathvariant="normal">g</mi><mo stretchy="false">)</mo></mrow><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}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">18</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathrm">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">25</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm" style="margin-right:0.01389em;">g</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">16</span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm" style="margin-right:0.01389em;">g</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">18</span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathrm">O</span><span class="mopen">(</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>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.</p><p>The overall sequence of elementary reactions is the mechanism of the reaction.</p><h2 id="_12-5-2-molecularity-and-the-rate-determining-step" tabindex="-1">12-5-2: Molecularity and the Rate-Determining Step <a class="header-anchor" href="#_12-5-2-molecularity-and-the-rate-determining-step" aria-hidden="true">#</a></h2><p>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.</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">N</mi><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo>→</mo><mn>2</mn><mi mathvariant="normal">C</mi><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{2CO + 2NO_2 \rightarrow 2CO_2 + N_2O_4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">2CO</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">2N</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">2C</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>From the balanced chemical equation, one might expect the reaction to occur via a collision of one molecule of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with a molecule of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> 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:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_{2}\right]^{2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>The fact that the reaction is second order in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[\mathrm{NO}_{2}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> and independent of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{CO}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span></span></span></span> tells us that it does not occur by the simple collision model outlined previously. If it did, its predicted rate law would be</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_{2}\right][\mathrm{CO}] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><p>The following two-step mechanism is consistent with the rate law if step 1 is much slower than step 2:</p><h3 id="_15-2-1-two-step-mechanism" tabindex="-1">15-2-1: Two-Step Mechanism <a class="header-anchor" href="#_15-2-1-two-step-mechanism" aria-hidden="true">#</a></h3><table><thead><tr><th>Steps</th><th>Reaction</th><th>Reaction Type</th></tr></thead><tbody><tr><td>step 1</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo><mover><mo><mo>⟶</mo></mo><mtext> slow </mtext></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}+\mathrm{NO}_{2} \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}_{3}+\mathrm{NO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.4471em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2971em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">⟶</span></span></span><span style="top:-3.711em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight"> slow </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span></span></span></span></td><td>elementary reaction</td></tr><tr><td>step 2</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo>→</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\mathrm{NO}_{3}+\mathrm{CO} \rightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0333em;vertical-align:-0.35em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.69em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span></span></td><td>elementary reaction</td></tr><tr><td>sum</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo>→</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}+\mathrm{CO} \rightarrow \mathrm{NO}+\mathrm{CO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td>overall reaction</td></tr></tbody></table><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO_3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> 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.</p><p>The sum of the elementary reactions in a reaction mechanism must give the overall balanced chemical equation of the reaction.</p><h2 id="_12-5-3-using-molecularity-to-describe-a-rate-law" tabindex="-1">12-5-3: Using Molecularity to Describe a Rate Law <a class="header-anchor" href="#_12-5-3-using-molecularity-to-describe-a-rate-law" aria-hidden="true">#</a></h2><p>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!)</p><div class="warning custom-block"><p class="custom-block-title">About the image</p><p>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)</p></div><p>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 ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\mathrm{A} \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span> products) is</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k[A] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B} \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span> products, the general rate law is</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>B</mi><mo stretchy="false">]</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\text { rate }=k[A][B] . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathnormal">A</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">]</span><span class="mord">.</span></span></span></span></span></p><table><thead><tr><th>Elementary Reaction</th><th>Molecularity</th><th>Rate</th><th>Reaction Order</th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Unimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span></span></span></span></td><td>first</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">2\mathrm{~A} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">2</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Bimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td>second</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Bimolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}][\mathrm{B}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">B</span><span class="mclose">]</span></span></span></span></td><td>second</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>+</mo><mi mathvariant="normal">B</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">2 \mathrm{~A}+\mathrm{B} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Termolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">B</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}]^2[\mathrm{~B}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">B</span></span><span class="mclose">]</span></span></span></span></td><td>third</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mo>+</mo><mi mathvariant="normal">B</mi><mo>+</mo><mi mathvariant="normal">C</mi><mo>→</mo><mtext> products </mtext></mrow><annotation encoding="application/x-tex">\mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow \text { products }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathrm">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> products </span></span></span></span></span></td><td>Termolecular</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi mathvariant="normal">C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k[\mathrm{~A}][\mathrm{B}][\mathrm{C}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">B</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathrm">C</span><span class="mclose">]</span></span></span></span></td><td>third</td></tr></tbody></table><blockquote><p>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).</p></blockquote><h2 id="_12-5-4-identifying-the-rate-determining-step" tabindex="-1">12-5-4: Identifying the Rate-Determining Step <a class="header-anchor" href="#_12-5-4-identifying-the-rate-determining-step" aria-hidden="true">#</a></h2><p>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.</p><p>Look at the rate laws for each elementary reaction in the example as well as for the overall reaction.</p><p>Rate laws for each elementary reaction in our example as well as for the overall reaction</p><table><thead><tr><th>Steps</th><th>Reaction</th><th>Rate</th></tr></thead><tbody><tr><td>step 1</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo><mover><mo><mo>→</mo></mo><msub><mi mathvariant="normal">k</mi><mn>1</mn></msub></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{NO}_2+\mathrm{NO}_2 \stackrel{\mathrm{k}_1}{\rightarrow} \mathrm{NO}_3+\mathrm{NO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3141em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.578em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mtext> predicted) </mtext></mrow><annotation encoding="application/x-tex">\text { rate }=k_1\left[\mathrm{NO}_2\right]^2(\text { predicted) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord"> predicted) </span></span></span></span></span></td></tr><tr><td>step 2</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo><mover><mo><mo>→</mo></mo><msub><mi>k</mi><mn>2</mn></msub></mover></mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\mathrm{NO}_3+\mathrm{CO} \stackrel{k_2}{\rightarrow} \mathrm{NO}_2+\mathrm{CO}_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5141em;vertical-align:-0.35em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-2.8541em;"><span class="pstrut" style="height:3.1641em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.1641em;"><span class="pstrut" style="height:3.1641em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1641em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.578em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mtext> predicted </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text { rate }=k_2\left[\mathrm{NO}_3\right][\mathrm{CO}](\text { predicted })</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span><span class="mopen">(</span><span class="mord text"><span class="mord"> predicted </span></span><span class="mclose">)</span></span></span></span></td></tr><tr><td>step 3</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo>+</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo><mover><mo><mo>→</mo></mo><mi>k</mi></mover></mo><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mo>+</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_2+\mathrm{CO} \stackrel{k}{\rightarrow} \mathrm{NO}+\mathrm{CO}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">→</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">NO</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mi>k</mi><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mtext> observed) </mtext></mrow><annotation encoding="application/x-tex">\text { rate }=k\left[\mathrm{NO}_2\right]^2(\text { observed) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord"> observed) </span></span></span></span></span></td></tr></tbody></table><p>The experimentally determined rate law for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">C O</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">CO</span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> for the overall reaction must equal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">k_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. That is, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is formed slowly in step 1, but once it is formed, it reacts very rapidly with CO in step 2.</p><p>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.</p><h3 id="_12-5-4-1-example-a-reaction-with-an-intermediate" tabindex="-1">12-5-4-1: Example-A Reaction with an Intermediate <a class="header-anchor" href="#_12-5-4-1-example-a-reaction-with-an-intermediate" aria-hidden="true">#</a></h3><p>In an alternative mechanism for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> appearing as an intermediate.</p><p>alternative mechanism for the reaction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\mathrm{CO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm">CO</span></span></span></span></span> with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> appearing as an intermediate.</p><p>Write the rate law for each elementary reaction. Is this mechanism consistent with the experimentally determined rate law (rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mrow><mi mathvariant="normal">k</mi><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mi mathvariant="normal">t</mi></mrow></mrow><annotation encoding="application/x-tex">=\mathrm{k[{NO}_{2}]^{2}t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">k</span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">2</span></span></span></span></span></span></span></span></span><span class="mord mathrm">t</span></span></span></span></span>)</p><p>Given: elementary reactions Asked for: rate law for each elementary reaction and overall rate law</p><h4 id="strategy" tabindex="-1">Strategy <a class="header-anchor" href="#strategy" aria-hidden="true">#</a></h4><ul><li><p>Determine the rate law for each elementary reaction in the reaction.</p></li><li><p>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.</p></li></ul><h4 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h4><details class="details custom-block"><summary>View solution</summary><p>A The rate law for step 1 is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">=k_{1}\left[\mathrm{NO}_{2}\right]^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>; for step 2 , it is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo fence="true">[</mo><msub><mrow><mtext> </mtext><mi mathvariant="normal">N</mi></mrow><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub><mo fence="true">]</mo></mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">=k_{2}\left[\mathrm{~N}_{2} \mathrm{O}_{4}\right][\mathrm{CO}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">N</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">CO</span></span><span class="mclose">]</span></span></span></span>.</p><p>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><msup><mrow><mo fence="true">[</mo><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">k_{1}\left[\mathrm{NO}_{2}\right]^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>. This is the same as the experimentally determined rate law. Hence this mechanism, with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as an intermediate, and the one described previously, with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{NO}_{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_{2} \mathrm{O}_{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, directly.</p></details></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/notes/12-5.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><!----></div><div class="pager" data-v-e033cd21><a class="pager-link next" href="/academic/chemistry/problems/02-20" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Problem: 02-20</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 02-20</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-2</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_chemistry_problems_02-20" data-v-c5936a1e><div><h1 id="presentation-problem-02-20" tabindex="-1">Presentation problem: 02-20 <a class="header-anchor" href="#presentation-problem-02-20" aria-hidden="true">#</a></h1><h2 id="question" tabindex="-1">Question <a class="header-anchor" href="#question" aria-hidden="true">#</a></h2><p>At <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>500</mn><mrow><mtext> </mtext><mi mathvariant="normal">K</mi></mrow></mrow><annotation encoding="application/x-tex">500 \mathrm{~K}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">500</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">K</span></span></span></span></span> in the presence of a copper surface, ethanol decomposes according to the equation</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">g</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">H</mi></mrow><mn>3</mn></msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><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) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mopen">(</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">CHO</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The pressure of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> was measured as a function of time and the following data were obtained:</p><table><thead><tr><th>Time (s)</th><th><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow></msub><mtext> (torr) </mtext></mrow><annotation encoding="application/x-tex">P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathrm mtight">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathrm mtight">OH</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> (torr) </span></span></span></span></span></th></tr></thead><tbody><tr><td>0</td><td>250.</td></tr><tr><td>100.</td><td>237</td></tr><tr><td>200.</td><td>224</td></tr><tr><td>300.</td><td>211</td></tr><tr><td>400.</td><td>198</td></tr><tr><td>500.</td><td>185</td></tr></tbody></table><p><strong>Since the pressure of a gas is directly proportional to the concentration of gas</strong>, 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> after <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>900.</mn><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">900 . \mathrm{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">900.</span><span class="mord mathrm">s</span></span></span></span> from the start of the reaction. (Hint: To determine the order of the reaction with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span>, compare how the pressure of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> decreases with each time listing.)</p><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/chemistry/solution1.png" alt="solution graph" data-zoomable=""></p><p>Due to the fact that the graph of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{p}\left[\mathrm{O}_2\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> over time is showing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathrm{R}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathrm">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> value of 1 we know we have a zero-order reaction. Therefore:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">k</mi><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">p</mi><mn>0</mn></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">]</mo></mrow><mo>−</mo><mi mathvariant="normal">p</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">]</mo></mrow></mrow><mi mathvariant="normal">t</mi></mfrac><mo>=</mo><mfrac><mrow><mn>250</mn><mtext> torr </mtext><mo>−</mo><mn>185</mn><mtext> torr </mtext></mrow><mrow><mn>500</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow></mrow></mfrac><mo>=</mo><mn>0.13</mn><msup><mtext> torr s </mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">k</mi><mo>=</mo><mn>0.13</mn><mtext> torr s </mtext><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mfrac><mrow><mn>1</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow></mrow><mrow><mn>760</mn><mtext> torr </mtext></mrow></mfrac><mo>=</mo><mn>1.71</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> rate </mtext><mo>=</mo><mi mathvariant="bold">k</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">t</mi></mrow><mo>+</mo><msub><mi mathvariant="normal">p</mi><mn>0</mn></msub><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo fence="true">(</mo><mn>1.71</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo fence="true">)</mo></mrow><mo>×</mo><mn>900</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo>+</mo><mn>0.33</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mn>0.176</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \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} \\ &amp; \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} \\ &amp; \text { rate }=\mathbf{k} \\ &amp; \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) \\ &amp; \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} \\ &amp; \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.2687em;vertical-align:-5.8843em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.3843em;"><span style="top:-8.3843em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-6.0769em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-4.2509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-2.7509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-1.2268em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:0.2973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:1.7973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.8843em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.3843em;"><span style="top:-8.3843em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">500</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">250</span><span class="mord text"><span class="mord"> torr </span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">185</span><span class="mord text"><span class="mord"> torr </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.13</span><span class="mord"><span class="mord text"><span class="mord"> torr s </span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.0769em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.13</span><span class="mord text"><span class="mord"> torr s </span></span><span class="mord"><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">760</span><span class="mord text"><span class="mord"> torr </span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1.71</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.2509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathbf">k</span></span></span><span style="top:-2.7509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathrm">kt</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-1.2268em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord">1.71</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">900</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.33</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span></span></span><span style="top:0.2973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.176</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span 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03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 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\mathrm{~K}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">500</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">K</span></span></span></span></span> in the presence of a copper surface, ethanol decomposes according to the equation</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">g</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">H</mi></mrow><mn>3</mn></msub><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">O</mi></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><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) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mopen">(</span><span class="mord mathrm" style="margin-right:0.01389em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">CH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">CHO</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The pressure of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> was measured as a function of time and the following data were obtained:</p><table><thead><tr><th>Time (s)</th><th><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow></msub><mtext> (torr) </mtext></mrow><annotation encoding="application/x-tex">P_{\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} \text { (torr) }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathrm mtight">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathrm mtight">OH</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> (torr) </span></span></span></span></span></th></tr></thead><tbody><tr><td>0</td><td>250.</td></tr><tr><td>100.</td><td>237</td></tr><tr><td>200.</td><td>224</td></tr><tr><td>300.</td><td>211</td></tr><tr><td>400.</td><td>198</td></tr><tr><td>500.</td><td>185</td></tr></tbody></table><p><strong>Since the pressure of a gas is directly proportional to the concentration of gas</strong>, 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> after <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>900.</mn><mi mathvariant="normal">s</mi></mrow><annotation encoding="application/x-tex">900 . \mathrm{s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">900.</span><span class="mord mathrm">s</span></span></span></span> from the start of the reaction. (Hint: To determine the order of the reaction with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span>, compare how the pressure of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow></mrow><annotation encoding="application/x-tex">\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span></span></span></span> decreases with each time listing.)</p><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/chemistry/solution1.png" alt="solution graph" data-zoomable=""></p><p>Due to the fact that the graph of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{p}\left[\mathrm{O}_2\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> over time is showing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathrm{R}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathrm">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> value of 1 we know we have a zero-order reaction. Therefore:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">k</mi><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">p</mi><mn>0</mn></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">]</mo></mrow><mo>−</mo><mi mathvariant="normal">p</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">]</mo></mrow></mrow><mi mathvariant="normal">t</mi></mfrac><mo>=</mo><mfrac><mrow><mn>250</mn><mtext> torr </mtext><mo>−</mo><mn>185</mn><mtext> torr </mtext></mrow><mrow><mn>500</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow></mrow></mfrac><mo>=</mo><mn>0.13</mn><msup><mtext> torr s </mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">k</mi><mo>=</mo><mn>0.13</mn><mtext> torr s </mtext><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mfrac><mrow><mn>1</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow></mrow><mrow><mn>760</mn><mtext> torr </mtext></mrow></mfrac><mo>=</mo><mn>1.71</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> rate </mtext><mo>=</mo><mi mathvariant="bold">k</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">t</mi></mrow><mo>+</mo><msub><mi mathvariant="normal">p</mi><mn>0</mn></msub><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo fence="true">(</mo><mn>1.71</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo fence="true">)</mo></mrow><mo>×</mo><mn>900</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo>+</mo><mn>0.33</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>5</mn></msub><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mn>0.176</mn><mrow><mtext> </mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">m</mi></mrow><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \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} \\ &amp; \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} \\ &amp; \text { rate }=\mathbf{k} \\ &amp; \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) \\ &amp; \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} \\ &amp; \mathrm{p}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)=0.176 \mathrm{~atm} \mathrm{~s}^{-1} \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.2687em;vertical-align:-5.8843em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.3843em;"><span style="top:-8.3843em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-6.0769em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-4.2509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-2.7509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-1.2268em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:0.2973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:1.7973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.8843em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.3843em;"><span style="top:-8.3843em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">500</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">250</span><span class="mord text"><span class="mord"> torr </span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">185</span><span class="mord text"><span class="mord"> torr </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.13</span><span class="mord"><span class="mord text"><span class="mord"> torr s </span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.0769em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.13</span><span class="mord text"><span class="mord"> torr s </span></span><span class="mord"><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">760</span><span class="mord text"><span class="mord"> torr </span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1.71</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.2509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathbf">k</span></span></span><span style="top:-2.7509em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathrm">kt</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathrm">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-1.2268em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord">1.71</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">900</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.33</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span></span></span><span style="top:0.2973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mord mathrm">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.176</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">atm</span></span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.7973em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.8843em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div 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xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><msub><mrow><mtext> </mtext><mi mathvariant="normal">N</mi></mrow><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mn>4</mn><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">N</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The decomposition of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> is represented by the equation above. A sample of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> is monitored as it decomposes, and the concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as a function of time is recorded. The results are shown in the table below.</p><table><thead><tr><th>Time (s)</th><th><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{[N_2O_5]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></th></tr></thead><tbody><tr><td>0</td><td>1.000</td></tr><tr><td>25.0</td><td>0.801</td></tr><tr><td>50.0</td><td>0.642</td></tr><tr><td>75.0</td><td>0.515</td></tr></tbody></table><p>Calculate the average rate of the reaction between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds.</p><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>+</mo><mi>b</mi><mrow><mtext> </mtext><mi mathvariant="normal">B</mi></mrow><mo>→</mo><mi>c</mi><mi mathvariant="normal">C</mi><mo>+</mo><mi>d</mi><mi mathvariant="normal">D</mi></mrow><annotation encoding="application/x-tex">a \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">B</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">c</span><span class="mord mathrm">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathrm">D</span></span></span></span></span></p><p>the rate of reaction is defined as</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathcolor="cyan"><mtext> rate </mtext><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">A</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>b</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">C</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>+</mo><mfrac><mn>1</mn><mi>d</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">D</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></mstyle></mrow><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text" style="color:cyan;"><span class="mord" style="color:cyan;"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">−</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">A</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">−</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">B</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">+</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">C</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">+</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">D</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b, c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span>, or <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> ). 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. With that in mind, let&#39;s write the reaction rate in terms of the rate of change in concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo fence="true">]</mo></mrow></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Since the coefficient for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in the balanced equation is 2 , we divided the rate of change in concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> by 2 . Additionally, since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is being consumed in the reaction, we included a negative sign in front of the expression.</p><p>Now, let&#39;s plug in the information from the table to calculate the average reaction rate between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext> rate </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mo stretchy="false">(</mo><mn>0.515</mn><mi>M</mi><mo>−</mo><mn>0.642</mn><mi>M</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>75.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo>−</mo><mn>50.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2.54</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mi>M</mi><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \text { rate } &amp; =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ &amp; =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.1871em;vertical-align:-1.8436em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3436em;"><span style="top:-4.3436em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord text"><span class="mord"> rate </span></span></span></span><span style="top:-2.2434em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8436em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3436em;"><span style="top:-4.3436em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">75.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">50.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0.515</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.642</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2434em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2.54</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8436em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>So, the average rate of the reaction between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">2.54</mn><mo>×</mo><mn mathvariant="bold">1</mn><msup><mn mathvariant="bold">0</mn><mrow><mo>−</mo><mn mathvariant="bold">3</mn></mrow></msup><mi mathvariant="normal">M</mi><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn mathvariant="bold">1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathbf">2.54</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">1</span><span class="mord"><span class="mord mathbf">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathbf mtight">3</span></span></span></span></span></span></span></span></span><span class="mord mathrm">M</span><span class="mord"><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span 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data-v-b05232f3>Problems & Solutions</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/02-20" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 02-20</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img 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mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mn>4</mn><msub><mrow><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 \mathrm{~N}_2 \mathrm{O}_5(g) \rightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">N</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord"><span class="mord"><span class="mord mathrm">NO</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The decomposition of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> is represented by the equation above. A sample of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span> is monitored as it decomposes, and the concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> as a function of time is recorded. The results are shown in the table below.</p><table><thead><tr><th>Time (s)</th><th><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{[N_2O_5]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></th></tr></thead><tbody><tr><td>0</td><td>1.000</td></tr><tr><td>25.0</td><td>0.801</td></tr><tr><td>50.0</td><td>0.642</td></tr><tr><td>75.0</td><td>0.515</td></tr></tbody></table><p>Calculate the average rate of the reaction between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds.</p><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mrow><mtext> </mtext><mi mathvariant="normal">A</mi></mrow><mo>+</mo><mi>b</mi><mrow><mtext> </mtext><mi mathvariant="normal">B</mi></mrow><mo>→</mo><mi>c</mi><mi mathvariant="normal">C</mi><mo>+</mo><mi>d</mi><mi mathvariant="normal">D</mi></mrow><annotation encoding="application/x-tex">a \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">A</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">B</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">c</span><span class="mord mathrm">C</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathrm">D</span></span></span></span></span></p><p>the rate of reaction is defined as</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathcolor="cyan"><mtext> rate </mtext><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">A</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>b</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">B</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">C</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>+</mo><mfrac><mn>1</mn><mi>d</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">[</mo><mi mathvariant="normal">D</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></mstyle></mrow><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text" style="color:cyan;"><span class="mord" style="color:cyan;"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">−</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">A</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">−</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">B</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">+</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">C</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:cyan;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord" style="color:cyan;">+</span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord mathnormal" style="color:cyan;">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord" style="color:cyan;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mord mathnormal" style="color:cyan;">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="color:cyan;border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:cyan;"><span class="mord" style="color:cyan;">Δ</span><span class="mopen" style="color:cyan;">[</span><span class="mord mathrm" style="color:cyan;">D</span><span class="mclose" style="color:cyan;">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Notice that the rate of change in concentration of each species is divided by its coefficient from the balanced chemical equation ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b, c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span>, or <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> ). 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. With that in mind, let&#39;s write the reaction rate in terms of the rate of change in concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> rate </mtext><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub><mo fence="true">]</mo></mrow></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text { rate }=-\frac{1}{2} \frac{\Delta\left[\mathrm{N}_2 \mathrm{O}_5\right]}{\Delta t} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> rate </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Since the coefficient for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> in the balanced equation is 2 , we divided the rate of change in concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> by 2 . Additionally, since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{N}_2 \mathrm{O}_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is being consumed in the reaction, we included a negative sign in front of the expression.</p><p>Now, let&#39;s plug in the information from the table to calculate the average reaction rate between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext> rate </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mo stretchy="false">(</mo><mn>0.515</mn><mi>M</mi><mo>−</mo><mn>0.642</mn><mi>M</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>75.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo>−</mo><mn>50.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2.54</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mi>M</mi><msup><mrow><mtext> </mtext><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \text { rate } &amp; =-\frac{1}{2} \frac{(0.515 M-0.642 M)}{(75.0 \mathrm{~s}-50.0 \mathrm{~s})} \\ &amp; =2.54 \times 10^{-3} M \mathrm{~s}^{-1} \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.1871em;vertical-align:-1.8436em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3436em;"><span style="top:-4.3436em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord text"><span class="mord"> rate </span></span></span></span><span style="top:-2.2434em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8436em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3436em;"><span style="top:-4.3436em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">75.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">50.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0.515</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">0.642</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2434em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2.54</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8436em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>So, the average rate of the reaction between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50.0</mn></mrow><annotation encoding="application/x-tex">50.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">50.0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75.0</mn></mrow><annotation encoding="application/x-tex">75.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">75.0</span></span></span></span> seconds is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">2.54</mn><mo>×</mo><mn mathvariant="bold">1</mn><msup><mn mathvariant="bold">0</mn><mrow><mo>−</mo><mn mathvariant="bold">3</mn></mrow></msup><mi mathvariant="normal">M</mi><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn mathvariant="bold">1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\bold{2.54 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathbf">2.54</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">1</span><span class="mord"><span class="mord mathbf">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathbf mtight">3</span></span></span></span></span></span></span></span></span><span class="mord mathrm">M</span><span class="mord"><span class="mord mathrm">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathbf mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/problems/03-02-1.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div 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xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">X</mi><mi mathvariant="normal">Y</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">=k[\mathrm{XY}]^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. In an experiment, the initial rate of the reaction is determined to be <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span> when the initial concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{XY}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span></span></span></span> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mi mathvariant="normal">L</mi></mrow><annotation encoding="application/x-tex">0.40 \mathrm{~mol} / \mathrm{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mord mathrm">L</span></span></span></span>.</p><ul><li>What is the value of the rate constant, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>, for the reaction? <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.10</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.10</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2.5</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2.5</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li></ul></li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p>lo tind the value of the rate constant for the reaction, let&#39;s first solve the rate law for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>k</mi><mo>=</mo><mfrac><mtext> rate </mtext><mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">X</mi><mi mathvariant="normal">Y</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">k=\frac{\text { rate }}{[\mathrm{XY}]^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2281em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2921em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> rate </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Next, let&#39;s plug in the initial rate and concentration given in the text:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>k</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mi mathvariant="normal">L</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><mrow><mn>0.16</mn><msup><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} k &amp; =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ &amp; =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ &amp; =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.9284em;vertical-align:-3.2142em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.7142em;"><span style="top:-5.7142em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0512em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-0.8728em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.2142em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.7142em;"><span style="top:-5.7142em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mord mathrm">L</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0512em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.2116em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathrm">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0384em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.8728em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.2142em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>So, the value of the rate constant for the reaction is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/problems/03-02-2.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 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mathvariant="normal">Y</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">=k[\mathrm{XY}]^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. In an experiment, the initial rate of the reaction is determined to be <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span> when the initial concentration of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{XY}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span></span></span></span> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mi mathvariant="normal">L</mi></mrow><annotation encoding="application/x-tex">0.40 \mathrm{~mol} / \mathrm{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mord mathrm">L</span></span></span></span>.</p><ul><li>What is the value of the rate constant, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>, for the reaction? <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.10</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.10 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.10</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0.40 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2.5</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2.5 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2.5</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></li></ul></li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p>lo tind the value of the rate constant for the reaction, let&#39;s first solve the rate law for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>k</mi><mo>=</mo><mfrac><mtext> rate </mtext><mrow><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">X</mi><mi mathvariant="normal">Y</mi></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">k=\frac{\text { rate }}{[\mathrm{XY}]^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2281em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2921em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathrm" style="margin-right:0.025em;">XY</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> rate </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>Next, let&#39;s plug in the initial rate and concentration given in the text:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>k</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0.40</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mi mathvariant="normal">L</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mn>0.16</mn><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi mathvariant="normal">L</mi><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><mrow><mn>0.16</mn><msup><mrow><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} k &amp; =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{(0.40 \mathrm{~mol} / \mathrm{L})^2} \\ &amp; =\frac{0.16 \mathrm{~mol} /(\mathrm{L} \cdot \mathrm{s})}{0.16 \mathrm{~mol}^2 / \mathrm{L}^2} \\ &amp; =1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.9284em;vertical-align:-3.2142em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.7142em;"><span style="top:-5.7142em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0512em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-0.8728em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.2142em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.7142em;"><span style="top:-5.7142em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0.40</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mord mathrm">L</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0512em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.2116em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathrm">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.16</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">mol</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathrm">L</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0384em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.8728em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.2142em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>So, the value of the rate constant for the reaction is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.0</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow><mo>⋅</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1.0 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.0</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">mol</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">s</span><span class="mclose">)</span></span></span></span></p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/problems/03-02-2.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div 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xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mi mathvariant="normal">X</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">Y</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Q</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm">X</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm">R</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The reaction represented above is found to be second order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> and first order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm" style="margin-right:0.025em;">Y</span></span></span></span>.</p><p>What happens to the rate of the reaction when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled?</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It increases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> It decreases by a factor of 2.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It decreases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It does not change.</li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p>To solve this problem, let&#39;s first write out the rate law for the reaction. According to the text, the reaction is second order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> and first order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm" style="margin-right:0.025em;">Y</span></span></span></span>, so the rate law is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">=k[\mathrm{X}]^2[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span></p><p>Now, let&#39;s think about how the rate changes when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled. Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is raised to the second power in the rate law, halving <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> decreases the reaction rate by a factor of 4 . Similarly, since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is raised to the first power, doubling <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled, the rate of the reaction decreases by a factor of 2 .</p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/problems/03-02-3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/academic/chemistry/problems/03-02-2" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Problem: 03-02-2</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-b05232f3>Problems & Solutions</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/02-20" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 02-20</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-2</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/chemistry/problems/03-02-3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Problem: 03-02-3</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img 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stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">Y</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">Q</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">R</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 \mathrm{X}(g)+2 \mathrm{Y}(g) \rightarrow \mathrm{Q}(g)+2 \mathrm{R}(g) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm">X</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathrm">R</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>The reaction represented above is found to be second order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> and first order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm" style="margin-right:0.025em;">Y</span></span></span></span>.</p><p>What happens to the rate of the reaction when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled?</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It increases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> It decreases by a factor of 2.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It decreases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It does not change.</li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2><p>To solve this problem, let&#39;s first write out the rate law for the reaction. According to the text, the reaction is second order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> and first order with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Y</mi></mrow><annotation encoding="application/x-tex">\mathrm{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm" style="margin-right:0.025em;">Y</span></span></span></span>, so the rate law is rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">=k[\mathrm{X}]^2[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span></p><p>Now, let&#39;s think about how the rate changes when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled. Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is raised to the second power in the rate law, halving <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> decreases the reaction rate by a factor of 4 . Similarly, since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is raised to the first power, doubling <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{X}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm">X</span><span class="mclose">]</span></span></span></span> is halved and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathrm{Y}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathrm" style="margin-right:0.025em;">Y</span><span class="mclose">]</span></span></span></span> is doubled, the rate of the reaction decreases by a factor of 2 .</p></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/chemistry/problems/03-02-3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/academic/chemistry/problems/03-02-2" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Problem: 03-02-2</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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of Development <a class="header-anchor" href="#patterns-of-organization-and-methods-of-development" aria-hidden="true">#</a></h1><blockquote><p>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.</p></blockquote><h2 id="major-patterns-of-organization" tabindex="-1">Major Patterns of Organization <a class="header-anchor" href="#major-patterns-of-organization" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/19875203514_543f7bff51_z-300x201.jpg" alt="A fruit pie."></p><p>Read the following sentences:</p><ul><li>Now take the pie out of the oven and let it cool on the stovetop.</li><li>Mix the dry ingredients with the liquid ingredients.</li><li>Set the pie crust aside while you make the filling.</li></ul><p>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.</p><p>First, let’s take a look at a couple of ways to think about organization.</p><h2 id="general-to-specific-or-specific-to-general" tabindex="-1">General to Specific or Specific to General <a class="header-anchor" href="#general-to-specific-or-specific-to-general" aria-hidden="true">#</a></h2><p>It might be useful to think about organizing your topic like a triangle:</p><p><img src="https://r2.toshiki.dev/image/Patterns_of_Organizaiton_Triangles.png" alt="Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General."></p><p>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.</p><p>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:</p><p>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.</p><p>An essay that uses the specific-to-general structure might start like this:</p><p>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.</p><p>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.</p><h2 id="does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" tabindex="-1">Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? <a class="header-anchor" href="#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" aria-hidden="true">#</a></h2><p>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.</p><h2 id="are-there-other-ways-to-think-about-organizing-my-ideas" tabindex="-1">Are There Other Ways to Think about Organizing My Ideas? <a class="header-anchor" href="#are-there-other-ways-to-think-about-organizing-my-ideas" aria-hidden="true">#</a></h2><p>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:</p><ul><li>Most important information first (consider what you want readers to focus on first)</li><li>Chronological order (the order in time that events take place)</li><li>Compare and contrast (ideas are organized together because of their relationship to each other)</li></ul><p>The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.</p><p>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.<br> Topics:</p><ul><li>Facing fears</li><li>Safety in sports</li><li>Community policing</li><li>Educating prisoners</li><li>Sex education</li></ul><h2 id="methods-of-development" tabindex="-1">Methods of Development <a class="header-anchor" href="#methods-of-development" aria-hidden="true">#</a></h2><p>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.</p><p>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.</p><p>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.</p><h3 id="cause-and-effect-or-effect-and-cause" tabindex="-1">Cause and Effect (or Effect and Cause) <a class="header-anchor" href="#cause-and-effect-or-effect-and-cause" aria-hidden="true">#</a></h3><p>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 <em>cause</em>, <em>effect</em>, and <em>result</em>, to cue the reader about your about the relationships that you’re establishing.</p><h3 id="problem-solution" tabindex="-1">Problem-Solution <a class="header-anchor" href="#problem-solution" aria-hidden="true">#</a></h3><p>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:</p><ul><li>Describe the problem, followed by the solution.</li><li>Propose the solution first and then describe the problems that motivated it.</li><li>Or a problem may be followed by several solutions, one of which is selected as the best.</li></ul><p>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 <em>problem</em> and <em>solution</em> to signal these sections of your paper for your reader.</p><h3 id="chronology-or-narrative" tabindex="-1">Chronology or Narrative <a class="header-anchor" href="#chronology-or-narrative" aria-hidden="true">#</a></h3><p>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 <em>then</em>, <em>next</em>, and <em>finally</em> to make the parts of the chronology clear.</p><h3 id="comparison-and-contrast" tabindex="-1">Comparison and Contrast <a class="header-anchor" href="#comparison-and-contrast" aria-hidden="true">#</a></h3><p>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 <em>by contrast</em>, <em>in comparison</em>, <em>while</em>, <em>some</em>, and <em>others</em>.</p><div class="tip custom-block"><p class="custom-block-title">Summary</p><p>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.</p></div><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li><p>This chapter was adapted from “<a href="https://openoregon.pressbooks.pub/wrd/chapter/patterns-of-organization-and-methods-of-development/" target="_blank" rel="noreferrer">Patterns of Organization and Methods of Development</a>” in <em>The Word on College Reading and Writing</em> by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>. Adapted by Allison Kilgannon.</p></li><li><p>“<a href="https://flic.kr/p/whiByE" target="_blank" rel="noreferrer">Peach and lavender pie</a>” by <a href="https://www.flickr.com/photos/heatherjoan/" target="_blank" rel="noreferrer">Heather Joan</a> is licensed under a <a href="https://creativecommons.org/licenses/by-nc-nd/2.0/" target="_blank" rel="noreferrer">CC BY-NC-ND 2.0 Licence</a>.</p></li><li><p>“General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>.</p></li><li><p>Kilgannon, Allison. “Patterns of Organization and Methods of Development.” <a href="http://Opentextbc.ca" target="_blank" rel="noreferrer">Opentextbc.ca</a>, 20 Aug. 2021, <a href="http://opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development" target="_blank" rel="noreferrer">opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These four methods of development</a>.</p></li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/literature/writing/methods-of-development.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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class="text" data-v-b05232f3>Patterns of Organization and Methods of Development</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of 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class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_literature_writing_methods-of-development" data-v-c5936a1e><div><h1 id="patterns-of-organization-and-methods-of-development" tabindex="-1">Patterns of Organization and Methods of Development <a class="header-anchor" href="#patterns-of-organization-and-methods-of-development" aria-hidden="true">#</a></h1><blockquote><p>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.</p></blockquote><h2 id="major-patterns-of-organization" tabindex="-1">Major Patterns of Organization <a class="header-anchor" href="#major-patterns-of-organization" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/19875203514_543f7bff51_z-300x201.jpg" alt="A fruit pie."></p><p>Read the following sentences:</p><ul><li>Now take the pie out of the oven and let it cool on the stovetop.</li><li>Mix the dry ingredients with the liquid ingredients.</li><li>Set the pie crust aside while you make the filling.</li></ul><p>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.</p><p>First, let’s take a look at a couple of ways to think about organization.</p><h2 id="general-to-specific-or-specific-to-general" tabindex="-1">General to Specific or Specific to General <a class="header-anchor" href="#general-to-specific-or-specific-to-general" aria-hidden="true">#</a></h2><p>It might be useful to think about organizing your topic like a triangle:</p><p><img src="https://r2.toshiki.dev/image/Patterns_of_Organizaiton_Triangles.png" alt="Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General."></p><p>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.</p><p>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:</p><p>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.</p><p>An essay that uses the specific-to-general structure might start like this:</p><p>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.</p><p>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.</p><h2 id="does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" tabindex="-1">Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? <a class="header-anchor" href="#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" aria-hidden="true">#</a></h2><p>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.</p><h2 id="are-there-other-ways-to-think-about-organizing-my-ideas" tabindex="-1">Are There Other Ways to Think about Organizing My Ideas? <a class="header-anchor" href="#are-there-other-ways-to-think-about-organizing-my-ideas" aria-hidden="true">#</a></h2><p>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:</p><ul><li>Most important information first (consider what you want readers to focus on first)</li><li>Chronological order (the order in time that events take place)</li><li>Compare and contrast (ideas are organized together because of their relationship to each other)</li></ul><p>The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.</p><p>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.<br> Topics:</p><ul><li>Facing fears</li><li>Safety in sports</li><li>Community policing</li><li>Educating prisoners</li><li>Sex education</li></ul><h2 id="methods-of-development" tabindex="-1">Methods of Development <a class="header-anchor" href="#methods-of-development" aria-hidden="true">#</a></h2><p>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.</p><p>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.</p><p>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.</p><h3 id="cause-and-effect-or-effect-and-cause" tabindex="-1">Cause and Effect (or Effect and Cause) <a class="header-anchor" href="#cause-and-effect-or-effect-and-cause" aria-hidden="true">#</a></h3><p>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 <em>cause</em>, <em>effect</em>, and <em>result</em>, to cue the reader about your about the relationships that you’re establishing.</p><h3 id="problem-solution" tabindex="-1">Problem-Solution <a class="header-anchor" href="#problem-solution" aria-hidden="true">#</a></h3><p>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:</p><ul><li>Describe the problem, followed by the solution.</li><li>Propose the solution first and then describe the problems that motivated it.</li><li>Or a problem may be followed by several solutions, one of which is selected as the best.</li></ul><p>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 <em>problem</em> and <em>solution</em> to signal these sections of your paper for your reader.</p><h3 id="chronology-or-narrative" tabindex="-1">Chronology or Narrative <a class="header-anchor" href="#chronology-or-narrative" aria-hidden="true">#</a></h3><p>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 <em>then</em>, <em>next</em>, and <em>finally</em> to make the parts of the chronology clear.</p><h3 id="comparison-and-contrast" tabindex="-1">Comparison and Contrast <a class="header-anchor" href="#comparison-and-contrast" aria-hidden="true">#</a></h3><p>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 <em>by contrast</em>, <em>in comparison</em>, <em>while</em>, <em>some</em>, and <em>others</em>.</p><div class="tip custom-block"><p class="custom-block-title">Summary</p><p>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.</p></div><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li><p>This chapter was adapted from “<a href="https://openoregon.pressbooks.pub/wrd/chapter/patterns-of-organization-and-methods-of-development/" target="_blank" rel="noreferrer">Patterns of Organization and Methods of Development</a>” in <em>The Word on College Reading and Writing</em> by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>. Adapted by Allison Kilgannon.</p></li><li><p>“<a href="https://flic.kr/p/whiByE" target="_blank" rel="noreferrer">Peach and lavender pie</a>” by <a href="https://www.flickr.com/photos/heatherjoan/" target="_blank" rel="noreferrer">Heather Joan</a> is licensed under a <a href="https://creativecommons.org/licenses/by-nc-nd/2.0/" target="_blank" rel="noreferrer">CC BY-NC-ND 2.0 Licence</a>.</p></li><li><p>“General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>.</p></li><li><p>Kilgannon, Allison. “Patterns of Organization and Methods of Development.” <a href="http://Opentextbc.ca" target="_blank" rel="noreferrer">Opentextbc.ca</a>, 20 Aug. 2021, <a href="http://opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development" target="_blank" rel="noreferrer">opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These four methods of development</a>.</p></li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/literature/writing/methods-of-development.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" 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collapsible collapsed" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div 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_academic_physics_ipho-formulas-jpn_1" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-1" tabindex="-1">Formulas for IPhO 日本語版: Section 1 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-1" aria-hidden="true">#</a></h1><h2 id="_1-数学" tabindex="-1">1: 数学 <a class="header-anchor" href="#_1-数学" aria-hidden="true">#</a></h2><h3 id="_1-1-taylor-展開" tabindex="-1">1.1: Taylor 展開 <a class="header-anchor" href="#_1-1-taylor-展開" aria-hidden="true">#</a></h3><ol><li><p>Taylor 展開（アバウトに切り捨てる:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mo>∑</mo><msup><mi>F</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><msup><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mi>n</mi></msup><mi mathvariant="normal">/</mi><mi>n</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span></span></span></p><p>線形近似（特別な場合）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>F</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msup><mi>F</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">F(x) \approx F\left(x_{0}\right)+F^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|x| \ll 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> のときの例 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo></mrow><annotation encoding="application/x-tex">:</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>≈</mo><mi>x</mi><mo separator="true">,</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo>≈</mo><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><mo separator="true">,</mo><msup><mi>e</mi><mi>x</mi></msup><mo>≈</mo><mn>1</mn><mo>+</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin x \approx x, \cos x \approx 1-x^{2} / 2, e^{x} \approx 1+x </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord">/2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>x</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>≈</mo><mn>1</mn><mo>+</mo><mi>n</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\ln (1+x) \approx x,(1+x)^{n} \approx 1+n x </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">x</span></span></span></span></span></p></li></ol><h3 id="_1-2-摂動法" tabindex="-1">1.2: 摂動法 <a class="header-anchor" href="#_1-2-摂動法" aria-hidden="true">#</a></h3><ol start="2"><li>摂動法：摂動のない（直接解ける）問題の解を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 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．</li></ol><h3 id="_1-3-定数係数線形微分方程式" tabindex="-1">1.3: 定数係数線形微分方程式 <a class="header-anchor" href="#_1-3-定数係数線形微分方程式" aria-hidden="true">#</a></h3><ol start="3"><li><p>定数係数線形微分方程式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mi>b</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>c</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a y^{\prime \prime}+b y^{\prime}+c y=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">cy</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の解:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mi>x</mi><mo fence="true">)</mo></mrow><mo>+</mo><mi>B</mi><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><msub><mi>λ</mi><mn>2</mn></msub><mi>x</mi><mo fence="true">)</mo></mrow><mtext>. </mtext></mrow><annotation encoding="application/x-tex">y=A \exp \left(\lambda_1 x\right)+B \exp \left(\lambda_2 x\right) \text {. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">. </span></span></span></span></span></span></p><p>ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{1,2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> は特性方程式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>λ</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a \lambda^2+b \lambda+c=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">bλ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の異な る 2 解. もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a, b, c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span> が実数で特性方程式の解が複素数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo>=</mo><mi>γ</mi><mo>±</mo><mi>i</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">\lambda_{1,2}=\gamma \pm i \omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span></span></span></span> ならば,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>C</mi><msup><mi>e</mi><mrow><mi>γ</mi><mi>x</mi></mrow></msup><mi>sin</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi>ω</mi><mi>x</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y=C e^{\gamma x} \sin \left(\omega x+\varphi_0\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p></li></ol><h3 id="_1-4-複素数" tabindex="-1">1.4: 複素数 <a class="header-anchor" href="#_1-4-複素数" aria-hidden="true">#</a></h3><ol start="4"><li><p>複素数</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>z</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo separator="true">,</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo>=</mo><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup><mo>=</mo><mi>z</mi><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>φ</mi><mo>=</mo><mi>arg</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mfrac><mi>b</mi><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">Re</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">Im</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn><mi>i</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo fence="true">∣</mo></mrow><mo>=</mo><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>1</mn></msub><mo fence="true">∣</mo></mrow><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>2</mn></msub><mo fence="true">∣</mo></mrow><mo separator="true">,</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.1644em;vertical-align:-5.3322em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8322em;"><span style="top:-8.4592em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.4278em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">arcsin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-4.3518em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mop"><span class="mord mathrm">Re</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">/2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mord mathrm">Im</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">/2</span><span class="mord mathnormal">i</span></span></span><span style="top:-2.8518em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.3171em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span><span style="top:0.8446em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5017em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5017em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.3322em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-5-ベクトルの内積と外積" tabindex="-1">1.5: ベクトルの内積と外積 <a class="header-anchor" href="#_1-5-ベクトルの内積と外積" aria-hidden="true">#</a></h3><ol start="5"><li><p>ベクトルの内積と外積は分配法則が成立する : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">a(b+c)=a b+a c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mi mathvariant="bold-italic">b</mi><mo>⋅</mo><mi mathvariant="bold-italic">a</mi><mo>=</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>+</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo>+</mo><mo>⋯</mo><mo>=</mo><mi>a</mi><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mo>−</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">a</mi><mo>⊥</mo><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mrow><mo fence="true">(</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo fence="true">)</mo></mrow><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo fence="true">)</mo></mrow><msub><mi mathvariant="bold-italic">e</mi><mi>y</mi></msub><mo>+</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">b</mi><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">c</mi></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>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）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">b</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">)</mo><mo>≡</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi mathvariant="bold-italic">c</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">b</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">c</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">a</mi><mo stretchy="false">)</mo></mrow><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}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ol><h3 id="_1-6-余弦定理と正弦定理" tabindex="-1">1.6: 余弦定理と正弦定理 <a class="header-anchor" href="#_1-6-余弦定理と正弦定理" aria-hidden="true">#</a></h3><ol start="6"><li>余弦定理と正弦定理：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>a</mi><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>A</mi><mo>=</mo><mi>b</mi><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>B</mi><mo>=</mo><mn>2</mn><mi>R</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; c^2=a^2+b^2-2 a b \cos C \\ &amp; a / \sin A=b / \sin B=2 R \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.7621em;"><span class="pstrut" style="height:2.8641em;"></span><span class="mord"></span></span><span style="top:-2.2621em;"><span class="pstrut" style="height:2.8641em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.8979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">a</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-7-三角法" tabindex="-1">1.7: 三角法 <a class="header-anchor" href="#_1-7-三角法" aria-hidden="true">#</a></h3><ol start="7"><li><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>±</mo><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>sin</mi><mo>⁡</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>∓</mo><mi>sin</mi><mo>⁡</mo><mi>α</mi><mi>sin</mi><mo>⁡</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>tan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>tan</mi><mo>⁡</mo><mi>α</mi><mo>±</mo><mi>tan</mi><mo>⁡</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mn>1</mn><mo>∓</mo><mi>tan</mi><mo>⁡</mo><mi>α</mi><mi>tan</mi><mo>⁡</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>α</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>α</mi></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>α</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>α</mi></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mfrac><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>+</mo><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mn>2</mn><mrow><mo fence="true">(</mo><mi>cos</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mi>cos</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mi>β</mi></mrow><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mo>…</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ &amp; \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ &amp; \tan (\alpha \pm \beta)=(\tan \alpha \pm \tan \beta) /(1 \mp \tan \alpha \tan \beta) \\ &amp; \cos ^2 \alpha=\frac{1+\cos 2 \alpha}{2}, \sin ^2 \alpha=\frac{1-\cos 2 \alpha}{2} \\ &amp; \cos \alpha \cos \beta=\frac{\cos (\alpha+\beta)+\cos (\alpha-\beta)}{2}, \ldots \\ &amp; \cos \alpha+\cos \beta=2\left(\cos \frac{\alpha+\beta}{2}+\cos \frac{\alpha-\beta}{2}\right), \ldots \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.9205em;vertical-align:-5.7102em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.2102em;"><span style="top:-8.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-7.3202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-5.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-3.8388em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-1.4258em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:1.0102em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.7102em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.2102em;"><span style="top:-8.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-7.3202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-5.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mord">/</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span><span style="top:-3.8388em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4258em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span><span style="top:1.0102em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.7102em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-8-円周角" tabindex="-1">1.8: 円周角 <a class="header-anchor" href="#_1-8-円周角" aria-hidden="true">#</a></h3><ol start="8"><li>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.</li></ol><h3 id="_1-9-三角形の面樍" tabindex="-1">1.9: 三角形の面樍 <a class="header-anchor" href="#_1-9-三角形の面樍" aria-hidden="true">#</a></h3><ol start="9"><li>三角形の面樍 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msub><mi>h</mi><mi>a</mi></msub><mo>=</mo><mi>p</mi><mi>r</mi><mo>=</mo><msqrt><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msqrt><mo>=</mo><mi>a</mi><mi>b</mi><mi>c</mi><mi mathvariant="normal">/</mi><mn>4</mn><mi>R</mi></mrow><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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-1" aria-hidden="true">#</a></h1><h2 id="_1-数学" tabindex="-1">1: 数学 <a class="header-anchor" href="#_1-数学" aria-hidden="true">#</a></h2><h3 id="_1-1-taylor-展開" tabindex="-1">1.1: Taylor 展開 <a class="header-anchor" href="#_1-1-taylor-展開" aria-hidden="true">#</a></h3><ol><li><p>Taylor 展開（アバウトに切り捨てる:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mo>∑</mo><msup><mi>F</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><msup><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mi>n</mi></msup><mi mathvariant="normal">/</mi><mi>n</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span></span></span></p><p>線形近似（特別な場合）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>F</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msup><mi>F</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">F(x) \approx F\left(x_{0}\right)+F^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|x| \ll 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> のときの例 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo></mrow><annotation encoding="application/x-tex">:</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>≈</mo><mi>x</mi><mo separator="true">,</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo>≈</mo><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><mo separator="true">,</mo><msup><mi>e</mi><mi>x</mi></msup><mo>≈</mo><mn>1</mn><mo>+</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin x \approx x, \cos x \approx 1-x^{2} / 2, e^{x} \approx 1+x </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord">/2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>x</mi><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>≈</mo><mn>1</mn><mo>+</mo><mi>n</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\ln (1+x) \approx x,(1+x)^{n} \approx 1+n x </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">x</span></span></span></span></span></p></li></ol><h3 id="_1-2-摂動法" tabindex="-1">1.2: 摂動法 <a class="header-anchor" href="#_1-2-摂動法" aria-hidden="true">#</a></h3><ol start="2"><li>摂動法：摂動のない（直接解ける）問題の解を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 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．</li></ol><h3 id="_1-3-定数係数線形微分方程式" tabindex="-1">1.3: 定数係数線形微分方程式 <a class="header-anchor" href="#_1-3-定数係数線形微分方程式" aria-hidden="true">#</a></h3><ol start="3"><li><p>定数係数線形微分方程式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mi>b</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>c</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a y^{\prime \prime}+b y^{\prime}+c y=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">cy</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の解:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mi>x</mi><mo fence="true">)</mo></mrow><mo>+</mo><mi>B</mi><mi>exp</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><msub><mi>λ</mi><mn>2</mn></msub><mi>x</mi><mo fence="true">)</mo></mrow><mtext>. </mtext></mrow><annotation encoding="application/x-tex">y=A \exp \left(\lambda_1 x\right)+B \exp \left(\lambda_2 x\right) \text {. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">. </span></span></span></span></span></span></p><p>ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{1,2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> は特性方程式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>λ</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a \lambda^2+b \lambda+c=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">bλ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の異な る 2 解. もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a, b, c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span> が実数で特性方程式の解が複素数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn></mrow></msub><mo>=</mo><mi>γ</mi><mo>±</mo><mi>i</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">\lambda_{1,2}=\gamma \pm i \omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span></span></span></span> ならば,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>C</mi><msup><mi>e</mi><mrow><mi>γ</mi><mi>x</mi></mrow></msup><mi>sin</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi>ω</mi><mi>x</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y=C e^{\gamma x} \sin \left(\omega x+\varphi_0\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p></li></ol><h3 id="_1-4-複素数" tabindex="-1">1.4: 複素数 <a class="header-anchor" href="#_1-4-複素数" aria-hidden="true">#</a></h3><ol start="4"><li><p>複素数</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>z</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo separator="true">,</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo>=</mo><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup><mo>=</mo><mi>z</mi><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>φ</mi><mo>=</mo><mi>arg</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mfrac><mi>b</mi><mrow><mi mathvariant="normal">∣</mi><mi>z</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">Re</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>+</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">Im</mi><mo>⁡</mo><mi>z</mi><mo>=</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mover accent="true"><mi>z</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn><mi>i</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo fence="true">∣</mo></mrow><mo>=</mo><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>1</mn></msub><mo fence="true">∣</mo></mrow><mrow><mo fence="true">∣</mo><msub><mi>z</mi><mn>2</mn></msub><mo fence="true">∣</mo></mrow><mo separator="true">,</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><mi>arg</mi><mo>⁡</mo><msub><mi>z</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>φ</mi></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>φ</mi></mrow></msup></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.1644em;vertical-align:-5.3322em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8322em;"><span style="top:-8.4592em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">bi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.4278em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">arcsin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-4.3518em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mop"><span class="mord mathrm">Re</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">/2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mord mathrm">Im</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">/2</span><span class="mord mathnormal">i</span></span></span><span style="top:-2.8518em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.044em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.3171em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span><span style="top:0.8446em;"><span class="pstrut" style="height:3.5017em;"></span><span class="mord"><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5017em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5017em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">i</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">φ</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.3322em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-5-ベクトルの内積と外積" tabindex="-1">1.5: ベクトルの内積と外積 <a class="header-anchor" href="#_1-5-ベクトルの内積と外積" aria-hidden="true">#</a></h3><ol start="5"><li><p>ベクトルの内積と外積は分配法則が成立する : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">a(b+c)=a b+a c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mi mathvariant="bold-italic">b</mi><mo>⋅</mo><mi mathvariant="bold-italic">a</mi><mo>=</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>+</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo>+</mo><mo>⋯</mo><mo>=</mo><mi>a</mi><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mo>−</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">a</mi><mo>⊥</mo><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mrow><mo fence="true">(</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo fence="true">)</mo></mrow><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo fence="true">)</mo></mrow><msub><mi mathvariant="bold-italic">e</mi><mi>y</mi></msub><mo>+</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">b</mi><mo>−</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">c</mi></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mclose">)</span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.75em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>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）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">b</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">)</mo><mo>≡</mo><mi mathvariant="bold-italic">a</mi><mo>⋅</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">b</mi><mo>×</mo><mi mathvariant="bold-italic">c</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi mathvariant="bold-italic">c</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">b</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">c</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">a</mi><mo stretchy="false">)</mo></mrow><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}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">c</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ol><h3 id="_1-6-余弦定理と正弦定理" tabindex="-1">1.6: 余弦定理と正弦定理 <a class="header-anchor" href="#_1-6-余弦定理と正弦定理" aria-hidden="true">#</a></h3><ol start="6"><li>余弦定理と正弦定理：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>C</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>a</mi><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>A</mi><mo>=</mo><mi>b</mi><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>B</mi><mo>=</mo><mn>2</mn><mi>R</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; c^2=a^2+b^2-2 a b \cos C \\ &amp; a / \sin A=b / \sin B=2 R \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.7621em;"><span class="pstrut" style="height:2.8641em;"></span><span class="mord"></span></span><span style="top:-2.2621em;"><span class="pstrut" style="height:2.8641em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.8979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">a</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-7-三角法" tabindex="-1">1.7: 三角法 <a class="header-anchor" href="#_1-7-三角法" aria-hidden="true">#</a></h3><ol start="7"><li><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>±</mo><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>sin</mi><mo>⁡</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>∓</mo><mi>sin</mi><mo>⁡</mo><mi>α</mi><mi>sin</mi><mo>⁡</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>tan</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>±</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>tan</mi><mo>⁡</mo><mi>α</mi><mo>±</mo><mi>tan</mi><mo>⁡</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mn>1</mn><mo>∓</mo><mi>tan</mi><mo>⁡</mo><mi>α</mi><mi>tan</mi><mo>⁡</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>α</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>α</mi></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>α</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>α</mi></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mfrac><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>+</mo><mi>cos</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mn>2</mn><mrow><mo fence="true">(</mo><mi>cos</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mi>cos</mi><mo>⁡</mo><mfrac><mrow><mi>α</mi><mo>−</mo><mi>β</mi></mrow><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mo>…</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ &amp; \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ &amp; \tan (\alpha \pm \beta)=(\tan \alpha \pm \tan \beta) /(1 \mp \tan \alpha \tan \beta) \\ &amp; \cos ^2 \alpha=\frac{1+\cos 2 \alpha}{2}, \sin ^2 \alpha=\frac{1-\cos 2 \alpha}{2} \\ &amp; \cos \alpha \cos \beta=\frac{\cos (\alpha+\beta)+\cos (\alpha-\beta)}{2}, \ldots \\ &amp; \cos \alpha+\cos \beta=2\left(\cos \frac{\alpha+\beta}{2}+\cos \frac{\alpha-\beta}{2}\right), \ldots \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.9205em;vertical-align:-5.7102em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.2102em;"><span style="top:-8.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-7.3202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-5.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-3.8388em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:-1.4258em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span><span style="top:1.0102em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.7102em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.2102em;"><span style="top:-8.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-7.3202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-5.8202em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mord">/</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span><span style="top:-3.8388em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4258em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span><span style="top:1.0102em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.7102em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-8-円周角" tabindex="-1">1.8: 円周角 <a class="header-anchor" href="#_1-8-円周角" aria-hidden="true">#</a></h3><ol start="8"><li>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.</li></ol><h3 id="_1-9-三角形の面樍" tabindex="-1">1.9: 三角形の面樍 <a class="header-anchor" href="#_1-9-三角形の面樍" aria-hidden="true">#</a></h3><ol start="9"><li>三角形の面樍 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msub><mi>h</mi><mi>a</mi></msub><mo>=</mo><mi>p</mi><mi>r</mi><mo>=</mo><msqrt><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msqrt><mo>=</mo><mi>a</mi><mi>b</mi><mi>c</mi><mi mathvariant="normal">/</mi><mn>4</mn><mi>R</mi></mrow><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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1777em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6491em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-13-積分" tabindex="-1">1.13: 積分 <a class="header-anchor" href="#_1-13-積分" aria-hidden="true">#</a></h3><ol start="13"><li>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,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><msup><mi>x</mi><mi>n</mi></msup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>x</mi><mo>=</mo><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\int x^n \mathrm{~d} x=x^{n+1} /(n+1) . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p> 置換積分の特別な場合 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>a</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\int f(a x+b) \mathrm{d} x=F(a x+b) / a . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">a</span><span class="mord">.</span></span></span></span></span></p></li></ol><h3 id="_1-14-円錐曲線" tabindex="-1">1.14: 円錐曲線 <a class="header-anchor" href="#_1-14-円錐曲線" aria-hidden="true">#</a></h3><ol start="14"><li>円錐曲線: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>a</mi><mn>12</mn></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mi>a</mi><mn>22</mn></msub><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9641em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> 0 で, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><mo>=</mo><msub><mi>a</mi><mn>22</mn></msub></mrow><annotation encoding="application/x-tex">a_{11}=a_{22}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ならば円, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><mrow><mo fence="true">(</mo><msub><mi>a</mi><mn>11</mn></msub><msub><mi>a</mi><mn>22</mn></msub><mo>−</mo><msubsup><mi>a</mi><mn>12</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_{11}\left(a_{11} a_{22}-a_{12}^2\right)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば楕円, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\cdots&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば双曲線, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><msub><mi>a</mi><mn>22</mn></msub><mo>−</mo><msubsup><mi>a</mi><mn>12</mn><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_{11} a_{22}-a_{12}^2=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば放物線. 楕円 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>1</mn></msub><mo>+</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mo separator="true">,</mo><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">l_1+l_2=2 a, \alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">A=\pi a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">πab</span></span></span></span>. 双曲線 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>1</mn></msub><mo>−</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mo separator="true">,</mo><msub><mi>α</mi><mn>1</mn></msub><mo>+</mo><msub><mi>α</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">l_1-l_2=2 a, \alpha_1+\alpha_2=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 放物線 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>l</mi><mo>+</mo><mi>h</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">: l+h=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const., <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_1-15-数值計算-台形規則" tabindex="-1">1.15: 数值計算 &amp; 台形規則 <a class="header-anchor" href="#_1-15-数值計算-台形規則" aria-hidden="true">#</a></h3><ol start="15"><li>数值計算. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の解を求めるニュートン法 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p> 近似積分の台形規則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>≈</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mrow><mo fence="true">[</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo fence="true">{</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mo>⋯</mo><mtext> </mtext></mrow></mrow><mrow><mrow><mo>+</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo fence="true">)</mo></mrow><mo fence="true">}</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.404em;vertical-align:-0.452em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.952em;"><span style="top:-2.952em;"><span class="pstrut" style="height:3.044em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.044em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose nulldelimiter"></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord">+</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">}</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.452em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-16-ベクトルの微分-積分" tabindex="-1">1.16: ベクトルの微分 &amp; 積分 <a class="header-anchor" href="#_1-16-ベクトルの微分-積分" aria-hidden="true">#</a></h3><ol start="16"><li>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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>つのベクトルの差を求めることで 微分す.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div 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class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><!----></div><div class="pager" data-v-e033cd21><a class="pager-link next" href="/academic/physics/ipho-formulas-jpn/2" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>2: 一般的な推奨事</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a 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+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1777em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6491em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-13-積分" tabindex="-1">1.13: 積分 <a class="header-anchor" href="#_1-13-積分" aria-hidden="true">#</a></h3><ol start="13"><li>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,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><msup><mi>x</mi><mi>n</mi></msup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>x</mi><mo>=</mo><msup><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\int x^n \mathrm{~d} x=x^{n+1} /(n+1) . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p> 置換積分の特別な場合 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>a</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\int f(a x+b) \mathrm{d} x=F(a x+b) / a . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">a</span><span class="mord">.</span></span></span></span></span></p></li></ol><h3 id="_1-14-円錐曲線" tabindex="-1">1.14: 円錐曲線 <a class="header-anchor" href="#_1-14-円錐曲線" aria-hidden="true">#</a></h3><ol start="14"><li>円錐曲線: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>a</mi><mn>12</mn></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mi>a</mi><mn>22</mn></msub><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">a_{11} x^2+2 a_{12} x y+a_{22} y^2+a_1 x+a_2 y+a_0=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9641em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> 0 で, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><mo>=</mo><msub><mi>a</mi><mn>22</mn></msub></mrow><annotation encoding="application/x-tex">a_{11}=a_{22}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ならば円, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><mrow><mo fence="true">(</mo><msub><mi>a</mi><mn>11</mn></msub><msub><mi>a</mi><mn>22</mn></msub><mo>−</mo><msubsup><mi>a</mi><mn>12</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_{11}\left(a_{11} a_{22}-a_{12}^2\right)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば楕円, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\cdots&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば双曲線, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>11</mn></msub><msub><mi>a</mi><mn>22</mn></msub><mo>−</mo><msubsup><mi>a</mi><mn>12</mn><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_{11} a_{22}-a_{12}^2=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">22</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> ならば放物線. 楕円 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>1</mn></msub><mo>+</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mo separator="true">,</mo><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">l_1+l_2=2 a, \alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> [訳 者注 : 焦点と曲線上の点を結ぶ直線と接線とのなす角 ], <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">A=\pi a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">πab</span></span></span></span>. 双曲線 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>1</mn></msub><mo>−</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mo separator="true">,</mo><msub><mi>α</mi><mn>1</mn></msub><mo>+</mo><msub><mi>α</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">l_1-l_2=2 a, \alpha_1+\alpha_2=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 放物線 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>l</mi><mo>+</mo><mi>h</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">: l+h=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const., <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_1-15-数值計算-台形規則" tabindex="-1">1.15: 数值計算 &amp; 台形規則 <a class="header-anchor" href="#_1-15-数值計算-台形規則" aria-hidden="true">#</a></h3><ol start="15"><li>数值計算. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> の解を求めるニュートン法 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">x_{n+1}=x_n-f\left(x_n\right) / f^{\prime}\left(x_n\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p> 近似積分の台形規則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>≈</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mrow><mo fence="true">[</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo fence="true">{</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mo>⋯</mo><mtext> </mtext></mrow></mrow><mrow><mrow><mo>+</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo fence="true">)</mo></mrow><mo fence="true">}</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.404em;vertical-align:-0.452em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.952em;"><span style="top:-2.952em;"><span class="pstrut" style="height:3.044em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.044em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose nulldelimiter"></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord">+</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">}</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.452em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p></li></ol><h3 id="_1-16-ベクトルの微分-積分" tabindex="-1">1.16: ベクトルの微分 &amp; 積分 <a class="header-anchor" href="#_1-16-ベクトルの微分-積分" aria-hidden="true">#</a></h3><ol start="16"><li>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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>つのベクトルの差を求めることで 微分す.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_10" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-10" tabindex="-1">Formulas for IPhO 日本語版: Section 10 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-10" aria-hidden="true">#</a></h1><h2 id="_10-熱力学" tabindex="-1">10: 熱力学 <a class="header-anchor" href="#_10-熱力学" aria-hidden="true">#</a></h2><h3 id="_10-1" tabindex="-1">10.1: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>V</mi><mo>=</mo><mfrac><mi>w</mi><mi>M</mi></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">p V=\frac{w}{M} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span> <a class="header-anchor" href="#_10-1" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>V</mi><mo>=</mo><mfrac><mi>w</mi><mi>M</mi></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">p V=\frac{w}{M} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span>.</li></ol><h3 id="_10-2-モルの気体の内部エネルギー" tabindex="-1">10.2: モルの気体の内部エネルギー <a class="header-anchor" href="#_10-2-モルの気体の内部エネルギー" aria-hidden="true">#</a></h3><ol start="2"><li>1 モルの気体の内部エネルギー: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U=\frac{i}{2} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2007em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8557em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span> [訳者注: 単 原子分子理想気体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">i=3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span>, 二原子分子理想気体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i=5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span>.</li></ol><h3 id="_10-3-標準状態" tabindex="-1">10.3: 標準状態 <a class="header-anchor" href="#_10-3-標準状態" aria-hidden="true">#</a></h3><ol start="3"><li>標準状態での 1 モルの気体の体積は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>22.4</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow></mrow><annotation encoding="application/x-tex">22.4 \mathrm{~L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">22.4</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span></span></span></span>.</li></ol><h3 id="_10-4-断熱過程" tabindex="-1">10.4: 断熱過程 <a class="header-anchor" href="#_10-4-断熱過程" aria-hidden="true">#</a></h3><ol start="4"><li>断熱過程: 音速に比べて遅く, 熱の出入りがない. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><msup><mi>V</mi><mi>γ</mi></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">p V^\gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mi>T</mi><msup><mi>V</mi><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">\left(T V^{\gamma-1}=\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mclose nulldelimiter"></span></span></span></span></span> const. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>.</li></ol><h3 id="_10-5-γ-cp-cv-i-2-i" tabindex="-1">10.5: γ=Cp/Cv=(i+2)/i <a class="header-anchor" href="#_10-5-γ-cp-cv-i-2-i" aria-hidden="true">#</a></h3><ol start="5"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><msub><mi>c</mi><mi>p</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>v</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">\gamma=c_p / c_v=(i+2) / i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">i</span></span></span></span>.</li></ol><h3 id="_10-6-boltzmann-分布" tabindex="-1">10.6: Boltzmann 分布 <a class="header-anchor" href="#_10-6-boltzmann-分布" aria-hidden="true">#</a></h3><ol start="6"><li>Boltzmann 分布 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>M</mi><mi>g</mi><mi>h</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>T</mi></mrow></msup><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>U</mi><mi mathvariant="normal">/</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\rho=\rho_0 e^{-M g h / R T}=\rho_0 e^{-U / k_B T} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1324em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mord mathnormal mtight">h</span><span class="mord mtight">/</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">RT</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1324em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_10-7-maxwell-分布" tabindex="-1">10.7: Maxwell 分布 <a class="header-anchor" href="#_10-7-maxwell-分布" aria-hidden="true">#</a></h3><ol start="7"><li>Maxwell 分布（v の速さをもつ分子の数）<div class="tip custom-block"><p class="custom-block-title">訳者注</p><p>位相空間で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}+\mathrm{d} \boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> 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] <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>m</mi><msup><mi mathvariant="bold-italic">v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\propto e^{-m \boldsymbol{v}^2 / 2 k_B T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9869em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">m</span><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mtight">/2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span></p></div></li></ol><h3 id="_10-8-大気圧" tabindex="-1">10.8: 大気圧 <a class="header-anchor" href="#_10-8-大気圧" aria-hidden="true">#</a></h3><ol start="8"><li>大気圧 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mo>≪</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\Delta p \ll p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mo>=</mo><mi>ρ</mi><mi>g</mi><mi mathvariant="normal">Δ</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\Delta p=\rho g \Delta h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">Δ</span><span class="mord mathnormal">h</span></span></span></span>.</li></ol><h3 id="_10-9-公式" tabindex="-1">10.9: 公式 <a class="header-anchor" href="#_10-9-公式" aria-hidden="true">#</a></h3><ol start="9"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>m</mi><mi>n</mi><mover accent="true"><msup><mi>v</mi><mn>2</mn></msup><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>n</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">p=\frac{1}{3} m n \overline{v^2}=n k_B T(n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2851em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">mn</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span></span></span></span> は数密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo><mo separator="true">,</mo><msqrt><mover accent="true"><mover accent="true"><msup><mi>v</mi><mn>2</mn></msup><mo stretchy="true">‾</mo></mover><mo stretchy="true">‾</mo></mover></msqrt><mo>=</mo></mrow><annotation encoding="application/x-tex">), \sqrt{\overline{\overline{v^2}}}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.2849em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5551em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-4.0601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-3.5151em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div 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is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-10" aria-hidden="true">#</a></h1><h2 id="_10-熱力学" tabindex="-1">10: 熱力学 <a class="header-anchor" href="#_10-熱力学" aria-hidden="true">#</a></h2><h3 id="_10-1" tabindex="-1">10.1: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>V</mi><mo>=</mo><mfrac><mi>w</mi><mi>M</mi></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">p V=\frac{w}{M} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span> <a class="header-anchor" href="#_10-1" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>V</mi><mo>=</mo><mfrac><mi>w</mi><mi>M</mi></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">p V=\frac{w}{M} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span>.</li></ol><h3 id="_10-2-モルの気体の内部エネルギー" tabindex="-1">10.2: モルの気体の内部エネルギー <a class="header-anchor" href="#_10-2-モルの気体の内部エネルギー" aria-hidden="true">#</a></h3><ol start="2"><li>1 モルの気体の内部エネルギー: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi>R</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U=\frac{i}{2} R T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2007em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8557em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">RT</span></span></span></span> [訳者注: 単 原子分子理想気体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">i=3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span>, 二原子分子理想気体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i=5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span>.</li></ol><h3 id="_10-3-標準状態" tabindex="-1">10.3: 標準状態 <a class="header-anchor" href="#_10-3-標準状態" aria-hidden="true">#</a></h3><ol start="3"><li>標準状態での 1 モルの気体の体積は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>22.4</mn><mrow><mtext> </mtext><mi mathvariant="normal">L</mi></mrow></mrow><annotation encoding="application/x-tex">22.4 \mathrm{~L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">22.4</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">L</span></span></span></span></span>.</li></ol><h3 id="_10-4-断熱過程" tabindex="-1">10.4: 断熱過程 <a class="header-anchor" href="#_10-4-断熱過程" aria-hidden="true">#</a></h3><ol start="4"><li>断熱過程: 音速に比べて遅く, 熱の出入りがない. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><msup><mi>V</mi><mi>γ</mi></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">p V^\gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mi>T</mi><msup><mi>V</mi><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">\left(T V^{\gamma-1}=\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mclose nulldelimiter"></span></span></span></span></span> const. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>.</li></ol><h3 id="_10-5-γ-cp-cv-i-2-i" tabindex="-1">10.5: γ=Cp/Cv=(i+2)/i <a class="header-anchor" href="#_10-5-γ-cp-cv-i-2-i" aria-hidden="true">#</a></h3><ol start="5"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><msub><mi>c</mi><mi>p</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>v</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">\gamma=c_p / c_v=(i+2) / i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">i</span></span></span></span>.</li></ol><h3 id="_10-6-boltzmann-分布" tabindex="-1">10.6: Boltzmann 分布 <a class="header-anchor" href="#_10-6-boltzmann-分布" aria-hidden="true">#</a></h3><ol start="6"><li>Boltzmann 分布 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>M</mi><mi>g</mi><mi>h</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi>T</mi></mrow></msup><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>U</mi><mi mathvariant="normal">/</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\rho=\rho_0 e^{-M g h / R T}=\rho_0 e^{-U / k_B T} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1324em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mord mathnormal mtight">h</span><span class="mord mtight">/</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">RT</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1324em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_10-7-maxwell-分布" tabindex="-1">10.7: Maxwell 分布 <a class="header-anchor" href="#_10-7-maxwell-分布" aria-hidden="true">#</a></h3><ol start="7"><li>Maxwell 分布（v の速さをもつ分子の数）<div class="tip custom-block"><p class="custom-block-title">訳者注</p><p>位相空間で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}+\mathrm{d} \boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> 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] <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>m</mi><msup><mi mathvariant="bold-italic">v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\propto e^{-m \boldsymbol{v}^2 / 2 k_B T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9869em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">m</span><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mtight">/2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567em;margin-left:-0.0315em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1433em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span></p></div></li></ol><h3 id="_10-8-大気圧" tabindex="-1">10.8: 大気圧 <a class="header-anchor" href="#_10-8-大気圧" aria-hidden="true">#</a></h3><ol start="8"><li>大気圧 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mo>≪</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\Delta p \ll p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mo>=</mo><mi>ρ</mi><mi>g</mi><mi mathvariant="normal">Δ</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\Delta p=\rho g \Delta h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">Δ</span><span class="mord mathnormal">h</span></span></span></span>.</li></ol><h3 id="_10-9-公式" tabindex="-1">10.9: 公式 <a class="header-anchor" href="#_10-9-公式" aria-hidden="true">#</a></h3><ol start="9"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>m</mi><mi>n</mi><mover accent="true"><msup><mi>v</mi><mn>2</mn></msup><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>n</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">p=\frac{1}{3} m n \overline{v^2}=n k_B T(n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2851em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">mn</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span></span></span></span> は数密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo><mo separator="true">,</mo><msqrt><mover accent="true"><mover accent="true"><msup><mi>v</mi><mn>2</mn></msup><mo stretchy="true">‾</mo></mover><mo stretchy="true">‾</mo></mover></msqrt><mo>=</mo></mrow><annotation encoding="application/x-tex">), \sqrt{\overline{\overline{v^2}}}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.2849em;"></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5551em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9401em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-4.0601em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-3.5151em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg 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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</li></ol><h3 id="_10-10-carnot-サイクル" tabindex="-1">10.10: Carnot サイクル <a class="header-anchor" href="#_10-10-carnot-サイクル" aria-hidden="true">#</a></h3><ol start="10"><li>Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>−</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S-T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> 座標を用いることにより <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi><mo>=</mo><mrow><mo fence="true">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo>−</mo><msub><mi>T</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><msub><mi>T</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\eta=\left(T_1-T_2\right) / T_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を得る.</li></ol><h3 id="_10-11-ヒートポンプ" tabindex="-1">10.11: ヒートポンプ <a class="header-anchor" href="#_10-11-ヒートポンプ" aria-hidden="true">#</a></h3><ol start="11"><li>ヒートポンプ: Carnot サイクルの逆. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi><mo>=</mo><mfrac><msub><mi>T</mi><mn>1</mn></msub><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>−</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\eta=\frac{T_1}{T_1-T_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3335em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8884em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_10-12-エントロピー" tabindex="-1">10.12: エントロピー <a class="header-anchor" href="#_10-12-エントロピー" aria-hidden="true">#</a></h3><ol start="12"><li>エントロピー <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi mathvariant="normal">d</mi><mi>S</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi>Q</mi><mi mathvariant="normal">/</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">: \mathrm{d} S=\mathrm{d} Q / T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">Q</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>.</li></ol><h3 id="_10-13-熱力学第一法則" tabindex="-1">10.13: 熱力学第一法則 <a class="header-anchor" href="#_10-13-熱力学第一法則" aria-hidden="true">#</a></h3><ol start="13"><li>熱力学第一法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>U</mi><mo>=</mo><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>A</mi><mo>+</mo><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>Q</mi></mrow><annotation encoding="application/x-tex">\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8352em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">Q</span></span></span></span></li></ol><h3 id="_10-14-熱力学第二法則" tabindex="-1">10.14: 熱力学第二法則 <a class="header-anchor" href="#_10-14-熱力学第二法則" aria-hidden="true">#</a></h3><ol start="14"><li>熱力学第二法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>S</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta S \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> (また <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mtext>real </mtext></msub><mo>≤</mo><msub><mi>η</mi><mtext>Carnot </mtext></msub><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">real </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">Carnot </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>.</li></ol><h3 id="_10-15-気体のする仕事" tabindex="-1">10.15: 気体のする仕事 <a class="header-anchor" href="#_10-15-気体のする仕事" aria-hidden="true">#</a></h3><ol start="15"><li>気体のする仕事（<a href="./10#_10-10-carnot-サイクル">ポイント 10</a> も参照）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mo>∫</mo><mi>p</mi><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>V</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mtext> 断熱過程: </mtext><mi>A</mi><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">断熱過程</span><span class="mord">: </span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0225em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3365em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mclose">)</span></span></span></span></span></p></li></ol><h3 id="_10-16-dalton-の法則" tabindex="-1">10.16: Dalton の法則 <a class="header-anchor" href="#_10-16-dalton-の法則" aria-hidden="true">#</a></h3><ol start="16"><li><p>Dalton の法則: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><div class="tip custom-block"><p class="custom-block-title">訳者注</p><p>理想気体のみ成立</p></div></li></ol><h3 id="_10-17-沸騰" tabindex="-1">10.17: 沸騰 <a class="header-anchor" href="#_10-17-沸騰" aria-hidden="true">#</a></h3><ol start="17"><li>沸騰: 飽和蒸気の圧力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>v</mi></msub><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub><mi mathvariant="normal">.</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">p_v=p_0 .2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.2</span></span></span></span> 液の界面では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>v</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>v</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p_{v 1}+p_{v 2}=p_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_10-18-熱流" tabindex="-1">10.18: 熱流 <a class="header-anchor" href="#_10-18-熱流" aria-hidden="true">#</a></h3><ol start="18"><li>熱流: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>k</mi><mi>S</mi><mi mathvariant="normal">Δ</mi><mi>T</mi><mi mathvariant="normal">/</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">P=k S \Delta T / l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> は熱伝導率). 直流回路に似て いる <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>↔</mo><mi>I</mi><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>T</mi><mo>↔</mo><mi>V</mi><mo separator="true">,</mo><mi>k</mi><mo>↔</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span>.</li></ol><h3 id="_10-19-熱容量" tabindex="-1">10.19: 熱容量 <a class="header-anchor" href="#_10-19-熱容量" aria-hidden="true">#</a></h3><ol start="19"><li>熱容量 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><mo>∫</mo><mi>c</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">Q=\int c(T) \mathrm{d} T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>. 固体では低温で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>∝</mo><msup><mi>T</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">c \propto T^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, 高温で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>3</mn><mi>N</mi><msub><mi>k</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">c=3 N k_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> （Dulong-Petit の法則. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> は結晶中の原子数）</li></ol><h3 id="_10-20-表面張力" tabindex="-1">10.20: 表面張力 <a class="header-anchor" href="#_10-20-表面張力" aria-hidden="true">#</a></h3><ol start="20"><li>表面張力 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>U</mi><mo>=</mo><mi>S</mi><mi>σ</mi><mo separator="true">,</mo><mi>F</mi><mo>=</mo><mi>l</mi><mi>σ</mi><mo separator="true">,</mo><mi>p</mi><mo>=</mo><mn>2</mn><mi>σ</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">U=S \sigma, F=l \sigma, p=2 \sigma / R </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></p></li></ol><h3 id="_10-21-stefan-boltzmann-の法則-灰色体" tabindex="-1">10.21: Stefan-Boltzmann の法則 (灰色体) <a class="header-anchor" href="#_10-21-stefan-boltzmann-の法則-灰色体" aria-hidden="true">#</a></h3><ol start="21"><li>Stefan-Boltzmann の法則 (灰色体) : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>ε</mi><mi>σ</mi><mi>A</mi><msup><mi>T</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">P=\varepsilon \sigma A T^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal">ε</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_10-22-wien-の変位則" tabindex="-1">10.22: Wien の変位則 <a class="header-anchor" href="#_10-22-wien-の変位則" aria-hidden="true">#</a></h3><ol start="22"><li>Wien の変位則: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ν</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>A</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mi mathvariant="normal">/</mi><mi>h</mi><mo stretchy="false">(</mo><mi>A</mi><mo>≈</mo></mrow><annotation encoding="application/x-tex">\nu_{\max }=A k_B T / h(A \approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span></span></span></span> 2.8), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>h</mi><mi>c</mi><mi mathvariant="normal">/</mi><msup><mi>A</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mrow><mo fence="true">(</mo><msup><mi>A</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>≈</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal">c</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">5</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/10.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</li></ol><h3 id="_10-10-carnot-サイクル" tabindex="-1">10.10: Carnot サイクル <a class="header-anchor" href="#_10-10-carnot-サイクル" aria-hidden="true">#</a></h3><ol start="10"><li>Carnot サイクル : 断熱過程 2 つと等温過程 2 つ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>−</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S-T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> 座標を用いることにより <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi><mo>=</mo><mrow><mo fence="true">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo>−</mo><msub><mi>T</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><msub><mi>T</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\eta=\left(T_1-T_2\right) / T_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を得る.</li></ol><h3 id="_10-11-ヒートポンプ" tabindex="-1">10.11: ヒートポンプ <a class="header-anchor" href="#_10-11-ヒートポンプ" aria-hidden="true">#</a></h3><ol start="11"><li>ヒートポンプ: Carnot サイクルの逆. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi><mo>=</mo><mfrac><msub><mi>T</mi><mn>1</mn></msub><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>−</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\eta=\frac{T_1}{T_1-T_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3335em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8884em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_10-12-エントロピー" tabindex="-1">10.12: エントロピー <a class="header-anchor" href="#_10-12-エントロピー" aria-hidden="true">#</a></h3><ol start="12"><li>エントロピー <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi mathvariant="normal">d</mi><mi>S</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi>Q</mi><mi mathvariant="normal">/</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">: \mathrm{d} S=\mathrm{d} Q / T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">Q</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>.</li></ol><h3 id="_10-13-熱力学第一法則" tabindex="-1">10.13: 熱力学第一法則 <a class="header-anchor" href="#_10-13-熱力学第一法則" aria-hidden="true">#</a></h3><ol start="13"><li>熱力学第一法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>U</mi><mo>=</mo><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>A</mi><mo>+</mo><msup><mi mathvariant="normal">d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>Q</mi></mrow><annotation encoding="application/x-tex">\mathrm{d}^{\prime} U=\mathrm{d}^{\prime} A+\mathrm{d}^{\prime} Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8352em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">Q</span></span></span></span></li></ol><h3 id="_10-14-熱力学第二法則" tabindex="-1">10.14: 熱力学第二法則 <a class="header-anchor" href="#_10-14-熱力学第二法則" aria-hidden="true">#</a></h3><ol start="14"><li>熱力学第二法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>S</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta S \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> (また <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>η</mi><mtext>real </mtext></msub><mo>≤</mo><msub><mi>η</mi><mtext>Carnot </mtext></msub><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left.\eta_{\text {real }} \leq \eta_{\text {Carnot }}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">real </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">Carnot </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>.</li></ol><h3 id="_10-15-気体のする仕事" tabindex="-1">10.15: 気体のする仕事 <a class="header-anchor" href="#_10-15-気体のする仕事" aria-hidden="true">#</a></h3><ol start="15"><li>気体のする仕事（<a href="./10#_10-10-carnot-サイクル">ポイント 10</a> も参照）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mo>∫</mo><mi>p</mi><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>V</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mtext> 断熱過程: </mtext><mi>A</mi><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A=\int p \mathrm{~d} V, \quad \text { 断熱過程: } A=\frac{i}{2} \Delta(p V) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">断熱過程</span><span class="mord">: </span></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0225em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3365em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mclose">)</span></span></span></span></span></p></li></ol><h3 id="_10-16-dalton-の法則" tabindex="-1">10.16: Dalton の法則 <a class="header-anchor" href="#_10-16-dalton-の法則" aria-hidden="true">#</a></h3><ol start="16"><li><p>Dalton の法則: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><div class="tip custom-block"><p class="custom-block-title">訳者注</p><p>理想気体のみ成立</p></div></li></ol><h3 id="_10-17-沸騰" tabindex="-1">10.17: 沸騰 <a class="header-anchor" href="#_10-17-沸騰" aria-hidden="true">#</a></h3><ol start="17"><li>沸騰: 飽和蒸気の圧力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>v</mi></msub><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub><mi mathvariant="normal">.</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">p_v=p_0 .2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.2</span></span></span></span> 液の界面では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>v</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>v</mi><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p_{v 1}+p_{v 2}=p_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_10-18-熱流" tabindex="-1">10.18: 熱流 <a class="header-anchor" href="#_10-18-熱流" aria-hidden="true">#</a></h3><ol start="18"><li>熱流: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>k</mi><mi>S</mi><mi mathvariant="normal">Δ</mi><mi>T</mi><mi mathvariant="normal">/</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">P=k S \Delta T / l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> は熱伝導率). 直流回路に似て いる <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>↔</mo><mi>I</mi><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>T</mi><mo>↔</mo><mi>V</mi><mo separator="true">,</mo><mi>k</mi><mo>↔</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P \leftrightarrow I, \Delta T \leftrightarrow V, k \leftrightarrow 1 / \rho)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span>.</li></ol><h3 id="_10-19-熱容量" tabindex="-1">10.19: 熱容量 <a class="header-anchor" href="#_10-19-熱容量" aria-hidden="true">#</a></h3><ol start="19"><li>熱容量 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><mo>∫</mo><mi>c</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">Q=\int c(T) \mathrm{d} T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>. 固体では低温で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>∝</mo><msup><mi>T</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">c \propto T^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, 高温で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>3</mn><mi>N</mi><msub><mi>k</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">c=3 N k_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> （Dulong-Petit の法則. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> は結晶中の原子数）</li></ol><h3 id="_10-20-表面張力" tabindex="-1">10.20: 表面張力 <a class="header-anchor" href="#_10-20-表面張力" aria-hidden="true">#</a></h3><ol start="20"><li>表面張力 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>U</mi><mo>=</mo><mi>S</mi><mi>σ</mi><mo separator="true">,</mo><mi>F</mi><mo>=</mo><mi>l</mi><mi>σ</mi><mo separator="true">,</mo><mi>p</mi><mo>=</mo><mn>2</mn><mi>σ</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">U=S \sigma, F=l \sigma, p=2 \sigma / R </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></p></li></ol><h3 id="_10-21-stefan-boltzmann-の法則-灰色体" tabindex="-1">10.21: Stefan-Boltzmann の法則 (灰色体) <a class="header-anchor" href="#_10-21-stefan-boltzmann-の法則-灰色体" aria-hidden="true">#</a></h3><ol start="21"><li>Stefan-Boltzmann の法則 (灰色体) : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>ε</mi><mi>σ</mi><mi>A</mi><msup><mi>T</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">P=\varepsilon \sigma A T^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal">ε</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_10-22-wien-の変位則" tabindex="-1">10.22: Wien の変位則 <a class="header-anchor" href="#_10-22-wien-の変位則" aria-hidden="true">#</a></h3><ol start="22"><li>Wien の変位則: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ν</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>A</mi><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mi mathvariant="normal">/</mi><mi>h</mi><mo stretchy="false">(</mo><mi>A</mi><mo>≈</mo></mrow><annotation encoding="application/x-tex">\nu_{\max }=A k_B T / h(A \approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span></span></span></span> 2.8), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>h</mi><mi>c</mi><mi mathvariant="normal">/</mi><msup><mi>A</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mrow><mo fence="true">(</mo><msup><mi>A</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>≈</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\lambda_{\max }=h c / A^{\prime} k_B T\left(A^{\prime} \approx 5\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal">c</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">5</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/10.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_11" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-11" tabindex="-1">Formulas for IPhO 日本語版: Section 11 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-11" aria-hidden="true">#</a></h1><h2 id="_11-量子力学" tabindex="-1">11: 量子力学 <a class="header-anchor" href="#_11-量子力学" aria-hidden="true">#</a></h2><h3 id="_11-1-p-hk" tabindex="-1">11.1:p=hk <a class="header-anchor" href="#_11-1-p-hk" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi mathvariant="normal">ℏ</mi><mi mathvariant="bold-italic">k</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">p</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi>h</mi><mi mathvariant="normal">/</mi><mi>λ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>E</mi><mo>=</mo><mi mathvariant="normal">ℏ</mi><mi>ω</mi><mo>=</mo><mi>h</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}=\hbar \boldsymbol{k}(|\boldsymbol{p}|=h / \lambda), E=\hbar \omega=h \nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℏ</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.01852em;">k</span></span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord">/</span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span>.</li></ol><h3 id="_11-2-干渉" tabindex="-1">11.2: 干渉 <a class="header-anchor" href="#_11-2-干渉" aria-hidden="true">#</a></h3><ol start="2"><li>干渉 : 波動光学のように.</li></ol><h3 id="_11-3-不確定性" tabindex="-1">11.3: 不確定性 <a class="header-anchor" href="#_11-3-不確定性" aria-hidden="true">#</a></h3><ol start="3"><li><p>不確定性（数学の定理）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≥</mo><mfrac><mi mathvariant="normal">ℏ</mi><mn>2</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>E</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>≥</mo><mfrac><mi mathvariant="normal">ℏ</mi><mn>2</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>ω</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>≥</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0519em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3659em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">ℏ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0519em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3659em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">ℏ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>滑らかでない場合の定性的な推定には <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> の方が適する <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>p</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≈</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">(\Delta p \Delta x \approx h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> など <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>.</p></li></ol><h3 id="_11-4-スペクトル" tabindex="-1">11.4: スペクトル <a class="header-anchor" href="#_11-4-スペクトル" aria-hidden="true">#</a></h3><ol start="4"><li>スペクトル : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mi>ν</mi><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><mo>−</mo><msub><mi>E</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">h \nu=E_n-E_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. スペクトル線の幅は寿 命に関係し, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mi>τ</mi><mo>≈</mo><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\Gamma \tau \approx \hbar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span></span></span></span>.</li></ol><h3 id="_11-5-振動子" tabindex="-1">11.5: 振動子 <a class="header-anchor" href="#_11-5-振動子" aria-hidden="true">#</a></h3><ol start="5"><li>振動子（例えば分子）のエネルギー準位（固有振動数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><msub><mi>ν</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>:</mo><msub><mi>E</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mi>n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mi>h</mi><msub><mi>ν</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\left.\nu_0\right): E_n=\left(n+\frac{1}{2}\right) h \nu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. 多数の固有振動数の場合, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mo>∑</mo><mi>h</mi><msub><mi>n</mi><mi>i</mi></msub><msub><mi>ν</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">E=\sum h n_i \nu_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_11-6-トンネル効果" tabindex="-1">11.6: トンネル効果 <a class="header-anchor" href="#_11-6-トンネル効果" aria-hidden="true">#</a></h3><ol start="6"><li>トンネル効果: 幅 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> の障壁 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span></span></span></span> は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mi>τ</mi><mo>≈</mo><mi mathvariant="normal">ℏ</mi><mo stretchy="false">(</mo><mi>τ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\Gamma \tau \approx \hbar(\tau=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℏ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi mathvariant="normal">/</mi><msqrt><mrow><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l / \sqrt{\Gamma / m})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">Γ/</span><span class="mord mathnormal">m</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-11" aria-hidden="true">#</a></h1><h2 id="_11-量子力学" tabindex="-1">11: 量子力学 <a class="header-anchor" href="#_11-量子力学" aria-hidden="true">#</a></h2><h3 id="_11-1-p-hk" tabindex="-1">11.1:p=hk <a class="header-anchor" href="#_11-1-p-hk" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi mathvariant="normal">ℏ</mi><mi mathvariant="bold-italic">k</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">p</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi>h</mi><mi mathvariant="normal">/</mi><mi>λ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>E</mi><mo>=</mo><mi mathvariant="normal">ℏ</mi><mi>ω</mi><mo>=</mo><mi>h</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}=\hbar \boldsymbol{k}(|\boldsymbol{p}|=h / \lambda), E=\hbar \omega=h \nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℏ</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.01852em;">k</span></span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord">/</span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span>.</li></ol><h3 id="_11-2-干渉" tabindex="-1">11.2: 干渉 <a class="header-anchor" href="#_11-2-干渉" aria-hidden="true">#</a></h3><ol start="2"><li>干渉 : 波動光学のように.</li></ol><h3 id="_11-3-不確定性" tabindex="-1">11.3: 不確定性 <a class="header-anchor" href="#_11-3-不確定性" aria-hidden="true">#</a></h3><ol start="3"><li><p>不確定性（数学の定理）:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≥</mo><mfrac><mi mathvariant="normal">ℏ</mi><mn>2</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>E</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>≥</mo><mfrac><mi mathvariant="normal">ℏ</mi><mn>2</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>ω</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>≥</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0519em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3659em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">ℏ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0519em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3659em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">ℏ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>滑らかでない場合の定性的な推定には <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> の方が適する <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>p</mi><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≈</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">(\Delta p \Delta x \approx h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">Δ</span><span class="mord mathnormal">p</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span> など <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>.</p></li></ol><h3 id="_11-4-スペクトル" tabindex="-1">11.4: スペクトル <a class="header-anchor" href="#_11-4-スペクトル" aria-hidden="true">#</a></h3><ol start="4"><li>スペクトル : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mi>ν</mi><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><mo>−</mo><msub><mi>E</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">h \nu=E_n-E_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. スペクトル線の幅は寿 命に関係し, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mi>τ</mi><mo>≈</mo><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\Gamma \tau \approx \hbar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span></span></span></span>.</li></ol><h3 id="_11-5-振動子" tabindex="-1">11.5: 振動子 <a class="header-anchor" href="#_11-5-振動子" aria-hidden="true">#</a></h3><ol start="5"><li>振動子（例えば分子）のエネルギー準位（固有振動数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><msub><mi>ν</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>:</mo><msub><mi>E</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mi>n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mi>h</mi><msub><mi>ν</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\left.\nu_0\right): E_n=\left(n+\frac{1}{2}\right) h \nu_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. 多数の固有振動数の場合, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mo>∑</mo><mi>h</mi><msub><mi>n</mi><mi>i</mi></msub><msub><mi>ν</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">E=\sum h n_i \nu_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_11-6-トンネル効果" tabindex="-1">11.6: トンネル効果 <a class="header-anchor" href="#_11-6-トンネル効果" aria-hidden="true">#</a></h3><ol start="6"><li>トンネル効果: 幅 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> の障壁 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span></span></span></span> は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mi>τ</mi><mo>≈</mo><mi mathvariant="normal">ℏ</mi><mo stretchy="false">(</mo><mi>τ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\Gamma \tau \approx \hbar(\tau=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">ℏ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi mathvariant="normal">/</mi><msqrt><mrow><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l / \sqrt{\Gamma / m})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">Γ/</span><span class="mord mathnormal">m</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mclose">)</span></span></span></span> であれば容易に透過する.</li></ol><h3 id="_11-7-bohr-モデル" tabindex="-1">11.7: Bohr モデル <a class="header-anchor" href="#_11-7-bohr-モデル" aria-hidden="true">#</a></h3><ol start="7"><li>Bohr モデル : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>∝</mo><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><msup><mi>n</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_n \propto-1 / n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">1/</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. (古典的に計算される) 円軌道では, 軌道の長さが波長 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mi>h</mi><mi mathvariant="normal">/</mi><mi>m</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">\lambda=h / m v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> の整数倍.</li></ol><h3 id="_11-8-compton-効果" tabindex="-1">11.8: Compton 効果 <a class="header-anchor" href="#_11-8-compton-効果" aria-hidden="true">#</a></h3><ol start="8"><li>Compton 効果: 光子が電子から散乱されると, 光子の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi><mo>=</mo><msub><mi>λ</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta \lambda=\lambda_C(1-\cos \theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_11-9-光電効果" tabindex="-1">11.9: 光電効果 <a class="header-anchor" href="#_11-9-光電効果" aria-hidden="true">#</a></h3><ol start="9"><li>光電効果: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo>+</mo><mi>m</mi><msubsup><mi>v</mi><mi>max</mi><mo>⁡</mo><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn><mo>=</mo><mi>h</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">W+m v_{\max }^2 / 2=h \nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span></span></span></span> は仕事関数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>−</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">I-V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> グラフ：光電流は阻止電圧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mi>h</mi><mi>ν</mi><mo>−</mo><mi>W</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">V=-(h \nu-W) / e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">e</span></span></span></span> で始まり, 正方向に電圧が大きくなると緩和する.</li></ol><h3 id="_11-10-stefan-boltzmann-の法則" tabindex="-1">11.10: Stefan-Boltzmann の法則 <a class="header-anchor" href="#_11-10-stefan-boltzmann-の法則" aria-hidden="true">#</a></h3><ol start="10"><li>Stefan-Boltzmann の法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>σ</mi><mi>A</mi><msup><mi>T</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">P=\sigma A T^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/11.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path 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data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>12: Keplerの法則</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span><span class="mclose">)</span></span></span></span> であれば容易に透過する.</li></ol><h3 id="_11-7-bohr-モデル" tabindex="-1">11.7: Bohr モデル <a class="header-anchor" href="#_11-7-bohr-モデル" aria-hidden="true">#</a></h3><ol start="7"><li>Bohr モデル : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>∝</mo><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><msup><mi>n</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_n \propto-1 / n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">1/</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. (古典的に計算される) 円軌道では, 軌道の長さが波長 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>=</mo><mi>h</mi><mi mathvariant="normal">/</mi><mi>m</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">\lambda=h / m v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> の整数倍.</li></ol><h3 id="_11-8-compton-効果" tabindex="-1">11.8: Compton 効果 <a class="header-anchor" href="#_11-8-compton-効果" aria-hidden="true">#</a></h3><ol start="8"><li>Compton 効果: 光子が電子から散乱されると, 光子の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi><mo>=</mo><msub><mi>λ</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta \lambda=\lambda_C(1-\cos \theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">Δ</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_11-9-光電効果" tabindex="-1">11.9: 光電効果 <a class="header-anchor" href="#_11-9-光電効果" aria-hidden="true">#</a></h3><ol start="9"><li>光電効果: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mo>+</mo><mi>m</mi><msubsup><mi>v</mi><mi>max</mi><mo>⁡</mo><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn><mo>=</mo><mi>h</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">W+m v_{\max }^2 / 2=h \nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> ( <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span></span></span></span> は仕事関数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span>. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>−</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">I-V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span> グラフ：光電流は阻止電圧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mi>h</mi><mi>ν</mi><mo>−</mo><mi>W</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">V=-(h \nu-W) / e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathnormal">e</span></span></span></span> で始まり, 正方向に電圧が大きくなると緩和する.</li></ol><h3 id="_11-10-stefan-boltzmann-の法則" tabindex="-1">11.10: Stefan-Boltzmann の法則 <a class="header-anchor" href="#_11-10-stefan-boltzmann-の法則" aria-hidden="true">#</a></h3><ol start="10"><li>Stefan-Boltzmann の法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mi>σ</mi><mi>A</mi><msup><mi>T</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">P=\sigma A T^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/11.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_12" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-12" tabindex="-1">Formulas for IPhO 日本語版: Section 12 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-12" aria-hidden="true">#</a></h1><h2 id="_12-kepler-の法則" tabindex="-1">12: Kepler の法則 <a class="header-anchor" href="#_12-kepler-の法則" aria-hidden="true">#</a></h2><h3 id="_12-1-f-u" tabindex="-1">12.1: F &amp; U <a class="header-anchor" href="#_12-1-f-u" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mi>r</mi><mtext>. </mtext></mrow><annotation encoding="application/x-tex">F=G M m / r^2, U=-G M m / r \text {. }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord text"><span class="mord">. </span></span></span></span></span></li></ol><h3 id="_12-2-kepler-の第一法則" tabindex="-1">12.2: Kepler の第一法則 <a class="header-anchor" href="#_12-2-kepler-の第一法則" aria-hidden="true">#</a></h3><ol start="2"><li>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).</li></ol><h3 id="_12-3-kepler-の第二法則" tabindex="-1">12.3: Kepler の第二法則 <a class="header-anchor" href="#_12-3-kepler-の第二法則" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_12-4-kepler-の第三法則" tabindex="-1">12.4: Kepler の第三法則 <a class="header-anchor" href="#_12-4-kepler-の第三法則" aria-hidden="true">#</a></h3><ol start="4"><li>Kepler の第三法則 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: r^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{3}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 乗 に比例する :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>T</mi><mn>1</mn><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>T</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>a</mi><mn>1</mn><mn>3</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>a</mi><mn>2</mn><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">T_1^2 / T_2^2=a_1^3 / a_2^3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_12-5-楕円軌道" tabindex="-1">12.5: 楕円軌道 <a class="header-anchor" href="#_12-5-楕円軌道" aria-hidden="true">#</a></h3><ol start="5"><li>重力場中で楕円軌道を描く質点の全エネルギー <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>+</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K+U)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">)</span></span></span></span> :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mn>2</mn><mi>a</mi></mrow><annotation encoding="application/x-tex">E=-G M m / 2 a </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/2</span><span class="mord mathnormal">a</span></span></span></span></span></p></li></ol><h3 id="_12-6-楕円率" tabindex="-1">12.6: 楕円率 <a class="header-anchor" href="#_12-6-楕円率" aria-hidden="true">#</a></h3><ol><li>楕円率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi><mo>=</mo><mi>d</mi><mi mathvariant="normal">/</mi><mi>a</mi><mo>≪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\varepsilon=d / a \ll 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 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.</li></ol><h3 id="_12-7-楕円の性質" tabindex="-1">12.7: 楕円の性質 <a class="header-anchor" href="#_12-7-楕円の性質" aria-hidden="true">#</a></h3><ol start="2"><li>楕円の性質 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>l</mi><mn>1</mn></msub><mo>+</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mtext>（</mtext><msub><mi>l</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">: l_1+l_2=2 a （ l_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">l_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は焦点までの距 離). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (一方の焦点から出た光は他方の焦点に 反射する). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">S=\pi a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">πab</span></span></span></span>.</li></ol><h3 id="_12-8-円の中心" tabindex="-1">12.8: 円の中心 <a class="header-anchor" href="#_12-8-円の中心" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_12-9-nge-lenz-ベクトル" tabindex="-1">12.9: nge-Lenz ベクトル <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_12-9-nge-lenz-ベクトル" aria-hidden="true">#</a></h3><ol start="4"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> に一致する.]：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">e</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow></mfrac><mo>−</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0491em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3631em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/12.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div 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data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-12" aria-hidden="true">#</a></h1><h2 id="_12-kepler-の法則" tabindex="-1">12: Kepler の法則 <a class="header-anchor" href="#_12-kepler-の法則" aria-hidden="true">#</a></h2><h3 id="_12-1-f-u" tabindex="-1">12.1: F &amp; U <a class="header-anchor" href="#_12-1-f-u" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mi>r</mi><mtext>. </mtext></mrow><annotation encoding="application/x-tex">F=G M m / r^2, U=-G M m / r \text {. }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord text"><span class="mord">. </span></span></span></span></span></li></ol><h3 id="_12-2-kepler-の第一法則" tabindex="-1">12.2: Kepler の第一法則 <a class="header-anchor" href="#_12-2-kepler-の第一法則" aria-hidden="true">#</a></h3><ol start="2"><li>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).</li></ol><h3 id="_12-3-kepler-の第二法則" tabindex="-1">12.3: Kepler の第二法則 <a class="header-anchor" href="#_12-3-kepler-の第二法則" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_12-4-kepler-の第三法則" tabindex="-1">12.4: Kepler の第三法則 <a class="header-anchor" href="#_12-4-kepler-の第三法則" aria-hidden="true">#</a></h3><ol start="4"><li>Kepler の第三法則 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: r^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> 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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{3}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 乗 に比例する :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>T</mi><mn>1</mn><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>T</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>a</mi><mn>1</mn><mn>3</mn></msubsup><mi mathvariant="normal">/</mi><msubsup><mi>a</mi><mn>2</mn><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">T_1^2 / T_2^2=a_1^3 / a_2^3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_12-5-楕円軌道" tabindex="-1">12.5: 楕円軌道 <a class="header-anchor" href="#_12-5-楕円軌道" aria-hidden="true">#</a></h3><ol start="5"><li>重力場中で楕円軌道を描く質点の全エネルギー <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>+</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K+U)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">)</span></span></span></span> :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo>=</mo><mo>−</mo><mi>G</mi><mi>M</mi><mi>m</mi><mi mathvariant="normal">/</mi><mn>2</mn><mi>a</mi></mrow><annotation encoding="application/x-tex">E=-G M m / 2 a </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span><span class="mord">/2</span><span class="mord mathnormal">a</span></span></span></span></span></p></li></ol><h3 id="_12-6-楕円率" tabindex="-1">12.6: 楕円率 <a class="header-anchor" href="#_12-6-楕円率" aria-hidden="true">#</a></h3><ol><li>楕円率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi><mo>=</mo><mi>d</mi><mi mathvariant="normal">/</mi><mi>a</mi><mo>≪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\varepsilon=d / a \ll 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 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.</li></ol><h3 id="_12-7-楕円の性質" tabindex="-1">12.7: 楕円の性質 <a class="header-anchor" href="#_12-7-楕円の性質" aria-hidden="true">#</a></h3><ol start="2"><li>楕円の性質 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>l</mi><mn>1</mn></msub><mo>+</mo><msub><mi>l</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mi>a</mi><mtext>（</mtext><msub><mi>l</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">: l_1+l_2=2 a （ l_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>l</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">l_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は焦点までの距 離). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><msub><mi>α</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_1=\alpha_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (一方の焦点から出た光は他方の焦点に 反射する). <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">S=\pi a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">πab</span></span></span></span>.</li></ol><h3 id="_12-8-円の中心" tabindex="-1">12.8: 円の中心 <a class="header-anchor" href="#_12-8-円の中心" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_12-9-nge-lenz-ベクトル" tabindex="-1">12.9: nge-Lenz ベクトル <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_12-9-nge-lenz-ベクトル" aria-hidden="true">#</a></h3><ol start="4"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> に一致する.]：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">e</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>G</mi><mi>M</mi><mi>m</mi></mrow></mfrac><mo>−</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\boldsymbol{e}=\frac{\boldsymbol{v} \times \boldsymbol{M}}{G M m}-\boldsymbol{e}_r=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0491em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3631em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span><span class="mord mathnormal">m</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_13" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-13" tabindex="-1">Formulas for IPhO 日本語版: Section 13 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-13" aria-hidden="true">#</a></h1><h2 id="_13-相対性理論" tabindex="-1">13: 相対性理論 <a class="header-anchor" href="#_13-相対性理論" aria-hidden="true">#</a></h2><h3 id="_13-1-lorentz-変換" tabindex="-1">13.1:Lorentz 変換 <a class="header-anchor" href="#_13-1-lorentz-変換" aria-hidden="true">#</a></h3><ol><li><p>Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">V</mi><mo>=</mo><mi>V</mi><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo>:</mo><mi>β</mi><mo>=</mo><mi>V</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo separator="true">,</mo><mi>γ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\left.\boldsymbol{V}=V \boldsymbol{e}_x\right): \beta=V / c, \gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.25555em;">V</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></msqrt><mtext> として, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="2em"></mspace><mi>c</mi><msup><mi>t</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>c</mi><mi>t</mi><mo>−</mo><mi>β</mi><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>β</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>−</mo><mi>β</mi><msub><mi>p</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi>p</mi><mi>x</mi></msub><mo>−</mo><mi>β</mi><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>p</mi><mi>y</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> ここで, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>E</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>p</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mi>m</mi><msub><mi>v</mi><mi>x</mi></msub></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo separator="true">,</mo><mtext> etc. </mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; 1 / \sqrt{1-\beta^2} \text { として, } \\ &amp; \qquad c t^{\prime}=\gamma(c t-\beta x), x^{\prime}=\gamma(x-\beta c t), y^{\prime}=y \\ &amp; 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 \\ &amp; \text { ここで, } \\ &amp; E=\frac{m c^2}{\sqrt{1-v^2 / c^2}}=m c^2+\frac{1}{2} m v^2+\cdots \\ &amp; p_x=\frac{m v_x}{\sqrt{1-v^2 / c^2}}, \text { etc. } \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.6485em;vertical-align:-5.5742em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-7.0586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-5.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-4.0355em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-1.8844em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:0.6531em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.5742em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0067em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9667em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-13" aria-hidden="true">#</a></h1><h2 id="_13-相対性理論" tabindex="-1">13: 相対性理論 <a class="header-anchor" href="#_13-相対性理論" aria-hidden="true">#</a></h2><h3 id="_13-1-lorentz-変換" tabindex="-1">13.1:Lorentz 変換 <a class="header-anchor" href="#_13-1-lorentz-変換" aria-hidden="true">#</a></h3><ol><li><p>Lorentz 変換 (Minkowski 幾何学の 4 次元時空の回 転）(慣性系間の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="bold-italic">V</mi><mo>=</mo><mi>V</mi><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo>:</mo><mi>β</mi><mo>=</mo><mi>V</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo separator="true">,</mo><mi>γ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\left.\boldsymbol{V}=V \boldsymbol{e}_x\right): \beta=V / c, \gamma=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.25555em;">V</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></msqrt><mtext> として, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="2em"></mspace><mi>c</mi><msup><mi>t</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>c</mi><mi>t</mi><mo>−</mo><mi>β</mi><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>β</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>−</mo><mi>β</mi><msub><mi>p</mi><mi>x</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi>p</mi><mi>x</mi></msub><mo>−</mo><mi>β</mi><mi>E</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><msubsup><mi>p</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>p</mi><mi>y</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> ここで, </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>E</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>p</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mi>m</mi><msub><mi>v</mi><mi>x</mi></msub></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>c</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo separator="true">,</mo><mtext> etc. </mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; 1 / \sqrt{1-\beta^2} \text { として, } \\ &amp; \qquad c t^{\prime}=\gamma(c t-\beta x), x^{\prime}=\gamma(x-\beta c t), y^{\prime}=y \\ &amp; 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 \\ &amp; \text { ここで, } \\ &amp; E=\frac{m c^2}{\sqrt{1-v^2 / c^2}}=m c^2+\frac{1}{2} m v^2+\cdots \\ &amp; p_x=\frac{m v_x}{\sqrt{1-v^2 / c^2}}, \text { etc. } \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.6485em;vertical-align:-5.5742em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-7.0586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-5.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-4.0355em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:-1.8844em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span><span style="top:0.6531em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.5742em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.0742em;"><span style="top:-8.5586em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0067em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9667em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span></li></ol><h3 id="_13-11-電場と磁場の-lorentz-変換" tabindex="-1">13.11: 電場と磁場の Lorentz 変換 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_13-11-電場と磁場の-lorentz-変換" aria-hidden="true">#</a></h3><ol start="11"><li>電場と磁場の Lorentz 変換 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi></msub><mo separator="true">,</mo><msubsup><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0413em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span>,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo>−</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/13.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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class="title" data-v-e033cd21>12: Keplerの法則</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span></li></ol><h3 id="_13-11-電場と磁場の-lorentz-変換" tabindex="-1">13.11: 電場と磁場の Lorentz 変換 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_13-11-電場と磁場の-lorentz-変換" aria-hidden="true">#</a></h3><ol start="11"><li>電場と磁場の Lorentz 変換 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">E</mi><mi mathvariant="normal">∥</mi></msub><mo separator="true">,</mo><msubsup><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">∥</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{E}_{\|}^{\prime}=\boldsymbol{E}_{\|}, \boldsymbol{B}_{\|}^{\prime}=\boldsymbol{B}_{\|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2499em;vertical-align:-0.422em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.422em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0413em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span>,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi mathvariant="normal">/</mi><mi>c</mi><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mi>γ</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">B</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mo>−</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">/</mi><mi>c</mi><mo>×</mo><msub><mi mathvariant="bold-italic">E</mi><mo lspace="0em" rspace="0em">⊥</mo></msub><mi mathvariant="normal">/</mi><mi>c</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8279em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">c</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/13.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a 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_academic_physics_ipho-formulas-jpn_2" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-2" tabindex="-1">Formulas for IPhO 日本語版: Section 2 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-2" aria-hidden="true">#</a></h1><h2 id="_2-一般的な推奨事" tabindex="-1">2: 一般的な推奨事 <a class="header-anchor" href="#_2-一般的な推奨事" aria-hidden="true">#</a></h2><ol><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> ). 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.</li><li>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.</li><li>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.</li><li>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.</li><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/2.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-2" aria-hidden="true">#</a></h1><h2 id="_2-一般的な推奨事" tabindex="-1">2: 一般的な推奨事 <a class="header-anchor" href="#_2-一般的な推奨事" aria-hidden="true">#</a></h2><ol><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> ). 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.</li><li>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.</li><li>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.</li><li>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.</li><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/2.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div 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_academic_physics_ipho-formulas-jpn_3" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-3" tabindex="-1">Formulas for IPhO 日本語版: Section 3 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-3" aria-hidden="true">#</a></h1><h2 id="_3-運動学" tabindex="-1">3: 運動学 <a class="header-anchor" href="#_3-運動学" aria-hidden="true">#</a></h2><h3 id="_3-1-質点" tabindex="-1">3.1: 質点 <a class="header-anchor" href="#_3-1-質点" aria-hidden="true">#</a></h3><ol><li>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）：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="bold-italic">x</mi><mo>=</mo><mo>∫</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">d</mi><mi>t</mi><mrow><mo fence="true">(</mo><mi>x</mi><mo>=</mo><mo>∫</mo><msub><mi>v</mi><mi>x</mi></msub><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mtext> など </mtext><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="bold-italic">v</mi><mo>=</mo><mo>∫</mo><mi mathvariant="bold-italic">a</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>t</mi><mo>=</mo><mo>∫</mo><msubsup><mi>v</mi><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>x</mi><mo>=</mo><mo>∫</mo><msubsup><mi>a</mi><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><msub><mi>v</mi><mi>x</mi></msub><mo separator="true">,</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mfrac><msub><mi>v</mi><mi>x</mi></msub><msub><mi>a</mi><mi>x</mi></msub></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><msub><mi>v</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8756em;vertical-align:-3.6878em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1878em;"><span style="top:-6.2289em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">など</span><span class="mord"> </span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span><span style="top:-3.4878em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-0.9655em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6878em;"><span></span></span></span></span></span></span></span></span></span></span></span></p> もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 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<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mi>t</mi><mo>+</mo><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><mo>=</mo><mrow><mo fence="true">(</mo><msup><mi>v</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><mn>2</mn><mi>a</mi><mtext>. </mtext></mrow><annotation encoding="application/x-tex">x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2141em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/2</span><span class="mord mathnormal">a</span><span class="mord text"><span class="mord">. </span></span></span></span></span></span></p></li></ol><h3 id="_3-2-回転運動" tabindex="-1">3.2: 回転運動 <a class="header-anchor" href="#_3-2-回転運動" aria-hidden="true">#</a></h3><ol start="2"><li>回転運動は, 並進運動と似ていて：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>ω</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">d</mi><mi>φ</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo separator="true">,</mo><mi>ε</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi>ω</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">τ</mi><mi mathvariant="normal">d</mi><mi>v</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi mathvariant="bold-italic">n</mi><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>R</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \omega &amp; =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} &amp; =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">φ</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.13472em;">τ</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">n</span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_3-3-曲線運動" tabindex="-1">3.3: 曲線運動 <a class="header-anchor" href="#_3-3-曲線運動" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_3-4-剛体の運動" tabindex="-1">3.4: 剛体の運動 <a class="header-anchor" href="#_3-4-剛体の運動" aria-hidden="true">#</a></h3><ol start="4"><li>剛体の運動: <ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>=</mo><msub><mi>v</mi><mi>B</mi></msub><mi>cos</mi><mo>⁡</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">v_A \cos \alpha=v_B \cos \beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> ここで, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は剛体上の点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> の速度, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が直線 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> となす角.</li><li>瞬間回転中心 (#質点の軌道 の曲率中心）は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">a</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">b</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span></span></span></span> に下ろした垂線の交点. 又は もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub><mo>⊥</mo><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> ならば, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の先端を結ぶ 直線と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> の交点.</li></ul></li></ol><h3 id="_3-5-非慣性系" tabindex="-1">3.5: 非慣性系 <a class="header-anchor" href="#_3-5-非慣性系" aria-hidden="true">#</a></h3><ol start="5"><li>非慣性系：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mspace width="1em"></mspace><msub><mi mathvariant="bold-italic">v</mi><mn>2</mn></msub><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">a</mi><mn>2</mn></msub><mo>=</mo><msub><mi mathvariant="bold-italic">a</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold-italic">a</mi><mn>1</mn></msub><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi><mo>+</mo><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> ここで, </mtext><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>⊥</mo><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mi mathvariant="normal">.</mi><mtext> もし </mtext><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mtext> なら </mtext><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.</mn></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">ここで</span><span class="mord">, </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">もし</span><span class="mord"> </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">なら</span><span class="mord"> </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p></li></ol><h3 id="_3-6-弾道問題" tabindex="-1">3.6: 弾道問題 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_3-6-弾道問題" aria-hidden="true">#</a></h3><ol start="6"><li>弾道問題：到達可能な範囲は<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>≤</mo><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mn>2</mn><mi>g</mi><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><msup><mi>x</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mn>2</mn><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2141em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></p> 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.</li></ol><h3 id="_3-7-最短経路" tabindex="-1">3.7: 最短経路 <a class="header-anchor" href="#_3-7-最短経路" aria-hidden="true">#</a></h3><ol start="7"><li>最短経路を求めるには，Fermat と Huygens の原理が 使える.</li></ol><h3 id="_3-8-ベクトル" tabindex="-1">3.8: ベクトル <a class="header-anchor" href="#_3-8-ベクトル" aria-hidden="true">#</a></h3><ol start="8"><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-3" aria-hidden="true">#</a></h1><h2 id="_3-運動学" tabindex="-1">3: 運動学 <a class="header-anchor" href="#_3-運動学" aria-hidden="true">#</a></h2><h3 id="_3-1-質点" tabindex="-1">3.1: 質点 <a class="header-anchor" href="#_3-1-質点" aria-hidden="true">#</a></h3><ol><li>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）：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="bold-italic">x</mi><mo>=</mo><mo>∫</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">d</mi><mi>t</mi><mrow><mo fence="true">(</mo><mi>x</mi><mo>=</mo><mo>∫</mo><msub><mi>v</mi><mi>x</mi></msub><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mtext> など </mtext><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="bold-italic">v</mi><mo>=</mo><mo>∫</mo><mi mathvariant="bold-italic">a</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>t</mi><mo>=</mo><mo>∫</mo><msubsup><mi>v</mi><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>x</mi><mo>=</mo><mo>∫</mo><msubsup><mi>a</mi><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><msub><mi>v</mi><mi>x</mi></msub><mo separator="true">,</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mfrac><msub><mi>v</mi><mi>x</mi></msub><msub><mi>a</mi><mi>x</mi></msub></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><msub><mi>v</mi><mi>x</mi></msub></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8756em;vertical-align:-3.6878em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1878em;"><span style="top:-6.2289em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">など</span><span class="mord"> </span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span><span style="top:-3.4878em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">d</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-0.9655em;"><span class="pstrut" style="height:3.4911em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6878em;"><span></span></span></span></span></span></span></span></span></span></span></span></p> もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 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<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mi>t</mi><mo>+</mo><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn><mo>=</mo><mrow><mo fence="true">(</mo><msup><mi>v</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><mn>2</mn><mi>a</mi><mtext>. </mtext></mrow><annotation encoding="application/x-tex">x=v_0 t+a t^2 / 2=\left(v^2-v_0^2\right) / 2 a \text {. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2141em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/2</span><span class="mord mathnormal">a</span><span class="mord text"><span class="mord">. </span></span></span></span></span></span></p></li></ol><h3 id="_3-2-回転運動" tabindex="-1">3.2: 回転運動 <a class="header-anchor" href="#_3-2-回転運動" aria-hidden="true">#</a></h3><ol start="2"><li>回転運動は, 並進運動と似ていて：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>ω</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">d</mi><mi>φ</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo separator="true">,</mo><mi>ε</mi><mo>=</mo><mi mathvariant="normal">d</mi><mi>ω</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">τ</mi><mi mathvariant="normal">d</mi><mi>v</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi mathvariant="bold-italic">n</mi><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>R</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \omega &amp; =\mathrm{d} \varphi / \mathrm{d} t, \varepsilon=\mathrm{d} \omega / \mathrm{d} t \\ \boldsymbol{a} &amp; =\boldsymbol{\tau} \mathrm{d} v / \mathrm{d} t+\boldsymbol{n} v^2 / R \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">φ</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.13472em;">τ</span></span></span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">n</span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_3-3-曲線運動" tabindex="-1">3.3: 曲線運動 <a class="header-anchor" href="#_3-3-曲線運動" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_3-4-剛体の運動" tabindex="-1">3.4: 剛体の運動 <a class="header-anchor" href="#_3-4-剛体の運動" aria-hidden="true">#</a></h3><ol start="4"><li>剛体の運動: <ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>A</mi></msub><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>=</mo><msub><mi>v</mi><mi>B</mi></msub><mi>cos</mi><mo>⁡</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">v_A \cos \alpha=v_B \cos \beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> ここで, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は剛体上の点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> の速度, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> が直線 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> となす角.</li><li>瞬間回転中心 (#質点の軌道 の曲率中心）は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">a</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">b</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">b</span></span></span></span></span></span> に下ろした垂線の交点. 又は もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub><mo>⊥</mo><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A, \boldsymbol{v}_B \perp A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> ならば, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の先端を結ぶ 直線と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">A B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> の交点.</li></ul></li></ol><h3 id="_3-5-非慣性系" tabindex="-1">3.5: 非慣性系 <a class="header-anchor" href="#_3-5-非慣性系" aria-hidden="true">#</a></h3><ol start="5"><li>非慣性系：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mspace width="1em"></mspace><msub><mi mathvariant="bold-italic">v</mi><mn>2</mn></msub><mo>=</mo><msub><mi mathvariant="bold-italic">v</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">a</mi><mn>2</mn></msub><mo>=</mo><msub><mi mathvariant="bold-italic">a</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold-italic">a</mi><mn>1</mn></msub><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi><mo>+</mo><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> ここで, </mtext><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>⊥</mo><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mi mathvariant="normal">.</mi><mtext> もし </mtext><msub><mi mathvariant="bold-italic">v</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mtext> なら </mtext><msub><mi mathvariant="bold-italic">a</mi><mrow><mi>C</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.</mn></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">ここで</span><span class="mord">, </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">もし</span><span class="mord"> </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">なら</span><span class="mord"> </span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">or</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p></li></ol><h3 id="_3-6-弾道問題" tabindex="-1">3.6: 弾道問題 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_3-6-弾道問題" aria-hidden="true">#</a></h3><ol start="6"><li>弾道問題：到達可能な範囲は<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>≤</mo><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mn>2</mn><mi>g</mi><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><msup><mi>x</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mn>2</mn><msubsup><mi>v</mi><mn>0</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y \leq v_0^2 /(2 g)-g x^2 /\left(2 v_0^2\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2141em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></p> 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.</li></ol><h3 id="_3-7-最短経路" tabindex="-1">3.7: 最短経路 <a class="header-anchor" href="#_3-7-最短経路" aria-hidden="true">#</a></h3><ol start="7"><li>最短経路を求めるには，Fermat と Huygens の原理が 使える.</li></ol><h3 id="_3-8-ベクトル" tabindex="-1">3.8: ベクトル <a class="header-anchor" href="#_3-8-ベクトル" aria-hidden="true">#</a></h3><ol start="8"><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div 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data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_4" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-4" tabindex="-1">Formulas for IPhO 日本語版: Section 4 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-4" aria-hidden="true">#</a></h1><h2 id="_4-力学" tabindex="-1">4: 力学 <a class="header-anchor" href="#_4-力学" aria-hidden="true">#</a></h2><h3 id="_4-1-剛体の二次元的な平衡" tabindex="-1">4.1: 剛体の二次元的な平衡 <a class="header-anchor" href="#_4-1-剛体の二次元的な平衡" aria-hidden="true">#</a></h3><ol><li>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.</li></ol><h3 id="_4-2-垂直抗力" tabindex="-1">4.2: 垂直抗力 <a class="header-anchor" href="#_4-2-垂直抗力" aria-hidden="true">#</a></h3><ol start="2"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度で接触点に加わる.</li></ol><h3 id="_4-3-並進運動と回転運動" tabindex="-1">4.3: 並進運動と回転運動 <a class="header-anchor" href="#_4-3-並進運動と回転運動" aria-hidden="true">#</a></h3><ol start="3"><li>並進運動と回転運動についての Newton の第二法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>m</mi><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">ε</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo>×</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=m \boldsymbol{a}, \boldsymbol{M}=I \boldsymbol{\varepsilon} \quad(\boldsymbol{M}=\boldsymbol{r} \times \boldsymbol{F}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol">ε</span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mclose">)</span></span></span></span></span></p> 二次元の場合には <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span> は本質的にスカラーで, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mi>F</mi><mi>l</mi><mo>=</mo><msub><mi>F</mi><mi>t</mi></msub><mi>r</mi><mo stretchy="false">(</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">M=F l=F_t r(l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> は力のうでの長さ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_4-4-一般化座標" tabindex="-1">4.4: 一般化座標 <a class="header-anchor" href="#_4-4-一般化座標" aria-hidden="true">#</a></h3><ol start="4"><li>一般化座標. 系の状態が 1 つの変数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span> とその時間微分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>ξ</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{\xi}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> で表され，ポテンシャルエネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U=U(\xi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span></span></span></span>, 運動エネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo>=</mo><mi>μ</mi><msup><mi>ξ</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">K=\mu \xi^2 / 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span> であるならば, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mover accent="true"><mi>ξ</mi><mo>¨</mo></mover><mo>=</mo></mrow><annotation encoding="application/x-tex">\mu \ddot{\xi}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">¨</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">d</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>ξ</mi></mrow><annotation encoding="application/x-tex">-\mathrm{d} U(\xi) / \mathrm{d} \xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span>. (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)</li></ol><h3 id="_4-5-系質点" tabindex="-1">4.5: 系質点 <a class="header-anchor" href="#_4-5-系質点" aria-hidden="true">#</a></h3><ol start="5"><li>系が質点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> で構成されているとき：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">r</mi><mi>c</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mi mathvariant="normal">/</mi><mo>∑</mo><msub><mi>m</mi><mi>j</mi></msub><mo separator="true">,</mo><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>I</mi><mi>z</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>=</mo><mo>∫</mo><mrow><mo fence="true">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi>m</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \boldsymbol{r}_c=\sum m_i \boldsymbol{r}_i / \sum m_j, \boldsymbol{P}=\sum m_i \boldsymbol{v}_i \\ &amp; \boldsymbol{L}=\sum m_i \boldsymbol{r}_i \times \boldsymbol{v}_i, K=\sum m_i v_i^2 / 2 \\ &amp; I_z=\sum m_i\left(x_i^2+y_i^2\right)=\int\left(x^2+y^2\right) \mathrm{d} m \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8223em;vertical-align:-3.6611em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">m</span></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-6-質量中心の速度" tabindex="-1">4.6: 質量中心の速度 <a class="header-anchor" href="#_4-6-質量中心の速度" aria-hidden="true">#</a></h3><ol start="6"><li>質量中心の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> であるような系 (添え字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> は質量 中心についての物理量であることを示す）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">R</mi><mi>c</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><msub><mi>K</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msubsup><mi>v</mi><mi>c</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">P</mi><mo>=</mo><msub><mi mathvariant="bold-italic">P</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.8979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-7-steiner-定理" tabindex="-1">4.7: Steiner 定理 <a class="header-anchor" href="#_4-7-steiner-定理" aria-hidden="true">#</a></h3><ol start="7"><li>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）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>c</mi></msub><mo>+</mo><mi>m</mi><msup><mi>b</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I=I_c+m b^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-8-ポイント-6" tabindex="-1">4.8: ポイント 6 <a class="header-anchor" href="#_4-8-ポイント-6" aria-hidden="true">#</a></h3><ol start="8"><li>ポイント 6 の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">P</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">L</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span></span></span></span> を用いて, Newton の第二法則 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">F</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">P</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">M</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">L</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span></span></span></p></li></ol><h3 id="_4-9-ポイント-5" tabindex="-1">4.9: ポイント 5 <a class="header-anchor" href="#_4-9-ポイント-5" aria-hidden="true">#</a></h3><ol start="9"><li>ポイント 5 にに加えて，質量中心を通る <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> 軸に対する慣性モ一メントは <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mrow><mi>z</mi><mn>0</mn></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">I_{z 0}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><msub><mi>m</mi><mi>i</mi></msub><msub><mi>m</mi><mi>j</mi></msub><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo fence="true">]</mo></mrow><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mn>2</mn><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><mo fence="true">)</mo></mrow></mrow><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)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></li></ol><h3 id="_4-10-原点に対する慣性" tabindex="-1">4.10: 原点に対する慣性 <a class="header-anchor" href="#_4-10-原点に対する慣性" aria-hidden="true">#</a></h3><ol start="10"><li>原点に対する慣性モ一メント <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>r</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\theta=\sum m_i r_i^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0728em;vertical-align:-0.2587em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>θ</mi><mo>=</mo><msub><mi>I</mi><mi>x</mi></msub><mo>+</mo><msub><mi>I</mi><mi>y</mi></msub><mo>+</mo><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">2 \theta=I_x+I_y+I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を用いることで二次元物体や等 方性のある物体の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を計算するのに有用.</li></ol><h3 id="_4-11-相当単振子の長" tabindex="-1">4.11: 相当単振子の長 <a class="header-anchor" href="#_4-11-相当単振子の長" aria-hidden="true">#</a></h3><ol start="11"><li>相当単振子の長さが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{l}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9313em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> である物理振子 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>ω</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>g</mi><mi mathvariant="normal">/</mi><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow></msqrt><mo separator="true">,</mo><mspace width="1em"></mspace><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>=</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \omega^2(l)=g /\left(l+I_c / m l\right) \\ &amp; \omega(l)=\omega(\tilde{l}-l)=\sqrt{g / \tilde{l}}, \quad \tilde{l}=l+I_c / m l \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.1641em;vertical-align:-2.3321em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3745em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span><span style="top:-3.3345em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-4" aria-hidden="true">#</a></h1><h2 id="_4-力学" tabindex="-1">4: 力学 <a class="header-anchor" href="#_4-力学" aria-hidden="true">#</a></h2><h3 id="_4-1-剛体の二次元的な平衡" tabindex="-1">4.1: 剛体の二次元的な平衡 <a class="header-anchor" href="#_4-1-剛体の二次元的な平衡" aria-hidden="true">#</a></h3><ol><li>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.</li></ol><h3 id="_4-2-垂直抗力" tabindex="-1">4.2: 垂直抗力 <a class="header-anchor" href="#_4-2-垂直抗力" aria-hidden="true">#</a></h3><ol start="2"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度で接触点に加わる.</li></ol><h3 id="_4-3-並進運動と回転運動" tabindex="-1">4.3: 並進運動と回転運動 <a class="header-anchor" href="#_4-3-並進運動と回転運動" aria-hidden="true">#</a></h3><ol start="3"><li>並進運動と回転運動についての Newton の第二法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>m</mi><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">ε</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo>×</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=m \boldsymbol{a}, \boldsymbol{M}=I \boldsymbol{\varepsilon} \quad(\boldsymbol{M}=\boldsymbol{r} \times \boldsymbol{F}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol">ε</span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mclose">)</span></span></span></span></span></p> 二次元の場合には <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span> は本質的にスカラーで, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mi>F</mi><mi>l</mi><mo>=</mo><msub><mi>F</mi><mi>t</mi></msub><mi>r</mi><mo stretchy="false">(</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">M=F l=F_t r(l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> は力のうでの長さ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_4-4-一般化座標" tabindex="-1">4.4: 一般化座標 <a class="header-anchor" href="#_4-4-一般化座標" aria-hidden="true">#</a></h3><ol start="4"><li>一般化座標. 系の状態が 1 つの変数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span> とその時間微分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>ξ</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{\xi}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> で表され，ポテンシャルエネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U=U(\xi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span></span></span></span>, 運動エネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo>=</mo><mi>μ</mi><msup><mi>ξ</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">K=\mu \xi^2 / 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span> であるならば, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mover accent="true"><mi>ξ</mi><mo>¨</mo></mover><mo>=</mo></mrow><annotation encoding="application/x-tex">\mu \ddot{\xi}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">¨</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">d</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>ξ</mi></mrow><annotation encoding="application/x-tex">-\mathrm{d} U(\xi) / \mathrm{d} \xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span>. (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)</li></ol><h3 id="_4-5-系質点" tabindex="-1">4.5: 系質点 <a class="header-anchor" href="#_4-5-系質点" aria-hidden="true">#</a></h3><ol start="5"><li>系が質点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> で構成されているとき：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">r</mi><mi>c</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mi mathvariant="normal">/</mi><mo>∑</mo><msub><mi>m</mi><mi>j</mi></msub><mo separator="true">,</mo><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>I</mi><mi>z</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>=</mo><mo>∫</mo><mrow><mo fence="true">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi>m</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \boldsymbol{r}_c=\sum m_i \boldsymbol{r}_i / \sum m_j, \boldsymbol{P}=\sum m_i \boldsymbol{v}_i \\ &amp; \boldsymbol{L}=\sum m_i \boldsymbol{r}_i \times \boldsymbol{v}_i, K=\sum m_i v_i^2 / 2 \\ &amp; I_z=\sum m_i\left(x_i^2+y_i^2\right)=\int\left(x^2+y^2\right) \mathrm{d} m \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8223em;vertical-align:-3.6611em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">m</span></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-6-質量中心の速度" tabindex="-1">4.6: 質量中心の速度 <a class="header-anchor" href="#_4-6-質量中心の速度" aria-hidden="true">#</a></h3><ol start="6"><li>質量中心の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> であるような系 (添え字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> は質量 中心についての物理量であることを示す）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">R</mi><mi>c</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><msub><mi>K</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msubsup><mi>v</mi><mi>c</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">P</mi><mo>=</mo><msub><mi mathvariant="bold-italic">P</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.8979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-7-steiner-定理" tabindex="-1">4.7: Steiner 定理 <a class="header-anchor" href="#_4-7-steiner-定理" aria-hidden="true">#</a></h3><ol start="7"><li>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）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>c</mi></msub><mo>+</mo><mi>m</mi><msup><mi>b</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I=I_c+m b^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-8-ポイント-6" tabindex="-1">4.8: ポイント 6 <a class="header-anchor" href="#_4-8-ポイント-6" aria-hidden="true">#</a></h3><ol start="8"><li>ポイント 6 の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">P</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">L</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span></span></span></span> を用いて, Newton の第二法則 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">F</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">P</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">M</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">L</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span></span></span></p></li></ol><h3 id="_4-9-ポイント-5" tabindex="-1">4.9: ポイント 5 <a class="header-anchor" href="#_4-9-ポイント-5" aria-hidden="true">#</a></h3><ol start="9"><li>ポイント 5 にに加えて，質量中心を通る <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> 軸に対する慣性モ一メントは <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mrow><mi>z</mi><mn>0</mn></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">I_{z 0}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><msub><mi>m</mi><mi>i</mi></msub><msub><mi>m</mi><mi>j</mi></msub><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo fence="true">]</mo></mrow><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mn>2</mn><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><mo fence="true">)</mo></mrow></mrow><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)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></li></ol><h3 id="_4-10-原点に対する慣性" tabindex="-1">4.10: 原点に対する慣性 <a class="header-anchor" href="#_4-10-原点に対する慣性" aria-hidden="true">#</a></h3><ol start="10"><li>原点に対する慣性モ一メント <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>r</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\theta=\sum m_i r_i^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0728em;vertical-align:-0.2587em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>θ</mi><mo>=</mo><msub><mi>I</mi><mi>x</mi></msub><mo>+</mo><msub><mi>I</mi><mi>y</mi></msub><mo>+</mo><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">2 \theta=I_x+I_y+I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を用いることで二次元物体や等 方性のある物体の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を計算するのに有用.</li></ol><h3 id="_4-11-相当単振子の長" tabindex="-1">4.11: 相当単振子の長 <a class="header-anchor" href="#_4-11-相当単振子の長" aria-hidden="true">#</a></h3><ol start="11"><li>相当単振子の長さが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{l}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9313em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> である物理振子 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>ω</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>g</mi><mi mathvariant="normal">/</mi><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow></msqrt><mo separator="true">,</mo><mspace width="1em"></mspace><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>=</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \omega^2(l)=g /\left(l+I_c / m l\right) \\ &amp; \omega(l)=\omega(\tilde{l}-l)=\sqrt{g / \tilde{l}}, \quad \tilde{l}=l+I_c / m l \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.1641em;vertical-align:-2.3321em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3745em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span><span style="top:-3.3345em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4655em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-12-慣性モーメントの係数" tabindex="-1">4.12: 慣性モーメントの係数 <a class="header-anchor" href="#_4-12-慣性モーメントの係数" aria-hidden="true">#</a></h3><ol start="12"><li>慣性モーメントの係数 : 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球殼 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 棒 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> （端に対しては <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ), 正方形 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_4-13-よく使われる保存則" tabindex="-1">4.13: よく使われる保存則 <a class="header-anchor" href="#_4-13-よく使われる保存則" aria-hidden="true">#</a></h3><ol start="13"><li>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))</li></ol><h3 id="_4-14-非慣性系における見かけの力" tabindex="-1">4.14: 非慣性系における見かけの力 <a class="header-anchor" href="#_4-14-非慣性系における見かけの力" aria-hidden="true">#</a></h3><ol start="14"><li>非慣性系における見かけの力 : 慣性力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>m</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">-m a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal">ma</span></span></span></span>, 遠心力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><msup><mi>ω</mi><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi></mrow><annotation encoding="application/x-tex">m \omega^2 \boldsymbol{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span></span></span></span>, Coriolis 力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>m</mi><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">2 m \boldsymbol{v} \times \Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> (避けた方がよい. 速 度に垂直なので仕事はしない).</li></ol><h3 id="_4-15-傾いた座標" tabindex="-1">4.15: 傾いた座標 <a class="header-anchor" href="#_4-15-傾いた座標" aria-hidden="true">#</a></h3><ol start="15"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 成分と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 成分をもつ. 軸は斜交することもある が, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><msub><mi>v</mi><mi>x</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo>+</mo><msub><mi>v</mi><mi>y</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> のとき <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">v_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 軸への射 影ではない.</li></ol><h3 id="_4-16-2-つの物体の衝突" tabindex="-1">4.16: 2 つの物体の衝突 <a class="header-anchor" href="#_4-16-2-つの物体の衝突" aria-hidden="true">#</a></h3><ol start="16"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度をなす.</li></ol><h3 id="_4-17-剛体のすべての運動" tabindex="-1">4.17: 剛体のすべての運動 <a class="header-anchor" href="#_4-17-剛体のすべての運動" aria-hidden="true">#</a></h3><ol start="17"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> まわりの回転として表せる. 物体上の点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> からの距離は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の軌跡の曲率半径とは異なる ことに注意せよ.</li></ol><h3 id="_4-18-紐の張力" tabindex="-1">4.18: 紐の張力 <a class="header-anchor" href="#_4-18-紐の張力" aria-hidden="true">#</a></h3><ol start="18"><li>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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi>T</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">N=T / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 似た場合と して, 表面張力による圧力は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mi>σ</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">p=2 \sigma / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 導出には直 径に沿った圧力を調ベる.</li></ol><h3 id="_4-19-液体の表面" tabindex="-1">4.19: 液体の表面 <a class="header-anchor" href="#_4-19-液体の表面" aria-hidden="true">#</a></h3><ol start="19"><li>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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span> をポテンシャルエ ネルギーの体積密度として, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><mo>−</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">p=P_0-w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span>.</li></ol><h3 id="_4-20-非圧縮性流体に対する-bernoulli-の法則" tabindex="-1">4.20: 非圧縮性流体に対する Bernoulli の法則 <a class="header-anchor" href="#_4-20-非圧縮性流体に対する-bernoulli-の法則" aria-hidden="true">#</a></h3><ol start="20"><li>非圧縮性流体に対する Bernoulli の法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mi>ρ</mi><mi>ϕ</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p> 一様な重力場では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\phi=g h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">h</span></span></span></span>. 比熱が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>p</mi></msub><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">J</mi></mrow><mi mathvariant="normal">/</mi><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">c_p[\mathrm{~J} / \mathrm{kg}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">J</span></span><span class="mord">/</span><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">kg</span></span><span class="mclose">]</span></span></span></span> である気 体では,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><msub><mi>c</mi><mi>p</mi></msub><mi>T</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\frac{1}{2} v^2+c_p T=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol><h3 id="_4-21-直線的な流線" tabindex="-1">4.21: 直線的な流線 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-21-直線的な流線" aria-hidden="true">#</a></h3><ol start="21"><li>直線的な流線に沾う運動量の連続性 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>+</mo><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">p+\rho v^2=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const.</li></ol><h3 id="_4-22-断熱不変量" tabindex="-1">4.22: 断熱不変量 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-22-断熱不変量" aria-hidden="true">#</a></h3><ol start="22"><li>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（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">x-p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 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.</li></ol><h3 id="_4-23-安定性" tabindex="-1">4.23: 安定性 <a class="header-anchor" href="#_4-23-安定性" aria-hidden="true">#</a></h3><ol start="23"><li>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.</li></ol><h3 id="_4-24-空間的に有限な運動に対する-virial-定理" tabindex="-1">4.24: 空間的に有限な運動に対する Virial 定理 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-24-空間的に有限な運動に対する-virial-定理" aria-hidden="true">#</a></h3><ol start="24"><li>空間的に有限な運動に対する Virial 定理：a) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord">∣</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle K\rangle=\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span> (時間平均). b) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><msup><mi mathvariant="normal">∣</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">2\langle K\rangle=-\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span>.</li></ol><h3 id="_4-25-tsiolkovsky-公式" tabindex="-1">4.25: Tsiolkovsky 公式 <a class="header-anchor" href="#_4-25-tsiolkovsky-公式" aria-hidden="true">#</a></h3><ol start="25"><li>Tsiolkovsky の公式（ロケット）: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>v</mi><mo>=</mo><mi>u</mi><mi>ln</mi><mo>⁡</mo><mfrac><mi>M</mi><mi>m</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta v=u \ln \frac{M}{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/4.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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class="title" data-v-e033cd21>3: 運動学</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/academic/physics/ipho-formulas-jpn/5" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>5: 振動と波</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4655em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-12-慣性モーメントの係数" tabindex="-1">4.12: 慣性モーメントの係数 <a class="header-anchor" href="#_4-12-慣性モーメントの係数" aria-hidden="true">#</a></h3><ol start="12"><li>慣性モーメントの係数 : 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球殼 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 棒 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> （端に対しては <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ), 正方形 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_4-13-よく使われる保存則" tabindex="-1">4.13: よく使われる保存則 <a class="header-anchor" href="#_4-13-よく使われる保存則" aria-hidden="true">#</a></h3><ol start="13"><li>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))</li></ol><h3 id="_4-14-非慣性系における見かけの力" tabindex="-1">4.14: 非慣性系における見かけの力 <a class="header-anchor" href="#_4-14-非慣性系における見かけの力" aria-hidden="true">#</a></h3><ol start="14"><li>非慣性系における見かけの力 : 慣性力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>m</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">-m a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal">ma</span></span></span></span>, 遠心力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><msup><mi>ω</mi><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi></mrow><annotation encoding="application/x-tex">m \omega^2 \boldsymbol{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span></span></span></span>, Coriolis 力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>m</mi><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">2 m \boldsymbol{v} \times \Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> (避けた方がよい. 速 度に垂直なので仕事はしない).</li></ol><h3 id="_4-15-傾いた座標" tabindex="-1">4.15: 傾いた座標 <a class="header-anchor" href="#_4-15-傾いた座標" aria-hidden="true">#</a></h3><ol start="15"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 成分と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 成分をもつ. 軸は斜交することもある が, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><msub><mi>v</mi><mi>x</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo>+</mo><msub><mi>v</mi><mi>y</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> のとき <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">v_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 軸への射 影ではない.</li></ol><h3 id="_4-16-2-つの物体の衝突" tabindex="-1">4.16: 2 つの物体の衝突 <a class="header-anchor" href="#_4-16-2-つの物体の衝突" aria-hidden="true">#</a></h3><ol start="16"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度をなす.</li></ol><h3 id="_4-17-剛体のすべての運動" tabindex="-1">4.17: 剛体のすべての運動 <a class="header-anchor" href="#_4-17-剛体のすべての運動" aria-hidden="true">#</a></h3><ol start="17"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> まわりの回転として表せる. 物体上の点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> からの距離は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の軌跡の曲率半径とは異なる ことに注意せよ.</li></ol><h3 id="_4-18-紐の張力" tabindex="-1">4.18: 紐の張力 <a class="header-anchor" href="#_4-18-紐の張力" aria-hidden="true">#</a></h3><ol start="18"><li>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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi>T</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">N=T / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 似た場合と して, 表面張力による圧力は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mi>σ</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">p=2 \sigma / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 導出には直 径に沿った圧力を調ベる.</li></ol><h3 id="_4-19-液体の表面" tabindex="-1">4.19: 液体の表面 <a class="header-anchor" href="#_4-19-液体の表面" aria-hidden="true">#</a></h3><ol start="19"><li>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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span> をポテンシャルエ ネルギーの体積密度として, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><mo>−</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">p=P_0-w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span>.</li></ol><h3 id="_4-20-非圧縮性流体に対する-bernoulli-の法則" tabindex="-1">4.20: 非圧縮性流体に対する Bernoulli の法則 <a class="header-anchor" href="#_4-20-非圧縮性流体に対する-bernoulli-の法則" aria-hidden="true">#</a></h3><ol start="20"><li>非圧縮性流体に対する Bernoulli の法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mi>ρ</mi><mi>ϕ</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p> 一様な重力場では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\phi=g h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">h</span></span></span></span>. 比熱が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>p</mi></msub><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">J</mi></mrow><mi mathvariant="normal">/</mi><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">c_p[\mathrm{~J} / \mathrm{kg}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">J</span></span><span class="mord">/</span><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">kg</span></span><span class="mclose">]</span></span></span></span> である気 体では,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><msub><mi>c</mi><mi>p</mi></msub><mi>T</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\frac{1}{2} v^2+c_p T=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol><h3 id="_4-21-直線的な流線" tabindex="-1">4.21: 直線的な流線 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-21-直線的な流線" aria-hidden="true">#</a></h3><ol start="21"><li>直線的な流線に沾う運動量の連続性 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>+</mo><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">p+\rho v^2=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const.</li></ol><h3 id="_4-22-断熱不変量" tabindex="-1">4.22: 断熱不変量 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-22-断熱不変量" aria-hidden="true">#</a></h3><ol start="22"><li>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（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">x-p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 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.</li></ol><h3 id="_4-23-安定性" tabindex="-1">4.23: 安定性 <a class="header-anchor" href="#_4-23-安定性" aria-hidden="true">#</a></h3><ol start="23"><li>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.</li></ol><h3 id="_4-24-空間的に有限な運動に対する-virial-定理" tabindex="-1">4.24: 空間的に有限な運動に対する Virial 定理 <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_4-24-空間的に有限な運動に対する-virial-定理" aria-hidden="true">#</a></h3><ol start="24"><li>空間的に有限な運動に対する Virial 定理：a) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord">∣</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle K\rangle=\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span> (時間平均). b) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><msup><mi mathvariant="normal">∣</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">2\langle K\rangle=-\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span>.</li></ol><h3 id="_4-25-tsiolkovsky-公式" tabindex="-1">4.25: Tsiolkovsky 公式 <a class="header-anchor" href="#_4-25-tsiolkovsky-公式" aria-hidden="true">#</a></h3><ol start="25"><li>Tsiolkovsky の公式（ロケット）: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>v</mi><mo>=</mo><mi>u</mi><mi>ln</mi><mo>⁡</mo><mfrac><mi>M</mi><mi>m</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta v=u \ln \frac{M}{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/4.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_5" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-5" tabindex="-1">Formulas for IPhO 日本語版: Section 5 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-5" aria-hidden="true">#</a></h1><h2 id="_5-振動と波" tabindex="-1">5. 振動と波 <a class="header-anchor" href="#_5-振動と波" aria-hidden="true">#</a></h2><h3 id="_5-1-減衰振動" tabindex="-1">5.1: 減衰振動 <a class="header-anchor" href="#_5-1-減衰振動" aria-hidden="true">#</a></h3><ol><li>減衰振動：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>2</mn><mi>γ</mi><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mi>x</mi><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>&lt;</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ddot{x}+2 \gamma \dot{x}+\omega_0^2 x=0(\gamma&lt;\omega) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7512em;vertical-align:-0.0833em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">¨</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span></span></span></span></span></p> 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) :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>γ</mi><mi>t</mi></mrow></msup><mi>sin</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi>t</mi><msqrt><mrow><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></msqrt><mo>−</mo><msub><mi>φ</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">x=x_0 e^{-\gamma t} \sin \left(t \sqrt{\omega_0^2-\gamma^2}-\varphi_0\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord mathnormal">t</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2987em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4337em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2587em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-5" aria-hidden="true">#</a></h1><h2 id="_5-振動と波" tabindex="-1">5. 振動と波 <a class="header-anchor" href="#_5-振動と波" aria-hidden="true">#</a></h2><h3 id="_5-1-減衰振動" tabindex="-1">5.1: 減衰振動 <a class="header-anchor" href="#_5-1-減衰振動" aria-hidden="true">#</a></h3><ol><li>減衰振動：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>2</mn><mi>γ</mi><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mi>x</mi><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>&lt;</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ddot{x}+2 \gamma \dot{x}+\omega_0^2 x=0(\gamma&lt;\omega) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7512em;vertical-align:-0.0833em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">¨</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span></span></span></span></span></p> 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) :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><msup><mi>e</mi><mrow><mo>−</mo><mi>γ</mi><mi>t</mi></mrow></msup><mi>sin</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mi>t</mi><msqrt><mrow><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></msqrt><mo>−</mo><msub><mi>φ</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">x=x_0 e^{-\gamma t} \sin \left(t \sqrt{\omega_0^2-\gamma^2}-\varphi_0\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord mathnormal">t</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2987em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4337em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2587em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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@@ -99,10 +99,10 @@ s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_5-10-doppler-効果" tabindex="-1">5.10: Doppler 効果 <a class="header-anchor" href="#_5-10-doppler-効果" aria-hidden="true">#</a></h3><ol start="10"><li>Doppler 効果 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo>=</mo><msub><mi>ν</mi><mn>0</mn></msub><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mi mathvariant="normal">∥</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>s</mi></msub></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>u</mi><mi mathvariant="normal">∥</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>s</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6973em;vertical-align:-0.6036em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0936em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5357em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3695em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.5686em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5357em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3695em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6036em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_5-11-huygens-の原理" tabindex="-1">5.11: Huygens の原理 <a class="header-anchor" href="#_5-11-huygens-の原理" aria-hidden="true">#</a></h3><ol start="11"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><msub><mi>c</mi><mi>s</mi></msub><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\Delta x=c_s \Delta t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span></span> で区切られた曲線（ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\Delta t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span></span> は時間 間隔， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">c_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は与えられた点の速度). 波は波面に垂直に 進む.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div 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+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_5-10-doppler-効果" tabindex="-1">5.10: Doppler 効果 <a class="header-anchor" href="#_5-10-doppler-効果" aria-hidden="true">#</a></h3><ol start="10"><li>Doppler 効果 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi><mo>=</mo><msub><mi>ν</mi><mn>0</mn></msub><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mi mathvariant="normal">∥</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>s</mi></msub></mrow><mrow><mn>1</mn><mo>−</mo><msub><mi>u</mi><mi mathvariant="normal">∥</mi></msub><mi mathvariant="normal">/</mi><msub><mi>c</mi><mi>s</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\nu=\nu_0 \frac{1+v_{\|} / c_s}{1-u_{\|} / c_s}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6973em;vertical-align:-0.6036em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0936em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5357em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3695em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.5686em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5357em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3695em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6036em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_5-11-huygens-の原理" tabindex="-1">5.11: Huygens の原理 <a class="header-anchor" href="#_5-11-huygens-の原理" aria-hidden="true">#</a></h3><ol start="11"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><msub><mi>c</mi><mi>s</mi></msub><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\Delta x=c_s \Delta t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span></span> で区切られた曲線（ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\Delta t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span></span> は時間 間隔， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">c_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は与えられた点の速度). 波は波面に垂直に 進む.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div 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data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_6" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-6" tabindex="-1">Formulas for IPhO 日本語版: Section 6 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-6" aria-hidden="true">#</a></h1><h2 id="_6-幾何光学-測光" tabindex="-1">6: 幾何光学，測光 <a class="header-anchor" href="#_6-幾何光学-測光" aria-hidden="true">#</a></h2><h3 id="_6-1-fermat-原理" tabindex="-1">6.1: Fermat 原理 <a class="header-anchor" href="#_6-1-fermat-原理" aria-hidden="true">#</a></h3><ol><li>Fermat の原理 : 点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> から <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> への波の経路は波の移動 時間が最も短いもの.</li></ol><h3 id="_6-2-snell-法則" tabindex="-1">6.2: Snell 法則 <a class="header-anchor" href="#_6-2-snell-法則" aria-hidden="true">#</a></h3><ol start="2"><li>Snell の法則 :</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><msub><mi>α</mi><mn>1</mn></msub><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><msub><mi>α</mi><mn>2</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>v</mi><mn>1</mn></msub><mi mathvariant="normal">/</mi><msub><mi>v</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><h3 id="_6-3" tabindex="-1">6.3: <a class="header-anchor" href="#_6-3" aria-hidden="true">#</a></h3><ol start="3"><li>屈折率が連続的に変化するならば，媒質を屈折率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>n</mi><mi mathvariant="normal">/</mi><mi>r</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">n^{\prime}=n / r \quad(r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> は曲率半径 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_6-4" tabindex="-1">6.4: <a class="header-anchor" href="#_6-4" aria-hidden="true">#</a></h3><ol start="4"><li>屈折率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">z</mi></mrow><annotation encoding="application/x-tex">\mathrm{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathrm">z</span></span></span></span> 座標にのみ依存するならば, 光子の運動量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>p</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">p_x, p_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> とエネルギーは保存される:</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mtext> const., </mtext><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">k</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">/</mi><mi>n</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord"> const., </span></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.01852em;">k</span></span></span><span class="mord">∣/</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p><h3 id="_6-5-薄いレンズの式" tabindex="-1">6.5:薄いレンズの式 <a class="header-anchor" href="#_6-5-薄いレンズの式" aria-hidden="true">#</a></h3><ol start="6"><li>薄いレンズの式（符号に注意する）：</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>a</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>b</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>f</mi><mo>≡</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">1 / a+1 / b=1 / f \equiv D </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></span></p><h3 id="_6-6-newton-の式" tabindex="-1">6.6: Newton の式 <a class="header-anchor" href="#_6-6-newton-の式" aria-hidden="true">#</a></h3><ol start="6"><li>Newton の式 : 物体側焦点から物体までの距離を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, 像側焦点から像までの距離を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> とすると, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><msup><mi>f</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x_1 x_2=f^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_6-7-像の位置を求める視差法" tabindex="-1">6.7: 像の位置を求める視差法 <a class="header-anchor" href="#_6-7-像の位置を求める視差法" aria-hidden="true">#</a></h3><ol start="7"><li>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.</li></ol><h3 id="_6-8-レンズを通る光線の経路の幾何学的な描き方" tabindex="-1">6.8: レンズを通る光線の経路の幾何学的な描き方 <a class="header-anchor" href="#_6-8-レンズを通る光線の経路の幾何学的な描き方" aria-hidden="true">#</a></h3><ol start="8"><li>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．</li></ol><h3 id="_6-9-光束" tabindex="-1">6.9: 光束 <a class="header-anchor" href="#_6-9-光束" aria-hidden="true">#</a></h3><ol start="9"><li>光束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span> [単位: lumen <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">lm</mi><mo>⁡</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\operatorname{lm})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop"><span class="mord mathrm">lm</span></span><span class="mclose">)</span></span></span></span>] 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">I=\Phi / \Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/Ω</span></span></span></span>. 照度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">lux</mi><mo>⁡</mo><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">x</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\operatorname{lux}(\mathrm{lx})]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mop"><span class="mord mathrm">lux</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">lx</span></span><span class="mclose">)]</span></span></span></span> は（面に入射する） 面積あたりの光束で, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">E=\Phi / S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</li></ol><h3 id="_6-10-gauss-定理" tabindex="-1">6.10: Gauss 定理 <a class="header-anchor" href="#_6-10-gauss-定理" aria-hidden="true">#</a></h3><ol start="10"><li>光束についての Gauss の定理 : 光度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">I_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の点光源を囲 む閉曲面を通って外に出る光束は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mn>4</mn><mi>π</mi><mo>∑</mo><msub><mi>I</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Phi=4 \pi \sum I_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. 光 源が 1 つで距離が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> のとき <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mi>I</mi><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E=I / r^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_6-11-実験のヒント" tabindex="-1">6.11: 実験のヒント <a class="header-anchor" href="#_6-11-実験のヒント" aria-hidden="true">#</a></h3><ol start="11"><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/6.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path 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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-6" aria-hidden="true">#</a></h1><h2 id="_6-幾何光学-測光" tabindex="-1">6: 幾何光学，測光 <a class="header-anchor" href="#_6-幾何光学-測光" aria-hidden="true">#</a></h2><h3 id="_6-1-fermat-原理" tabindex="-1">6.1: Fermat 原理 <a class="header-anchor" href="#_6-1-fermat-原理" aria-hidden="true">#</a></h3><ol><li>Fermat の原理 : 点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> から <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> への波の経路は波の移動 時間が最も短いもの.</li></ol><h3 id="_6-2-snell-法則" tabindex="-1">6.2: Snell 法則 <a class="header-anchor" href="#_6-2-snell-法則" aria-hidden="true">#</a></h3><ol start="2"><li>Snell の法則 :</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>sin</mi><mo>⁡</mo><msub><mi>α</mi><mn>1</mn></msub><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><msub><mi>α</mi><mn>2</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>v</mi><mn>1</mn></msub><mi mathvariant="normal">/</mi><msub><mi>v</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\sin \alpha_1 / \sin \alpha_2=n_2 / n_1=v_1 / v_2 . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p><h3 id="_6-3" tabindex="-1">6.3: <a class="header-anchor" href="#_6-3" aria-hidden="true">#</a></h3><ol start="3"><li>屈折率が連続的に変化するならば，媒質を屈折率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>n</mi><mi mathvariant="normal">/</mi><mi>r</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">n^{\prime}=n / r \quad(r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> は曲率半径 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_6-4" tabindex="-1">6.4: <a class="header-anchor" href="#_6-4" aria-hidden="true">#</a></h3><ol start="4"><li>屈折率が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">z</mi></mrow><annotation encoding="application/x-tex">\mathrm{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathrm">z</span></span></span></span> 座標にのみ依存するならば, 光子の運動量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>p</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">p_x, p_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> とエネルギーは保存される:</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mtext> const., </mtext><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">k</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">/</mi><mi>n</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">k_x, k_y=\text { const., }|\boldsymbol{k}| / n=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord"> const., </span></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.01852em;">k</span></span></span><span class="mord">∣/</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p><h3 id="_6-5-薄いレンズの式" tabindex="-1">6.5:薄いレンズの式 <a class="header-anchor" href="#_6-5-薄いレンズの式" aria-hidden="true">#</a></h3><ol start="6"><li>薄いレンズの式（符号に注意する）：</li></ol><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>a</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>b</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>f</mi><mo>≡</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">1 / a+1 / b=1 / f \equiv D </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></span></p><h3 id="_6-6-newton-の式" tabindex="-1">6.6: Newton の式 <a class="header-anchor" href="#_6-6-newton-の式" aria-hidden="true">#</a></h3><ol start="6"><li>Newton の式 : 物体側焦点から物体までの距離を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, 像側焦点から像までの距離を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> とすると, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><msup><mi>f</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x_1 x_2=f^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_6-7-像の位置を求める視差法" tabindex="-1">6.7: 像の位置を求める視差法 <a class="header-anchor" href="#_6-7-像の位置を求める視差法" aria-hidden="true">#</a></h3><ol start="7"><li>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.</li></ol><h3 id="_6-8-レンズを通る光線の経路の幾何学的な描き方" tabindex="-1">6.8: レンズを通る光線の経路の幾何学的な描き方 <a class="header-anchor" href="#_6-8-レンズを通る光線の経路の幾何学的な描き方" aria-hidden="true">#</a></h3><ol start="8"><li>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．</li></ol><h3 id="_6-9-光束" tabindex="-1">6.9: 光束 <a class="header-anchor" href="#_6-9-光束" aria-hidden="true">#</a></h3><ol start="9"><li>光束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span> [単位: lumen <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">lm</mi><mo>⁡</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\operatorname{lm})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop"><span class="mord mathrm">lm</span></span><span class="mclose">)</span></span></span></span>] 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">I=\Phi / \Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/Ω</span></span></span></span>. 照度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">lux</mi><mo>⁡</mo><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">x</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\operatorname{lux}(\mathrm{lx})]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mop"><span class="mord mathrm">lux</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">lx</span></span><span class="mclose">)]</span></span></span></span> は（面に入射する） 面積あたりの光束で, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">E=\Phi / S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>.</li></ol><h3 id="_6-10-gauss-定理" tabindex="-1">6.10: Gauss 定理 <a class="header-anchor" href="#_6-10-gauss-定理" aria-hidden="true">#</a></h3><ol start="10"><li>光束についての Gauss の定理 : 光度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">I_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> の点光源を囲 む閉曲面を通って外に出る光束は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mn>4</mn><mi>π</mi><mo>∑</mo><msub><mi>I</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Phi=4 \pi \sum I_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. 光 源が 1 つで距離が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> のとき <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mi>I</mi><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E=I / r^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_6-11-実験のヒント" tabindex="-1">6.11: 実験のヒント <a class="header-anchor" href="#_6-11-実験のヒント" aria-hidden="true">#</a></h3><ol start="11"><li>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.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_7" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-7" tabindex="-1">Formulas for IPhO 日本語版: Section 7 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-7" aria-hidden="true">#</a></h1><h2 id="_7-波動光学" tabindex="-1">7: 波動光学 <a class="header-anchor" href="#_7-波動光学" aria-hidden="true">#</a></h2><h3 id="_7-1-huygens-の原理に基づいた回折" tabindex="-1">7.1: Huygens の原理に基づいた回折 <a class="header-anchor" href="#_7-1-huygens-の原理に基づいた回折" aria-hidden="true">#</a></h3><ol><li>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.</li></ol><h3 id="_7-2-二重スリット" tabindex="-1">7.2: 二重スリット <a class="header-anchor" href="#_7-2-二重スリット" aria-hidden="true">#</a></h3><ol start="2"><li>二重スリット（幅は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>≪</mo><mi>a</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \ll a, \lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span> による干渉:強 め合う角 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>φ</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">.</mi><mi>I</mi><mo>∝</mo></mrow><annotation encoding="application/x-tex">\varphi_{\max }=\arcsin (n \lambda / d), n \in \mathbb{Z} . I \propto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arcsin</span><span class="mopen">(</span><span class="mord mathnormal">nλ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">Z</span><span class="mord">.</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mrow><mo fence="true">(</mo><mi>k</mi><mfrac><mi>a</mi><mn>2</mn></mfrac><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos ^2\left(k \frac{a}{2} \sin \varphi\right),(k=2 \pi / \lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord">/</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_7-3-単スリット-弱め合う角" tabindex="-1">7.3: 単スリット-弱め合う角 <a class="header-anchor" href="#_7-3-単スリット-弱め合う角" aria-hidden="true">#</a></h3><ol start="3"><li>単スリット:弱め合う角: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>φ</mi><mtext>min </mtext></msub><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>n</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\varphi_{\text {min }}=\arcsin (n \lambda / d), n \in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3175em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arcsin</span><span class="mopen">(</span><span class="mord mathnormal">nλ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo separator="true">,</mo><mi>n</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}, n \neq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbb">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 中央の強め合う部分は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=\pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span> の間である ことに注意せよ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>∝</mo><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mrow><mo fence="true">(</mo><mi>k</mi><mfrac><mi>d</mi><mn>2</mn></mfrac><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">I \propto \sin ^2\left(k \frac{d}{2} \sin \varphi\right) / \sin \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2301em;vertical-align:-0.35em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span></li></ol><h3 id="_7-4-回折格子" tabindex="-1">7.4: 回折格子 <a class="header-anchor" href="#_7-4-回折格子" aria-hidden="true">#</a></h3><ol start="4"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> を回折格子の正味の長さとすれ ばポイント 3 と同じ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 番目の明線のスペクトルの分 解能は，溝の総数を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> 本として <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mi>n</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\frac{\lambda}{\Delta \lambda}=n N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</li></ol><h3 id="_7-5-分光器の分解能" tabindex="-1">7.5: 分光器の分解能 <a class="header-anchor" href="#_7-5-分光器の分解能" aria-hidden="true">#</a></h3><ol start="5"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> として, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mfrac><mi>L</mi><mi>λ</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\lambda}{\Delta \lambda}=\frac{L}{\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_7-6-プリズムの分解能" tabindex="-1">7.6: プリズムの分解能 <a class="header-anchor" href="#_7-6-プリズムの分解能" aria-hidden="true">#</a></h3><ol start="7"><li>プリズムの分解能 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>n</mi></mrow><mrow><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>λ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">: \frac{\lambda}{\Delta \lambda}=a \frac{\mathrm{d} n}{\mathrm{~d} \lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mspace nobreak mtight"><span class="mtight"> </span></span><span class="mord mathrm mtight">d</span></span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li></ol><h3 id="_7-7-角度距離" tabindex="-1">7.7: 角度距離 <a class="header-anchor" href="#_7-7-角度距離" aria-hidden="true">#</a></h3><ol start="7"><li>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 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi><mo>≈</mo><mn>1.22</mn><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">\varphi \approx 1.22 \lambda / d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.22</span><span class="mord mathnormal">λ</span><span class="mord">/</span><span class="mord mathnormal">d</span></span></span></span>. 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.</li></ol><h3 id="_7-8-bragg-の法則" tabindex="-1">7.8: Bragg の法則 <a class="header-anchor" href="#_7-8-bragg-の法則" aria-hidden="true">#</a></h3><ol start="8"><li>Bragg の法則：間隔が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> の平行な結晶面の組は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>d</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mi>n</mi><mi>λ</mi></mrow><annotation encoding="application/x-tex">2 d \sin \theta=n \lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">2</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">nλ</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> 線を反射する. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> は結 晶面と X 線がなす角 (かすめ角).</li></ol><h3 id="_7-9-高密度電体媒質反射" tabindex="-1">7.9: 高密度電体媒質反射 <a class="header-anchor" href="#_7-9-高密度電体媒質反射" aria-hidden="true">#</a></h3><ol start="9"><li>光学的に高密度な誘電体媒質による反射 : 位相が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span> ず れる. 半透明の薄膜では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">→</mo></msub><mo>+</mo><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">←</mo></msub><mo>=</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">\phi_{\rightarrow}+\phi_{\leftarrow}=\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">→</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">←</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span>. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">→</mo></msub></mrow><annotation encoding="application/x-tex">\phi_{\rightarrow}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">→</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">←</mo></msub></mrow><annotation encoding="application/x-tex">\phi_{\leftarrow}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">←</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は反射波と透過波の位相差（矢印は入射方向を 示す)</li></ol><h3 id="_7-10-fabry-perot-干渉計" tabindex="-1">7.10: Fabry-Pérot 干渉計 <a class="header-anchor" href="#_7-10-fabry-perot-干渉計" aria-hidden="true">#</a></h3><ol start="10"><li>Fabry-Pérot 干渉計 : 高い反射率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo>≪</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(1-r \ll 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> を持 つ 2 枚の平行な半透明の鏡. 分解能は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>ν</mi><mrow><mi mathvariant="normal">Δ</mi><mi>ν</mi></mrow></mfrac><mo>≈</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><mrow><mi>λ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\nu}{\Delta \nu} \approx \frac{2 a}{\lambda(1-r)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3651em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 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.</li></ol><h3 id="_7-11-コヒーレントな電磁波" tabindex="-1">7.11: コヒーレントな電磁波 <a class="header-anchor" href="#_7-11-コヒーレントな電磁波" aria-hidden="true">#</a></h3><ol start="11"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">n=n(\omega)=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\varepsilon(\omega)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal">ε</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-7" aria-hidden="true">#</a></h1><h2 id="_7-波動光学" tabindex="-1">7: 波動光学 <a class="header-anchor" href="#_7-波動光学" aria-hidden="true">#</a></h2><h3 id="_7-1-huygens-の原理に基づいた回折" tabindex="-1">7.1: Huygens の原理に基づいた回折 <a class="header-anchor" href="#_7-1-huygens-の原理に基づいた回折" aria-hidden="true">#</a></h3><ol><li>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.</li></ol><h3 id="_7-2-二重スリット" tabindex="-1">7.2: 二重スリット <a class="header-anchor" href="#_7-2-二重スリット" aria-hidden="true">#</a></h3><ol start="2"><li>二重スリット（幅は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>≪</mo><mi>a</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \ll a, \lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span> による干渉:強 め合う角 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>φ</mi><mi>max</mi><mo>⁡</mo></msub><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">.</mi><mi>I</mi><mo>∝</mo></mrow><annotation encoding="application/x-tex">\varphi_{\max }=\arcsin (n \lambda / d), n \in \mathbb{Z} . I \propto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arcsin</span><span class="mopen">(</span><span class="mord mathnormal">nλ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">Z</span><span class="mord">.</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mrow><mo fence="true">(</mo><mi>k</mi><mfrac><mi>a</mi><mn>2</mn></mfrac><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos ^2\left(k \frac{a}{2} \sin \varphi\right),(k=2 \pi / \lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord">/</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_7-3-単スリット-弱め合う角" tabindex="-1">7.3: 単スリット-弱め合う角 <a class="header-anchor" href="#_7-3-単スリット-弱め合う角" aria-hidden="true">#</a></h3><ol start="3"><li>単スリット:弱め合う角: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>φ</mi><mtext>min </mtext></msub><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>n</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\varphi_{\text {min }}=\arcsin (n \lambda / d), n \in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3175em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">min </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">arcsin</span><span class="mopen">(</span><span class="mord mathnormal">nλ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo separator="true">,</mo><mi>n</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}, n \neq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathbb">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 中央の強め合う部分は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=\pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span> の間である ことに注意せよ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>∝</mo><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mrow><mo fence="true">(</mo><mi>k</mi><mfrac><mi>d</mi><mn>2</mn></mfrac><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">I \propto \sin ^2\left(k \frac{d}{2} \sin \varphi\right) / \sin \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2301em;vertical-align:-0.35em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span></li></ol><h3 id="_7-4-回折格子" tabindex="-1">7.4: 回折格子 <a class="header-anchor" href="#_7-4-回折格子" aria-hidden="true">#</a></h3><ol start="4"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> を回折格子の正味の長さとすれ ばポイント 3 と同じ. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 番目の明線のスペクトルの分 解能は，溝の総数を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> 本として <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mi>n</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">\frac{\lambda}{\Delta \lambda}=n N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>.</li></ol><h3 id="_7-5-分光器の分解能" tabindex="-1">7.5: 分光器の分解能 <a class="header-anchor" href="#_7-5-分光器の分解能" aria-hidden="true">#</a></h3><ol start="5"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> として, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mfrac><mi>L</mi><mi>λ</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\lambda}{\Delta \lambda}=\frac{L}{\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_7-6-プリズムの分解能" tabindex="-1">7.6: プリズムの分解能 <a class="header-anchor" href="#_7-6-プリズムの分解能" aria-hidden="true">#</a></h3><ol start="7"><li>プリズムの分解能 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mfrac><mi>λ</mi><mrow><mi mathvariant="normal">Δ</mi><mi>λ</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>n</mi></mrow><mrow><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>λ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">: \frac{\lambda}{\Delta \lambda}=a \frac{\mathrm{d} n}{\mathrm{~d} \lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mspace nobreak mtight"><span class="mtight"> </span></span><span class="mord mathrm mtight">d</span></span><span class="mord mathnormal mtight">λ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">d</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li></ol><h3 id="_7-7-角度距離" tabindex="-1">7.7: 角度距離 <a class="header-anchor" href="#_7-7-角度距離" aria-hidden="true">#</a></h3><ol start="7"><li>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 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi><mo>≈</mo><mn>1.22</mn><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">\varphi \approx 1.22 \lambda / d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1.22</span><span class="mord mathnormal">λ</span><span class="mord">/</span><span class="mord mathnormal">d</span></span></span></span>. 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.</li></ol><h3 id="_7-8-bragg-の法則" tabindex="-1">7.8: Bragg の法則 <a class="header-anchor" href="#_7-8-bragg-の法則" aria-hidden="true">#</a></h3><ol start="8"><li>Bragg の法則：間隔が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> の平行な結晶面の組は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>d</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mi>n</mi><mi>λ</mi></mrow><annotation encoding="application/x-tex">2 d \sin \theta=n \lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">2</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">nλ</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> 線を反射する. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> は結 晶面と X 線がなす角 (かすめ角).</li></ol><h3 id="_7-9-高密度電体媒質反射" tabindex="-1">7.9: 高密度電体媒質反射 <a class="header-anchor" href="#_7-9-高密度電体媒質反射" aria-hidden="true">#</a></h3><ol start="9"><li>光学的に高密度な誘電体媒質による反射 : 位相が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span> ず れる. 半透明の薄膜では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">→</mo></msub><mo>+</mo><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">←</mo></msub><mo>=</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">\phi_{\rightarrow}+\phi_{\leftarrow}=\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">→</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">←</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span>. ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">→</mo></msub></mrow><annotation encoding="application/x-tex">\phi_{\rightarrow}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">→</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϕ</mi><mo lspace="0em" rspace="0em">←</mo></msub></mrow><annotation encoding="application/x-tex">\phi_{\leftarrow}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1068em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">←</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は反射波と透過波の位相差（矢印は入射方向を 示す)</li></ol><h3 id="_7-10-fabry-perot-干渉計" tabindex="-1">7.10: Fabry-Pérot 干渉計 <a class="header-anchor" href="#_7-10-fabry-perot-干渉計" aria-hidden="true">#</a></h3><ol start="10"><li>Fabry-Pérot 干渉計 : 高い反射率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo>≪</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(1-r \ll 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≪</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> を持 つ 2 枚の平行な半透明の鏡. 分解能は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>ν</mi><mrow><mi mathvariant="normal">Δ</mi><mi>ν</mi></mrow></mfrac><mo>≈</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><mrow><mi>λ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\nu}{\Delta \nu} \approx \frac{2 a}{\lambda(1-r)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3651em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">λ</span><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 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.</li></ol><h3 id="_7-11-コヒーレントな電磁波" tabindex="-1">7.11: コヒーレントな電磁波 <a class="header-anchor" href="#_7-11-コヒーレントな電磁波" aria-hidden="true">#</a></h3><ol start="11"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">n=n(\omega)=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\varepsilon(\omega)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.305em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal">ε</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span> (普通 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>≈</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mu \approx 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> ) 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）: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><mi>c</mi><mi>n</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>E</mi><mn>2</mn></msup><mo>=</mo><mfrac><mi>c</mi><mrow><mi>n</mi><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><msup><mi>B</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">I=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9641em;vertical-align:-0.15em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2952em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> は実 効值)</li></ol><h3 id="_7-12-malus-の法則" tabindex="-1">7.12: Malus の法則 <a class="header-anchor" href="#_7-12-malus-の法則" aria-hidden="true">#</a></h3><ol start="12"><li>Malus の法則 : 直線偏光が角度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span> で偏光板を通過する と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi><mn>0</mn></msub><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>φ</mi></mrow><annotation encoding="application/x-tex">I=I_0 \cos ^2 \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span></li></ol><h3 id="_7-13-1-4-波長版" tabindex="-1">7.13: 1/4 波長版 <a class="header-anchor" href="#_7-13-1-4-波長版" aria-hidden="true">#</a></h3><ol start="13"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1 / 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span></span></span></span> 波長版 : 直線偏光成分間の位相が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\pi / 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord">/2</span></span></span></span> ずれる.</li></ol><h3 id="_7-14-brewster-角" tabindex="-1">7.14: Brewster 角 <a class="header-anchor" href="#_7-14-brewster-角" aria-hidden="true">#</a></h3><ol start="14"><li>Brewster 角: 入射角が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\tan \varphi=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 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.</li></ol><h3 id="_7-15-光学素子による回折" tabindex="-1">7.15: 光学素子による回折 <a class="header-anchor" href="#_7-15-光学素子による回折" aria-hidden="true">#</a></h3><ol start="15"><li>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.</li></ol><h3 id="_7-16-光ファイバー" tabindex="-1">7.16: 光ファイバー <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_7-16-光ファイバー" aria-hidden="true">#</a></h3><ol start="17"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>n</mi><mi mathvariant="normal">/</mi><mi>n</mi><mo>≈</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Delta n / n \approx \frac{1}{2}(\lambda / d)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">n</span><span class="mord">/</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/7.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/academic/physics/ipho-formulas-jpn/6" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>6: 幾何光学，測光</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/academic/physics/ipho-formulas-jpn/8" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>8: 電気回路</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span> (普通 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>≈</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mu \approx 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6776em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> ) 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）: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><mi>c</mi><mi>n</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>E</mi><mn>2</mn></msup><mo>=</mo><mfrac><mi>c</mi><mrow><mi>n</mi><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><msup><mi>B</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">I=c n \varepsilon_0 E^2=\frac{c}{n \mu_0} B^2(E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9641em;vertical-align:-0.15em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2952em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> は実 効值)</li></ol><h3 id="_7-12-malus-の法則" tabindex="-1">7.12: Malus の法則 <a class="header-anchor" href="#_7-12-malus-の法則" aria-hidden="true">#</a></h3><ol start="12"><li>Malus の法則 : 直線偏光が角度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span> で偏光板を通過する と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi><mn>0</mn></msub><msup><mrow><mi>cos</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>φ</mi></mrow><annotation encoding="application/x-tex">I=I_0 \cos ^2 \varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span></li></ol><h3 id="_7-13-1-4-波長版" tabindex="-1">7.13: 1/4 波長版 <a class="header-anchor" href="#_7-13-1-4-波長版" aria-hidden="true">#</a></h3><ol start="13"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1 / 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span></span></span></span> 波長版 : 直線偏光成分間の位相が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\pi / 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord">/2</span></span></span></span> ずれる.</li></ol><h3 id="_7-14-brewster-角" tabindex="-1">7.14: Brewster 角 <a class="header-anchor" href="#_7-14-brewster-角" aria-hidden="true">#</a></h3><ol start="14"><li>Brewster 角: 入射角が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\tan \varphi=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 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.</li></ol><h3 id="_7-15-光学素子による回折" tabindex="-1">7.15: 光学素子による回折 <a class="header-anchor" href="#_7-15-光学素子による回折" aria-hidden="true">#</a></h3><ol start="15"><li>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.</li></ol><h3 id="_7-16-光ファイバー" tabindex="-1">7.16: 光ファイバー <span class="VPBadge tip" data-v-350d3852><!--[-->supplemental<!--]--></span> <a class="header-anchor" href="#_7-16-光ファイバー" aria-hidden="true">#</a></h3><ol start="17"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\mathrm{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> 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, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>n</mi><mi mathvariant="normal">/</mi><mi>n</mi><mo>≈</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>λ</mi><mi mathvariant="normal">/</mi><mi>d</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Delta n / n \approx \frac{1}{2}(\lambda / d)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">n</span><span class="mord">/</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord">/</span><span class="mord mathnormal">d</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>.</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/7.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path 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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_8" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-8" tabindex="-1">Formulas for IPhO 日本語版: Section 8 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-8" aria-hidden="true">#</a></h1><h2 id="_8-電気回路" tabindex="-1">8: 電気回路 <a class="header-anchor" href="#_8-電気回路" aria-hidden="true">#</a></h2><h3 id="_8-1-v-i-r-p-v-i" tabindex="-1">8.1: V=I R, P=V I <a class="header-anchor" href="#_8-1-v-i-r-p-v-i" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mi>I</mi><mi>R</mi><mo separator="true">,</mo><mi>P</mi><mo>=</mo><mi>V</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">V=I R, P=V I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>R</mi><mtext>直列 </mtext></msub><mo>=</mo><mo>∑</mo><msub><mi>R</mi><mi>i</mi></msub><mo separator="true">,</mo><msubsup><mi>R</mi><mtext>並列 </mtext><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mo>∑</mo><msubsup><mi>R</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">R_{\text {直列 }}=\sum R_i, R_{\text {並列 }}^{-1}=\sum R_i^{-1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">直列</span><span class="mord mtight"> </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4163em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">並列</span><span class="mord mtight"> </span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2837em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.433em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.267em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_8-2-kirchhoff-の法則" tabindex="-1">8.2: Kirchhoff の法則 <a class="header-anchor" href="#_8-2-kirchhoff-の法則" aria-hidden="true">#</a></h3><ol start="2"><li>Kirchhoff の法則 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo>∑</mo><mstyle scriptlevel="1"><mtable rowspacing="0.1em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mtext> 節点 </mtext></mstyle></mtd></mtr></mtable></mstyle></munder><mi>I</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><munder><mo>∑</mo><mtext>閉路 </mtext></munder><mi>V</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{\substack{\text { 節点 }}} I=0, \sum_{\text {閉路 }} V=0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4703em;vertical-align:-1.4203em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8568em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6817em;"><span style="top:-2.6983em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight"> </span><span class="mord cjk_fallback mtight">節点</span><span class="mord mtight"> </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1817em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4203em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3443em;vertical-align:-1.2943em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">閉路</span><span class="mord mtight"> </span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2943em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ol><h3 id="_8-3-ポイント-2-の方程式を減らすために" tabindex="-1">8.3: ポイント 2 の方程式を減らすために <a class="header-anchor" href="#_8-3-ポイント-2-の方程式を減らすために" aria-hidden="true">#</a></h3><ol start="3"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo><mi mathvariant="normal">△</mi><mtext>又</mtext></mrow><annotation encoding="application/x-tex">\Rightarrow \triangle 又</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord cjk_fallback">又</span></span></span></span> 又 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> の形, 起電力のある 2 端子の場合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> 抵抗と電池の直列)</li></ol><h3 id="_8-4-無限につながる抵抗" tabindex="-1">8.4: 無限につながる抵抗 <a class="header-anchor" href="#_8-4-無限につながる抵抗" aria-hidden="true">#</a></h3><ol start="4"><li>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.</li></ol><h3 id="_8-5-交流回路" tabindex="-1">8.5: 交流回路 <a class="header-anchor" href="#_8-5-交流回路" aria-hidden="true">#</a></h3><ol start="5"><li>交流回路: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span> を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> に置き換えてポイント <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>∼</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">1 \sim 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> を用 いる.<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>Z</mi><mi>R</mi></msub><mo>=</mo><mi>R</mi><mo separator="true">,</mo><msub><mi>Z</mi><mi>C</mi></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>i</mi><mi>ω</mi><mi>C</mi><mo separator="true">,</mo><msub><mi>Z</mi><mi>L</mi></msub><mo>=</mo><mi>i</mi><mi>ω</mi><mi>L</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>φ</mi><mo>=</mo><mi>arg</mi><mo>⁡</mo><mi>Z</mi><mo separator="true">,</mo><msub><mi>V</mi><mtext>eff </mtext></msub><mo>=</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><msub><mi>I</mi><mtext>eff </mtext></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>V</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>I</mi><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>arg</mi><mo>⁡</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∑</mo><msubsup><mi>I</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>R</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.9em;vertical-align:-2.2em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7em;"><span style="top:-4.91em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span><span class="mord mathnormal">L</span></span></span><span style="top:-3.41em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">eff </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">eff </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.7em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_8-6-特性時間" tabindex="-1">8.6: 特性時間 <a class="header-anchor" href="#_8-6-特性時間" aria-hidden="true">#</a></h3><ol start="6"><li>特性時間: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mrow><mi>R</mi><mi>C</mi></mrow></msub><mo>=</mo><mi>R</mi><mi>C</mi><mo separator="true">,</mo><msub><mi>τ</mi><mrow><mi>L</mi><mi>R</mi></mrow></msub><mo>=</mo><mi>L</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi mathvariant="normal">.</mi><msub><mi>ω</mi><mrow><mi>L</mi><mi>C</mi></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\tau_{R C}=R C, \tau_{L R}=L / R . \omega_{L C}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">RC</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord">.</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mi>L</mi><mi>C</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">1 / \sqrt{L C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1767em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 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href="#formulas-for-ipho-日本語版-section-8" aria-hidden="true">#</a></h1><h2 id="_8-電気回路" tabindex="-1">8: 電気回路 <a class="header-anchor" href="#_8-電気回路" aria-hidden="true">#</a></h2><h3 id="_8-1-v-i-r-p-v-i" tabindex="-1">8.1: V=I R, P=V I <a class="header-anchor" href="#_8-1-v-i-r-p-v-i" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mi>I</mi><mi>R</mi><mo separator="true">,</mo><mi>P</mi><mo>=</mo><mi>V</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">V=I R, P=V I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>R</mi><mtext>直列 </mtext></msub><mo>=</mo><mo>∑</mo><msub><mi>R</mi><mi>i</mi></msub><mo separator="true">,</mo><msubsup><mi>R</mi><mtext>並列 </mtext><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mo>∑</mo><msubsup><mi>R</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">R_{\text {直列 }}=\sum R_i, R_{\text {並列 }}^{-1}=\sum R_i^{-1} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">直列</span><span class="mord mtight"> </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4163em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">並列</span><span class="mord mtight"> </span></span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2837em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.433em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.267em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_8-2-kirchhoff-の法則" tabindex="-1">8.2: Kirchhoff の法則 <a class="header-anchor" href="#_8-2-kirchhoff-の法則" aria-hidden="true">#</a></h3><ol start="2"><li>Kirchhoff の法則 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo>∑</mo><mstyle scriptlevel="1"><mtable rowspacing="0.1em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mtext> 節点 </mtext></mstyle></mtd></mtr></mtable></mstyle></munder><mi>I</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><munder><mo>∑</mo><mtext>閉路 </mtext></munder><mi>V</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{\substack{\text { 節点 }}} I=0, \sum_{\text {閉路 }} V=0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4703em;vertical-align:-1.4203em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8568em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6817em;"><span style="top:-2.6983em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight"> </span><span class="mord cjk_fallback mtight">節点</span><span class="mord mtight"> </span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1817em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4203em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3443em;vertical-align:-1.2943em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">閉路</span><span class="mord mtight"> </span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2943em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ol><h3 id="_8-3-ポイント-2-の方程式を減らすために" tabindex="-1">8.3: ポイント 2 の方程式を減らすために <a class="header-anchor" href="#_8-3-ポイント-2-の方程式を減らすために" aria-hidden="true">#</a></h3><ol start="3"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo><mi mathvariant="normal">△</mi><mtext>又</mtext></mrow><annotation encoding="application/x-tex">\Rightarrow \triangle 又</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span><span class="mord cjk_fallback">又</span></span></span></span> 又 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> の形, 起電力のある 2 端子の場合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> 抵抗と電池の直列)</li></ol><h3 id="_8-4-無限につながる抵抗" tabindex="-1">8.4: 無限につながる抵抗 <a class="header-anchor" href="#_8-4-無限につながる抵抗" aria-hidden="true">#</a></h3><ol start="4"><li>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.</li></ol><h3 id="_8-5-交流回路" tabindex="-1">8.5: 交流回路 <a class="header-anchor" href="#_8-5-交流回路" aria-hidden="true">#</a></h3><ol start="5"><li>交流回路: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span> を <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> に置き換えてポイント <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>∼</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">1 \sim 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> を用 いる.<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>Z</mi><mi>R</mi></msub><mo>=</mo><mi>R</mi><mo separator="true">,</mo><msub><mi>Z</mi><mi>C</mi></msub><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>i</mi><mi>ω</mi><mi>C</mi><mo separator="true">,</mo><msub><mi>Z</mi><mi>L</mi></msub><mo>=</mo><mi>i</mi><mi>ω</mi><mi>L</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>φ</mi><mo>=</mo><mi>arg</mi><mo>⁡</mo><mi>Z</mi><mo separator="true">,</mo><msub><mi>V</mi><mtext>eff </mtext></msub><mo>=</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><msub><mi>I</mi><mtext>eff </mtext></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>V</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>I</mi><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>arg</mi><mo>⁡</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∑</mo><msubsup><mi>I</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>R</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable><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} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.9em;vertical-align:-2.2em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7em;"><span style="top:-4.91em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">iω</span><span class="mord mathnormal">L</span></span></span><span style="top:-3.41em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord mathnormal">φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">eff </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">eff </span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.7em;"><span class="pstrut" style="height:3.05em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_8-6-特性時間" tabindex="-1">8.6: 特性時間 <a class="header-anchor" href="#_8-6-特性時間" aria-hidden="true">#</a></h3><ol start="6"><li>特性時間: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mrow><mi>R</mi><mi>C</mi></mrow></msub><mo>=</mo><mi>R</mi><mi>C</mi><mo separator="true">,</mo><msub><mi>τ</mi><mrow><mi>L</mi><mi>R</mi></mrow></msub><mo>=</mo><mi>L</mi><mi mathvariant="normal">/</mi><mi>R</mi><mi mathvariant="normal">.</mi><msub><mi>ω</mi><mrow><mi>L</mi><mi>C</mi></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\tau_{R C}=R C, \tau_{L R}=L / R . \omega_{L C}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">RC</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">RC</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord">.</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mi>L</mi><mi>C</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">1 / \sqrt{L C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1767em;vertical-align:-0.25em;"></span><span class="mord">1/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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+++ b/academic/physics/ipho-formulas-jpn/9.html
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     <title>Formulas for IPhO 日本語版: Section 9 | Toshiki's Note</title>
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aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 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運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" 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style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_9" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-9" tabindex="-1">Formulas for IPhO 日本語版: Section 9 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-9" aria-hidden="true">#</a></h1><h2 id="_9-電磁気学" tabindex="-1">9: 電磁気学 <a class="header-anchor" href="#_9-電磁気学" aria-hidden="true">#</a></h2><h3 id="_9-1-coulomb-の法則" tabindex="-1">9.1: Coulomb の法則 <a class="header-anchor" href="#_9-1-coulomb-の法則" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">F=k q_1 q_2 / r^2, U=k q_1 q_2 / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> で, Kepler の法則が 使える (<a href="./12">Section 12 参照</a>).</li></ol><h3 id="_9-2-gauss-の法則" tabindex="-1">9.2: Gauss の法則 <a class="header-anchor" href="#_9-2-gauss-の法則" aria-hidden="true">#</a></h3><ol start="2"><li><p>Gauss の法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi>ε</mi><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mi>Q</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>π</mi><mi>G</mi><mi>M</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ε</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal">Q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span></span></span></span></span></p></li></ol><h3 id="_9-3-循環定理" tabindex="-1">9.3: 循環定理 <a class="header-anchor" href="#_9-3-循環定理" aria-hidden="true">#</a></h3><ol start="3"><li><p>循環定理 :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mo>=</mo><mover accent="true"><mi mathvariant="normal">Φ</mi><mo>˙</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>∮</mo><mfrac><mrow><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi></mrow><mi>μ</mi></mfrac><mo>=</mo><mi>I</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">Φ</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ol><h3 id="_9-4-電流素片により生じる磁束密度" tabindex="-1">9.4: 電流素片により生じる磁束密度 <a class="header-anchor" href="#_9-4-電流素片により生じる磁束密度" aria-hidden="true">#</a></h3><ol start="4"><li>電流素片により生じる磁束密度 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>I</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>×</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mi mathvariant="normal">.</mi></mrow><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} . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">.</span></span></span></span></span></p> したがって電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> が流れる円形回路の中心では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_9-5-ローレンツ力" tabindex="-1">9.5: ローレンツ力 <a class="header-anchor" href="#_9-5-ローレンツ力" aria-hidden="true">#</a></h3><ol start="5"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>e</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">E</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7694em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span>.</li></ol><h3 id="_9-6-gauss-の定理と循環定理より" tabindex="-1">9.6: Gauss の定理と循環定理より <a class="header-anchor" href="#_9-6-gauss-の定理と循環定理より" aria-hidden="true">#</a></h3><ol start="6"><li>Gauss の定理と循環定理より：帯電した導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \pi \varepsilon_0 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 \pi r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 帯電した面について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \varepsilon_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる面につい て <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>i</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 i}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2528em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9078em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 一様に帯電した球殼（又は無限に長い円 筒）の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 軸に沿って表面に電流が流れる 円筒の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> で一様に帯電, 又は一様 な電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">i</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6933em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span></span></span></span> が流れる, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d=3) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d-2) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 平面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d=1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> の内部で,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">E</mi><mo>=</mo><mfrac><mi>ρ</mi><mrow><mi>ε</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">r</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>μ</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">i</mi><mo>×</mo><mi mathvariant="bold-italic">r</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ε</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ρ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-7-長いソレノイド" tabindex="-1">9.7: 長いソレノイド <a class="header-anchor" href="#_9-7-長いソレノイド" aria-hidden="true">#</a></h3><ol start="7"><li>長いソレノイド: 内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mi>μ</mi><mi>n</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">B=\mu n I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>, 外部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 磁束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mi>N</mi><mi>B</mi><mi>S</mi><mrow><mo fence="true">(</mo><mi>n</mi><mo>=</mo><mfrac><mi>N</mi><mi>l</mi></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\Phi=N B S\left(n=\frac{N}{l}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2223em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NBS</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>. インダクタンス <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">L=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi>I</mi><mo>=</mo><mi>μ</mi><msup><mi>n</mi><mn>2</mn></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\Phi / I=\mu n^2 V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>. 短いソレノイド <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>B</mi><mi mathvariant="normal">∥</mi></msub><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>n</mi><mi>I</mi><mi mathvariant="normal">Ω</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span><span class="mord mtight">Ω</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord">Ω</span></span></span></span> は 立体角).</li></ol><h3 id="_9-8-磁場を小型コイルや衝撃検流計で測定する" tabindex="-1">9.8: 磁場を小型コイルや衝撃検流計で測定する <a class="header-anchor" href="#_9-8-磁場を小型コイルや衝撃検流計で測定する" aria-hidden="true">#</a></h3><ol start="9"><li>磁場を小型コイルや衝撃検流計で測定する： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">q=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mfrac><mi>V</mi><mi>R</mi></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mi>N</mi><mi>S</mi><mi mathvariant="normal">Δ</mi><mi>B</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NS</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>.</li></ol><h3 id="_9-9-静電場のエネルギー" tabindex="-1">9.9: 静電場のエネルギー <a class="header-anchor" href="#_9-9-静電場のエネルギー" aria-hidden="true">#</a></h3><ol start="9"><li>静電場のエネルギー：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>U</mi><mo>=</mo><mi>k</mi><munder><mo>∑</mo><mrow><mi>i</mi><mo>&lt;</mo><mi>j</mi></mrow></munder><mfrac><mrow><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub></mrow><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>q</mi><mo separator="true">,</mo><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>q</mi><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">U=k \sum_{i&lt;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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5213em;vertical-align:-1.4138em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ρ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span></span></p></li></ol><h3 id="_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" tabindex="-1">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 <a class="header-anchor" href="#_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" aria-hidden="true">#</a></h3><ol start="10"><li>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.</li></ol><h3 id="_9-11-全ての電荷" tabindex="-1">9.11: 全ての電荷 <a class="header-anchor" href="#_9-11-全ての電荷" aria-hidden="true">#</a></h3><ol start="12"><li>全ての電荷が距離 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> にある場合（例えば，不均一に帯 電した球やリングの中心） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi>Q</mi><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\phi \phi=k Q / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">Q</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></li></ol><h3 id="_9-12-外部電荷" tabindex="-1">9.12: 外部電荷 <a class="header-anchor" href="#_9-12-外部電荷" aria-hidden="true">#</a></h3><ol start="13"><li>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.</li></ol><h3 id="_9-14-導体" tabindex="-1">9.14: 導体 <a class="header-anchor" href="#_9-14-導体" aria-hidden="true">#</a></h3><ol start="14"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> という 電荷を持った導電性の球があるように見える).</li></ol><h3 id="_9-15-静電容量" tabindex="-1">9.15: 静電容量 <a class="header-anchor" href="#_9-15-静電容量" aria-hidden="true">#</a></h3><ol start="15"><li>静電容量: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mo>=</mo><mi>ε</mi><mi>S</mi><mi mathvariant="normal">/</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">C=\varepsilon S / d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">εS</span><span class="mord">/</span><span class="mord mathnormal">d</span></span></span></span> (平板), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>π</mi><mi>ε</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">4 \pi \varepsilon r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal">ε</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> (球), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><mi>ε</mi><mi>l</mi><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mi>R</mi><mi mathvariant="normal">/</mi><mi>r</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2 \pi \varepsilon l(\ln R / r)^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.01968em;">εl</span><span class="mopen">(</span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> (同軸円筒).</li></ol><h3 id="_9-16-双極子モーメント" tabindex="-1">9.16: 双極子モーメント <a class="header-anchor" href="#_9-16-双極子モーメント" aria-hidden="true">#</a></h3><ol start="16"><li><p>双極子モーメント：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>q</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>=</mo><mi>q</mi><mi mathvariant="bold-italic">d</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">S</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.3802em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-17-双極子場" tabindex="-1">9.17: 双極子場 <a class="header-anchor" href="#_9-17-双極子場" aria-hidden="true">#</a></h3><ol start="17"><li>双極子場 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi mathvariant="bold-italic">p</mi><mo>⋅</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>E</mi><mo separator="true">,</mo><mi>B</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_9-18-双極子に働く力" tabindex="-1">9.18: 双極子に働く力 <a class="header-anchor" href="#_9-18-双極子に働く力" aria-hidden="true">#</a></h3><ol start="19"><li>双極子に働く力 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">E</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">B</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><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}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1418em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8918em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.372em;vertical-align:-0.3802em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9918em;"><span style="top:-3.3029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> [訳 者注 : ここの微分はむしろ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">grad</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\operatorname{grad}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">grad</span></span></span></span></span> である]. 2 つの双極 子間の相互作用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>F</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: F \propto r^{-4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_9-19-磁気双極子としての点電荷" tabindex="-1">9.19: 磁気双極子としての点電荷 <a class="header-anchor" href="#_9-19-磁気双極子としての点電荷" aria-hidden="true">#</a></h3><ol start="19"><li>磁気双極子としての点電荷 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>μ</mi></msub><mo>∝</mo><mi mathvariant="normal">Φ</mi><mo>∝</mo><msubsup><mi>v</mi><mo lspace="0em" rspace="0em">⊥</mo><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">p_\mu \propto \Phi \propto v_{\perp}^2 / B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> は断熱 不変量 (<a href="./4#_4-22-断熱不変量">Section 4: #22</a> 参照).</li></ol><h3 id="_9-20-鏡像法" tabindex="-1">9.20: 鏡像法 <a class="header-anchor" href="#_9-20-鏡像法" aria-hidden="true">#</a></h3><ol start="20"><li>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.</li></ol><h3 id="_9-21-一様-電-場中の球-円柱-の分極" tabindex="-1">9.21: 一様（電）場中の球 (円柱) の分極 <a class="header-anchor" href="#_9-21-一様-電-場中の球-円柱-の分極" aria-hidden="true">#</a></h3><ol start="21"><li>一様（電）場中の球 (円柱) の分極 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>+</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">(+\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">+</span><span class="mord mathnormal">ρ</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">-\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">ρ</span></span></span></span> に一 様に帯電した球 (円柱) の重ね合わせで， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>∝</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">d \propto E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span>.</li></ol><h3 id="_9-22-渦電流" tabindex="-1">9.22: 渦電流 <a class="header-anchor" href="#_9-22-渦電流" aria-hidden="true">#</a></h3><ol start="22"><li>渦電流: 電流損失密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>ρ</mi><mi mathvariant="normal">.</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">\approx B^2 v^2 / \rho .1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">ρ</span><span class="mord">.1</span></span></span></span> 回の通過で与え られる運動量 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>τ</mi><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>a</mi><mn>3</mn></msup><mi>d</mi><mi mathvariant="normal">/</mi><mi>ρ</mi></mrow><annotation encoding="application/x-tex">F \tau \approx B^2 a^3 d / \rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">ρ</span></span></span></span> (ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> は厚さ, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> は大きさ).</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/9.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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href="#formulas-for-ipho-日本語版-section-9" aria-hidden="true">#</a></h1><h2 id="_9-電磁気学" tabindex="-1">9: 電磁気学 <a class="header-anchor" href="#_9-電磁気学" aria-hidden="true">#</a></h2><h3 id="_9-1-coulomb-の法則" tabindex="-1">9.1: Coulomb の法則 <a class="header-anchor" href="#_9-1-coulomb-の法則" aria-hidden="true">#</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">F=k q_1 q_2 / r^2, U=k q_1 q_2 / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> で, Kepler の法則が 使える (<a href="./12">Section 12 参照</a>).</li></ol><h3 id="_9-2-gauss-の法則" tabindex="-1">9.2: Gauss の法則 <a class="header-anchor" href="#_9-2-gauss-の法則" aria-hidden="true">#</a></h3><ol start="2"><li><p>Gauss の法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi>ε</mi><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mi>Q</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>π</mi><mi>G</mi><mi>M</mi></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ε</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal">Q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span></span></span></span></span></p></li></ol><h3 id="_9-3-循環定理" tabindex="-1">9.3: 循環定理 <a class="header-anchor" href="#_9-3-循環定理" aria-hidden="true">#</a></h3><ol start="3"><li><p>循環定理 :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mo>=</mo><mover accent="true"><mi mathvariant="normal">Φ</mi><mo>˙</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>∮</mo><mfrac><mrow><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi></mrow><mi>μ</mi></mfrac><mo>=</mo><mi>I</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn></mrow><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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">Φ</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ol><h3 id="_9-4-電流素片により生じる磁束密度" tabindex="-1">9.4: 電流素片により生じる磁束密度 <a class="header-anchor" href="#_9-4-電流素片により生じる磁束密度" aria-hidden="true">#</a></h3><ol start="4"><li>電流素片により生じる磁束密度 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>I</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>×</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mi mathvariant="normal">.</mi></mrow><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} . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">.</span></span></span></span></span></p> したがって電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> が流れる円形回路の中心では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_9-5-ローレンツ力" tabindex="-1">9.5: ローレンツ力 <a class="header-anchor" href="#_9-5-ローレンツ力" aria-hidden="true">#</a></h3><ol start="5"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>e</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">E</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7694em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span>.</li></ol><h3 id="_9-6-gauss-の定理と循環定理より" tabindex="-1">9.6: Gauss の定理と循環定理より <a class="header-anchor" href="#_9-6-gauss-の定理と循環定理より" aria-hidden="true">#</a></h3><ol start="6"><li>Gauss の定理と循環定理より：帯電した導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \pi \varepsilon_0 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 \pi r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 帯電した面について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \varepsilon_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる面につい て <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>i</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 i}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2528em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9078em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 一様に帯電した球殼（又は無限に長い円 筒）の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 軸に沿って表面に電流が流れる 円筒の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> で一様に帯電, 又は一様 な電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">i</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6933em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span></span></span></span> が流れる, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d=3) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d-2) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 平面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d=1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> の内部で,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">E</mi><mo>=</mo><mfrac><mi>ρ</mi><mrow><mi>ε</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">r</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>μ</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">i</mi><mo>×</mo><mi mathvariant="bold-italic">r</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ε</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ρ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-7-長いソレノイド" tabindex="-1">9.7: 長いソレノイド <a class="header-anchor" href="#_9-7-長いソレノイド" aria-hidden="true">#</a></h3><ol start="7"><li>長いソレノイド: 内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mi>μ</mi><mi>n</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">B=\mu n I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>, 外部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 磁束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mi>N</mi><mi>B</mi><mi>S</mi><mrow><mo fence="true">(</mo><mi>n</mi><mo>=</mo><mfrac><mi>N</mi><mi>l</mi></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\Phi=N B S\left(n=\frac{N}{l}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2223em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NBS</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>. インダクタンス <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">L=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi>I</mi><mo>=</mo><mi>μ</mi><msup><mi>n</mi><mn>2</mn></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\Phi / I=\mu n^2 V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>. 短いソレノイド <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>B</mi><mi mathvariant="normal">∥</mi></msub><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>n</mi><mi>I</mi><mi mathvariant="normal">Ω</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span><span class="mord mtight">Ω</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord">Ω</span></span></span></span> は 立体角).</li></ol><h3 id="_9-8-磁場を小型コイルや衝撃検流計で測定する" tabindex="-1">9.8: 磁場を小型コイルや衝撃検流計で測定する <a class="header-anchor" href="#_9-8-磁場を小型コイルや衝撃検流計で測定する" aria-hidden="true">#</a></h3><ol start="9"><li>磁場を小型コイルや衝撃検流計で測定する： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">q=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mfrac><mi>V</mi><mi>R</mi></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mi>N</mi><mi>S</mi><mi mathvariant="normal">Δ</mi><mi>B</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NS</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>.</li></ol><h3 id="_9-9-静電場のエネルギー" tabindex="-1">9.9: 静電場のエネルギー <a class="header-anchor" href="#_9-9-静電場のエネルギー" aria-hidden="true">#</a></h3><ol start="9"><li>静電場のエネルギー：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>U</mi><mo>=</mo><mi>k</mi><munder><mo>∑</mo><mrow><mi>i</mi><mo>&lt;</mo><mi>j</mi></mrow></munder><mfrac><mrow><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub></mrow><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>q</mi><mo separator="true">,</mo><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>q</mi><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">U=k \sum_{i&lt;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 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5213em;vertical-align:-1.4138em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ρ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span></span></p></li></ol><h3 id="_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" tabindex="-1">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 <a class="header-anchor" href="#_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" aria-hidden="true">#</a></h3><ol start="10"><li>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.</li></ol><h3 id="_9-11-全ての電荷" tabindex="-1">9.11: 全ての電荷 <a class="header-anchor" href="#_9-11-全ての電荷" aria-hidden="true">#</a></h3><ol start="12"><li>全ての電荷が距離 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> にある場合（例えば，不均一に帯 電した球やリングの中心） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi>Q</mi><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\phi \phi=k Q / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">Q</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></li></ol><h3 id="_9-12-外部電荷" tabindex="-1">9.12: 外部電荷 <a class="header-anchor" href="#_9-12-外部電荷" aria-hidden="true">#</a></h3><ol start="13"><li>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.</li></ol><h3 id="_9-14-導体" tabindex="-1">9.14: 導体 <a class="header-anchor" href="#_9-14-導体" aria-hidden="true">#</a></h3><ol start="14"><li>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 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> という 電荷を持った導電性の球があるように見える).</li></ol><h3 id="_9-15-静電容量" tabindex="-1">9.15: 静電容量 <a class="header-anchor" href="#_9-15-静電容量" aria-hidden="true">#</a></h3><ol start="15"><li>静電容量: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mo>=</mo><mi>ε</mi><mi>S</mi><mi mathvariant="normal">/</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">C=\varepsilon S / d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">εS</span><span class="mord">/</span><span class="mord mathnormal">d</span></span></span></span> (平板), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>π</mi><mi>ε</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">4 \pi \varepsilon r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal">ε</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> (球), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><mi>ε</mi><mi>l</mi><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mi>R</mi><mi mathvariant="normal">/</mi><mi>r</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2 \pi \varepsilon l(\ln R / r)^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.01968em;">εl</span><span class="mopen">(</span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> (同軸円筒).</li></ol><h3 id="_9-16-双極子モーメント" tabindex="-1">9.16: 双極子モーメント <a class="header-anchor" href="#_9-16-双極子モーメント" aria-hidden="true">#</a></h3><ol start="16"><li><p>双極子モーメント：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>q</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>=</mo><mi>q</mi><mi mathvariant="bold-italic">d</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">S</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.3802em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-17-双極子場" tabindex="-1">9.17: 双極子場 <a class="header-anchor" href="#_9-17-双極子場" aria-hidden="true">#</a></h3><ol start="17"><li>双極子場 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi mathvariant="bold-italic">p</mi><mo>⋅</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>E</mi><mo separator="true">,</mo><mi>B</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_9-18-双極子に働く力" tabindex="-1">9.18: 双極子に働く力 <a class="header-anchor" href="#_9-18-双極子に働く力" aria-hidden="true">#</a></h3><ol start="19"><li>双極子に働く力 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">E</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">B</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><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}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1418em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8918em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.372em;vertical-align:-0.3802em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9918em;"><span style="top:-3.3029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> [訳 者注 : ここの微分はむしろ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">grad</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\operatorname{grad}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">grad</span></span></span></span></span> である]. 2 つの双極 子間の相互作用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>F</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: F \propto r^{-4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_9-19-磁気双極子としての点電荷" tabindex="-1">9.19: 磁気双極子としての点電荷 <a class="header-anchor" href="#_9-19-磁気双極子としての点電荷" aria-hidden="true">#</a></h3><ol start="19"><li>磁気双極子としての点電荷 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>μ</mi></msub><mo>∝</mo><mi mathvariant="normal">Φ</mi><mo>∝</mo><msubsup><mi>v</mi><mo lspace="0em" rspace="0em">⊥</mo><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">p_\mu \propto \Phi \propto v_{\perp}^2 / B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> は断熱 不変量 (<a href="./4#_4-22-断熱不変量">Section 4: #22</a> 参照).</li></ol><h3 id="_9-20-鏡像法" tabindex="-1">9.20: 鏡像法 <a class="header-anchor" href="#_9-20-鏡像法" aria-hidden="true">#</a></h3><ol start="20"><li>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.</li></ol><h3 id="_9-21-一様-電-場中の球-円柱-の分極" tabindex="-1">9.21: 一様（電）場中の球 (円柱) の分極 <a class="header-anchor" href="#_9-21-一様-電-場中の球-円柱-の分極" aria-hidden="true">#</a></h3><ol start="21"><li>一様（電）場中の球 (円柱) の分極 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>+</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">(+\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">+</span><span class="mord mathnormal">ρ</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">-\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">ρ</span></span></span></span> に一 様に帯電した球 (円柱) の重ね合わせで， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>∝</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">d \propto E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span>.</li></ol><h3 id="_9-22-渦電流" tabindex="-1">9.22: 渦電流 <a class="header-anchor" href="#_9-22-渦電流" aria-hidden="true">#</a></h3><ol start="22"><li>渦電流: 電流損失密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>ρ</mi><mi mathvariant="normal">.</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">\approx B^2 v^2 / \rho .1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">ρ</span><span class="mord">.1</span></span></span></span> 回の通過で与え られる運動量 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>τ</mi><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>a</mi><mn>3</mn></msup><mi>d</mi><mi mathvariant="normal">/</mi><mi>ρ</mi></mrow><annotation encoding="application/x-tex">F \tau \approx B^2 a^3 d / \rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">ρ</span></span></span></span> (ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> は厚さ, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> は大きさ).</li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/9.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_vocabulary_2023_02_2023-02-27" data-v-c5936a1e><div><h1 id="_2023-2-27-vocabulary" tabindex="-1">2023-2-27: Vocabulary <a class="header-anchor" href="#_2023-2-27-vocabulary" aria-hidden="true">#</a></h1><blockquote><p>This table of vocabularies are from &quot;Sadlier Vocabulary Workshop-Level G (Unit 4)&quot; with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.</p></blockquote><table><thead><tr><th>Vocabulary</th><th>Definition</th></tr></thead><tbody><tr><td>Atrophy</td><td>(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away</td></tr><tr><td>Bastion</td><td>A fortified place, stronghold</td></tr><tr><td>Concord</td><td>A state of agreement, harmony, unanimity; a treaty, pact, covenant</td></tr><tr><td>Consummate</td><td>(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection</td></tr><tr><td>Disarray</td><td>(n.) disorder, confusion; (v.) to throw into disorder</td></tr><tr><td>Exigency</td><td>Urgency, pressure; urgent demand, pressing need; an emergency</td></tr><tr><td>Flotsam</td><td>Floating debris; homeless, impoverished people</td></tr><tr><td>Frenetic</td><td>Frenzied, highly agitated</td></tr><tr><td>Glean</td><td>To gather bit by bit; to gather small quantities of grain left in a field by the reapers.</td></tr><tr><td>Grouse</td><td>(n.) A type of game bird; a complaint; (v.) to complain, grumble</td></tr><tr><td>Incarcerate</td><td>To imprison, confine, jail</td></tr><tr><td>Incumbent</td><td>(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of</td></tr><tr><td>Jocular</td><td>Humorous, jesting, jolly, joking</td></tr><tr><td>Ludicrous</td><td>Ridiculous, laughable, absurd</td></tr><tr><td>Mordant</td><td>Biting or caustic in thought, manner, or style; sharply or bitterly harsh.</td></tr><tr><td>Nettle</td><td>(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely</td></tr><tr><td>Pecuniary</td><td>Consisting of or measured in money; of or related to money</td></tr><tr><td>Pusillanimous</td><td>Contemptibly cowardly or mean-spirited</td></tr><tr><td>Recumbent</td><td>In a reclining position, lying down, in the posture of one sleeping or resting.</td></tr><tr><td>Stratagem</td><td>A scheme to outwit or deceive an opponent or to gain and end.</td></tr></tbody></table><div class="tip custom-block"><p class="custom-block-title">Reference</p><ul><li>&quot;Sadlier Vocabulary Workshop- Level G (Unit 4).&quot; kensiezoe. Quizlet, n.d. <a href="https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/" target="_blank" rel="noreferrer">https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/</a></li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/vocabulary/2023/02/2023-02-27.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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tissue; any progressive decline or failure; (v.) to waste away</td></tr><tr><td>Bastion</td><td>A fortified place, stronghold</td></tr><tr><td>Concord</td><td>A state of agreement, harmony, unanimity; a treaty, pact, covenant</td></tr><tr><td>Consummate</td><td>(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection</td></tr><tr><td>Disarray</td><td>(n.) disorder, confusion; (v.) to throw into disorder</td></tr><tr><td>Exigency</td><td>Urgency, pressure; urgent demand, pressing need; an emergency</td></tr><tr><td>Flotsam</td><td>Floating debris; homeless, impoverished people</td></tr><tr><td>Frenetic</td><td>Frenzied, highly agitated</td></tr><tr><td>Glean</td><td>To gather bit by bit; to gather small quantities of grain left in a field by the reapers.</td></tr><tr><td>Grouse</td><td>(n.) A type of game bird; a complaint; (v.) to complain, grumble</td></tr><tr><td>Incarcerate</td><td>To imprison, confine, jail</td></tr><tr><td>Incumbent</td><td>(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of</td></tr><tr><td>Jocular</td><td>Humorous, jesting, jolly, joking</td></tr><tr><td>Ludicrous</td><td>Ridiculous, laughable, absurd</td></tr><tr><td>Mordant</td><td>Biting or caustic in thought, manner, or style; sharply or bitterly harsh.</td></tr><tr><td>Nettle</td><td>(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely</td></tr><tr><td>Pecuniary</td><td>Consisting of or measured in money; of or related to money</td></tr><tr><td>Pusillanimous</td><td>Contemptibly cowardly or mean-spirited</td></tr><tr><td>Recumbent</td><td>In a reclining position, lying down, in the posture of one sleeping or resting.</td></tr><tr><td>Stratagem</td><td>A scheme to outwit or deceive an opponent or to gain and end.</td></tr></tbody></table><div class="tip custom-block"><p class="custom-block-title">Reference</p><ul><li>&quot;Sadlier Vocabulary Workshop- Level G (Unit 4).&quot; kensiezoe. Quizlet, n.d. <a href="https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/" target="_blank" rel="noreferrer">https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/</a></li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/vocabulary/2023/02/2023-02-27.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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to use?</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/application/markdown-it-katex/support-function" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: KaTeX supported functions</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/application/markdown-it-katex/support-table" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: KaTeX support tables</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar 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data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _application_markdown-it-katex_index" data-v-c5936a1e><div><h1 id="andatoshiki-markdown-it-katex" tabindex="-1">@andatoshiki/markdown-it-katex <a class="header-anchor" href="#andatoshiki-markdown-it-katex" aria-hidden="true">#</a></h1><blockquote><p>Add graceful <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>KaTeX</mtext></mrow><annotation encoding="application/x-tex">\KaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">K</span><span class="mspace" style="margin-right:-0.17em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span> rendering to your Markdown like a charm with <code>markdown-it</code> plugin.</p></blockquote><h2 id="_1-installation" tabindex="-1">1: Installation <a class="header-anchor" href="#_1-installation" aria-hidden="true">#</a></h2><p>Before you start using this plugin, make sure you have already installed the default <code>markdown-it</code> parser; if not, please run the following command or refer to the official <a href="https://github.com/markdown-it/markdown-it#install" target="_blank" rel="noreferrer"><code>markdown-it</code></a> documentation.</p><div class="language-sh line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">npm</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">install</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">markdown-it</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">--save</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">npm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">install</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">markdown-it</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">--save</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing <code>&lt;/body&gt;</code> for markdown-it-katex&#39;s core utils to be loaded for the static page.</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-aIjCY" id="tab-bwhEgf2" checked="checked"><label for="tab-bwhEgf2">npm</label><input type="radio" name="group-aIjCY" id="tab-nG7ua0K"><label for="tab-nG7ua0K">yarn</label><input type="radio" name="group-aIjCY" id="tab-zzmD0cp"><label for="tab-zzmD0cp">pnpm</label><input type="radio" name="group-aIjCY" id="tab-zzTwXAT"><label for="tab-zzTwXAT">cdn</label></div><div class="blocks"><div class="language-sh active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">npm</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">install</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">-D</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">npm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">install</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">-D</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-sh line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">yarn</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">add</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">--dev</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">yarn</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">add</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">--dev</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-sh line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">pnpm</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">add</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">-D</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">pnpm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">add</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">-D</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#676E95;font-style:italic;">&lt;!-- your other body contents ... --&gt;</span></span>
+<span class="line"><span style="color:#89DDFF;">    &lt;</span><span style="color:#F07178;">script</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">src</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;">&gt;&lt;/</span><span style="color:#F07178;">script</span><span style="color:#89DDFF;">&gt;</span></span>
+<span class="line"><span style="color:#89DDFF;">&lt;/</span><span style="color:#F07178;">body</span><span style="color:#89DDFF;">&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#7F848E;font-style:italic;">&lt;!-- your other body contents ... --&gt;</span></span>
+<span class="line"><span style="color:#ABB2BF;">    &lt;</span><span style="color:#E06C75;">script</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">src</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js&quot;</span><span style="color:#ABB2BF;">&gt;&lt;/</span><span style="color:#E06C75;">script</span><span style="color:#ABB2BF;">&gt;</span></span>
+<span class="line"><span style="color:#ABB2BF;">&lt;/</span><span style="color:#E06C75;">body</span><span style="color:#ABB2BF;">&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br></div></div></div></div><p>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 <code>&lt;head&gt;</code> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,</p><p>Or, you could clone or download the entire repository source of <a href="https://github.com/katex/katex" target="_blank" rel="noreferrer">KaTeX</a> 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.</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-u_1ZO" id="tab-KuQtFtF" checked="checked"><label for="tab-KuQtFtF">jsdelivr</label><input type="radio" name="group-u_1ZO" id="tab-HpGkgcI"><label for="tab-HpGkgcI">jsd-gcore</label><input type="radio" name="group-u_1ZO" id="tab-NQlwrlC"><label for="tab-NQlwrlC">jsd-fastly</label><input type="radio" name="group-u_1ZO" id="tab-e_ZECnH"><label for="tab-e_ZECnH">cdnjs</label><input type="radio" name="group-u_1ZO" id="tab-yyOd22U"><label for="tab-yyOd22U">unpkg</label><input type="radio" name="group-u_1ZO" id="tab-_-J8wjY"><label for="tab-_-J8wjY">scss or css</label></div><div class="blocks"><div class="language-html active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://unpkg.com/katex@0.16.0/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://unpkg.com/katex@0.16.0/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-scss line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">scss</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#676E95;font-style:italic;">// ... your other styles</span></span>
+<span class="line"><span style="color:#89DDFF;font-style:italic;">@import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css</span><span style="color:#89DDFF;">&#39;</span><span style="color:#89DDFF;">;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#7F848E;font-style:italic;">// ... your other styles</span></span>
+<span class="line"><span style="color:#C678DD;">@import</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css&#39;</span><span style="color:#ABB2BF;">;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br></div></div></div></div><p>If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (<a href="https://github.com/sindresorhus/github-markdown-css/" target="_blank" rel="noreferrer">github-markdown-css</a>) for styling your HTML output with a similar style replica of GitHub&#39;s markdown styling to your familiarity.</p><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>This forked project maintained by <a href="https://github.com/andatoshiki" target="_blank" rel="noreferrer">Anda Toshiki</a> comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version <code>16.0</code>, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, <code>16.0</code> may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.</p><div class="danger custom-block"><p class="custom-block-title">Warning</p><p>Since this project is a fork of the <a href="https://www.npmjs.com/package/markdown-it-katex" target="_blank" rel="noreferrer">original</a> markdown-it-katex project that hasn&#39;t been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around <code>ver. 0.9.0</code> which clearly is outdated and results in broken styles with overflowing and other potential bug presents.</p></div><h2 id="_2-usage" tabindex="-1">2: Usage <a class="header-anchor" href="#_2-usage" aria-hidden="true">#</a></h2><p>RTo render equations, you need to include the <code>markdown-it-katex</code> plugin in the <code>markdown-it</code> components in your JavaScript or TypeScript file as follows,</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-4yYUV" id="tab-APG9Jxl" checked="checked"><label for="tab-APG9Jxl">js</label><input type="radio" name="group-4yYUV" id="tab-GexAwq9"><label for="tab-GexAwq9">ts</label></div><div class="blocks"><div class="language-js active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#C792EA;">var</span><span style="color:#A6ACCD;"> md </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">require</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">markdown-it</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)()</span><span style="color:#89DDFF;">,</span></span>
+<span class="line"><span style="color:#A6ACCD;">    mk </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">require</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">andatoshiki/markdown-it-katex</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#676E95;font-style:italic;">// double backslash is required for javascript strings, but not html input</span></span>
+<span class="line"><span style="color:#C792EA;">var</span><span style="color:#A6ACCD;"> result </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">render</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"># Math Rulez! </span><span style="color:#A6ACCD;">\n</span><span style="color:#C3E88D;">  $</span><span style="color:#A6ACCD;">\\</span><span style="color:#C3E88D;">sqrt{3x-1}+(1+x)^2$</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#C678DD;">var</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">md</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">require</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;markdown-it&#39;</span><span style="color:#ABB2BF;">)(),</span></span>
+<span class="line"><span style="color:#ABB2BF;">    </span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">require</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;andatoshiki/markdown-it-katex&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#7F848E;font-style:italic;">// double backslash is required for javascript strings, but not html input</span></span>
+<span class="line"><span style="color:#C678DD;">var</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">result</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">render</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;# Math Rulez! </span><span style="color:#56B6C2;">\n</span><span style="color:#98C379;">  $</span><span style="color:#56B6C2;">\\</span><span style="color:#98C379;">sqrt{3x-1}+(1+x)^2$&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br></div></div><div class="language-ts line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">ts</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;font-style:italic;">import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">*</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;font-style:italic;">as</span><span style="color:#A6ACCD;"> mk </span><span style="color:#89DDFF;font-style:italic;">from</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">markdown-it-katex</span><span style="color:#89DDFF;">&#39;</span></span>
+<span class="line"><span style="color:#89DDFF;font-style:italic;">import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">*</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;font-style:italic;">as</span><span style="color:#A6ACCD;"> MarkdownIt </span><span style="color:#89DDFF;font-style:italic;">from</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">@andatoshiki/markdown-it</span><span style="color:#89DDFF;">&#39;</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#C792EA;">const</span><span style="color:#A6ACCD;"> md </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">new</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">MarkdownIt</span><span style="color:#A6ACCD;">()</span></span>
+<span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#676E95;font-style:italic;">// double backslash is not required for TypeScript strings or template literals</span></span>
+<span class="line"><span style="color:#C792EA;">const</span><span style="color:#A6ACCD;"> result </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">render</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"># Math Rulez! </span><span style="color:#A6ACCD;">\n</span><span style="color:#C3E88D;">  $</span><span style="color:#A6ACCD;">\\</span><span style="color:#C3E88D;">sqrt{3x-1}+(1+x)^2$</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#C678DD;">import</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">*</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">as</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">from</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;markdown-it-katex&#39;</span></span>
+<span class="line"><span style="color:#C678DD;">import</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">*</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">as</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">MarkdownIt</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">from</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;@andatoshiki/markdown-it&#39;</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#C678DD;">const</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">new</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">MarkdownIt</span><span style="color:#ABB2BF;">()</span></span>
+<span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#7F848E;font-style:italic;">// double backslash is not required for TypeScript strings or template literals</span></span>
+<span class="line"><span style="color:#C678DD;">const</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">result</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">render</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;# Math Rulez! </span><span style="color:#56B6C2;">\n</span><span style="color:#98C379;">  $</span><span style="color:#56B6C2;">\\</span><span style="color:#98C379;">sqrt{3x-1}+(1+x)^2$&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br></div></div></div></div><h3 id="_3-configuration" tabindex="-1">3: Configuration <a class="header-anchor" href="#_3-configuration" aria-hidden="true">#</a></h3><p>The following list of variable are the customizable components of <code>markdown-it-katex</code>, adjust to your needs,</p><ul><li><code>katex</code>: You can change KaTeX version by passing the instance.</li><li><code>blockClass</code>: Class added to KaTeX block.</li><li>Apply any other <a href="https://katex.org/docs/options.html" target="_blank" rel="noreferrer">KaTeX options</a> if needed as a regard to the docs</li></ul><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk</span><span style="color:#89DDFF;">,</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">{</span><span style="color:#A6ACCD;"> </span><span style="color:#F07178;">blockClass</span><span style="color:#89DDFF;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">math-block</span><span style="color:#89DDFF;">&#39;</span><span style="color:#89DDFF;">,</span><span style="color:#A6ACCD;"> </span><span style="color:#F07178;">errorColor</span><span style="color:#89DDFF;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"> #cc0000</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">}</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">, { </span><span style="color:#E06C75;">blockClass</span><span style="color:#ABB2BF;">: </span><span style="color:#98C379;">&#39;math-block&#39;</span><span style="color:#ABB2BF;">, </span><span style="color:#E06C75;">errorColor</span><span style="color:#ABB2BF;">: </span><span style="color:#98C379;">&#39; #cc0000&#39;</span><span style="color:#ABB2BF;"> })</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><h2 id="_4-examples" tabindex="-1">4: Examples <a class="header-anchor" href="#_4-examples" aria-hidden="true">#</a></h2><h3 id="_4-1-inline-math" tabindex="-1">4.1: Inline math <a class="header-anchor" href="#_4-1-inline-math" aria-hidden="true">#</a></h3><p>To render your LaTeX equations inline, enclose them with a single dollar sign <code>$</code> on each side of the equation as follows,</p><div class="language-tex line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">tex</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">$</span><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">sqrt</span><span style="color:#89DDFF;">{</span><span style="color:#F78C6C;">3</span><span style="color:#FFCB6B;">x</span><span style="color:#89DDFF;">-</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">}+(</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">+</span><span style="color:#FFCB6B;">x</span><span style="color:#89DDFF;">)^</span><span style="color:#F78C6C;">2</span><span style="color:#89DDFF;">$</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">$</span><span style="color:#D19A66;">\sqrt</span><span style="color:#E5C07B;">{</span><span style="color:#D19A66;">3</span><span style="color:#E5C07B;">x-</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">}+(</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">+x)^</span><span style="color:#D19A66;">2</span><span style="color:#E5C07B;">$</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>rendered output equation as follows, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sqrt{3x-1}+(1+x)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.1744em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8656em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-2.8256em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><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
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+M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1744em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>.</p><h3 id="_4-2-blocked-math" tabindex="-1">4.2: Blocked math <a class="header-anchor" href="#_4-2-blocked-math" aria-hidden="true">#</a></h3><p>To render blocks, use double dollar sign (<code>$$</code>). This mode utilizes larger symbols and centers the result as a block displayed as a <code>&lt;div&gt;</code> block in HTML output, as follows,</p><div class="language-tex line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">tex</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">$$</span></span>
+<span class="line"><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{\</span><span style="color:#FFCB6B;">partial</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\</span><span style="color:#FFCB6B;">partial </span><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\left(\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{</span><span style="color:#FFCB6B;">y</span><span style="color:#89DDFF;">^{\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}\right)</span></span>
+<span class="line"><span style="color:#FFCB6B;">= </span><span style="color:#89DDFF;">\left(\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{</span><span style="color:#FFCB6B;">y</span><span style="color:#89DDFF;">^{\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}\right)</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\left\{(\</span><span style="color:#FFCB6B;">log y</span><span style="color:#89DDFF;">)^</span><span style="color:#FFCB6B;">r </span><span style="color:#89DDFF;">+</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">sum</span><span style="color:#89DDFF;">_{</span><span style="color:#FFCB6B;">i=</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">}^</span><span style="color:#FFCB6B;">r </span><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{(-</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">)^</span><span style="color:#FFCB6B;"> Ir </span><span style="color:#89DDFF;">\</span><span style="color:#FFCB6B;">cdots </span><span style="color:#89DDFF;">(</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">-</span><span style="color:#FFCB6B;">i</span><span style="color:#89DDFF;">+</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">)</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">(\</span><span style="color:#FFCB6B;">log y</span><span style="color:#89DDFF;">)^{</span><span style="color:#FFCB6B;">ri</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">i</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\right\}</span></span>
+<span class="line"><span style="color:#89DDFF;">$$</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">$$</span></span>
+<span class="line"><span style="color:#D19A66;">\frac</span><span style="color:#E5C07B;"> {</span><span style="color:#D19A66;">\partial</span><span style="color:#E5C07B;">^r} {</span><span style="color:#D19A66;">\partial</span><span style="color:#E5C07B;"> </span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">^r} \left(</span><span style="color:#D19A66;">\frac</span><span style="color:#E5C07B;"> {y^{</span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">}} {</span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">}\right)</span></span>
+<span class="line"><span style="color:#E5C07B;">= \left(</span><span style="color:#D19A66;">\frac</span><span style="color:#E5C07B;"> {y^{</span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">}} {</span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">}\right) \left\{(</span><span style="color:#D19A66;">\log</span><span style="color:#E5C07B;"> y)^r + </span><span style="color:#D19A66;">\sum</span><span style="color:#E5C07B;">_{i=</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">}^r </span><span style="color:#D19A66;">\frac</span><span style="color:#E5C07B;"> {(-</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">)^ Ir </span><span style="color:#D19A66;">\cdots</span><span style="color:#E5C07B;"> (r-i+</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">) (</span><span style="color:#D19A66;">\log</span><span style="color:#E5C07B;"> y)^{ri}} {</span><span style="color:#D19A66;">\omega</span><span style="color:#E5C07B;">^i} \right\}</span></span>
+<span class="line"><span style="color:#E5C07B;">$$</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br></div></div><p>the above block equation renders the LaTeX equation as a block with output as follows,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msup><mi mathvariant="normal">∂</mi><mi>r</mi></msup><mrow><mi mathvariant="normal">∂</mi><msup><mi>ω</mi><mi>r</mi></msup></mrow></mfrac><mrow><mo fence="true">(</mo><mfrac><msup><mi>y</mi><mi>ω</mi></msup><mi>ω</mi></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><msup><mi>y</mi><mi>ω</mi></msup><mi>ω</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">{</mo><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mi>r</mi></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>r</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>I</mi></msup><mi>r</mi><mo>⋯</mo><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mrow><mi>r</mi><mi>i</mi></mrow></msup></mrow><msup><mi>ω</mi><mi>i</mi></msup></mfrac><mo fence="true">}</mo></mrow></mrow><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\} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3414em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">ω</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0277em;vertical-align:-1.2777em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3414em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">ω</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7507em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">}</span></span></span></span></span></span></span></p><h2 id="_5-syntax" tabindex="-1">5: Syntax <a class="header-anchor" href="#_5-syntax" aria-hidden="true">#</a></h2><p>Math parsing in markdown is designed to comply with the &quot;latex-in-markdown&quot; conventions set by <a href="https://pandoc.org" target="_blank" rel="noreferrer">Pandoc</a>,</p><p>Anything between two <code>$</code> characters will be treated as TeX math. The opening <code>$</code> 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, <code>$20,000</code> and <code>$30,000</code> won’t parse as math. If for some reason you need to enclose text in literal <code>$</code> characters, backslash-escape them and they won’t be treated as math delimiters.</p><blockquote><ul><li>Pandoc. “Pandoc - Pandoc User’s Guide.” <a href="http://Pandoc.org" target="_blank" rel="noreferrer">Pandoc.org</a>, <a href="http://pandoc.org/MANUAL.html#math" target="_blank" rel="noreferrer">pandoc.org/MANUAL.html#math</a>. Accessed 2 Mar. 2023.</li></ul></blockquote><h2 id="_6-supported-math-syntax" tabindex="-1">6: Supported math syntax <a class="header-anchor" href="#_6-supported-math-syntax" aria-hidden="true">#</a></h2><p>KaTeX is a popular, open-source math typesetting library that is based on TeX and LaTeX. 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class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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on behalf of this part of the documentation</strong>, the table might be outdated as versions migrates, please also refer to <a href="https://katex.org/docs/supported.html" target="_blank" rel="noreferrer">KaTeX&#39;s official documentation</a>.</p></div><p>This following is a list of TeX functions supported by KaTeX. It is sorted into logical groups.</p><p>There is a similar <a href="./support-table">Support Table</a> 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.</p><h2 id="_1-accents" tabindex="-1">1: Accents <a class="header-anchor" href="#_1-accents" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">a&#39;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a&#39;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> <code>\tilde{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>g</mi><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\mathring{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.3472em;"><span class="mord">˚</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> <code>\mathring{g}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">a&#39;&#39;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a&#39;&#39;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">~</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7166em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7166em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><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
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widetilde{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏠</mo></mover></mrow><annotation encoding="application/x-tex">\overgroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0253em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 80
+H435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0
+ 435 0h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0
+ 3-1 3-3v-38c-76-158-257-219-435-219H0z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overgroup{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">a^{\prime}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a^{\prime}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">~</mo></munder></mrow><annotation encoding="application/x-tex">\utilde{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0893em;vertical-align:-0.406em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.594em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><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
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.406em;"><span></span></span></span></span></span></span></span></span> <code>\utilde{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏡</mo></munder></mrow><annotation encoding="application/x-tex">\undergroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;vertical-align:-0.342em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.658em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 262
+H435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219
+ 435 219h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18
+ 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.342em;"><span></span></span></span></span></span></span></span></span> <code>\undergroup{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\acute{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span> <code>\acute{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>F</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em;" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
+3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
+10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
+-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
+-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
+H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
+c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\vec{F}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⇒</mo></mover></mrow><annotation encoding="application/x-tex">\Overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2433em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2433em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.56em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\Overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7622em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> <code>\bar{y}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></mover></mrow><annotation encoding="application/x-tex">\overleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overleftarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\breve{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span> <code>\breve{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></munder></mrow><annotation encoding="application/x-tex">\underleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underleftarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></munder></mrow><annotation encoding="application/x-tex">\underrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\check{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span> <code>\check{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">↼</mo></mover></mrow><annotation encoding="application/x-tex">\overleftharpoon{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9526em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9526em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overleftharpoon{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">⇀</mo></mover></mrow><annotation encoding="application/x-tex">\overrightharpoon{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9526em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9526em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overrightharpoon{ac}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span> <code>\dot{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></mover></mrow><annotation encoding="application/x-tex">\overleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overleftrightarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏞</mo></mover></mrow><annotation encoding="application/x-tex">\overbrace{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3313em;"></span><span class="mord mover"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.7833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
+-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
+ 5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
+c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
+-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overbrace{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\ddot{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span> <code>\ddot{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></munder></mrow><annotation encoding="application/x-tex">\underleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underleftrightarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏟</mo></munder></mrow><annotation encoding="application/x-tex">\underbrace{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3313em;vertical-align:-0.648em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.352em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
+ 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
+ 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
+-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
+c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
+ 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
+ 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
+-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
+ 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
+-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.648em;"><span></span></span></span></span></span></span></span></span> <code>\underbrace{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\grave{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span> <code>\grave{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> <code>\overline{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></mover></mrow><annotation encoding="application/x-tex">\overlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overlinesegment{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> <code>\hat{\theta}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\underline{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></munder></mrow><annotation encoding="application/x-tex">\underlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underlinesegment{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7306em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widehat{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\widecheck{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7306em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widecheck{ac}</code></td><td style="text-align:left;"></td></tr></tbody></table><p><strong><em>Accent functions inside \text{…}</em></strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&#39;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span></span> <code>\&#39;{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\text{\~{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˜</span></span></span></span></span></span></span></span></span></span></span> <code>\~{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\text{\.{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span></span> <code>\.{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˝</mo></mover></mrow><annotation encoding="application/x-tex">\text{\H{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˝</span></span></span></span></span></span></span></span></span></span></span> <code>\H{a}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\`{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span></span> <code>\`{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\={a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span></span> <code>\={a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&quot;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span></span> <code>\&quot;{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\v{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span></span> <code>\v{a}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\^{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span></span></span></span></span> <code>\^{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\text{\u{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span></span> <code>\u{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\r{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\r{a}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>See also <a href="#_5-letters-and-unicode">letters</a>.</p><h2 id="_2-delimiters" tabindex="-1">2: Delimiters <a class="header-anchor" href="#_2-delimiters" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(~)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mclose">)</span></span></span></span> <code>( )</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lparen~\rparen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mclose">)</span></span></span></span> <code>\lparen</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rparen</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mtext> </mtext><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">⌈~⌉</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mspace nobreak"> </span><span class="mclose">⌉</span></span></span></span> <code>⌈ ⌉</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mtext> </mtext><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">\lceil~\rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mspace nobreak"> </span><span class="mclose">⌉</span></span></span></span> <code>\lceil</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rceil</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mtext> </mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[~]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace nobreak"> </span><span class="mclose">]</span></span></span></span> <code>[ ]</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mtext> </mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\lbrack~\rbrack</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace nobreak"> </span><span class="mclose">]</span></span></span></span> <code>\lbrack</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rbrack</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><mtext> </mtext><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">⌊~⌋</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mspace nobreak"> </span><span class="mclose">⌋</span></span></span></span> <code>⌊ ⌋</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><mtext> </mtext><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\lfloor~\rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mspace nobreak"> </span><span class="mclose">⌋</span></span></span></span> <code>\lfloor</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rfloor</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\downarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mclose">}</span></span></span></span> <code>\{ \}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\lbrace \rbrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mclose">}</span></span></span></span> <code>\lbrace</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rbrace</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎰</mo><mo stretchy="false">⎱</mo></mrow><annotation encoding="application/x-tex">⎰⎱</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⎰</span><span class="mclose">⎱</span></span></span></span> <code>⎰⎱</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎰</mo><mo stretchy="false">⎱</mo></mrow><annotation encoding="application/x-tex">\lmoustache \rmoustache</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⎰</span><span class="mclose">⎱</span></span></span></span> <code>\lmoustache</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rmoustache</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↕</mo></mrow><annotation encoding="application/x-tex">\updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">↕</span></span></span></span> <code>\updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">⟨~⟩</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>⟨ ⟩</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle~\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>\langle</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟮</mo><mtext> </mtext><mo stretchy="false">⟯</mo></mrow><annotation encoding="application/x-tex">⟮~⟯</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⟮</span><span class="mspace nobreak"> </span><span class="mclose">⟯</span></span></span></span> <code>⟮ ⟯</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟮</mo><mtext> </mtext><mo stretchy="false">⟯</mo></mrow><annotation encoding="application/x-tex">\lgroup~\rgroup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⟮</span><span class="mspace nobreak"> </span><span class="mclose">⟯</span></span></span></span> <code>\lgroup</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rgroup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span> <code>|</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span> <code>\vert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">┌</mo><mo stretchy="false">┐</mo></mrow><annotation encoding="application/x-tex">┌ ┐</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mopen amsrm">┌</span><span class="mclose amsrm">┐</span></span></span></span> <code>┌ ┐</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌜</mi></mo><mo><mi mathvariant="normal">⌝</mi></mo></mrow><annotation encoding="application/x-tex">\ulcorner \urcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mopen amsrm">┌</span><span class="mclose amsrm">┐</span></span></span></span> <code>\ulcorner</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\urcorner</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Downarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span> <code>\|</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span> <code>\Vert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">└</mo><mo stretchy="false">┘</mo></mrow><annotation encoding="application/x-tex">└ ┘</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mopen amsrm">└</span><span class="mclose amsrm">┘</span></span></span></span> <code>└ ┘</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌞</mi></mo><mo><mi mathvariant="normal">⌟</mi></mo></mrow><annotation encoding="application/x-tex">\llcorner \lrcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mopen amsrm">└</span><span class="mclose amsrm">┘</span></span></span></span> <code>\llcorner</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\lrcorner</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇕</mo></mrow><annotation encoding="application/x-tex">\Updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇕</span></span></span></span> <code>\Updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo><mtext> </mtext><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">\lvert~\rvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∣</span><span class="mspace nobreak"> </span><span class="mclose">∣</span></span></span></span> <code>\lvert</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rvert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo><mtext> </mtext><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\lVert~\rVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span><span class="mspace nobreak"> </span><span class="mclose">∥</span></span></span></span> <code>\lVert</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rVert</code></td><td style="text-align:left;"><code>\left.</code></td><td style="text-align:left;"><code>\right.</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">\</mi></mrow><annotation encoding="application/x-tex">\backslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">\</span></span></span></span> <code>\backslash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\lang~\rang</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>\lang</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rang</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mtext> </mtext><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\lt~\gt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace nobreak"> </span><span class="mrel">&gt;</span></span></span></span> <code>\lt \gt</code></td><td style="text-align:left;"></td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr></tbody></table><p><strong>Delimiter Sizing</strong></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle mathsize="1.728em"><mrow><mi>A</mi><mi>B</mi></mrow></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\LARGE{AB}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8308em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord sizing reset-size6 size9"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span> <code>\left(\LARGE{AB}\right)</code></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo><mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">(</mo><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo></mrow><annotation encoding="application/x-tex">( \big( \Big( \bigg( \Bigg(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mopen">(</span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="delimsizing size4">(</span></span></span></span></span> <code>( \big( \Big( \bigg( \Bigg(</code></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><code>\left</code></td><td style="text-align:left;"><code>\big</code></td><td style="text-align:left;"><code>\bigl</code></td><td style="text-align:left;"><code>\bigm</code></td><td style="text-align:left;"><code>\bigr</code></td></tr><tr><td style="text-align:left;"><code>\middle</code></td><td style="text-align:left;"><code>\Big</code></td><td style="text-align:left;"><code>\Bigl</code></td><td style="text-align:left;"><code>\Bigm</code></td><td style="text-align:left;"><code>\Bigr</code></td></tr><tr><td style="text-align:left;"><code>\right</code></td><td style="text-align:left;"><code>\bigg</code></td><td style="text-align:left;"><code>\biggl</code></td><td style="text-align:left;"><code>\biggm</code></td><td style="text-align:left;"><code>\biggr</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\Bigg</code></td><td style="text-align:left;"><code>\Biggl</code></td><td style="text-align:left;"><code>\Biggm</code></td><td style="text-align:left;"><code>\Biggr</code></td></tr></tbody></table><h2 id="_3-environments" tabindex="-1">3: Environments <a class="header-anchor" href="#_3-environments" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b \\ c &amp; d \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{cc}a &amp; b\\c &amp; d\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{array}{cc}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{pmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{pmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{bmatrix}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><annotation encoding="application/x-tex">\begin{vmatrix} a &amp; b \\ c &amp; d \end{vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><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"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><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"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{vmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∥</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∥</mo></mrow><annotation encoding="application/x-tex">\begin{Vmatrix} a &amp; b \\ c &amp; d \end{Vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><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
+M367 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 M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><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
+M367 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 M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{Vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Vmatrix}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\begin{Bmatrix} a &amp; b \\ c &amp; d \end{Bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">}</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{Bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Bmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.66em" columnalign="center center center" columnlines="dashed dashed" columnspacing="1em" rowlines="solid dashed"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>e</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>f</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>g</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>h</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>i</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\def\arraystretch{1.5}\begin{array}{c:c:c} a &amp; b &amp; c \\ \hline d &amp; e &amp; f \\ \hdashline g &amp; h &amp; i \end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.4em;vertical-align:-2.45em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:4.95em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:5.4em;border-right-width:0.04em;border-right-style:dashed;margin:0 -0.02em;vertical-align:-2.45em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:5.4em;border-right-width:0.04em;border-right-style:dashed;margin:0 -0.02em;vertical-align:-2.45em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:4.95em;"></span><span class="hdashline" style="border-bottom-width:0.04em;"></span></span><span style="top:-6.1em;"><span class="pstrut" style="height:4.95em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\arraystretch{1.5}</code><br>   <code>\begin{array}{c:c:c}</code><br>   <code>a &amp; b &amp; c \\ \hline</code><br>   <code>d &amp; e &amp; f \\</code><br>   <code>\hdashline</code><br>   <code>g &amp; h &amp; i</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>d</mi><mo>+</mo><mi>e</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>f</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} a&amp;=b+c \\ d+e&amp;=f \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{aligned}</code><br>   <code>a&amp;=b+c \\</code><br>   <code>d+e&amp;=f</code><br><code>\end{aligned}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>10</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>13</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>4</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{alignedat}{2}10&amp;x+&amp;3&amp;y=2\\3&amp;x+&amp;13&amp;y=4\end{alignedat}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">13</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{alignedat}{2}</code><br>   <code>10&amp;x+ &amp;3&amp;y = 2 \\</code><br>   <code> 3&amp;x+&amp;13&amp;y = 4</code><br><code>\end{alignedat}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>e</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered} a=b \\ e=b+c \end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{gathered}</code><br>   <code>a=b \\ </code><br>   <code>e=b+c</code><br><code>\end{gathered}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">x = \begin{cases} a &amp;\text{if } b \\ c &amp;\text{if } d \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>x = \begin{cases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{cases}</code></td></tr></tbody></table><p>KaTeX also supports <code>darray</code> and <code>dcases</code>.</p><p>Acceptable line separators include: <code>\\</code>, <code>\cr</code>, <code>\\[distance]</code>, and <code>\cr[distance]</code>. <em>Distance</em> can be written with any of the <a href="#units">KaTeX units</a>.</p><p>The <code>{array}</code> environment supports <code>|</code> and <code>:</code> vertical separators.</p><p>The <code>{array}</code> environment does not yet support <code>\cline</code> or <code>\multicolumn</code>.</p><h2 id="_4-html" tabindex="-1">4: HTML <a class="header-anchor" href="#_4-html" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\href</mtext></mstyle></mrow><annotation encoding="application/x-tex">\href{https://katex.org/}{\KaTeX}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\href</span></span></span></span></span></td><td style="text-align:left;"><code>\href{https://katex.org/}{\KaTeX}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\url</mtext></mstyle></mrow><annotation encoding="application/x-tex">\url{https://katex.org/}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\url</span></span></span></span></span></td><td style="text-align:left;"><code>\url{https://katex.org/}</code></td></tr></tbody></table><h2 id="_5-letters-and-unicode" tabindex="-1">5: Letters and Unicode <a class="header-anchor" href="#_5-letters-and-unicode" aria-hidden="true">#</a></h2><p><strong>Greek Letters</strong></p><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">Z</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">Θ</mi><mi mathvariant="normal">I</mi><mspace></mspace><mi mathvariant="normal">K</mi><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">Ξ</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">Π</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">Σ</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">Υ</mi><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">X</mi><mi mathvariant="normal">Ψ</mi><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">Α Β Γ Δ Ε Ζ Η Θ Ι \allowbreak Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">ABΓΔEZHΘI</span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">KΛMNΞOΠPΣTΥΦXΨΩ</span></span></span></span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace></mspace><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi><mi>ϵ</mi><mi>ζ</mi><mi>η</mi><mi>θ</mi><mi>ι</mi><mi>κ</mi><mi>λ</mi><mi>μ</mi><mi>ν</mi><mi>ξ</mi><mi>o</mi><mi>π</mi><mspace></mspace><mi>ρ</mi><mi>σ</mi><mi>τ</mi><mi>υ</mi><mi>ϕ</mi><mi>χ</mi><mi>ψ</mi><mi>ω</mi><mi>ε</mi><mi>ϑ</mi><mi>ϖ</mi><mi>ϱ</mi><mi>ς</mi><mi>φ</mi></mrow><annotation encoding="application/x-tex">\allowbreak α β γ δ ϵ ζ η θ ι κ λ μ ν ξ o π \allowbreak ρ σ τ υ ϕ χ ψ ω ε ϑ ϖ ϱ ς φ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal">δϵ</span><span class="mord mathnormal" style="margin-right:0.07378em;">ζ</span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mord mathnormal">ι</span><span class="mord mathnormal">κλ</span><span class="mord mathnormal" style="margin-right:0.06366em;">μν</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal" style="margin-right:0.03588em;">τυ</span><span class="mord mathnormal">ϕ</span><span class="mord mathnormal">χ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord mathnormal">εϑ</span><span class="mord mathnormal" style="margin-right:0.02778em;">ϖ</span><span class="mord mathnormal">ϱςφ</span></span></span></span></p><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi></mrow><annotation encoding="application/x-tex">\Alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span></span></span></span> <code>\Alpha</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">B</mi></mrow><annotation encoding="application/x-tex">\Beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span></span></span></span> <code>\Beta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span></span></span></span> <code>\Gamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> <code>\Delta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">E</mi></mrow><annotation encoding="application/x-tex">\Epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">E</span></span></span></span> <code>\Epsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Z</mi></mrow><annotation encoding="application/x-tex">\Zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">Z</span></span></span></span> <code>\Zeta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">H</mi></mrow><annotation encoding="application/x-tex">\Eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">H</span></span></span></span> <code>\Eta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Θ</span></span></span></span> <code>\Theta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">I</mi></mrow><annotation encoding="application/x-tex">\Iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">I</span></span></span></span> <code>\Iota</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">K</mi></mrow><annotation encoding="application/x-tex">\Kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">K</span></span></span></span> <code>\Kappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Λ</span></span></span></span> <code>\Lambda</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">M</mi></mrow><annotation encoding="application/x-tex">\Mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">M</span></span></span></span> <code>\Mu</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi></mrow><annotation encoding="application/x-tex">\Nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">N</span></span></span></span> <code>\Nu</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ξ</span></span></span></span> <code>\Xi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\Omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">O</span></span></span></span> <code>\Omicron</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> <code>\Pi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> <code>\Sigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">T</mi></mrow><annotation encoding="application/x-tex">\Tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">T</span></span></span></span> <code>\Tau</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Υ</mi></mrow><annotation encoding="application/x-tex">\Upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Υ</span></span></span></span> <code>\Upsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span> <code>\Phi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\Chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> <code>\Chi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ψ</span></span></span></span> <code>\Psi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> <code>\Omega</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\varGamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Γ</span></span></span></span> <code>\varGamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\varDelta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Δ</span></span></span></span> <code>\varDelta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\varTheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Θ</span></span></span></span> <code>\varTheta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\varLambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Λ</span></span></span></span> <code>\varLambda</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\varXi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ξ</span></span></span></span> <code>\varXi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\varPi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Π</span></span></span></span> <code>\varPi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\varSigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Σ</span></span></span></span> <code>\varSigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Υ</mi></mrow><annotation encoding="application/x-tex">\varUpsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Υ</span></span></span></span> <code>\varUpsilon</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\varPhi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Φ</span></span></span></span> <code>\varPhi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\varPsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ψ</span></span></span></span> <code>\varPsi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\varOmega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ω</span></span></span></span> <code>\varOmega</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> <code>\alpha</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> <code>\beta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span> <code>\gamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> <code>\delta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> <code>\epsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07378em;">ζ</span></span></span></span> <code>\zeta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span></span></span></span> <code>\eta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> <code>\theta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ι</span></span></span></span> <code>\iota</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> <code>\kappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> <code>\lambda</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> <code>\mu</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> <code>\nu</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span> <code>\xi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ο</mi></mrow><annotation encoding="application/x-tex">\omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ο</span></span></span></span> <code>\omicron</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span> <code>\pi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> <code>\rho</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <code>\sigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> <code>\tau</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>υ</mi></mrow><annotation encoding="application/x-tex">\upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">υ</span></span></span></span> <code>\upsilon</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span> <code>\phi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">χ</span></span></span></span> <code>\chi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> <code>\psi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span> <code>\omega</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span> <code>\varepsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϰ</mi></mrow><annotation encoding="application/x-tex">\varkappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span></span></span></span> <code>\varkappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\vartheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span> <code>\vartheta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\thetasym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span> <code>\thetasym</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϖ</mi></mrow><annotation encoding="application/x-tex">\varpi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">ϖ</span></span></span></span> <code>\varpi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi></mrow><annotation encoding="application/x-tex">\varrho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϱ</span></span></span></span> <code>\varrho</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ς</mi></mrow><annotation encoding="application/x-tex">\varsigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5278em;vertical-align:-0.0972em;"></span><span class="mord mathnormal" style="margin-right:0.07986em;">ς</span></span></span></span> <code>\varsigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span> <code>\varphi</code></td></tr></tbody></table><p>| <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϝ</mi></mrow><annotation encoding="application/x-tex">\digamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">ϝ</span></span></span></span> <code>\digamma</code></p><p><strong>Other Letters</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ı</mi></mrow><annotation encoding="application/x-tex">\imath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal"></span></span></span></span> <code>\imath</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> <code>\nabla</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span> <code>\Im</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\Reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\Reals</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Œ</mtext></mrow><annotation encoding="application/x-tex">\text{\OE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord latin_fallback">Œ</span></span></span></span></span> <code>\text{\OE}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ȷ</mi></mrow><annotation encoding="application/x-tex">\jmath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal"></span></span></span></span> <code>\jmath</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∂</mi></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="margin-right:0.05556em;">∂</span></span></span></span> <code>\partial</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\image</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span> <code>\image</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\wp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span> <code>\wp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ø</mtext></mrow><annotation encoding="application/x-tex">\text{\o}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.0972em;"></span><span class="mord text"><span class="mord">ø</span></span></span></span></span> <code>\text{\o}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\aleph</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\aleph</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⅁</mi></mrow><annotation encoding="application/x-tex">\Game</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">⅁</span></span></span></span> <code>\Game</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">k</mi></mrow><annotation encoding="application/x-tex">\Bbbk</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">k</span></span></span></span> <code>\Bbbk</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\weierp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span> <code>\weierp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ø</mtext></mrow><annotation encoding="application/x-tex">\text{\O}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7806em;vertical-align:-0.0486em;"></span><span class="mord text"><span class="mord">Ø</span></span></span></span></span> <code>\text{\O}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alef</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\alef</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⅎ</mi></mrow><annotation encoding="application/x-tex">\Finv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">Ⅎ</span></span></span></span> <code>\Finv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> <code>\N</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex">\Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">Z</span></span></span></span> <code>\Z</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ß</mtext></mrow><annotation encoding="application/x-tex">\text{\ss}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">ß</span></span></span></span></span> <code>\text{\ss}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alefsym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\alefsym</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\cnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> <code>\cnums</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\natnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> <code>\natnums</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\aa}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\aa}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ı</mtext></mrow><annotation encoding="application/x-tex">\text{\i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">ı</span></span></span></span></span> <code>\text{\i}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℶ</mi></mrow><annotation encoding="application/x-tex">\beth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℶ</span></span></span></span> <code>\beth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\Complex</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> <code>\Complex</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\R</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>A</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\AA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9468em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\AA}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ȷ</mtext></mrow><annotation encoding="application/x-tex">\text{\j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord latin_fallback">ȷ</span></span></span></span></span> <code>\text{\j}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℷ</mi></mrow><annotation encoding="application/x-tex">\gimel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℷ</span></span></span></span> <code>\gimel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> <code>\ell</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\Re</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span> <code>\Re</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>æ</mtext></mrow><annotation encoding="application/x-tex">\text{\ae}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">æ</span></span></span></span></span> <code>\text{\ae}</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℸ</mi></mrow><annotation encoding="application/x-tex">\daleth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℸ</span></span></span></span> <code>\daleth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span></span></span></span> <code>\hbar</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\real</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span> <code>\real</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Æ</mtext></mrow><annotation encoding="application/x-tex">\text{\AE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Æ</span></span></span></span></span> <code>\text{\AE}</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ð</mi></mrow><annotation encoding="application/x-tex">\eth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ð</span></span></span></span> <code>\eth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℏ</span></span></span></span> <code>\hslash</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\reals</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>œ</mtext></mrow><annotation encoding="application/x-tex">\text{\oe}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">œ</span></span></span></span></span> <code>\text{\oe}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∂</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">ℑ</mi><mi mathvariant="normal">Ⅎ</mi><mi mathvariant="normal">ℵ</mi><mi mathvariant="normal">ℶ</mi><mi mathvariant="normal">ℷ</mi><mi mathvariant="normal">ℸ</mi><mi mathvariant="normal">⅁</mi><mi mathvariant="normal">ℏ</mi><mi mathvariant="normal">ð</mi></mrow><annotation encoding="application/x-tex">∂ ∇ ℑ Ⅎ ℵ ℶ ℷ ℸ ⅁ ℏ ð</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord">∇ℑ</span><span class="mord amsrm">Ⅎ</span><span class="mord">ℵ</span><span class="mord amsrm">ℶℷℸ⅁</span><span class="mord">ℏ</span><span class="mord amsrm">ð</span></span></span></span> ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖÙÚÛÜÝÞßàáâãäåçèéêëìíîïðñòóôöùúûüýþÿ</p><p><strong>Unicode Mathematical Alphanumeric Symbols</strong></p><table><thead><tr><th>Item</th><th>Range</th><th>Item</th><th>Range</th></tr></thead><tbody><tr><td>Bold</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝐀-𝐙 𝐚-𝐳 𝟎-𝟗</mtext></mrow><annotation encoding="application/x-tex">\text{𝐀-𝐙 𝐚-𝐳 𝟎-𝟗}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord text"><span class="mord textbf">A</span><span class="mord">-</span><span class="mord textbf">Z</span><span class="mord"> </span><span class="mord textbf">a</span><span class="mord">-</span><span class="mord textbf">z</span><span class="mord"> </span><span class="mord textbf">0</span><span class="mord">-</span><span class="mord textbf">9</span></span></span></span></span></td><td>Double-struck</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝔸-</mtext><mi mathvariant="normal">Z</mi><mtext> </mtext><mi>𝕜</mi></mrow><annotation encoding="application/x-tex">\text{𝔸-}ℤ\ 𝕜</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord text"><span class="mord textbb">A</span><span class="mord">-</span></span><span class="mord amsrm">Z</span><span class="mspace"> </span><span class="mord mathbb">k</span></span></span></span></td></tr><tr><td>Italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝐴-𝑍 𝑎-𝑧</mtext></mrow><annotation encoding="application/x-tex">\text{𝐴-𝑍 𝑎-𝑧}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord textit">A</span><span class="mord">-</span><span class="mord textit">Z</span><span class="mord"> </span><span class="mord textit">a</span><span class="mord">-</span><span class="mord textit">z</span></span></span></span></span></td><td>Sans serif</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝖠-𝖹 𝖺-𝗓 𝟢-𝟫</mtext></mrow><annotation encoding="application/x-tex">\text{𝖠-𝖹 𝖺-𝗓 𝟢-𝟫}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">A</span><span class="mord">-</span><span class="mord textsf">Z</span><span class="mord"> </span><span class="mord textsf">a</span><span class="mord">-</span><span class="mord textsf">z</span><span class="mord"> </span><span class="mord textsf">0</span><span class="mord">-</span><span class="mord textsf">9</span></span></span></span></span></td></tr><tr><td>Bold Italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝑨-𝒁 𝒂-𝒛</mtext></mrow><annotation encoding="application/x-tex">\text{𝑨-𝒁 𝒂-𝒛}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord text"><span class="mord boldsymbol">A</span><span class="mord">-</span><span class="mord boldsymbol">Z</span><span class="mord"> </span><span class="mord boldsymbol">a</span><span class="mord">-</span><span class="mord boldsymbol">z</span></span></span></span></span></td><td>Sans serif bold</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝗔-𝗭 𝗮-𝘇 𝟬-𝟵</mtext></mrow><annotation encoding="application/x-tex">\text{𝗔-𝗭 𝗮-𝘇 𝟬-𝟵}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textboldsf">A</span><span class="mord">-</span><span class="mord textboldsf">Z</span><span class="mord"> </span><span class="mord textboldsf">a</span><span class="mord">-</span><span class="mord textboldsf">z</span><span class="mord"> </span><span class="mord textboldsf">0</span><span class="mord">-</span><span class="mord textboldsf">9</span></span></span></span></span></td></tr><tr><td>Script</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝒜-𝒵</mtext></mrow><annotation encoding="application/x-tex">\text{𝒜-𝒵}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;"></span><span class="mord text"><span class="mord textscr">A</span><span class="mord">-</span><span class="mord textscr">Z</span></span></span></span></span></td><td>Sans serif italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝘈-𝘡 𝘢-𝘻</mtext></mrow><annotation encoding="application/x-tex">\text{𝘈-𝘡 𝘢-𝘻}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textitsf">A</span><span class="mord">-</span><span class="mord textitsf">Z</span><span class="mord"> </span><span class="mord textitsf">a</span><span class="mord">-</span><span class="mord textitsf">z</span></span></span></span></span></td></tr><tr><td>Fractur</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝔄-</mtext><mi mathvariant="fraktur">Z</mi><mtext> 𝔞-𝔷</mtext></mrow><annotation encoding="application/x-tex">\text{𝔄-}ℨ\text{ 𝔞-𝔷}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1891em;"></span><span class="mord text"><span class="mord textfrak">A</span><span class="mord">-</span></span><span class="mord mathfrak">Z</span><span class="mord text"><span class="mord"> </span><span class="mord textfrak">a</span><span class="mord">-</span><span class="mord textfrak">z</span></span></span></span></span></td><td>Monospace</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝙰-𝚉 𝚊-𝚣 𝟶-𝟿</mtext></mrow><annotation encoding="application/x-tex">\text{𝙰-𝚉 𝚊-𝚣 𝟶-𝟿}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">A</span><span class="mord">-</span><span class="mord texttt">Z</span><span class="mord"> </span><span class="mord texttt">a</span><span class="mord">-</span><span class="mord texttt">z</span><span class="mord"> </span><span class="mord texttt">0</span><span class="mord">-</span><span class="mord texttt">9</span></span></span></span></span></td></tr></tbody></table><p><strong>Unicode</strong></p><p>The letters listed above will render in any KaTeX rendering mode.</p><p>If the KaTeX rendering mode is set to <code>strict: false</code> or <code>strict:&quot;warn&quot;</code> (default), then KaTeX will accept all Unicode letters. The letters not listed above will be rendered from system fonts, not KaTeX-supplied fonts, so their typography may clash. They may also cause small vertical alignment issues. KaTeX has detailed metrics for glyphs in Latin, Greek, and Cyrillic, but other glyphs are treated as if they are each as tall as the letter M.</p><p>For Persian composite characters, a user-supplied <a href="https://github.com/HosseinAgha/persian-katex-plugin" target="_blank" rel="noreferrer">plug-in</a> is under development.</p><h2 id="_6-layout" tabindex="-1">6: Layout <a class="header-anchor" href="#_6-layout" aria-hidden="true">#</a></h2><h3 id="_6-1-annotation" tabindex="-1">6.1: Annotation <a class="header-anchor" href="#_6-1-annotation" aria-hidden="true">#</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\cancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\cancel{5}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mover><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mo stretchy="true">⏞</mo></mover><mtext>note</mtext></mover></mrow><annotation encoding="application/x-tex">\overbrace{a+b+c}^{\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0563em;vertical-align:-0.0833em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.973em;"><span style="top:-3.3424em;"><span class="pstrut" style="height:3.3424em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3424em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.7944em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
+-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
+ 5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
+c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
+-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span><span style="top:-4.8849em;"><span class="pstrut" style="height:3.3424em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span> <code>\overbrace{a+b+c}^{\text{note}}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="downdiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\bcancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\bcancel{5}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><munder><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mo stretchy="true">⏟</mo></munder><mtext>note</mtext></munder></mrow><annotation encoding="application/x-tex">\underbrace{a+b+c}_{\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0563em;vertical-align:-1.3619em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-1.6381em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span class="svg-align" style="top:-2.2687em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
+ 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
+ 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
+-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
+c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
+ 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
+ 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
+-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
+ 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
+-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7313em;"><span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3619em;"><span></span></span></span></span></span></span></span></span> <code>\underbrace{a+b+c}_{\text{note}}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike downdiagonalstrike"><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\xcancel{ABC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cancel-lap"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord cancel-pad"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.6833em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.6833em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span></span></span></span></span></span></span> <code>\xcancel{ABC}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\not =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>\not =</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="horizontalstrike"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\sout{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ab</span><span class="mord mathnormal">c</span></span></span><span style="top:-3.2155em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy sout"></span></span></span></span></span></span></span></span></span> <code>\sout{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>π</mi><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></mstyle></mstyle></mstyle></menclose></mrow><annotation encoding="application/x-tex">\boxed{\pi=\frac c d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4736em;vertical-align:-1.026em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4476em;"><span style="top:-4.4736em;"><span class="pstrut" style="height:4.4736em;"></span><span class="boxpad"><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><span style="top:-3.4476em;"><span class="pstrut" style="height:4.4736em;"></span><span class="stretchy fbox" style="height:2.4736em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.026em;"><span></span></span></span></span></span></span></span></span> <code>\boxed{\pi=\frac c d}</code></td></tr></tbody></table><p><code>\tag{hi} x+y^{2x}</code></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>x</mi><mo>+</mo><msup><mi>y</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(hi)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag{hi} x+y^{2x} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">hi</span></span><span class="mord">)</span></span></span></span></span></span></p><p><code>\tag*{hi} x+y^{2x}</code></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>x</mi><mo>+</mo><msup><mi>y</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>hi</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{hi} x+y^{2x} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">hi</span></span></span></span></span></span></p><h3 id="_6-2-line-breaks" tabindex="-1">6.2: Line Breaks <a class="header-anchor" href="#_6-2-line-breaks" aria-hidden="true">#</a></h3><p>KaTeX 0.10.0+ will insert automatic line breaks in inline math after relations or binary operators such as “=” or “+”. These can be suppressed by <code>\nobreak</code> or by placing math inside a pair of braces, as in <code>{F=ma}</code>. <code>\allowbreak</code> will allow automatic line breaks at locations other than relations or operators.</p><p>Hard line breaks are <code>\\</code> and <code>\newline</code>.</p><p>In display math, KaTeX does not insert automatic line breaks. It ignores display math hard line breaks when rendering option <code>strict: true</code>.</p><h3 id="_6-3-vertical-layout" tabindex="-1">6.3: Vertical Layout <a class="header-anchor" href="#_6-3-vertical-layout" aria-hidden="true">#</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <code>x_n</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\stackrel{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\stackrel{!}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0px"><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">a \atop b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0904em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>a \atop b</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">e^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> <code>e^x</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\overset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\overset{!}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mpadded voffset="0.25em"><mstyle scriptlevel="0" displaystyle="false"><mtext>b</mtext></mstyle></mpadded><mi>c</mi></mrow><annotation encoding="application/x-tex">a\raisebox{0.25em}{b}c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9444em;"></span><span class="mord mathnormal">a</span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9444em;"><span style="top:-3.25em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">b</span></span></span></span></span></span><span class="mord mathnormal">c</span></span></span></span> <code>a\raisebox{0.25em}{b}c</code></td></tr><tr><td style="text-align:left;">$_u^o $ <code>_u^o </code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><munder><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></munder></mo></mrow><annotation encoding="application/x-tex">\underset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.119em;vertical-align:-0.7521em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3669em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span></span></span></span></span> <code>\underset{!}{=}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>The second argument of <code>\raisebox</code> can contain math if it is nested within <code>$…$</code> delimiters, as in <code>\raisebox{0.25em}{$\frac a b$}</code></p><h3 id="_6-4-overlap-and-spacing" tabindex="-1">6.4: Overlap and Spacing <a class="header-anchor" href="#_6-4-overlap-and-spacing" aria-hidden="true">#</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo lspace="0em" rspace="0em">=</mo><mpadded lspace="-1width" width="0px"><mrow><mi mathvariant="normal">/</mi><mtext> </mtext></mrow></mpadded></mrow><annotation encoding="application/x-tex">{=}\mathllap{/\,}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mrel">=</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span> <code>{=}\mathllap{/\,}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><msup><mi>x</mi><mpadded height="0px" depth="0px"><mn>2</mn></mpadded></msup><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(x^{\smash{2}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.363em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> <code>\left(x^{\smash{2}}\right)</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="0px"><mrow><mtext> </mtext><mi mathvariant="normal">/</mi></mrow></mpadded><mo lspace="0em" rspace="0em">=</mo></mrow><annotation encoding="application/x-tex">\mathrlap{\,/}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span></span></span><span class="fix"></span></span></span></span><span class="mord"><span class="mrel">=</span></span></span></span></span> <code>\mathrlap{\,/}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mpadded depth="0px"><mi>y</mi></mpadded></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\smash[b]{y}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></span></span></span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg 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+M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt{\smash[b]{y}} </code></td></tr></tbody></table><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mo>∑</mo><mpadded lspace="-0.5width" width="0px"><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></mpadded></munder><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_{\mathclap{1\le i\le j\le n}} x_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4638em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="clap mtight"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="inner mtight"><span><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">n</span></span></span></span><span class="fix"></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <code>\sum_{\mathclap{1\le i\le j\le n}} x_{ij}</code></p><p>KaTeX also supports <code>\llap</code>, <code>\rlap</code>, and <code>\clap</code>, but they will take only text, not math, as arguments.</p><p><strong>Spacing</strong></p><table><thead><tr><th style="text-align:left;">Function</th><th style="text-align:left;">Produces</th><th style="text-align:left;">Function</th><th style="text-align:left;">Produces</th></tr></thead><tbody><tr><td style="text-align:left;"><code>\,</code></td><td style="text-align:left;">³∕₁₈ em space</td><td style="text-align:left;"><code>\kern{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\thinspace</code></td><td style="text-align:left;">³∕₁₈ em space</td><td style="text-align:left;"><code>\mkern{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\:</code></td><td style="text-align:left;">⁴∕₁₈ em space</td><td style="text-align:left;"><code>\mskip{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\medspace</code></td><td style="text-align:left;">⁴∕₁₈ em space</td><td style="text-align:left;"><code>\hskip{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\;</code></td><td style="text-align:left;">⁵∕₁₈ em space</td><td style="text-align:left;"><code>\hspace{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\thickspace</code></td><td style="text-align:left;">⁵∕₁₈ em space</td><td style="text-align:left;"><code>\hspace*{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\enspace</code></td><td style="text-align:left;">½ em space</td><td style="text-align:left;"><code>\phantom{content}</code></td><td style="text-align:left;">space the width and height of content</td></tr><tr><td style="text-align:left;"><code>\quad</code></td><td style="text-align:left;">1 em space</td><td style="text-align:left;"><code>\hphantom{content}</code></td><td style="text-align:left;">space the width of content</td></tr><tr><td style="text-align:left;"><code>\qquad</code></td><td style="text-align:left;">2 em space</td><td style="text-align:left;"><code>\vphantom{content}</code></td><td style="text-align:left;">a strut the height of content</td></tr><tr><td style="text-align:left;"><code>~</code></td><td style="text-align:left;">non-breaking space</td><td style="text-align:left;"><code>\!</code></td><td style="text-align:left;">– ³∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\&lt;space&gt;</code></td><td style="text-align:left;">space</td><td style="text-align:left;"><code>\negthinspace</code></td><td style="text-align:left;">– ³∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\nobreakspace</code></td><td style="text-align:left;">non-breaking space</td><td style="text-align:left;"><code>\negmedspace</code></td><td style="text-align:left;">– ⁴∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\space</code></td><td style="text-align:left;">space</td><td style="text-align:left;"><code>\negthickspace</code></td><td style="text-align:left;">– ⁵∕₁₈ em space</td></tr></tbody></table><p><strong>Notes:</strong></p><p><code>distance</code> will accept any of the <a href="#units">KaTeX units</a>.</p><p><code>\kern</code>, <code>\mkern</code>, <code>\mskip</code>, and <code>\hspace</code> accept unbraced distances, as in: <code>\kern1em</code>.</p><p><code>\mkern</code> and <code>\mskip</code> will not work in text mode and both will write a console warning for any unit except <code>mu</code>.</p><h2 id="_7-logic-and-set-theory" tabindex="-1">7: Logic and Set Theory <a class="header-anchor" href="#_7-logic-and-set-theory" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span></span></span></span> <code>\forall</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∁</mi></mrow><annotation encoding="application/x-tex">\complement</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;"></span><span class="mord amsrm">∁</span></span></span></span> <code>\complement</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∴</mo></mrow><annotation encoding="application/x-tex">\therefore</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∴</span></span></span></span> <code>\therefore</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span> <code>\emptyset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span> <code>\exists</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> <code>\subset</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∵</mo></mrow><annotation encoding="application/x-tex">\because</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∵</span></span></span></span> <code>\because</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\empty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span> <code>\empty</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exist</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span> <code>\exist</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊃</span></span></span></span> <code>\supset</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦</span></span></span></span> <code>\mapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\varnothing</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∅</span></span></span></span> <code>\varnothing</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∄</mi></mrow><annotation encoding="application/x-tex">\nexists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">∄</span></span></span></span> <code>\nexists</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∣</span></span></span></span> <code>\mid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\to</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\implies</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\implies</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\in</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\land</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\land</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\gets</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\gets</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟸</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\impliedby</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟸</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\impliedby</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\isin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\isin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\lor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\lor</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\iff</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\iff</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></span> <code>\notin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\ni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\ni</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∌</mo></mrow><annotation encoding="application/x-tex">\notni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\notni</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">\neg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span> <code>\neg</code> or <code>\lnot</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mo>∴</mo><mi mathvariant="normal">∁</mi><mo>∵</mo><mi mathvariant="normal">∃</mi><mo>∣</mo><mo>∈</mo><mo mathvariant="normal">∉</mo><mo>∋</mo><mo>⊂</mo><mo>⊃</mo><mo>∧</mo><mo>∨</mo><mo>↦</mo><mo>→</mo><mo>←</mo><mo>↔</mo><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">∀ ∴ ∁ ∵ ∃ ∣ ∈ ∉ ∋ ⊂ ⊃ ∧ ∨ ↦ → ← ↔ ¬</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.825em;"></span><span class="mord amsrm">∁</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">∵</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∃</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣∈</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋⊂⊃</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5666em;vertical-align:-0.011em;"></span><span class="mord">∧</span><span class="mord">∨</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↦→←↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span> ℂ ℍ ℕ ℙ ℚ ℝ</p><h2 id="_8-macros" tabindex="-1">8: Macros <a class="header-anchor" href="#_8-macros" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\def\foo{x^2} \foo + \foo</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\foo{x^2} \foo + \foo</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\gdef\bar#1{#1^2} \bar{y} + \bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\gdef\bar#1{#1^2} \bar{y} + \bar{y}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\global\def\macroname#1#2…{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\newcommand\macroname[numargs]{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\renewcommand\macroname[numargs]{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\providecommand\macroname[numargs]{definition}</code></td></tr></tbody></table><p>Macros can also be defined in the KaTeX rendering options.</p><p>Macros accept up to nine arguments: #1, #2, etc.</p><p><code>\gdef</code> and <code>\global\def</code> macros will persist between math expressions.</p><p>Available functions include:</p><p><code>\char</code> <code>\mathchoice</code> <code>\TextOrMath</code> <code>\@ifstar</code> <code>\@ifnextchar</code> <code>\@firstoftwo</code> <code>\@secondoftwo</code> <code>\relax</code></p><p>@ is a valid character for commands, as if <code>\makeatletter</code> were in effect.</p><h2 id="_9-operators" tabindex="-1">9: Operators <a class="header-anchor" href="#_9-operators" aria-hidden="true">#</a></h2><h3 id="_9-1-big-operators" tabindex="-1">9.1: Big Operators <a class="header-anchor" href="#_9-1-big-operators" aria-hidden="true">#</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span></span></span></span> <code>\sum</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∏</mo></mrow><annotation encoding="application/x-tex">\prod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span></span></span></span> <code>\prod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨂</mo></mrow><annotation encoding="application/x-tex">\bigotimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨂</span></span></span></span> <code>\bigotimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋁</span></span></span></span> <code>\bigvee</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\int</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\int</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∐</span></span></span></span> <code>\coprod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨁</mo></mrow><annotation encoding="application/x-tex">\bigoplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨁</span></span></span></span> <code>\bigoplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋀</mo></mrow><annotation encoding="application/x-tex">\bigwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋀</span></span></span></span> <code>\bigwedge</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∬</mo></mrow><annotation encoding="application/x-tex">\iint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∬</span></span></span></span> <code>\iint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\intop</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\intop</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨀</mo></mrow><annotation encoding="application/x-tex">\bigodot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨀</span></span></span></span> <code>\bigodot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋂</mo></mrow><annotation encoding="application/x-tex">\bigcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋂</span></span></span></span> <code>\bigcap</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∭</mo></mrow><annotation encoding="application/x-tex">\iiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∭</span></span></span></span> <code>\iiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo largeop="false">∫</mo></mrow><annotation encoding="application/x-tex">\smallint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\smallint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨄</mo></mrow><annotation encoding="application/x-tex">\biguplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨄</span></span></span></span> <code>\biguplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋃</mo></mrow><annotation encoding="application/x-tex">\bigcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋃</span></span></span></span> <code>\bigcup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo></mrow><annotation encoding="application/x-tex">\oint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span></span></span></span> <code>\oint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∯</mo></mrow><annotation encoding="application/x-tex">\oiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∬</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:0.957em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.957em" height="0.499em" style="width:0.957em;" viewBox="0 0 957 499" preserveAspectRatio="xMinYMin"><path d="M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6
+-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z
+m368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8
+60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span> <code>\oiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∰</mo></mrow><annotation encoding="application/x-tex">\oiiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∭</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:1.304em;"><svg xmlns="http://www.w3.org/2000/svg" width="1.304em" height="0.499em" style="width:1.304em;" viewBox="0 0 1304 499" preserveAspectRatio="xMinYMin"><path d="M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6
+-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z
+m525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0
+85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span> <code>\oiiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨆</mo></mrow><annotation encoding="application/x-tex">\bigsqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨆</span></span></span></span> <code>\bigsqcup</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mo>∬</mo><mo>∭</mo><mo>∮</mo><mo>∏</mo><mo>∐</mo><mo>∑</mo><mo>⋀</mo><mo>⋁</mo><mo>⋂</mo><mo>⋃</mo><mo>⨀</mo><mo>⨁</mo><mo>⨂</mo><mo>⨄</mo><mo>⨆</mo></mrow><annotation encoding="application/x-tex">∫ ∬ ∭ ∮ ∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ ⨄ ⨆</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∬∭</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏∐∑⋀⋁⋂⋃⨀⨁⨂⨄⨆</span></span></span></span></p><h3 id="_9-2-binary-operators" tabindex="-1">9.2: Binary Operators <a class="header-anchor" href="#_9-2-binary-operators" aria-hidden="true">#</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> <code>+</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span> <code>\cdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋗</mo></mrow><annotation encoding="application/x-tex">\gtrdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋗</span></span></span></span> <code>\gtrdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \pmod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.3333em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> <code>x \pmod a</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span> <code>-</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">⋅</mo></mrow><annotation encoding="application/x-tex">\cdotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mpunct">⋅</span></span></span></span> <code>\cdotp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊺</mo></mrow><annotation encoding="application/x-tex">\intercal</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊺</span></span></span></span> <code>\intercal</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \pod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> <code>x \pod a</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">/</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">/</span></span></span></span> <code>/</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\centerdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋅</span></span></span></span> <code>\centerdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\land</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\land</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\rhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊳</span></span></span></span> <code>\rhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> <code>*</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∘</span></span></span></span> <code>\circ</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋋</mo></mrow><annotation encoding="application/x-tex">\leftthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋋</span></span></span></span> <code>\leftthreetimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋌</mo></mrow><annotation encoding="application/x-tex">\rightthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋌</span></span></span></span> <code>\rightthreetimes</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">⨿</span></span></span></span> <code>\amalg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊛</mo></mrow><annotation encoding="application/x-tex">\circledast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊛</span></span></span></span> <code>\circledast</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">.</mo></mrow><annotation encoding="application/x-tex">\ldotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em;"></span><span class="mpunct">.</span></span></span></span> <code>\ldotp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋊</mo></mrow><annotation encoding="application/x-tex">\rtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋊</span></span></span></span> <code>\rtimes</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&amp;</mo></mrow><annotation encoding="application/x-tex">\And</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span> <code>\And</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊚</mo></mrow><annotation encoding="application/x-tex">\circledcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊚</span></span></span></span> <code>\circledcirc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\lor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\lor</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\setminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∖</span></span></span></span> <code>\setminus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> <code>\ast</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊝</mo></mrow><annotation encoding="application/x-tex">\circleddash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊝</span></span></span></span> <code>\circleddash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋖</mo></mrow><annotation encoding="application/x-tex">\lessdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋖</span></span></span></span> <code>\lessdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\smallsetminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∖</span></span></span></span> <code>\smallsetminus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊼</mo></mrow><annotation encoding="application/x-tex">\barwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊼</span></span></span></span> <code>\barwedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\Cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span> <code>\Cup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\lhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊲</span></span></span></span> <code>\lhd</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊓</mo></mrow><annotation encoding="application/x-tex">\sqcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊓</span></span></span></span> <code>\sqcap</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◯</mo></mrow><annotation encoding="application/x-tex">\bigcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">◯</span></span></span></span> <code>\bigcirc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∪</mo></mrow><annotation encoding="application/x-tex">\cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∪</span></span></span></span> <code>\cup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋉</mo></mrow><annotation encoding="application/x-tex">\ltimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋉</span></span></span></span> <code>\ltimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊔</span></span></span></span> <code>\sqcup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> </mtext><mo lspace="0.22em" rspace="0.22em"><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow></mo><mtext> </mtext></mrow><annotation encoding="application/x-tex">\bmod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;"></span></span></span></span> <code>\bmod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋎</mo></mrow><annotation encoding="application/x-tex">\curlyvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋎</span></span></span></span> <code>\curlyvee</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>a</mi></mrow><annotation encoding="application/x-tex">x \mod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span> <code>x\mod a</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> <code>\times</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊡</mo></mrow><annotation encoding="application/x-tex">\boxdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊡</span></span></span></span> <code>\boxdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋏</mo></mrow><annotation encoding="application/x-tex">\curlywedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋏</span></span></span></span> <code>\curlywedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∓</mo></mrow><annotation encoding="application/x-tex">\mp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">∓</span></span></span></span> <code>\mp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\unlhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊴</span></span></span></span> <code>\unlhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊟</mo></mrow><annotation encoding="application/x-tex">\boxminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊟</span></span></span></span> <code>\boxminus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>÷</mo></mrow><annotation encoding="application/x-tex">\div</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">÷</span></span></span></span> <code>\div</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊙</mo></mrow><annotation encoding="application/x-tex">\odot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊙</span></span></span></span> <code>\odot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\unrhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊵</span></span></span></span> <code>\unrhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊞</mo></mrow><annotation encoding="application/x-tex">\boxplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊞</span></span></span></span> <code>\boxplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋇</mo></mrow><annotation encoding="application/x-tex">\divideontimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋇</span></span></span></span> <code>\divideontimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊖</mo></mrow><annotation encoding="application/x-tex">\ominus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊖</span></span></span></span> <code>\ominus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊎</mo></mrow><annotation encoding="application/x-tex">\uplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊎</span></span></span></span> <code>\uplus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊠</mo></mrow><annotation encoding="application/x-tex">\boxtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊠</span></span></span></span> <code>\boxtimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∔</mo></mrow><annotation encoding="application/x-tex">\dotplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7742em;vertical-align:-0.082em;"></span><span class="mord amsrm">∔</span></span></span></span> <code>\dotplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span></span></span></span> <code>\oplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\vee</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∙</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∙</span></span></span></span> <code>\bullet</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩞</mo></mrow><annotation encoding="application/x-tex">\doublebarwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9503em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⩞</span></span></span></span> <code>\doublebarwedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊗</span></span></span></span> <code>\otimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊻</mo></mrow><annotation encoding="application/x-tex">\veebar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊻</span></span></span></span> <code>\veebar</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\Cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span> <code>\Cap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\doublecap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span> <code>\doublecap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊘</mo></mrow><annotation encoding="application/x-tex">\oslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊘</span></span></span></span> <code>\oslash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\wedge</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∩</mo></mrow><annotation encoding="application/x-tex">\cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∩</span></span></span></span> <code>\cap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\doublecup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span> <code>\doublecup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo></mrow><annotation encoding="application/x-tex">\pm</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">±</span></span></span></span> <code>\pm</code> or <code>\plusmn</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≀</mo></mrow><annotation encoding="application/x-tex">\wr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">≀</span></span></span></span> <code>\wr</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo><mo>−</mo><mi mathvariant="normal">/</mi><mo>∗</mo><mo>⋅</mo><mo>±</mo><mo>×</mo><mo>÷</mo><mo>∓</mo><mo>∔</mo><mo>∧</mo><mo>∨</mo><mo>∩</mo><mo>∪</mo><mo>≀</mo><mo>⊎</mo><mo>⊓</mo><mo>⊔</mo><mo>⊕</mo><mo>⊖</mo><mo>⊗</mo><mo>⊘</mo><mo>⊙</mo><mo>⊚</mo><mo>⊛</mo><mo>⊝</mo></mrow><annotation encoding="application/x-tex">+ - / * ⋅ ± × ÷ ∓ ∔ ∧ ∨ ∩ ∪ ≀ ⊎ ⊓ ⊔ ⊕ ⊖ ⊗ ⊘ ⊙ ⊚ ⊛ ⊝</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">÷</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7756em;vertical-align:-0.0833em;"></span><span class="mord">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin amsrm">∔</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">≀</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊎</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊔</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊘</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊙</span><span class="mord amsrm">⊚⊛⊝</span></span></span></span></p><h3 id="_9-3-fractions-and-binomials" tabindex="-1">9.3: Fractions and Binomials <a class="header-anchor" href="#_9-3-fractions-and-binomials" aria-hidden="true">#</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\frac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\tfrac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\tfrac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\genfrac ( ] {2pt}{1}a{a+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4108em;vertical-align:-0.5594em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8514em;"><span style="top:-2.4989em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.55em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5594em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span> <code>\genfrac ( ] {2pt}{1}a{a+1}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">{a \over b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> <code>{a \over b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mi>b</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\dfrac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">{a \above{2pt} b+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4458em;vertical-align:-0.5944em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8514em;"><span style="top:-2.4639em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.55em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5944em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> <code>{a \above{2pt} b+1}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi mathvariant="normal">/</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">a/b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord">/</span><span class="mord mathnormal">b</span></span></span></span> <code>a/b</code></td><td style="text-align:left;"></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>b</mi></mfrac></mstyle></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\cfrac{a}{1 + \cfrac{1}{b}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.756em;vertical-align:-2.166em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-3.82em;"><span class="pstrut" style="height:3.59em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.33em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.166em;"><span></span></span></span></span></span><span></span></span></span></span></span> <code>\cfrac{a}{1 + \cfrac{1}{b}}</code></td></tr></tbody></table><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> <code>\binom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\dbinom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span> <code>\dbinom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">{n\brace k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">{</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">}</span></span></span></span></span></span></span> <code>{n\brace k}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">{n \choose k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span> <code>{n \choose k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\tbinom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> <code>\tbinom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">{n\brack k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span> <code>{n\brack k}</code></td></tr></tbody></table><h3 id="_9-4-math-operators" tabindex="-1">9.4: Math Operators <a class="header-anchor" href="#_9-4-math-operators" aria-hidden="true">#</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcsin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcsin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">arcsin</span></span></span></span> <code>\arcsin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cotg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cotg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">cot<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\cotg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ln</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ln</span></span></span></span> <code>\ln</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\det</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">det</span></span></span></span> <code>\det</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arccos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arccos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">arccos</span></span></span></span> <code>\arccos</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>coth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\coth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">coth</span></span></span></span> <code>\coth</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\log</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>gcd</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\gcd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span style="margin-right:0.01389em;">g</span>cd</span></span></span></span> <code>\gcd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">arctan</span></span></span></span> <code>\arctan</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>csc</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\csc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">csc</span></span></span></span> <code>\csc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">sec</span></span></span></span> <code>\sec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\inf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">in<span style="margin-right:0.07778em;">f</span></span></span></span></span> <code>\inf</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">ct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\ctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span></span></span></span> <code>\sin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">lim</span></span></span></span> <code>\lim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arcct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arcctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cth</span></span></span></span> <code>\cth</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sinh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sinh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sinh</span></span></span></span> <code>\sinh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\liminf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm" style="margin-right:0.07778em;">inf</span></span></span></span></span> <code>\liminf</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>deg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\deg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">de<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\deg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sh</span></span></span></span> <code>\sh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\limsup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">sup</span></span></span></span></span> <code>\limsup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ch</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ch</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ch</span></span></span></span> <code>\ch</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>dim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\dim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">dim</span></span></span></span> <code>\dim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">tan</span></span></span></span> <code>\tan</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\max</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">max</span></span></span></span> <code>\max</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cos</span></span></span></span> <code>\cos</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>exp</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\exp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">exp</span></span></span></span> <code>\exp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tanh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tanh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">tanh</span></span></span></span> <code>\tanh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">min</span></span></span></span> <code>\min</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cosec</span></span></span></span> <code>\cosec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>hom</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\hom</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">hom</span></span></span></span> <code>\hom</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">t<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\tg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Pr</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\Pr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">Pr</span></span></span></span> <code>\Pr</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cosh</span></span></span></span> <code>\cosh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ker</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ker</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ker</span></span></span></span> <code>\ker</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>th</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\th</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">th</span></span></span></span> <code>\th</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">sup</span></span></span></span> <code>\sup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cot</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">cot</span></span></span></span> <code>\cot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">l<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\lg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">f</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\operatorname{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.07778em;">f</span></span></span></span></span> <code>\operatorname{f}</code></td><td></td></tr></tbody></table><p>Functions on the right column of this table can take <code>\limits</code>.</p><h3 id="_9-5-sqrt" tabindex="-1">9.5: <code>\sqrt</code> <a class="header-anchor" href="#_9-5-sqrt" aria-hidden="true">#</a></h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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+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
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+M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt{x}</code></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mroot><mi>x</mi><mn>3</mn></mroot></mrow><annotation encoding="application/x-tex">\sqrt[3]{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6586em;"><span style="top:-2.8363em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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+M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt[3]{x}</code></p><h2 id="_10-relations" tabindex="-1">10: Relations <a class="header-anchor" href="#_10-relations" aria-hidden="true">#</a></h2><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\stackrel{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\stackrel{!}{=}</code></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>=</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≖</mo></mrow><annotation encoding="application/x-tex">\eqcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">≖</span></span></span></span> <code>\eqcirc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋚</mo></mrow><annotation encoding="application/x-tex">\lesseqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋚</span></span></span></span> <code>\lesseqgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊐</mo></mrow><annotation encoding="application/x-tex">\sqsupset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊐</span></span></span></span> <code>\sqsupset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">&lt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> <code>&lt;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∹</mi></mo></mrow><annotation encoding="application/x-tex">\eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\eqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪋</mo></mrow><annotation encoding="application/x-tex">\lesseqqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪋</span></span></span></span> <code>\lesseqqgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊒</mo></mrow><annotation encoding="application/x-tex">\sqsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊒</span></span></span></span> <code>\sqsupseteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> <code>&gt;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">−</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span> <code>\Eqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≶</mo></mrow><annotation encoding="application/x-tex">\lessgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≶</span></span></span></span> <code>\lessgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋐</mo></mrow><annotation encoding="application/x-tex">\Subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋐</span></span></span></span> <code>\Subset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo></mrow><annotation encoding="application/x-tex">:</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span></span></span></span> <code>:</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≕</mi></mo></mrow><annotation encoding="application/x-tex">\eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\eqqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≲</mo></mrow><annotation encoding="application/x-tex">\lesssim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≲</span></span></span></span> <code>\lesssim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> <code>\subset</code> or <code>\sub</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span></span></span></span> <code>\approx</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">=</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span> <code>\Eqqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≪</mo></mrow><annotation encoding="application/x-tex">\ll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≪</span></span></span></span> <code>\ll</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊆</mo></mrow><annotation encoding="application/x-tex">\subseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊆</span></span></span></span> <code>\subseteq</code> or <code>\sube</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≊</mo></mrow><annotation encoding="application/x-tex">\approxeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">≊</span></span></span></span> <code>\approxeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≂</mo></mrow><annotation encoding="application/x-tex">\eqsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel amsrm">≂</span></span></span></span> <code>\eqsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\lll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span> <code>\lll</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫅</mo></mrow><annotation encoding="application/x-tex">\subseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫅</span></span></span></span> <code>\subseteqq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≍</mo></mrow><annotation encoding="application/x-tex">\asymp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≍</span></span></span></span> <code>\asymp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪖</mo></mrow><annotation encoding="application/x-tex">\eqslantgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪖</span></span></span></span> <code>\eqslantgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\llless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span> <code>\llless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≻</mo></mrow><annotation encoding="application/x-tex">\succ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≻</span></span></span></span> <code>\succ</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∍</mo></mrow><annotation encoding="application/x-tex">\backepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">∍</span></span></span></span> <code>\backepsilon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪕</mo></mrow><annotation encoding="application/x-tex">\eqslantless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪕</span></span></span></span> <code>\eqslantless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">\lt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> <code>\lt</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪸</mo></mrow><annotation encoding="application/x-tex">\succapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪸</span></span></span></span> <code>\succapprox</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∽</mo></mrow><annotation encoding="application/x-tex">\backsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">∽</span></span></span></span> <code>\backsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≡</mo></mrow><annotation encoding="application/x-tex">\equiv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≡</span></span></span></span> <code>\equiv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∣</span></span></span></span> <code>\mid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≽</mo></mrow><annotation encoding="application/x-tex">\succcurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≽</span></span></span></span> <code>\succcurlyeq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋍</mo></mrow><annotation encoding="application/x-tex">\backsimeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.464em;"></span><span class="mrel amsrm">⋍</span></span></span></span> <code>\backsimeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≒</mo></mrow><annotation encoding="application/x-tex">\fallingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≒</span></span></span></span> <code>\fallingdotseq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\models</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.999em;vertical-align:-0.249em;"></span><span class="mrel">⊨</span></span></span></span> <code>\models</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪰</mo></mrow><annotation encoding="application/x-tex">\succeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span></span></span></span> <code>\succeq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≬</mo></mrow><annotation encoding="application/x-tex">\between</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≬</span></span></span></span> <code>\between</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\frown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌢</span></span></span></span> <code>\frown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">⊸</span></span></span></span> <code>\multimap</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≿</mo></mrow><annotation encoding="application/x-tex">\succsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≿</span></span></span></span> <code>\succsim</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\bowtie</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span> <code>\bowtie</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> <code>\ge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\owns</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\owns</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋑</mo></mrow><annotation encoding="application/x-tex">\Supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋑</span></span></span></span> <code>\Supset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≏</mo></mrow><annotation encoding="application/x-tex">\bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≏</span></span></span></span> <code>\bumpeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\geq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> <code>\geq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\parallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∥</span></span></span></span> <code>\parallel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊃</span></span></span></span> <code>\supset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≎</mo></mrow><annotation encoding="application/x-tex">\Bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≎</span></span></span></span> <code>\Bumpeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≧</mo></mrow><annotation encoding="application/x-tex">\geqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≧</span></span></span></span> <code>\geqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\perp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span></span></span> <code>\perp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊇</mo></mrow><annotation encoding="application/x-tex">\supseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊇</span></span></span></span> <code>\supseteq</code> or <code>\supe</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≗</mo></mrow><annotation encoding="application/x-tex">\circeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≗</span></span></span></span> <code>\circeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩾</mo></mrow><annotation encoding="application/x-tex">\geqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩾</span></span></span></span> <code>\geqslant</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋔</mo></mrow><annotation encoding="application/x-tex">\pitchfork</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⋔</span></span></span></span> <code>\pitchfork</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫆</mo></mrow><annotation encoding="application/x-tex">\supseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫆</span></span></span></span> <code>\supseteqq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span> <code>\colonapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≫</mo></mrow><annotation encoding="application/x-tex">\gg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≫</span></span></span></span> <code>\gg</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≺</mo></mrow><annotation encoding="application/x-tex">\prec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≺</span></span></span></span> <code>\prec</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\thickapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel amsrm">≈</span></span></span></span> <code>\thickapprox</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\Colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span> <code>\Colonapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\ggg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span> <code>\ggg</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪷</mo></mrow><annotation encoding="application/x-tex">\precapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪷</span></span></span></span> <code>\precapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\thicksim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">∼</span></span></span></span> <code>\thicksim</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span> <code>\coloneq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\gggtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span> <code>\gggtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≼</mo></mrow><annotation encoding="application/x-tex">\preccurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≼</span></span></span></span> <code>\preccurlyeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\trianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊴</span></span></span></span> <code>\trianglelefteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span> <code>\Coloneq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\gt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> <code>\gt</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪯</mo></mrow><annotation encoding="application/x-tex">\preceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span></span></span></span> <code>\preceq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≜</mo></mrow><annotation encoding="application/x-tex">\triangleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9987em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≜</span></span></span></span> <code>\triangleq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≔</mi></mo></mrow><annotation encoding="application/x-tex">\coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span> <code>\coloneqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪆</mo></mrow><annotation encoding="application/x-tex">\gtrapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪆</span></span></span></span> <code>\gtrapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≾</mo></mrow><annotation encoding="application/x-tex">\precsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≾</span></span></span></span> <code>\precsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\trianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊵</span></span></span></span> <code>\trianglerighteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">=</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span> <code>\Coloneqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋛</mo></mrow><annotation encoding="application/x-tex">\gtreqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋛</span></span></span></span> <code>\gtreqless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\propto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">∝</span></span></span></span> <code>\propto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\varpropto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∝</span></span></span></span> <code>\varpropto</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span> <code>\colonsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪌</mo></mrow><annotation encoding="application/x-tex">\gtreqqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪌</span></span></span></span> <code>\gtreqqless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≓</mo></mrow><annotation encoding="application/x-tex">\risingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≓</span></span></span></span> <code>\risingdotseq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\vartriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">△</span></span></span></span> <code>\vartriangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\Colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span> <code>\Colonsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≷</mo></mrow><annotation encoding="application/x-tex">\gtrless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≷</span></span></span></span> <code>\gtrless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\shortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∣</span></span></span></span> <code>\shortmid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\vartriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊲</span></span></span></span> <code>\vartriangleleft</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≅</mo></mrow><annotation encoding="application/x-tex">\cong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.589em;"></span><span class="mrel">≅</span></span></span></span> <code>\cong</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≳</mo></mrow><annotation encoding="application/x-tex">\gtrsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≳</span></span></span></span> <code>\gtrsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\shortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∥</span></span></span></span> <code>\shortparallel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\vartriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊳</span></span></span></span> <code>\vartriangleright</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋞</mo></mrow><annotation encoding="application/x-tex">\curlyeqprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋞</span></span></span></span> <code>\curlyeqprec</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\in</code> or <code>\isin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">∼</span></span></span></span> <code>\sim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mo><mo>:</mo></mo></mo></mrow><annotation encoding="application/x-tex">\vcentcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span> <code>\vcentcolon</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋟</mo></mrow><annotation encoding="application/x-tex">\curlyeqsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋟</span></span></span></span> <code>\curlyeqsucc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\Join</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span> <code>\Join</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≃</span></span></span></span> <code>\simeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊢</span></span></span></span> <code>\vdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊣</span></span></span></span> <code>\dashv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> <code>\le</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\smallfrown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌢</span></span></span></span> <code>\smallfrown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\vDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊨</span></span></span></span> <code>\vDash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\dblcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\dblcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> <code>\leq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smallsmile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌣</span></span></span></span> <code>\smallsmile</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊩</mo></mrow><annotation encoding="application/x-tex">\Vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊩</span></span></span></span> <code>\Vdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≐</mo></mrow><annotation encoding="application/x-tex">\doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.673em;"></span><span class="mrel">≐</span></span></span></span> <code>\doteq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≦</mo></mrow><annotation encoding="application/x-tex">\leqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≦</span></span></span></span> <code>\leqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌣</span></span></span></span> <code>\smile</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊪</mo></mrow><annotation encoding="application/x-tex">\Vvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊪</span></span></span></span> <code>\Vvdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\Doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span> <code>\Doteq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩽</mo></mrow><annotation encoding="application/x-tex">\leqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩽</span></span></span></span> <code>\leqslant</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊏</mo></mrow><annotation encoding="application/x-tex">\sqsubset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊏</span></span></span></span> <code>\sqsubset</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\doteqdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span> <code>\doteqdot</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪅</mo></mrow><annotation encoding="application/x-tex">\lessapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪅</span></span></span></span> <code>\lessapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊑</mo></mrow><annotation encoding="application/x-tex">\sqsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊑</span></span></span></span> <code>\sqsubseteq</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mo>&lt;</mo><mo>&gt;</mo><mo>:</mo><mo>∈</mo><mo>∋</mo><mo>∝</mo><mo>∼</mo><mo>∽</mo><mo>≂</mo><mo>≃</mo><mo>≅</mo><mo>≈</mo><mo>≊</mo><mo>≍</mo><mo>≎</mo><mo>≏</mo><mo>≐</mo><mo>≑</mo><mo>≒</mo><mo>≓</mo><mo>≖</mo><mo>≗</mo><mo>≜</mo><mo>≡</mo><mo>≤</mo><mo>≥</mo><mo>≦</mo><mo>≧</mo><mo>≫</mo><mo>≬</mo><mo>≳</mo><mo>≷</mo><mo>≺</mo><mo>≻</mo><mo>≼</mo><mo>≽</mo><mo>≾</mo><mo>≿</mo><mo>⊂</mo><mo>⊃</mo><mo>⊆</mo><mo>⊇</mo><mo>⊏</mo><mo>⊐</mo><mo>⊑</mo><mo>⊒</mo><mo>⊢</mo><mo>⊣</mo><mo>⊩</mo><mo>⊪</mo><mo>⊸</mo><mo>⋈</mo><mo>⋍</mo><mo>⋐</mo><mo>⋑</mo><mo>⋔</mo><mo>⋙</mo><mo>⋛</mo><mo>⋞</mo><mo>⋟</mo><mo>⌢</mo><mo>⌣</mo><mo>⩾</mo><mo>⪆</mo><mo>⪌</mo><mo>⪕</mo><mo>⪖</mo><mo>⪯</mo><mo>⪰</mo><mo>⪷</mo><mo>⪸</mo><mo>⫅</mo><mo>⫆</mo><mo>≲</mo><mo>⩽</mo><mo>⪅</mo><mo>≶</mo><mo>⋚</mo><mo>⪋</mo><mo>⊥</mo><mo>⊨</mo></mrow><annotation encoding="application/x-tex">= &lt; &gt; : ∈ ∋ ∝ ∼ ∽ ≂ ≃ ≅ ≈ ≊ ≍ ≎ ≏ ≐ ≑ ≒ ≓ ≖ ≗ ≜ ≡ ≤ ≥ ≦ ≧ ≫ ≬ ≳ ≷ ≺ ≻ ≼ ≽ ≾ ≿ ⊂ ⊃ ⊆ ⊇ ⊏ ⊐ ⊑ ⊒ ⊢ ⊣ ⊩ ⊪ ⊸ ⋈ ⋍ ⋐ ⋑ ⋔ ⋙ ⋛ ⋞ ⋟ ⌢ ⌣ ⩾ ⪆ ⪌ ⪕ ⪖ ⪯ ⪰ ⪷ ⪸ ⫅ ⫆ ≲ ⩽ ⪅ ≶ ⋚ ⪋ ⟂ ⊨</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">=&lt;&gt;:∈∋∝∼</span></span><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel amsrm">∽≂</span></span><span class="base"><span class="strut" style="height:0.589em;"></span><span class="mrel">≃≅≈</span></span><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">≊</span></span><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≍</span></span><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≎≏</span></span><span class="base"><span class="strut" style="height:0.673em;"></span><span class="mrel">≐</span></span><span class="base"><span class="strut" style="height:1.1462em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≑≒≓≖≗≜</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≡≤≥</span></span><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≦≧</span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≫</span></span><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≬≳≷</span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≺≻</span></span><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≼≽≾≿</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊂⊃⊆⊇</span></span><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊏⊐</span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mrel">⊑⊒⊢⊣</span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊩⊪⊸</span></span><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋍⋐⋑⋔⋙⋛⋞⋟</span></span><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌢⌣</span></span><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⩾⪆⪌⪕⪖</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯⪰</span></span><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪷⪸⫅⫆≲⩽⪅≶⋚⪋</span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊨</span></span></span></span> <code>≔ ≕ ⩴</code></p><h3 id="_10-1-negated-relations" tabindex="-1">10.1: Negated Relations <a class="header-anchor" href="#_10-1-negated-relations" aria-hidden="true">#</a></h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\not =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>\not =</code></p><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪊</mo></mrow><annotation encoding="application/x-tex">\gnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪊</span></span></span></span> <code>\gnapprox</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\ngeqslant</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊈</span></span></span></span> <code>\nsubseteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪵</mo></mrow><annotation encoding="application/x-tex">\precneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪵</span></span></span></span> <code>\precneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪈</mo></mrow><annotation encoding="application/x-tex">\gneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪈</span></span></span></span> <code>\gneq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≯</mo></mrow><annotation encoding="application/x-tex">\ngtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≯</span></span></span></span> <code>\ngtr</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nsubseteqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋨</mo></mrow><annotation encoding="application/x-tex">\precnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋨</span></span></span></span> <code>\precnsim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≩</span></span></span></span> <code>\gneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≰</span></span></span></span> <code>\nleq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊁</mo></mrow><annotation encoding="application/x-tex">\nsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊁</span></span></span></span> <code>\nsucc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\subsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊊</span></span></span></span> <code>\subsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋧</mo></mrow><annotation encoding="application/x-tex">\gnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋧</span></span></span></span> <code>\gnsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nleqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋡</mo></mrow><annotation encoding="application/x-tex">\nsucceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋡</span></span></span></span> <code>\nsucceq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\subsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫋</span></span></span></span> <code>\subsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\gvertneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nleqslant</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊉</span></span></span></span> <code>\nsupseteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪺</mo></mrow><annotation encoding="application/x-tex">\succnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪺</span></span></span></span> <code>\succnapprox</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪉</mo></mrow><annotation encoding="application/x-tex">\lnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪉</span></span></span></span> <code>\lnapprox</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≮</mo></mrow><annotation encoding="application/x-tex">\nless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≮</span></span></span></span> <code>\nless</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nsupseteqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪶</mo></mrow><annotation encoding="application/x-tex">\succneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪶</span></span></span></span> <code>\succneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪇</mo></mrow><annotation encoding="application/x-tex">\lneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪇</span></span></span></span> <code>\lneq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∤</span></span></span></span> <code>\nmid</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋪</mo></mrow><annotation encoding="application/x-tex">\ntriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋪</span></span></span></span> <code>\ntriangleleft</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋩</mo></mrow><annotation encoding="application/x-tex">\succnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋩</span></span></span></span> <code>\succnsim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≨</span></span></span></span> <code>\lneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></span> <code>\notin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋬</mo></mrow><annotation encoding="application/x-tex">\ntrianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋬</span></span></span></span> <code>\ntrianglelefteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\supsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊋</span></span></span></span> <code>\supsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋦</mo></mrow><annotation encoding="application/x-tex">\lnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋦</span></span></span></span> <code>\lnsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∌</mo></mrow><annotation encoding="application/x-tex">\notni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\notni</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋫</mo></mrow><annotation encoding="application/x-tex">\ntriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋫</span></span></span></span> <code>\ntriangleright</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\supsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫌</span></span></span></span> <code>\supsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\lvertneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∦</span></span></span></span> <code>\nparallel</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋭</mo></mrow><annotation encoding="application/x-tex">\ntrianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋭</span></span></span></span> <code>\ntrianglerighteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\varsubsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsubsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≆</mo></mrow><annotation encoding="application/x-tex">\ncong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≆</span></span></span></span> <code>\ncong</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊀</mo></mrow><annotation encoding="application/x-tex">\nprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊀</span></span></span></span> <code>\nprec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊬</mo></mrow><annotation encoding="application/x-tex">\nvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊬</span></span></span></span> <code>\nvdash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\varsubsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsubsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> <code>\ne</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋠</mo></mrow><annotation encoding="application/x-tex">\npreceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋠</span></span></span></span> <code>\npreceq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊭</mo></mrow><annotation encoding="application/x-tex">\nvDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊭</span></span></span></span> <code>\nvDash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\varsupsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsupsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> <code>\neq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nshortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nshortmid</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊯</mo></mrow><annotation encoding="application/x-tex">\nVDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊯</span></span></span></span> <code>\nVDash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\varsupsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsupsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≱</span></span></span></span> <code>\ngeq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nshortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nshortparallel</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊮</mo></mrow><annotation encoding="application/x-tex">\nVdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊮</span></span></span></span> <code>\nVdash</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\ngeqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≁</mo></mrow><annotation encoding="application/x-tex">\nsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">≁</span></span></span></span> <code>\nsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪹</mo></mrow><annotation encoding="application/x-tex">\precnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪹</span></span></span></span> <code>\precnapprox</code></td><td></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo><mo mathvariant="normal">∌</mo><mo>∤</mo><mo>∦</mo><mo>≁</mo><mo>≆</mo><mo mathvariant="normal">≠</mo><mo>≨</mo><mo>≩</mo><mo>≮</mo><mo>≯</mo><mo>≰</mo><mo>≱</mo><mo>⊀</mo><mo>⊁</mo><mo>⊈</mo><mo>⊉</mo><mo>⊊</mo><mo>⊋</mo><mo>⊬</mo><mo>⊭</mo><mo>⊮</mo><mo>⊯</mo><mo>⋠</mo><mo>⋡</mo><mo>⋦</mo><mo>⋧</mo><mo>⋨</mo><mo>⋩</mo><mo>⋬</mo><mo>⋭</mo><mo>⪇</mo><mo>⪈</mo><mo>⪉</mo><mo>⪊</mo><mo>⪵</mo><mo>⪶</mo><mo>⪹</mo><mo>⪺</mo><mo>⫋</mo><mo>⫌</mo></mrow><annotation encoding="application/x-tex">∉ ∌ ∤ ∦ ≁ ≆ ≠ ≨ ≩ ≮ ≯ ≰ ≱ ⊀ ⊁ ⊈ ⊉ ⊊ ⊋ ⊬ ⊭ ⊮ ⊯ ⋠ ⋡ ⋦ ⋧ ⋨ ⋩ ⋬ ⋭ ⪇ ⪈ ⪉ ⪊ ⪵ ⪶ ⪹ ⪺ ⫋ ⫌</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">∤∦≁≆</span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≨≩≮≯≰≱⊀⊁⊈⊉⊊⊋⊬⊭⊮⊯⋠⋡⋦⋧⋨⋩⋬⋭⪇⪈⪉⪊⪵⪶⪹⪺⫋⫌</span></span></span></span></p><h3 id="_10-2-arrows" tabindex="-1">10.2: Arrows <a class="header-anchor" href="#_10-2-arrows" aria-hidden="true">#</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↺</mo></mrow><annotation encoding="application/x-tex">\circlearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↺</span></span></span></span> <code>\circlearrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↼</mo></mrow><annotation encoding="application/x-tex">\leftharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↼</span></span></span></span> <code>\leftharpoonup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\rArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\rArr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↻</mo></mrow><annotation encoding="application/x-tex">\circlearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↻</span></span></span></span> <code>\circlearrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇇</mo></mrow><annotation encoding="application/x-tex">\leftleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇇</span></span></span></span> <code>\leftleftarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\rarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↶</mo></mrow><annotation encoding="application/x-tex">\curvearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↶</span></span></span></span> <code>\curvearrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\restriction</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span> <code>\restriction</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↷</mo></mrow><annotation encoding="application/x-tex">\curvearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↷</span></span></span></span> <code>\curvearrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\rightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Darr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇆</mo></mrow><annotation encoding="application/x-tex">\leftrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇆</span></span></span></span> <code>\leftrightarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\Rightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\dArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\dArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇋</mo></mrow><annotation encoding="application/x-tex">\leftrightharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">⇋</span></span></span></span> <code>\leftrightharpoons</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↣</mo></mrow><annotation encoding="application/x-tex">\rightarrowtail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↣</span></span></span></span> <code>\rightarrowtail</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\darr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↭</mo></mrow><annotation encoding="application/x-tex">\leftrightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">↭</span></span></span></span> <code>\leftrightsquigarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇁</mo></mrow><annotation encoding="application/x-tex">\rightharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇁</span></span></span></span> <code>\rightharpoondown</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇠</mo></mrow><annotation encoding="application/x-tex">\dashleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇠</span></span></span></span> <code>\dashleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇚</mo></mrow><annotation encoding="application/x-tex">\Lleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇚</span></span></span></span> <code>\Lleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇀</mo></mrow><annotation encoding="application/x-tex">\rightharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇀</span></span></span></span> <code>\rightharpoonup</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇢</mo></mrow><annotation encoding="application/x-tex">\dashrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇢</span></span></span></span> <code>\dashrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟵</mo></mrow><annotation encoding="application/x-tex">\longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟵</span></span></span></span> <code>\longleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇄</mo></mrow><annotation encoding="application/x-tex">\rightleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇄</span></span></span></span> <code>\rightleftarrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\downarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟸</mo></mrow><annotation encoding="application/x-tex">\Longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟸</span></span></span></span> <code>\Longleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇌</mo></mrow><annotation encoding="application/x-tex">\rightleftharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.682em;vertical-align:-0.011em;"></span><span class="mrel">⇌</span></span></span></span> <code>\rightleftharpoons</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Downarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟷</mo></mrow><annotation encoding="application/x-tex">\longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟷</span></span></span></span> <code>\longleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇉</mo></mrow><annotation encoding="application/x-tex">\rightrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇉</span></span></span></span> <code>\rightrightarrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇊</mo></mrow><annotation encoding="application/x-tex">\downdownarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇊</span></span></span></span> <code>\downdownarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟺</mo></mrow><annotation encoding="application/x-tex">\Longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟺</span></span></span></span> <code>\Longleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span> <code>\rightsquigarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇃</mo></mrow><annotation encoding="application/x-tex">\downharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇃</span></span></span></span> <code>\downharpoonleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟼</mo></mrow><annotation encoding="application/x-tex">\longmapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟼</span></span></span></span> <code>\longmapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇛</mo></mrow><annotation encoding="application/x-tex">\Rrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇛</span></span></span></span> <code>\Rrightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇂</mo></mrow><annotation encoding="application/x-tex">\downharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇂</span></span></span></span> <code>\downharpoonright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span> <code>\longrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↱</mo></mrow><annotation encoding="application/x-tex">\Rsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↱</span></span></span></span> <code>\Rsh</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\gets</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\gets</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟹</mo></mrow><annotation encoding="application/x-tex">\Longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟹</span></span></span></span> <code>\Longrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↘</mo></mrow><annotation encoding="application/x-tex">\searrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↘</span></span></span></span> <code>\searrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Harr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↫</mo></mrow><annotation encoding="application/x-tex">\looparrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↫</span></span></span></span> <code>\looparrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↙</mo></mrow><annotation encoding="application/x-tex">\swarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↙</span></span></span></span> <code>\swarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\hArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\hArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↬</mo></mrow><annotation encoding="application/x-tex">\looparrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↬</span></span></span></span> <code>\looparrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\to</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\harr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Lrarr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↞</mo></mrow><annotation encoding="application/x-tex">\twoheadleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↞</span></span></span></span> <code>\twoheadleftarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↩</mo></mrow><annotation encoding="application/x-tex">\hookleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↩</span></span></span></span> <code>\hookleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\lrArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\lrArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↠</mo></mrow><annotation encoding="application/x-tex">\twoheadrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↠</span></span></span></span> <code>\twoheadrightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↪</span></span></span></span> <code>\hookrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\lrarr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\iff</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\iff</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↰</mo></mrow><annotation encoding="application/x-tex">\Lsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↰</span></span></span></span> <code>\Lsh</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\uArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\uArr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟸</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\impliedby</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟸</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\impliedby</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦</span></span></span></span> <code>\mapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\implies</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\implies</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↗</mo></mrow><annotation encoding="application/x-tex">\nearrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↗</span></span></span></span> <code>\nearrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\Larr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↚</mo></mrow><annotation encoding="application/x-tex">\nleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↚</span></span></span></span> <code>\nleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\lArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\lArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇍</mo></mrow><annotation encoding="application/x-tex">\nLeftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇍</span></span></span></span> <code>\nLeftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↕</mo></mrow><annotation encoding="application/x-tex">\updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">↕</span></span></span></span> <code>\updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\larr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↮</mo></mrow><annotation encoding="application/x-tex">\nleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↮</span></span></span></span> <code>\nleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇕</mo></mrow><annotation encoding="application/x-tex">\Updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇕</span></span></span></span> <code>\Updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\leadsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span> <code>\leadsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇎</mo></mrow><annotation encoding="application/x-tex">\nLeftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇎</span></span></span></span> <code>\nLeftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↿</mo></mrow><annotation encoding="application/x-tex">\upharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↿</span></span></span></span> <code>\upharpoonleft</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\leftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↛</mo></mrow><annotation encoding="application/x-tex">\nrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↛</span></span></span></span> <code>\nrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\upharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span> <code>\upharpoonright</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\Leftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇏</mo></mrow><annotation encoding="application/x-tex">\nRightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇏</span></span></span></span> <code>\nRightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇈</mo></mrow><annotation encoding="application/x-tex">\upuparrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇈</span></span></span></span> <code>\upuparrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↢</mo></mrow><annotation encoding="application/x-tex">\leftarrowtail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↢</span></span></span></span> <code>\leftarrowtail</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↖</mo></mrow><annotation encoding="application/x-tex">\nwarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↖</span></span></span></span> <code>\nwarrow</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↽</mo></mrow><annotation encoding="application/x-tex">\leftharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↽</span></span></span></span> <code>\leftharpoondown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\Rarr</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo><mo>↑</mo><mo>→</mo><mo>↓</mo><mo>↔</mo><mo>↕</mo><mo>↖</mo><mo>↗</mo><mo>↘</mo><mo>↙</mo><mo>↚</mo><mo>↛</mo><mo>↞</mo><mo>↠</mo><mo>↢</mo><mo>↣</mo><mo>↦</mo><mo>↩</mo><mo>↪</mo><mo>↫</mo><mo>↬</mo><mo>↭</mo><mo>↮</mo><mo>↰</mo><mo>↱</mo><mo>↶</mo><mo>↷</mo><mo>↺</mo><mo>↻</mo><mo>↼</mo><mo>↽</mo><mo>↾</mo><mo>↾</mo><mo>↿</mo><mo>⇀</mo><mo>⇁</mo><mo>⇂</mo><mo>⇃</mo><mo>⇄</mo><mo>⇆</mo><mo>⇇</mo><mo>⇈</mo><mo>⇉</mo><mo>⇊</mo><mo>⇋</mo><mo>⇌</mo><mo>⇍</mo><mo>⇎</mo><mo>⇏</mo><mo>⇐</mo><mo>⇑</mo><mo>⇒</mo><mo>⇓</mo><mo>⇔</mo><mo>⇕</mo><mo>⇚</mo><mo>⇛</mo><mo>⇝</mo><mo>⇠</mo><mo>⇢</mo><mo>⟵</mo><mo>⟶</mo><mo>⟷</mo><mo>⟸</mo><mo>⟹</mo><mo>⟺</mo><mo>⟼</mo></mrow><annotation encoding="application/x-tex">← ↑ → ↓ ↔ ↕ ↖ ↗ ↘ ↙ ↚ ↛ ↞ ↠ ↢ ↣ ↦ ↩ ↪ ↫ ↬ ↭ ↮ ↰ ↱↶ ↷ ↺ ↻ ↼ ↽ ↾ ↾ ↿ ⇀ ⇁ ⇂ ⇃ ⇄ ⇆ ⇇ ⇈ ⇉ ⇊ ⇋ ⇌⇍ ⇎ ⇏ ⇐ ⇑ ⇒ ⇓ ⇔ ⇕ ⇚ ⇛ ⇝ ⇠ ⇢ ⟵ ⟶ ⟷ ⟸ ⟹ ⟺ ⟼</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">←↑→↓↔↕↖↗↘↙</span></span><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↚↛↞↠↢↣</span></span><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦↩↪</span></span><span class="base"><span class="strut" style="height:0.7742em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↫↬↭↮↰↱↶↷↺↻</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↼↽</span></span><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾↾↿</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇀⇁</span></span><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇂⇃⇄⇆⇇⇈⇉⇊⇋</span></span><span class="base"><span class="strut" style="height:0.682em;vertical-align:-0.011em;"></span><span class="mrel">⇌</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇍⇎⇏</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇐⇑⇒⇓⇔⇕</span></span><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇚⇛⇝⇠⇢</span></span><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟵⟶⟷⟸⟹⟺⟼</span></span></span></span> ↽</p><p><strong>Extensible Arrows</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">←</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
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+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>u</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>r</mi></mrow></mpadded><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>o</mi><mi>v</mi><mi>e</mi><mi>r</mi></mrow></mpadded></munderover></mrow><annotation encoding="application/x-tex">\xrightarrow[under]{over}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5315em;vertical-align:-0.6081em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9234em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">o</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">er</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-2.0919em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">er</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6081em;"><span></span></span></span></span></span></span></span></span> <code>\xrightarrow[under]{over}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇐</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
+c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
+ 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
+-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
+-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
+m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xLeftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇒</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xRightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xRightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftrightarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.56em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
+c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
+ 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
+-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
+-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
+m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xLeftrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↩</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3
+ 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0
+-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21
+ 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xhookleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↪</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5
+-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3
+-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21
+ 71.5 23h399859zM103 281v-40h399897v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
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+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xhookrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↞</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 167c68 40
+ 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69
+-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3
+-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19
+-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101
+ 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xtwoheadleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↠</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167
+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"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xtwoheadrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↼</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftharpoonup{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇀</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xrightharpoonup{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↽</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333
+ 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5
+ 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667
+-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftharpoondown{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇁</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8
+ 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5
+-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95
+-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xrightharpoondown{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇋</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5
+ 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3
+-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7
+-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z
+m0 0v40h400000v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8
+ 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3
+ 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3
+-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z
+m0-194v40h400000v-40zm0 0v40h400000v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span> <code>\xleftrightharpoons{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇌</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightleftharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12
+ 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7
+-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0
+v40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11
+-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7
+ 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z
+m0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span> <code>\xrightleftharpoons{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇄</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtofrom{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1251em;vertical-align:-0.014em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1111em;"><span style="top:-3.325em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.686em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.528em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMinYMin slice"><path d="M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23
+-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8
+c28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3
+ 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23
+ 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32
+-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142
+-167z M100 147v40h399900v-40zM0 341v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.014em;"><span></span></span></span></span></span></span></span></span> <code>\xtofrom{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↦</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xmapsto{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.5em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V448H0V74H40V241H400000v40z
+M40 281 V448H0V74H40V241H400000v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
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+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xmapsto{abc}</code></td></tr></tbody></table><p>|<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">=</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xlongequal{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 50 h400000 v40H0z m0 194h40000v40H0z
+M0 50 h400000 v40H0z m0 194h40000v40H0z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xlongequal{abc}</code></p><p>Extensible arrows all can take an optional argument in the same manner, as <code>\xrightarrow[under]{over}</code>.</p><h2 id="_11-style-color-size-and-font" tabindex="-1">11: Style, Color, Size, and Font <a class="header-anchor" href="#_11-style-color-size-and-font" aria-hidden="true">#</a></h2><p><strong>Class Assignment</strong></p><ul><li><p><code>\mathbin</code> <code>\mathclose</code> <code>\mathinner</code> <code>\mathop</code></p></li><li><p><code>\mathopen</code> <code>\mathord</code> <code>\mathpunct</code> <code>\mathrel</code></p></li></ul><p><strong>Color</strong></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\color{blue} F=ma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:blue;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:blue;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:blue;">ma</span></span></span></span> <code>\color{blue} F=ma</code></p><p>Note that KaTeX <code>\color</code> acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{F=ma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:blue;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:blue;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:blue;">ma</span></span></span></span> <code>\textcolor{blue}{F=ma}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#228B22"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{#228B22}{F=ma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:#228B22;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:#228B22;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:#228B22;">ma</span></span></span></span> <code>\textcolor{#228B22}{F=ma}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" mathbackground="aqua"><mtext>A</mtext></mpadded></mrow><annotation encoding="application/x-tex">\colorbox{aqua}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2833em;vertical-align:-0.3em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9833em;"><span style="top:-2.9833em;"><span class="pstrut" style="height:3.2833em;"></span><span class="stretchy colorbox" style="height:1.2833em;background-color:aqua;"></span></span><span style="top:-3.2833em;"><span class="pstrut" style="height:3.2833em;"></span><span class="mord boxpad"><span class="mord">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3em;"><span></span></span></span></span></span></span></span></span> <code>\colorbox{aqua}{A}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" style="border:0.04em solid red;" mathbackground="aqua"><mtext>A</mtext></mpadded></mrow><annotation encoding="application/x-tex">\fcolorbox{red}{aqua}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3633em;vertical-align:-0.34em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0233em;"><span style="top:-3.0233em;"><span class="pstrut" style="height:3.3633em;"></span><span class="stretchy fcolorbox" style="height:1.3633em;border-style:solid;border-width:0.04em;background-color:aqua;border-color:red;"></span></span><span style="top:-3.3633em;"><span class="pstrut" style="height:3.3633em;"></span><span class="mord boxpad"><span class="mord">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34em;"><span></span></span></span></span></span></span></span></span> <code>\fcolorbox{red}{aqua}{A}</code></li></ul><p>For color definition, KaTeX color functions will accept the standard HTML <a href="https://developer.mozilla.org/en-US/docs/Web/CSS/color_value#Color_keywords" target="_blank" rel="noreferrer">predefined color names</a>. They will also accept an RGB argument in CSS hexa­decimal style. The &quot;#&quot; is optional before a six-digit specification.</p><p><strong>Font</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathrm{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">Ab0</span></span></span></span></span> <code>\mathrm{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\mathbf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\mathbf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\mathit{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathit">Ab</span></span></span></span></span> <code>\mathit{Ab}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathnormal{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal">b0</span></span></span></span></span> <code>\mathnormal{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="bold">Ab0</mtext></mrow><annotation encoding="application/x-tex">\textbf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textbf">Ab0</span></span></span></span></span> <code>\textbf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="italic">Ab</mtext></mrow><annotation encoding="application/x-tex">\textit{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textit">Ab</span></span></span></span></span> <code>\textit{Ab}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\textrm{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textrm">Ab0</span></span></span></span></span> <code>\textrm{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\bf Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\bf Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\it Ab</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathit">Ab</span></span></span></span></span> <code>\it Ab</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\rm Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">Ab0</span></span></span></span></span> <code>\rm Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\bold{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\bold{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi></mrow><annotation encoding="application/x-tex">\Bbb{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">AB</span></span></span></span></span> <code>\Bbb{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\textnormal{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textrm">Ab0</span></span></span></span></span> <code>\textnormal{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\boldsymbol{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">Ab</span></span></span></span></span></span> <code>\boldsymbol{Ab}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi></mrow><annotation encoding="application/x-tex">\mathbb{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">AB</span></span></span></span></span> <code>\mathbb{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\text{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Ab0</span></span></span></span></span> <code>\text{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\bm{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">Ab</span></span></span></span></span></span> <code>\bm{Ab}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">b</mi><mn mathvariant="fraktur">0</mn></mrow><annotation encoding="application/x-tex">\frak{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">Ab0</span></span></span></span></span> <code>\frak{Ab0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">0</mn></mrow><annotation encoding="application/x-tex">\mathsf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Ab0</span></span></span></span></span> <code>\mathsf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">0</mn></mrow><annotation encoding="application/x-tex">\mathtt{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord mathtt">Ab0</span></span></span></span></span> <code>\mathtt{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">b</mi><mn mathvariant="fraktur">0</mn></mrow><annotation encoding="application/x-tex">\mathfrak{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">Ab0</span></span></span></span></span> <code>\mathfrak{Ab0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="sans-serif">Ab0</mtext></mrow><annotation encoding="application/x-tex">\textsf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">Ab0</span></span></span></span></span> <code>\textsf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Ab0</mtext></mrow><annotation encoding="application/x-tex">\texttt{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">Ab0</span></span></span></span></span> <code>\texttt{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">B</mi><mn mathvariant="script">0</mn></mrow><annotation encoding="application/x-tex">\mathcal{AB0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="mord mathcal" style="margin-right:0.03041em;">B</span><span class="mord">0</span></span></span></span></span> <code>\mathcal{AB0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">0</mn></mrow><annotation encoding="application/x-tex">\sf Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Ab0</span></span></span></span></span> <code>\sf Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">0</mn></mrow><annotation encoding="application/x-tex">\tt Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord mathtt">Ab0</span></span></span></span></span> <code>\tt Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">B</mi></mrow><annotation encoding="application/x-tex">\mathscr{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.04087em;">AB</span></span></span></span></span> <code>\mathscr{AB}</code></td></tr></tbody></table><p>One can stack font family, font weight, and font shape by using the <code>\textXX</code> versions of the font functions. So <code>\textsf{\textbf{H}}</code> will produce <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="bold-sans-serif">H</mtext></mrow><annotation encoding="application/x-tex">\textsf{\textbf{H}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord text"><span class="mord textsf textbf">H</span></span></span></span></span></span>. The other versions do not stack, e.g., <code>\mathsf{\mathbf{H}}</code> will produce <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">H</mi></mrow><annotation encoding="application/x-tex">\mathsf{\mathbf{H}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">H</span></span></span></span>.</p><p>In cases where KaTeX fonts do not have a bold glyph, <code>\pmb</code> can simulate one. For example, <code>\pmb{\mu}</code> renders as : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle style="text-shadow:0.02em 0.01em 0.04px;"><mi>μ</mi></mstyle></mrow><annotation encoding="application/x-tex">\pmb{\mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord" style="text-shadow:0.02em 0.01em 0.04px;"><span class="mord mathnormal">μ</span></span></span></span></span></p><p><strong>Size</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.488em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\Huge AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7001em;"></span><span class="mord mathnormal sizing reset-size6 size11">A</span><span class="mord mathnormal sizing reset-size6 size11" style="margin-right:0.05017em;">B</span></span></span></span> <code>\Huge AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\normalsize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> <code>\normalsize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\huge AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4172em;"></span><span class="mord mathnormal sizing reset-size6 size10">A</span><span class="mord mathnormal sizing reset-size6 size10" style="margin-right:0.05017em;">B</span></span></span></span> <code>\huge AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\small AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.615em;"></span><span class="mord mathnormal sizing reset-size6 size5">A</span><span class="mord mathnormal sizing reset-size6 size5" style="margin-right:0.05017em;">B</span></span></span></span> <code>\small AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.728em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\LARGE AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1808em;"></span><span class="mord mathnormal sizing reset-size6 size9">A</span><span class="mord mathnormal sizing reset-size6 size9" style="margin-right:0.05017em;">B</span></span></span></span> <code>\LARGE AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.8em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\footnotesize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5467em;"></span><span class="mord mathnormal sizing reset-size6 size4">A</span><span class="mord mathnormal sizing reset-size6 size4" style="margin-right:0.05017em;">B</span></span></span></span> <code>\footnotesize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.44em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\Large AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.984em;"></span><span class="mord mathnormal sizing reset-size6 size8">A</span><span class="mord mathnormal sizing reset-size6 size8" style="margin-right:0.05017em;">B</span></span></span></span> <code>\Large AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.7em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptsize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4783em;"></span><span class="mord mathnormal sizing reset-size6 size3">A</span><span class="mord mathnormal sizing reset-size6 size3" style="margin-right:0.05017em;">B</span></span></span></span> <code>\scriptsize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.2em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\large AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.82em;"></span><span class="mord mathnormal sizing reset-size6 size7">A</span><span class="mord mathnormal sizing reset-size6 size7" style="margin-right:0.05017em;">B</span></span></span></span> <code>\large AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.5em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\tiny AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3417em;"></span><span class="mord mathnormal sizing reset-size6 size1">A</span><span class="mord mathnormal sizing reset-size6 size1" style="margin-right:0.05017em;">B</span></span></span></span> <code>\tiny AB</code></td></tr></tbody></table><p><strong>Style</strong></p><table><thead><tr><th style="text-align:left;"></th></tr></thead></table><p>|<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_{i=1}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span></span></span></span> <code>\displaystyle\sum_{i=1}^n</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="false"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mstyle></mrow><annotation encoding="application/x-tex">\textstyle\sum_{i=1}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span></span></span></span> <code>\textstyle\sum_{i=1}^n</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="1" displaystyle="false"><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptstyle x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3014em;"></span><span class="mord mathnormal mtight sizing reset-size6 size3">x</span></span></span></span> <code>\scriptstyle x</code>         (The size of a first sub/superscript) |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.2153em;"></span><span class="mord mathnormal mtight sizing reset-size6 size1">x</span></span></span></span> <code>\scriptscriptstyle x</code> (The size of subsequent sub/superscripts) |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></munder></mrow><annotation encoding="application/x-tex">\lim\limits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span></span></span></span> <code>\lim\limits_x</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\lim\nolimits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <code>\lim\nolimits_x</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">x^2</mtext></mrow><annotation encoding="application/x-tex">\verb!x^2!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">x^2</span></span></span></span></span> <code>\verb!x^2!</code></p><p><code>\text{…}</code> will accept nested <code>$…$</code> fragments and render them in math mode.</p><h2 id="_12-symbols-and-punctuation" tabindex="-1">12: Symbols and Punctuation <a class="header-anchor" href="#_12-symbols-and-punctuation" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><code>% comment</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>KaTeX</mtext></mrow><annotation encoding="application/x-tex">\KaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">K</span><span class="mspace" style="margin-right:-0.17em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span> <code>\KaTeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">%</span></span></span></span> <code>\%</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\cdots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>LaTeX</mtext></mrow><annotation encoding="application/x-tex">\LaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">L</span><span class="mspace" style="margin-right:-0.36em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span> <code>\LaTeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">#</mi></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">#</span></span></span></span> <code>\#</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋱</mo></mrow><annotation encoding="application/x-tex">\ddots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.823em;"></span><span class="minner">⋱</span></span></span></span> <code>\ddots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>TeX</mtext></mrow><annotation encoding="application/x-tex">\TeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span> <code>\TeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&amp;</mi></mrow><annotation encoding="application/x-tex">\&amp;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span> <code>\&amp;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\ldots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> <code>\nabla</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">_</mi></mrow><annotation encoding="application/x-tex">\_</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord" style="margin-right:0.02778em;">_</span></span></span></span> <code>\_</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.03em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span> <code>\vdots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> <code>\infty</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>_</mtext></mrow><annotation encoding="application/x-tex">\text{\textunderscore}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">_</span></span></span></span></span> <code>\text{\textunderscore}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsb</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> <code>\infin</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>–</mtext></mrow><annotation encoding="application/x-tex">\text{--}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">–</span></span></span></span></span> <code>\text{--}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dotsc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dotsc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\checkmark</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✓</span></span></span></span> <code>\checkmark</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>–</mtext></mrow><annotation encoding="application/x-tex">\text{\textendash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">–</span></span></span></span></span> <code>\text{\textendash}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> ⁣</mtext><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsi</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">†</mi></mrow><annotation encoding="application/x-tex">\dag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span> <code>\dag</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>—</mtext></mrow><annotation encoding="application/x-tex">\text{---}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">—</span></span></span></span></span> <code>\text{---}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsm</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsm</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span> <code>\dagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>—</mtext></mrow><annotation encoding="application/x-tex">\text{\textemdash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">—</span></span></span></span></span> <code>\text{\textemdash}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dotso</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dotso</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>†</mtext></mrow><annotation encoding="application/x-tex">\text{\textdagger}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">†</span></span></span></span></span> <code>\text{\textdagger}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>~</mtext></mrow><annotation encoding="application/x-tex">\text{\textasciitilde}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord">~</span></span></span></span></span> <code>\text{\textasciitilde}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\sdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span> <code>\sdot</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‡</mi></mrow><annotation encoding="application/x-tex">\ddag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\ddag</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‘</mi></mrow><annotation encoding="application/x-tex">`</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">‘</span></span></span></span> <code>`</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\mathellipsis</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\mathellipsis</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\ddagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\ddagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‘</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">‘</span></span></span></span></span> <code>text{\textquoteleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\text{\textellipsis}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="mord text"><span class="minner">…</span></span></span></span></span> <code>\text{\textellipsis}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‡</mtext></mrow><annotation encoding="application/x-tex">\text{\textdaggerdbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">‡</span></span></span></span></span> <code>\text{\textdaggerdbl}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‘</mi></mrow><annotation encoding="application/x-tex">\lq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">‘</span></span></span></span> <code>\lq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\Box</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span> <code>\Box</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\Dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\Dagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>’</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">’</span></span></span></span></span> <code>\text{\textquoteright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\square</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span> <code>\square</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi></mrow><annotation encoding="application/x-tex">\angle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span></span></span></span> <code>\angle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\rq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>\rq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">■</mi></mrow><annotation encoding="application/x-tex">\blacksquare</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">■</span></span></span></span> <code>\blacksquare</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∡</mi></mrow><annotation encoding="application/x-tex">\measuredangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">∡</span></span></span></span> <code>\measuredangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>“</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">“</span></span></span></span></span> <code>\text{\textquotedblleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi></mrow><annotation encoding="application/x-tex">\triangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span> <code>\triangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∢</mi></mrow><annotation encoding="application/x-tex">\sphericalangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5576em;vertical-align:-0.0352em;"></span><span class="mord amsrm">∢</span></span></span></span> <code>\sphericalangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&quot;</mi></mrow><annotation encoding="application/x-tex">&quot;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&quot;</span></span></span></span> <code>&quot;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▽</mi></mrow><annotation encoding="application/x-tex">\triangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▽</span></span></span></span> <code>\triangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊤</mi></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊤</span></span></span></span> <code>\top</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>”</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">”</span></span></span></span></span> <code>\text{\textquotedblright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◃</mo></mrow><annotation encoding="application/x-tex">\triangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">◃</span></span></span></span> <code>\triangleleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊥</mi></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊥</span></span></span></span> <code>\bot</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace></mspace><mspace width="0.1111em"></mspace><mo lspace="0em" rspace="0.17em"></mo><mtext> ⁣</mtext><mo lspace="0em" rspace="0em">:</mo><mspace width="0.3333em"></mspace></mrow><annotation encoding="application/x-tex">\colon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mspace nobreak"></span><span class="mspace" style="margin-right:0.1111em;"></span><span class="mpunct"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mrel">:</span></span><span class="mspace" style="margin-right:0.3333em;"></span></span></span></span> <code>\colon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▹</mo></mrow><annotation encoding="application/x-tex">\triangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">▹</span></span></span></span> <code>\triangleright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">$</span></span></span></span> <code>\$</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‵</mi></mrow><annotation encoding="application/x-tex">\backprime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">‵</span></span></span></span> <code>\backprime</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▽</mo></mrow><annotation encoding="application/x-tex">\bigtriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">▽</span></span></span></span> <code>\bigtriangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>$</mtext></mrow><annotation encoding="application/x-tex">\text{\textdollar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord text"><span class="mord">$</span></span></span></span></span> <code>\text{\textdollar}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">′</mo></mrow><annotation encoding="application/x-tex">\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">′</span></span></span></span> <code>\prime</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\bigtriangleup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span> <code>\bigtriangleup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\pounds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span> <code>\pounds</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&lt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textless}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&lt;</span></span></span></span></span> <code>\text{\textless}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▲</mi></mrow><annotation encoding="application/x-tex">\blacktriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▲</span></span></span></span> <code>\blacktriangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\mathsterling</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span> <code>\mathsterling</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&gt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textgreater}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&gt;</span></span></span></span></span> <code>\text{\textgreater}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▼</mi></mrow><annotation encoding="application/x-tex">\blacktriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▼</span></span></span></span> <code>\blacktriangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>£</mtext></mrow><annotation encoding="application/x-tex">\text{\textsterling}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">£</span></span></span></span></span> <code>\text{\textsterling}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>|</mtext></mrow><annotation encoding="application/x-tex">\text{\textbar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">|</span></span></span></span></span> <code>\text{\textbar}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◀</mo></mrow><annotation encoding="application/x-tex">\blacktriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">◀</span></span></span></span> <code>\blacktriangleleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¥</mi></mrow><annotation encoding="application/x-tex">\yen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm" style="margin-right:0.025em;">¥</span></span></span></span> <code>\yen</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>∥</mtext></mrow><annotation encoding="application/x-tex">\text{\textbardbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">∥</span></span></span></span></span> <code>\text{\textbardbl}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▶</mo></mrow><annotation encoding="application/x-tex">\blacktriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">▶</span></span></span></span> <code>\blacktriangleright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">√</mi></mrow><annotation encoding="application/x-tex">\surd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.2em;"></span><span class="mord">√</span></span></span></span> <code>\surd</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>{</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">{</span></span></span></span></span> <code>\text{\textbraceleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋄</mo></mrow><annotation encoding="application/x-tex">\diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋄</span></span></span></span> <code>\diamond</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">°</mi></mrow><annotation encoding="application/x-tex">\degree</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">°</span></span></span></span> <code>\degree</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>}</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">}</span></span></span></span></span> <code>\text{\textbraceright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\Diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span> <code>\Diamond</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>°</mtext></mrow><annotation encoding="application/x-tex">\text{\textdegree}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">°</span></span></span></span></span> <code>\text{\textdegree}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>¶</mtext></mrow><annotation encoding="application/x-tex">\text{\P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">¶</span></span></span></span></span> <code>\text{\P}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\lozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span> <code>\lozenge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℧</mi></mrow><annotation encoding="application/x-tex">\mho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">℧</span></span></span></span> <code>\mho</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span> <code>\text{\S}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⧫</mi></mrow><annotation encoding="application/x-tex">\blacklozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">⧫</span></span></span></span> <code>\blacklozenge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╲</mi></mrow><annotation encoding="application/x-tex">\diagdown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9356em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╲</span></span></span></span> <code>\diagdown</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\sect}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span> <code>\text{\sect}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">⋆</span></span></span></span> <code>\star</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╱</mi></mrow><annotation encoding="application/x-tex">\diagup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╱</span></span></span></span> <code>\diagup</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>©</mtext></mrow><annotation encoding="application/x-tex">\copyright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">c</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\copyright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">★</mi></mrow><annotation encoding="application/x-tex">\bigstar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">★</span></span></span></span> <code>\bigstar</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♭</mi></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.75em;"></span><span class="mord">♭</span></span></span></span> <code>\flat</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">®</mi></mrow><annotation encoding="application/x-tex">\circledR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">®</span></span></span></span> <code>\circledR</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span> <code>\clubsuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♮</mi></mrow><annotation encoding="application/x-tex">\natural</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♮</span></span></span></span> <code>\natural</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>®</mtext></mrow><annotation encoding="application/x-tex">\text{\textregistered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sizing reset-size6 size3">R</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\textregistered}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span> <code>\clubs</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♯</mi></mrow><annotation encoding="application/x-tex">\sharp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♯</span></span></span></span> <code>\sharp</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⓢ</mi></mrow><annotation encoding="application/x-tex">\circledS</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">Ⓢ</span></span></span></span> <code>\circledS</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamondsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span> <code>\diamondsuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\heartsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span> <code>\heartsuit</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>◯</mo></mover></mrow><annotation encoding="application/x-tex">\text{\textcircled a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\textcircled a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamonds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span> <code>\diamonds</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\hearts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span> <code>\hearts</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✠</mi></mrow><annotation encoding="application/x-tex">\maltese</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✠</span></span></span></span> <code>\maltese</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spadesuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span> <code>\spadesuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spades</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span> <code>\spades</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi><mi mathvariant="normal">¥</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">∞</mi><mo separator="true">⋅</mo><mi mathvariant="normal">∠</mi><mi mathvariant="normal">∡</mi><mi mathvariant="normal">∢</mi><mi mathvariant="normal">♠</mi><mi mathvariant="normal">♡</mi><mi mathvariant="normal">♢</mi><mi mathvariant="normal">♣</mi><mi mathvariant="normal">♭</mi><mi mathvariant="normal">♮</mi><mi mathvariant="normal">♯</mi><mi mathvariant="normal">✓</mi><mo>…</mo><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi><mo>⋯</mo><mo>⋱</mo><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ !</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.6944em;vertical-align:-0.1944em;"></span><span class="mord">£</span><span class="mord amsrm" style="margin-right:0.025em;">¥</span><span class="mord">∇∞</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∠</span><span class="mord amsrm">∡∢</span><span class="mord">♠♡♢♣♭♮♯</span><span class="mord amsrm">✓</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯⋱</span><span class="mclose">!</span></span></span></span> ‼</p><h2 id="_13-units" tabindex="-1">13: Units <a class="header-anchor" href="#_13-units" aria-hidden="true">#</a></h2><p>In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.</p><table><thead><tr><th style="text-align:left;">KaTeX Unit</th><th style="text-align:left;">Value</th><th style="text-align:left;">KaTeX Unit</th><th style="text-align:left;">Value</th></tr></thead><tbody><tr><td style="text-align:left;">em</td><td style="text-align:left;">CSS em</td><td style="text-align:left;">bp</td><td style="text-align:left;">1/72​ inch × F × G</td></tr><tr><td style="text-align:left;">ex</td><td style="text-align:left;">CSS ex</td><td style="text-align:left;">pc</td><td style="text-align:left;">12 KaTeX pt</td></tr><tr><td style="text-align:left;">mu</td><td style="text-align:left;">1/18 CSS em</td><td style="text-align:left;">dd</td><td style="text-align:left;">1238/1157​ KaTeX pt</td></tr><tr><td style="text-align:left;">pt</td><td style="text-align:left;">1/72.27 inch × F × G</td><td style="text-align:left;">cc</td><td style="text-align:left;">14856/1157 KaTeX pt</td></tr><tr><td style="text-align:left;">mm</td><td style="text-align:left;">1 mm × F × G</td><td style="text-align:left;">nd</td><td style="text-align:left;">685/642 KaTeX pt</td></tr><tr><td style="text-align:left;">cm</td><td style="text-align:left;">1 cm × F × G</td><td style="text-align:left;">nc</td><td style="text-align:left;">1370/107​ KaTeX pt</td></tr><tr><td style="text-align:left;">in</td><td style="text-align:left;">1 inch × F × G</td><td style="text-align:left;">sp</td><td style="text-align:left;">1/65536 KaTeX pt</td></tr></tbody></table><p>where:</p><ul><li><p>F = (font size of surrounding HTML text)/(10 pt)</p></li><li><p>G = 1.21 by default, because KaTeX font-size is normally 1.21 × the surrounding font size. This value can be overridden by the CSS of an HTML page.</p></li></ul><p>The effect of style and size:</p><table><thead><tr><th style="text-align:center;">Unit</th><th style="text-align:center;">textstyle</th><th style="text-align:center;">scriptscript</th><th style="text-align:center;">huge</th></tr></thead><tbody><tr><td style="text-align:center;">em or ex</td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mrow><annotation encoding="application/x-tex">\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="2em" height="2em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule mtight sizing reset-size6 size1" style="border-right-width:2em;border-top-width:2em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\huge\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.074em;"></span><span class="mord rule sizing reset-size6 size10" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td></tr><tr><td style="text-align:center;">mu</td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mrow><annotation encoding="application/x-tex">\rule{18mu}{18mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1.472em" height="1.472em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle\rule{18mu}{18mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.736em;"></span><span class="mord rule mtight sizing reset-size6 size1" style="border-right-width:1.472em;border-top-width:1.472em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\huge\rule{18mu}{18mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.074em;"></span><span class="mord rule sizing reset-size6 size10" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td></tr><tr><td style="text-align:center;">others</td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mrow><annotation encoding="application/x-tex">\rule{10pt}{10pt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="2em" height="2em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle\rule{10pt}{10pt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule mtight sizing reset-size6 size1" style="border-right-width:2em;border-top-width:2em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0.4822em" height="0.4822em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\huge\rule{10pt}{10pt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule sizing reset-size6 size10" style="border-right-width:0.4822em;border-top-width:0.4822em;bottom:0em;"></span></span></span></span></td></tr></tbody></table></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/application/markdown-it-katex/support-function.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" 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to use?</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/application/markdown-it-katex/support-function" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: KaTeX supported functions</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/application/markdown-it-katex/support-table" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: KaTeX support tables</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div 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class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" 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versions migrates, please also refer to <a href="https://katex.org/docs/supported.html" target="_blank" rel="noreferrer">KaTeX&#39;s official documentation</a>.</p></div><h2 id="symbols" tabindex="-1">Symbols <a class="header-anchor" href="#symbols" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">!</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">n!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">n</span><span class="mclose">!</span></span></span></span></td><td style="text-align:left;"><code>n!</code></td></tr><tr><td style="text-align:left;">\!</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> ⁣</mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\!b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\!b</code></td></tr><tr><td style="text-align:left;">#</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\def\bar#1{#1^2} \bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\bar#1{#1^2} \bar{y}</code></td></tr><tr><td style="text-align:left;">\#</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">#</mi></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">#</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">%</td><td style="text-align:left;"></td><td style="text-align:left;"><code>%this is a comment</code></td></tr><tr><td style="text-align:left;">\%</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">%</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">&amp;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b\cr c &amp; d \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\&amp;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&amp;</mi></mrow><annotation encoding="application/x-tex">\&amp;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">&#39;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">&#39;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\&#39;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&#39;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\&#39;{a}}</code></td></tr><tr><td style="text-align:left;">(</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo></mrow><annotation encoding="application/x-tex">(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">)</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\(…\)</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="false"><mfrac><mi>a</mi><mi>b</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\text{\(\frac a b\)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord text"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\(\frac a b\)}</code></td></tr><tr><td style="text-align:left;">\</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\ b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\ b</code></td></tr><tr><td style="text-align:left;">\&quot;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&quot;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\&quot;{a}}</code></td></tr><tr><td style="text-align:left;">\$</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>$</mtext></mrow><annotation encoding="application/x-tex">\text{\textdollar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord text"><span class="mord">$</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\,</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\,\,{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>a\,\,{b}</code></td></tr><tr><td style="text-align:left;">\.</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\text{\.{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\.{a}}</code></td></tr><tr><td style="text-align:left;">\:</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\:\:{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>a\:\:{b}</code></td></tr><tr><td style="text-align:left;">\;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext>  </mtext><mtext>  </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\;\;{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;">a<code>\;\;{b}</code></td></tr><tr><td style="text-align:left;">_</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>x_i</code></td></tr><tr><td style="text-align:left;">\_</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">_</mi></mrow><annotation encoding="application/x-tex">\_</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord" style="margin-right:0.02778em;">_</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\<code>`</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\`{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\&#39;{a}}</code></td></tr><tr><td style="text-align:left;">&lt;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">&lt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\=</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\={a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\\={a}}</code></td></tr><tr><td style="text-align:left;">&gt;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\&gt;</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\&gt;\&gt;{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>a\&gt;\&gt;{b}</code></td></tr><tr><td style="text-align:left;">[</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">]</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">]</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">{</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span></span></td><td style="text-align:left;"><code>{a}</code></td></tr><tr><td style="text-align:left;">}</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span></span></td><td style="text-align:left;"><code>{a}</code></td></tr><tr><td style="text-align:left;">\{</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\}</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">}</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">|</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\|</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">~</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>no no no breaks</mtext></mrow><annotation encoding="application/x-tex">\text{no~no~no~breaks}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">no</span><span class="mord nobreak"> </span><span class="mord">no</span><span class="mord nobreak"> </span><span class="mord">no</span><span class="mord nobreak"> </span><span class="mord">breaks</span></span></span></span></span></td><td style="text-align:left;"><code>\text{no~no~no~breaks}</code></td></tr><tr><td style="text-align:left;">\~</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\text{\~{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˜</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\\~{a}}</code></td></tr><tr><td style="text-align:left;">\\</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b\\ c &amp; d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">^</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>x^i</code></td></tr><tr><td style="text-align:left;">\^</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\^{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\\^{a}}</code></td></tr></tbody></table><h2 id="a" tabindex="-1">A <a class="header-anchor" href="#a" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\AA</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>A</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\AA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9468em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\AA}</code></td></tr><tr><td style="text-align:left;">\aa</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\aa}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\aa}</code></td></tr><tr><td style="text-align:left;">\above</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">{a \above{2pt} b+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4458em;vertical-align:-0.5944em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8514em;"><span style="top:-2.4639em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.55em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5944em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td style="text-align:left;"><code>{a \above{2pt} b+1}</code></td></tr><tr><td style="text-align:left;">\abovewithdelims</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\acute</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>e</mi><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\acute e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">e</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\acute e</code></td></tr><tr><td style="text-align:left;">\AE</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Æ</mtext></mrow><annotation encoding="application/x-tex">\text{\AE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Æ</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\AE}</code></td></tr><tr><td style="text-align:left;">\ae</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>æ</mtext></mrow><annotation encoding="application/x-tex">\text{\ae}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">æ</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\ae}</code></td></tr><tr><td style="text-align:left;">\alef</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alef</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\alefsym</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alefsym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\aleph</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\aleph</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" align=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>{aligned}</code></td></tr><tr><td style="text-align:left;" aligned=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>d</mi><mo>+</mo><mi>e</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>f</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}a&amp;=b+c\\d+e&amp;=f\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{aligned}</code><br>   <code>a&amp;=b+c \\</code><br>   <code>d+e&amp;=f</code><br><code>\end{aligned}</code></td></tr><tr><td style="text-align:left;" alignat=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>{alignedat}</code></td></tr><tr><td style="text-align:left;" alignedat=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>10</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>13</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>4</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{alignedat}{2}10&amp;x+&amp;3&amp;y=2\\3&amp;x+&amp;13&amp;y=4\end{alignedat}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">13</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{alignedat}{2}</code><br>   <code>10&amp;x+ &amp;3&amp;y = 2 \\</code><br>   <code> 3&amp;x+&amp;13&amp;y = 4</code><br><code>\end{alignedat}</code></td></tr><tr><td style="text-align:left;">\allowbreak</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Alpha</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi></mrow><annotation encoding="application/x-tex">\Alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\alpha</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\amalg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">⨿</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\And</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&amp;</mo></mrow><annotation encoding="application/x-tex">\And</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\and</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax" target="_blank" rel="noreferrer">Deprecated</a></td></tr><tr><td style="text-align:left;">\ang</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax" target="_blank" rel="noreferrer">Deprecated</a></td></tr><tr><td style="text-align:left;">\angl</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\angle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi></mrow><annotation encoding="application/x-tex">\angle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\approx</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\approxeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≊</mo></mrow><annotation encoding="application/x-tex">\approxeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">≊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arccos</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arccos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arccos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">arccos</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arcctg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arcct<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arcsin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcsin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcsin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">arcsin</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arctan</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">arctan</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arctg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arct<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\argmax</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">arg max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\argmax</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">arg</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">max</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\argmin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">arg min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\argmin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">arg</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">min</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" array=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{cc}a&amp;b\\c&amp;d\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{array}{cc}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;">\array</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>{array}</code></td></tr><tr><td style="text-align:left;">\arraystretch</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.66em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\def\arraystretch{1.5}\begin{array}{cc}a&amp;b\\c&amp;d\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\arraystretch{1.5}</code><br><code>\begin{array}{cc}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;">\Arrowvert</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\arrowvert</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ast</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\asymp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≍</mo></mrow><annotation encoding="application/x-tex">\asymp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≍</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\atop</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0px"><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">{a \atop b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0904em;vertical-align:-0.345em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td style="text-align:left;"><code>{a \atop b}</code></td></tr><tr><td style="text-align:left;">\atopwithdelims</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="b" tabindex="-1">B <a class="header-anchor" href="#b" aria-hidden="true">#</a></h2><p>| Symbol/Function | Rendered | Source or Comment | | :------------------ | :-------------------------------------------------- | :------------------------------------------------------------------------------------------------ | ------ | | \backepsilon | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∍</mo></mrow><annotation encoding="application/x-tex">\backepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">∍</span></span></span></span> | | | \backprime | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‵</mi></mrow><annotation encoding="application/x-tex">\backprime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">‵</span></span></span></span> | | | \backsim | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∽</mo></mrow><annotation encoding="application/x-tex">\backsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">∽</span></span></span></span> | | | \backsimeq | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋍</mo></mrow><annotation encoding="application/x-tex">\backsimeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.464em;"></span><span class="mrel amsrm">⋍</span></span></span></span> | | | \backslash | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">\</mi></mrow><annotation encoding="application/x-tex">\backslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">\</span></span></span></span> | | | \bar | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7622em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> | <code>\bar{y}</code> | | \barwedge | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊼</mo></mrow><annotation encoding="application/x-tex">\barwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊼</span></span></span></span> | | | \Bbb | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\Bbb{ABC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">ABC</span></span></span></span></span> | <code>\Bbb{ABC}</code><br>KaTeX supports A-Z &amp; k | | \Bbbk | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">k</mi></mrow><annotation encoding="application/x-tex">\Bbbk</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">k</span></span></span></span> | | | \bbox | <span style="color:firebrick;">Not supported</span> | | | \bcancel | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="downdiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\bcancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> | <code>\bcancel{5}</code> | | \because | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∵</mo></mrow><annotation encoding="application/x-tex">\because</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∵</span></span></span></span> | | | \begin | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b\\ c &amp; d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span> | <code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code> | | \begingroup | <span class="katex-error" title="ParseError: KaTeX parse error: Expected &#39;\endgroup&#39;, got &#39;}&#39; at position 14: \begingroup a}̲">\begingroup a}</span> | <code>\begingroup a}</code> | | \Beta | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">B</mi></mrow><annotation encoding="application/x-tex">\Beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span></span></span></span> | | | \beta | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> | | | \beth | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℶ</mi></mrow><annotation encoding="application/x-tex">\beth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℶ</span></span></span></span> | | | \between | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≬</mo></mrow><annotation encoding="application/x-tex">\between</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≬</span></span></span></span> | | | \bf | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">B</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">12</mn></mrow><annotation encoding="application/x-tex">\bf AaBb12</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">AaBb12</span></span></span></span></span> | <code>\bf AaBb12</code> | | \bfseries | <span style="color:firebrick;">Not supported</span> | | | \big | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(\big)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord"><span class="delimsizing size1">)</span></span></span></span></span> | <code>\big(\big)</code> | | \Big | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\Big(\Big)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span> | <code>\Big(\Big)</code> | | \bigcap | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋂</mo></mrow><annotation encoding="application/x-tex">\bigcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋂</span></span></span></span> | | | \bigcirc | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◯</mo></mrow><annotation encoding="application/x-tex">\bigcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">◯</span></span></span></span> | | | \bigcup | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋃</mo></mrow><annotation encoding="application/x-tex">\bigcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋃</span></span></span></span> | | | \bigg | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">(</mo><mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">)</mo></mrow><annotation encoding="application/x-tex">\bigg(\bigg)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="delimsizing size3">)</span></span></span></span></span> | <code>\bigg(\bigg)</code> | | \Bigg | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><annotation encoding="application/x-tex">\Bigg(\Bigg)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord"><span class="delimsizing size4">)</span></span></span></span></span> | <code>\Bigg(\Bigg)</code> | | \biggl | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="2.4em" maxsize="2.4em">(</mo></mrow><annotation encoding="application/x-tex">\biggl(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mopen"><span class="delimsizing size3">(</span></span></span></span></span> | <code>\biggl(</code> | | \Biggl | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="3em" maxsize="3em">(</mo></mrow><annotation encoding="application/x-tex">\Biggl(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mopen"><span class="delimsizing size4">(</span></span></span></span></span> | <code>\Biggl(</code> | | \biggm | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">∣</mo></mrow><annotation encoding="application/x-tex">\biggm\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mrel"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
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+c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span> | <code>\biggm\vert</code> | | \Biggm | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">∣</mo></mrow><annotation encoding="application/x-tex">\Biggm\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mrel"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:5em;"></span><span style="width:0.333em;height:3.000em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3.000em" viewBox="0 0 333 3000"><path d="M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15
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+c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span> | <code>\Biggm\vert</code> | | \biggr | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="2.4em" maxsize="2.4em">)</mo></mrow><annotation encoding="application/x-tex">\biggr)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mclose"><span class="delimsizing size3">)</span></span></span></span></span> | <code>\biggr)</code> | | \Biggr | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><annotation encoding="application/x-tex">\Biggr)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mclose"><span class="delimsizing size4">)</span></span></span></span></span> | <code>\Biggr)</code> | | \bigl | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo></mrow><annotation encoding="application/x-tex">\bigl(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mopen"><span class="delimsizing size1">(</span></span></span></span></span> | <code>\bigl(</code> | | \Bigl | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo></mrow><annotation encoding="application/x-tex">\Bigl(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mopen"><span class="delimsizing size2">(</span></span></span></span></span> | <code>\Bigl(</code> | | \bigm | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">∣</mo></mrow><annotation encoding="application/x-tex">\bigm\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mrel"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-2.85em;"><span class="pstrut" style="height:3.2em;"></span><span style="width:0.333em;height:1.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.200em" viewBox="0 0 333 1200"><path d="M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15
+c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15
+c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span></span></span> | <code>\bigm\vert</code> | | \Bigm | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">∣</mo></mrow><annotation encoding="application/x-tex">\Bigm\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mrel"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.15em;"><span style="top:-3.15em;"><span class="pstrut" style="height:3.8em;"></span><span style="width:0.333em;height:1.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="1.800em" viewBox="0 0 333 1800"><path d="M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15
+c10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15
+c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.65em;"><span></span></span></span></span></span></span></span></span></span> | <code>\Bigm\vert</code> | | \bigodot | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨀</mo></mrow><annotation encoding="application/x-tex">\bigodot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨀</span></span></span></span> | | | \bigominus | <span style="color:firebrick;">Not supported</span> | <a href="https://github.com/KaTeX/KaTeX/issues/1222" target="_blank" rel="noreferrer">Issue #1222</a> | | \bigoplus | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨁</mo></mrow><annotation encoding="application/x-tex">\bigoplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨁</span></span></span></span> | | | \bigoslash | <span style="color:firebrick;">Not supported</span> | <a href="https://github.com/KaTeX/KaTeX/issues/1222" target="_blank" rel="noreferrer">Issue #1222</a> | | \bigotimes | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨂</mo></mrow><annotation encoding="application/x-tex">\bigotimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨂</span></span></span></span> | | | \bigr | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\bigr)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mclose"><span class="delimsizing size1">)</span></span></span></span></span> | <code>\bigr)</code> | | \Bigr | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true" stretchy="true" minsize="1.8em" maxsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\Bigr)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mclose"><span class="delimsizing size2">)</span></span></span></span></span> | <code>\Bigr)</code> | | \bigsqcap | <span style="color:firebrick;">Not supported</span> | <a href="https://github.com/KaTeX/KaTeX/issues/1222" target="_blank" rel="noreferrer">Issue #1222</a> | | \bigsqcup | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨆</mo></mrow><annotation encoding="application/x-tex">\bigsqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨆</span></span></span></span> | | | \bigstar | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">★</mi></mrow><annotation encoding="application/x-tex">\bigstar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">★</span></span></span></span> | | | \bigtriangledown | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▽</mo></mrow><annotation encoding="application/x-tex">\bigtriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">▽</span></span></span></span> | | | \bigtriangleup | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\bigtriangleup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span> | | | \biguplus | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨄</mo></mrow><annotation encoding="application/x-tex">\biguplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨄</span></span></span></span> | | | \bigvee | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋁</span></span></span></span> | | | \bigwedge | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋀</mo></mrow><annotation encoding="application/x-tex">\bigwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋀</span></span></span></span> | | | \binom | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom n k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> | <code>\binom n k</code> | | \blacklozenge | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⧫</mi></mrow><annotation encoding="application/x-tex">\blacklozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">⧫</span></span></span></span> | | | \blacksquare | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">■</mi></mrow><annotation encoding="application/x-tex">\blacksquare</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">■</span></span></span></span> | | | \blacktriangle | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▲</mi></mrow><annotation encoding="application/x-tex">\blacktriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▲</span></span></span></span> | | | \blacktriangledown | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▼</mi></mrow><annotation encoding="application/x-tex">\blacktriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▼</span></span></span></span> | | | \blacktriangleleft | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◀</mo></mrow><annotation encoding="application/x-tex">\blacktriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">◀</span></span></span></span> | | | \blacktriangleright | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▶</mo></mrow><annotation encoding="application/x-tex">\blacktriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">▶</span></span></span></span> | | | \bm | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\bm{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">AaBb</span></span></span></span></span></span> | <code>\bm{AaBb}</code> | | {Bmatrix} | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\begin{Bmatrix}a&amp;b\\c&amp;d\end{Bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">}</span></span></span></span></span></span> | <code>\begin{Bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Bmatrix}</code> | | {bmatrix} | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\begin{bmatrix}a&amp;b\\c&amp;d\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span> | <code>\begin{bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{bmatrix}</code> | | \bmod | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mo lspace="0.22em" rspace="0.22em"><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow></mo><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a \bmod b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> | <code>a \bmod b</code> | | \bold | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">B</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">123</mn></mrow><annotation encoding="application/x-tex">\bold{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">AaBb123</span></span></span></span></span> | <code>\bold{AaBb123}</code> | | \boldsymbol | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">a</mi><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\boldsymbol{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">AaBb</span></span></span></span></span></span> | <code>\boldsymbol{AaBb}</code> | | \bot | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊥</mi></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊥</span></span></span></span> | | | \bowtie | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\bowtie</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span> | | | \Box | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\Box</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span> | | | \boxdot | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊡</mo></mrow><annotation encoding="application/x-tex">\boxdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊡</span></span></span></span> | | | \boxed | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>a</mi><mi>b</mi></mrow></mstyle></mstyle></mstyle></menclose></mrow><annotation encoding="application/x-tex">\boxed{ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3744em;vertical-align:-0.34em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0344em;"><span style="top:-3.3744em;"><span class="pstrut" style="height:3.3744em;"></span><span class="boxpad"><span class="mord"><span class="mord"><span class="mord mathnormal">ab</span></span></span></span></span><span style="top:-3.0344em;"><span class="pstrut" style="height:3.3744em;"></span><span class="stretchy fbox" style="height:1.3744em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34em;"><span></span></span></span></span></span></span></span></span> | <code>\boxed{ab}</code> | | \boxminus | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊟</mo></mrow><annotation encoding="application/x-tex">\boxminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊟</span></span></span></span> | | | \boxplus | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊞</mo></mrow><annotation encoding="application/x-tex">\boxplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊞</span></span></span></span> | | | \boxtimes | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊠</mo></mrow><annotation encoding="application/x-tex">\boxtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊠</span></span></span></span> | | | \Bra | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">⟨</mo><mi>ψ</mi><mo fence="true">∥</mo></mrow><annotation encoding="application/x-tex">\left\langle\psi\right\|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">⟨</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose delimcenter" style="top:0em;">∥</span></span></span></span></span> | <code>\Bra{\psi}</code> | | \bra | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded><mo stretchy="false">⟨</mo><mi>ψ</mi><mi mathvariant="normal">∥</mi></mpadded></mrow><annotation encoding="application/x-tex">\mathinner{\langle{\psi}\|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span><span class="mord">∥</span></span></span></span></span> | <code>\bra{\psi}</code> | | \braket | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded><mo stretchy="false">⟨</mo><mrow><mi>ϕ</mi><mi mathvariant="normal">∥</mi><mi>ψ</mi></mrow><mo stretchy="false">⟩</mo></mpadded></mrow><annotation encoding="application/x-tex">\mathinner{\langle{\phi\|\psi}\rangle}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="mord">∥</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span><span class="mclose">⟩</span></span></span></span></span> | <code>\braket{\phi | \psi}</code> | | \brace | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">{n\brace k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">{</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">}</span></span></span></span></span></span></span> | <code>{n\brace k}</code> | | \bracevert | <span style="color:firebrick;">Not supported</span> | | | \brack | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">{n\brack k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span> | <code>{n\brack k}</code> | | \breve | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>e</mi><mi>u</mi></mrow><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\breve{eu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mord mathnormal">u</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span> | <code>\breve{eu}</code> | | \buildrel | <span style="color:firebrick;">Not supported</span> | | | \bull | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∙</mo></mrow><annotation encoding="application/x-tex">\bull</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∙</span></span></span></span> | | | \bullet | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∙</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∙</span></span></span></span> | | | \Bumpeq | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≎</mo></mrow><annotation encoding="application/x-tex">\Bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≎</span></span></span></span> | | | \bumpeq | <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≏</mo></mrow><annotation encoding="application/x-tex">\bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≏</span></span></span></span> | |</p><h2 id="c" tabindex="-1">C <a class="header-anchor" href="#c" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\C</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax" target="_blank" rel="noreferrer">Deprecated</a></td></tr><tr><td style="text-align:left;">\cal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">a</mi><mi mathvariant="script">B</mi><mi mathvariant="script">b</mi><mn mathvariant="script">123</mn></mrow><annotation encoding="application/x-tex">\cal AaBb123</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="mord mathnormal">a</span><span class="mord mathcal" style="margin-right:0.03041em;">B</span><span class="mord mathnormal">b</span><span class="mord">123</span></span></span></span></span></td><td style="text-align:left;"><code>\cal AaBb123</code></td></tr><tr><td style="text-align:left;">\cancel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\cancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\cancel{5}</code></td></tr><tr><td style="text-align:left;">\cancelto</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Cap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\Cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∩</mo></mrow><annotation encoding="application/x-tex">\cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∩</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" cases=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}a&amp;\text{if }b\\c&amp;\text{if }d\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{cases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{cases}</code></td></tr><tr><td style="text-align:left;">\cases</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>{cases}</code></td></tr><tr><td style="text-align:left;" CD=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cdotp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">⋅</mo></mrow><annotation encoding="application/x-tex">\cdotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mpunct">⋅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cdots</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ce</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">C</mi><msub><mpadded width="0px"><mphantom><mi>X</mi></mphantom></mpadded><mpadded height="0px"><mn>6</mn></mpadded></msub><mi mathvariant="normal">H</mi><msub><mpadded width="0px"><mphantom><mi>X</mi></mphantom></mpadded><mpadded height="0px"><mn>5</mn></mpadded></msub><mo lspace="0em" rspace="0em">−</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">O</mi></mrow></mrow><annotation encoding="application/x-tex">{\mathrm{C}{\vphantom{X}}_{\smash[t]{6}}\mathrm{H}{\vphantom{X}}_{\smash[t]{5}}{-}\mathrm{CHO}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathrm">C</span><span class="mord"><span class="mord"><span class="mord rlap"><span class="inner"><span class="mord" style="color:transparent;"><span class="mord mathnormal" style="margin-right:0.07847em;color:transparent;">X</span></span></span><span class="fix"></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.15em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathrm">H</span><span class="mord"><span class="mord"><span class="mord rlap"><span class="inner"><span class="mord" style="color:transparent;"><span class="mord mathnormal" style="margin-right:0.07847em;color:transparent;">X</span></span></span><span class="fix"></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.15em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord">−</span></span><span class="mord"><span class="mord mathrm">CHO</span></span></span></span></span></span> <br> <span class="katex-error" title="ParseError: KaTeX parse error: Undefined control sequence: \ce at position 1: \̲c̲e̲{C6H5-CHO}">\ce{C6H5-CHO}</span></td><td style="text-align:left;"><code>{\mathrm{C}{\vphantom{X}}_{\smash[t]{6}}\mathrm{H}{\vphantom{X}}_{\smash[t]{5}}{-}\mathrm{CHO}}</code> <br> or <code>\ce{C6H5-CHO}</code> Requires an <a href="https://github.com/KaTeX/KaTeX/tree/master/contrib/mhchem/" target="_blank" rel="noreferrer">extension</a></td></tr><tr><td style="text-align:left;">\cee</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Deprecated by mhchem</td></tr><tr><td style="text-align:left;">\centerdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\centerdot b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin amsrm">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\centerdot b</code></td></tr><tr><td style="text-align:left;">\cf</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Deprecated by mhchem;</td></tr><tr><td style="text-align:left;">use <code>\ce</code> instead</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cfrac</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>2</mn><mn>1</mn></mfrac></mstyle></mrow></mfrac></mstyle></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.236em;vertical-align:-3.646em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-3.82em;"><span class="pstrut" style="height:3.59em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.33em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.166em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-3.82em;"><span class="pstrut" style="height:3.59em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.33em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.646em;"><span></span></span></span></span></span><span></span></span></span></span></span></td><td style="text-align:left;"><code>\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}</code></td></tr><tr><td style="text-align:left;">\check</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>o</mi><mi>e</mi></mrow><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\check{oe}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">oe</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\check{oe}</code></td></tr><tr><td style="text-align:left;">\ch</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ch</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ch</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ch</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\checkmark</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\checkmark</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Chi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\Chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\chi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">χ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\choose</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">{n+1 \choose k+2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2984em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8951em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></td><td style="text-align:left;"><code>{n+1 \choose k+2}</code></td></tr><tr><td style="text-align:left;">\circ</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≗</mo></mrow><annotation encoding="application/x-tex">\circeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≗</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circlearrowleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↺</mo></mrow><annotation encoding="application/x-tex">\circlearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↺</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circlearrowright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↻</mo></mrow><annotation encoding="application/x-tex">\circlearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↻</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circledast</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊛</mo></mrow><annotation encoding="application/x-tex">\circledast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊛</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circledcirc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊚</mo></mrow><annotation encoding="application/x-tex">\circledcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊚</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circleddash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊝</mo></mrow><annotation encoding="application/x-tex">\circleddash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊝</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circledR</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">®</mi></mrow><annotation encoding="application/x-tex">\circledR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">®</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\circledS</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⓢ</mi></mrow><annotation encoding="application/x-tex">\circledS</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">Ⓢ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\class</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">A PR is pending.</td></tr><tr><td style="text-align:left;">\cline</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/269" target="_blank" rel="noreferrer">Issue #269</a></td></tr><tr><td style="text-align:left;">\clubs</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\clubsuit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cnums</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\cnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\colon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace></mspace><mspace width="0.1111em"></mspace><mo lspace="0em" rspace="0.17em"></mo><mtext> ⁣</mtext><mo lspace="0em" rspace="0em">:</mo><mspace width="0.3333em"></mspace></mrow><annotation encoding="application/x-tex">\colon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mspace nobreak"></span><span class="mspace" style="margin-right:0.1111em;"></span><span class="mpunct"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mrel">:</span></span><span class="mspace" style="margin-right:0.3333em;"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Colonapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\Colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\colonapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Coloneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\coloneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Coloneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">=</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\coloneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≔</mi></mo></mrow><annotation encoding="application/x-tex">\coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Colonsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\Colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\colonsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\color</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#0000FF"><mi>A</mi><mi>a</mi><mi>B</mi><mi>b</mi><mn>123</mn></mstyle></mrow><annotation encoding="application/x-tex">\color{#0000FF} AaBb123</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="color:#0000FF;">A</span><span class="mord mathnormal" style="color:#0000FF;">a</span><span class="mord mathnormal" style="margin-right:0.05017em;color:#0000FF;">B</span><span class="mord mathnormal" style="color:#0000FF;">b</span><span class="mord" style="color:#0000FF;">123</span></span></span></span></td><td style="text-align:left;"><code>\color{#0000FF} AaBb123</code></td></tr><tr><td style="text-align:left;">\colorbox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" mathbackground="red"><mtext>Black on red</mtext></mpadded></mrow><annotation encoding="application/x-tex">\colorbox{red}{Black on red}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2944em;vertical-align:-0.3em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9944em;"><span style="top:-2.9944em;"><span class="pstrut" style="height:3.2944em;"></span><span class="stretchy colorbox" style="height:1.2944em;background-color:red;"></span></span><span style="top:-3.2944em;"><span class="pstrut" style="height:3.2944em;"></span><span class="mord boxpad"><span class="mord">Black on red</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\colorbox{red}{Black on red}</code></td></tr><tr><td style="text-align:left;">\complement</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∁</mi></mrow><annotation encoding="application/x-tex">\complement</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;"></span><span class="mord amsrm">∁</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Complex</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\Complex</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cong</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≅</mo></mrow><annotation encoding="application/x-tex">\cong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.589em;"></span><span class="mrel">≅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Coppa</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\coppa</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\coprod</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\copyright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>©</mtext></mrow><annotation encoding="application/x-tex">\copyright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">c</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cos</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cos</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cosec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cosec</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cosh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cosh</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cot</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">cot</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cotg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cotg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cotg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">cot<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\coth</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>coth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\coth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">coth</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b\cr c &amp; d \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \cr</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\csc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>csc</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\csc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">csc</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cssId</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">A PR is pending.</td></tr><tr><td style="text-align:left;">\ctg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">ct<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cth</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cth</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Cup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\Cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\cup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∪</mo></mrow><annotation encoding="application/x-tex">\cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∪</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curlyeqprec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋞</mo></mrow><annotation encoding="application/x-tex">\curlyeqprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋞</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curlyeqsucc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋟</mo></mrow><annotation encoding="application/x-tex">\curlyeqsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋟</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curlyvee</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋎</mo></mrow><annotation encoding="application/x-tex">\curlyvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋎</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curlywedge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋏</mo></mrow><annotation encoding="application/x-tex">\curlywedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋏</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curvearrowleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↶</mo></mrow><annotation encoding="application/x-tex">\curvearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↶</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\curvearrowright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↷</mo></mrow><annotation encoding="application/x-tex">\curvearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↷</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="d" tabindex="-1">D <a class="header-anchor" href="#d" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\dag</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">†</mi></mrow><annotation encoding="application/x-tex">\dag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Dagger</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\Dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dagger</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\daleth</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℸ</mi></mrow><annotation encoding="application/x-tex">\daleth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℸ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Darr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\dArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\darr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" darray=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{darray}{cc}a&amp;b\\c&amp;d\end{darray}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{darray}{cc}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{darray}</code></td></tr><tr><td style="text-align:left;">\dashleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇠</mo></mrow><annotation encoding="application/x-tex">\dashleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dashrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇢</mo></mrow><annotation encoding="application/x-tex">\dashrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dashv</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dbinom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\dbinom n k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\dbinom n k</code></td></tr><tr><td style="text-align:left;">\dblcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\dblcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" dcases=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{dcases}a&amp;\text{if }b\\c&amp;\text{if }d\end{dcases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{dcases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{dcases}</code></td></tr><tr><td style="text-align:left;">\ddag</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‡</mi></mrow><annotation encoding="application/x-tex">\ddag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ddagger</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\ddagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ddddot</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dddot</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ddot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\ddot x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\ddot x</code></td></tr><tr><td style="text-align:left;">\ddots</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋱</mo></mrow><annotation encoding="application/x-tex">\ddots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.823em;"></span><span class="minner">⋱</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\DeclareMathOperator</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\def</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\def\foo{x^2} \foo + \foo</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\foo{x^2} \foo + \foo</code></td></tr><tr><td style="text-align:left;">\definecolor</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/750" target="_blank" rel="noreferrer">Issue #750</a></td></tr><tr><td style="text-align:left;">\deg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>deg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\deg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">de<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\degree</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">°</mi></mrow><annotation encoding="application/x-tex">\degree</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">°</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\delta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Delta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\det</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\det</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">det</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Digamma</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\digamma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϝ</mi></mrow><annotation encoding="application/x-tex">\digamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">ϝ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dfrac</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>b</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{a-1}{b-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\dfrac{a-1}{b-1}</code></td></tr><tr><td style="text-align:left;">\diagdown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╲</mi></mrow><annotation encoding="application/x-tex">\diagdown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9356em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╲</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\diagup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╱</mi></mrow><annotation encoding="application/x-tex">\diagup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╱</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Diamond</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\Diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\diamond</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋄</mo></mrow><annotation encoding="application/x-tex">\diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋄</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\diamonds</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamonds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\diamondsuit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamondsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>dim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\dim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">dim</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\displaylines</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\displaystyle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mn>0</mn><mi>n</mi></munderover></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_0^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\displaystyle\sum_0^n</code></td></tr><tr><td style="text-align:left;">\div</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>÷</mo></mrow><annotation encoding="application/x-tex">\div</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">÷</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\divideontimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋇</mo></mrow><annotation encoding="application/x-tex">\divideontimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\dot x</code></td></tr><tr><td style="text-align:left;">\Doteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\Doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\doteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≐</mo></mrow><annotation encoding="application/x-tex">\doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.673em;"></span><span class="mrel">≐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\doteqdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\doteqdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dotplus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∔</mo></mrow><annotation encoding="application/x-tex">\dotplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7742em;vertical-align:-0.082em;"></span><span class="mord amsrm">∔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\dots</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1 + \dots + x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>x_1 + \dots + x_n</code></td></tr><tr><td style="text-align:left;">\dotsb</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1 +\dotsb + x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>x_1 +\dotsb + x_n</code></td></tr><tr><td style="text-align:left;">\dotsc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,\dotsc,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></td><td style="text-align:left;"><code>x,\dotsc,y</code></td></tr><tr><td style="text-align:left;">\dotsi</td><td style="text-align:left;">$$\int_{A_1}\int_{A_2}\dotsi$$</td><td style="text-align:left;"><code>\int_{A_1}\int_{A_2}\dotsi</code></td></tr><tr><td style="text-align:left;">\dotsm</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1 x_2 \dotsm x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>$x_1 x_2 \dotsm x_n</code></td></tr><tr><td style="text-align:left;">\dotso</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dotso</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\doublebarwedge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩞</mo></mrow><annotation encoding="application/x-tex">\doublebarwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9503em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⩞</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\doublecap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\doublecap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\doublecup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\doublecup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Downarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\downarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\downdownarrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇊</mo></mrow><annotation encoding="application/x-tex">\downdownarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\downharpoonleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇃</mo></mrow><annotation encoding="application/x-tex">\downharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\downharpoonright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇂</mo></mrow><annotation encoding="application/x-tex">\downharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇂</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" drcases=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\begin{drcases}a&amp;\text{if }b\\c&amp;\text{if }d\end{drcases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">}</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{drcases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{drcases}</code></td></tr></tbody></table><h2 id="e" tabindex="-1">E <a class="header-anchor" href="#e" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\edef</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">\def\foo{a}\edef\bar{\foo}\def\foo{}\bar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></td><td style="text-align:left;"><code>\def\foo{a}\edef\bar{\foo}\def\foo{}\bar</code></td></tr><tr><td style="text-align:left;">\ell</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\else</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1003" target="_blank" rel="noreferrer">Issue #1003</a></td></tr><tr><td style="text-align:left;">\em</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\emph</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\empty</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\empty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\emptyset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\enclose</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Non standard</td></tr><tr><td style="text-align:left;">\end</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b\\ c &amp; d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\endgroup</td><td style="text-align:left;"><span class="katex-error" title="ParseError: KaTeX parse error: Expected &#39;}&#39;, got &#39;\endgroup&#39; at position 3: {a\̲e̲n̲d̲g̲r̲o̲u̲p̲">{a\endgroup</span></td><td style="text-align:left;"><code>{a\endgroup</code></td></tr><tr><td style="text-align:left;">\enspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace width="0.5em"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\enspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.5em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\enspace b</code></td></tr><tr><td style="text-align:left;">\Epsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">E</mi></mrow><annotation encoding="application/x-tex">\Epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">E</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\epsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqalign</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqalignno</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqcirc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≖</mo></mrow><annotation encoding="application/x-tex">\eqcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">≖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Eqcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">−</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∹</mi></mo></mrow><annotation encoding="application/x-tex">\eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" equation=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/445" target="_blank" rel="noreferrer">Issue #445</a></td></tr><tr><td style="text-align:left;" eqnarray=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Eqqcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">=</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqqcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≕</mi></mo></mrow><annotation encoding="application/x-tex">\eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqref</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/350" target="_blank" rel="noreferrer">Issue #350</a></td></tr><tr><td style="text-align:left;">\eqsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≂</mo></mrow><annotation encoding="application/x-tex">\eqsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel amsrm">≂</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqslantgtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪖</mo></mrow><annotation encoding="application/x-tex">\eqslantgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eqslantless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪕</mo></mrow><annotation encoding="application/x-tex">\eqslantless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪕</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\equiv</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≡</mo></mrow><annotation encoding="application/x-tex">\equiv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Eta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">H</mi></mrow><annotation encoding="application/x-tex">\Eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">H</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\eth</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ð</mi></mrow><annotation encoding="application/x-tex">\eth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ð</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\euro</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\exist</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exist</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\exists</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\exp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>exp</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\exp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">exp</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\expandafter</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="f" tabindex="-1">F <a class="header-anchor" href="#f" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\fallingdotseq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≒</mo></mrow><annotation encoding="application/x-tex">\fallingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\fbox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mtext>Hi there!</mtext></mstyle></menclose></mrow><annotation encoding="application/x-tex">\fbox{Hi there!}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3744em;vertical-align:-0.34em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0344em;"><span style="top:-3.3744em;"><span class="pstrut" style="height:3.3744em;"></span><span class="boxpad"><span class="mord"><span class="mord">Hi there!</span></span></span></span><span style="top:-3.0344em;"><span class="pstrut" style="height:3.3744em;"></span><span class="stretchy fbox" style="height:1.3744em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\fbox{Hi there!}</code></td></tr><tr><td style="text-align:left;">\fcolorbox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" style="border:0.04em solid red;" mathbackground="aqua"><mtext>A</mtext></mpadded></mrow><annotation encoding="application/x-tex">\fcolorbox{red}{aqua}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3633em;vertical-align:-0.34em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0233em;"><span style="top:-3.0233em;"><span class="pstrut" style="height:3.3633em;"></span><span class="stretchy fcolorbox" style="height:1.3633em;border-style:solid;border-width:0.04em;background-color:aqua;border-color:red;"></span></span><span style="top:-3.3633em;"><span class="pstrut" style="height:3.3633em;"></span><span class="mord boxpad"><span class="mord">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\fcolorbox{red}{aqua}{A}</code></td></tr><tr><td style="text-align:left;">\fi</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1003" target="_blank" rel="noreferrer">Issue #1003</a></td></tr><tr><td style="text-align:left;">\Finv</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⅎ</mi></mrow><annotation encoding="application/x-tex">\Finv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">Ⅎ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\flat</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♭</mi></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.75em;"></span><span class="mord">♭</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\footnotesize</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.8em"><mi>f</mi><mi>o</mi><mi>o</mi><mi>t</mi><mi>n</mi><mi>o</mi><mi>t</mi><mi>e</mi><mi>s</mi><mi>i</mi><mi>z</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\footnotesize footnotesize</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7111em;vertical-align:-0.1556em;"></span><span class="mord mathnormal sizing reset-size6 size4" style="margin-right:0.10764em;">f</span><span class="mord mathnormal sizing reset-size6 size4">oo</span><span class="mord mathnormal sizing reset-size6 size4">t</span><span class="mord mathnormal sizing reset-size6 size4">n</span><span class="mord mathnormal sizing reset-size6 size4">o</span><span class="mord mathnormal sizing reset-size6 size4">t</span><span class="mord mathnormal sizing reset-size6 size4">es</span><span class="mord mathnormal sizing reset-size6 size4">i</span><span class="mord mathnormal sizing reset-size6 size4">ze</span></span></span></span></td><td style="text-align:left;"><code>\footnotesize footnotesize</code></td></tr><tr><td style="text-align:left;">\forall</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\frac</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac a b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\frac a b</code></td></tr><tr><td style="text-align:left;">\frak</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">a</mi><mi mathvariant="fraktur">B</mi><mi mathvariant="fraktur">b</mi></mrow><annotation encoding="application/x-tex">\frak{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">AaBb</span></span></span></span></span></td><td style="text-align:left;"><code>\frak{AaBb}</code></td></tr><tr><td style="text-align:left;">\frown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\frown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\futurelet</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="g" tabindex="-1">G <a class="header-anchor" href="#g" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\Game</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⅁</mi></mrow><annotation encoding="application/x-tex">\Game</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">⅁</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Gamma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gamma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" gather=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>{gathered}</code></td></tr><tr><td style="text-align:left;" gathered=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>e</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}a=b\\e=b+c\end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{gathered}</code><br>   <code>a=b \\ </code><br>   <code>e=b+c</code><br><code>\end{gathered}</code></td></tr><tr><td style="text-align:left;">\gcd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>gcd</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\gcd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span style="margin-right:0.01389em;">g</span>cd</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gdef</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\gdef\bar#1{#1^2} \bar{y} + \bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\gdef\bar#1{#1^2} \bar{y} + \bar{y}</code></td></tr><tr><td style="text-align:left;">\ge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\geneuro</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\geneuronarrow</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\geneurowide</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\genfrac</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\genfrac ( ] {2pt}{0}a{a+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6278em;vertical-align:-1.1778em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3806em;"><span style="top:-1.9056em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.95em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1778em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></td><td style="text-align:left;"><code>\genfrac ( ] {2pt}{0}a{a+1}</code></td></tr><tr><td style="text-align:left;">\geq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\geq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\geqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≧</mo></mrow><annotation encoding="application/x-tex">\geqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≧</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\geqslant</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩾</mo></mrow><annotation encoding="application/x-tex">\geqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩾</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gets</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\gets</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≫</mo></mrow><annotation encoding="application/x-tex">\gg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ggg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\ggg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gggtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\gggtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gimel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℷ</mi></mrow><annotation encoding="application/x-tex">\gimel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℷ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\global</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\global\def\add#1#2{#1+#2} \add 2 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></td><td style="text-align:left;"><code>\global\def\add#1#2{#1+#2} \add 2 3</code></td></tr><tr><td style="text-align:left;">\gnapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪊</mo></mrow><annotation encoding="application/x-tex">\gnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪈</mo></mrow><annotation encoding="application/x-tex">\gneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≩</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gnsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋧</mo></mrow><annotation encoding="application/x-tex">\gnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋧</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\grave</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>e</mi><mi>u</mi></mrow><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\grave{eu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mord mathnormal">u</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\grave{eu}</code></td></tr><tr><td style="text-align:left;">\gt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \gt b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a \gt b</code></td></tr><tr><td style="text-align:left;">\gtrdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋗</mo></mrow><annotation encoding="application/x-tex">\gtrdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋗</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gtrapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪆</mo></mrow><annotation encoding="application/x-tex">\gtrapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gtreqless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋛</mo></mrow><annotation encoding="application/x-tex">\gtreqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋛</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gtreqqless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪌</mo></mrow><annotation encoding="application/x-tex">\gtreqqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪌</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gtrless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≷</mo></mrow><annotation encoding="application/x-tex">\gtrless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≷</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gtrsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≳</mo></mrow><annotation encoding="application/x-tex">\gtrsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≳</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\gvertneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="h" tabindex="-1">H <a class="header-anchor" href="#h" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\H</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˝</mo></mover></mrow><annotation encoding="application/x-tex">\text{\H{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˝</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\H{a}}</code></td></tr><tr><td style="text-align:left;">\Harr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\hArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\harr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hat</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\hat{\theta}</code></td></tr><tr><td style="text-align:left;">\hbar</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hbox</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hdashline</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em" rowlines="dashed"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix}a&amp;b\\ \hdashline c &amp;d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:3.45em;"></span><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span style="top:-3.7em;"><span class="pstrut" style="height:3.45em;"></span><span class="hdashline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>\hdashline</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\hearts</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\hearts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\heartsuit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\heartsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hfil</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hfill</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Issues <a href="https://github.com/KaTeX/KaTeX/issues/164" target="_blank" rel="noreferrer">#164</a> &amp; <a href="https://github.com/KaTeX/KaTeX/issues/269" target="_blank" rel="noreferrer">#269</a></td></tr><tr><td style="text-align:left;">\hline</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em" rowlines="solid"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix}a&amp;b\\ \hline c &amp;d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:3.45em;"></span><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span style="top:-3.7em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\ \hline</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\hom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>hom</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\hom</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">hom</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hookleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↩</mo></mrow><annotation encoding="application/x-tex">\hookleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↩</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hookrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↪</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hphantom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mpadded height="0px" depth="0px"><mphantom><mi>b</mi><mi>c</mi></mphantom></mpadded><mi>d</mi></mrow><annotation encoding="application/x-tex">a\hphantom{bc}d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord" style="color:transparent;"><span class="mord mathnormal" style="color:transparent;">b</span><span class="mord mathnormal" style="color:transparent;">c</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span></span></span></span></td><td style="text-align:left;"><code>a\hphantom{bc}d</code></td></tr><tr><td style="text-align:left;">\href</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\href</mtext></mstyle></mrow><annotation encoding="application/x-tex">\href{https://katex.org/}{\KaTeX}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\href</span></span></span></span></span></td><td style="text-align:left;"><code>\href{https://katex.org/}{\KaTeX}</code> Requires <code>trust</code></td></tr><tr><td style="text-align:left;">\hskip</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mspace width="1em"></mspace><mi>i</mi><mspace width="2em"></mspace><mi>d</mi></mrow><annotation encoding="application/x-tex">w\hskip1em i\hskip2em d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:2em;"></span><span class="mord mathnormal">d</span></span></span></span></td><td style="text-align:left;"><code>w\hskip1em i\hskip2em d</code></td></tr><tr><td style="text-align:left;">\hslash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℏ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\hspace</td><td style="text-align:left;"><span class="katex-error" title="ParseError: KaTeX parse error: Invalid size: &#39;7&#39; at position 9: s\hspace7̲ex k">s\hspace7ex k</span></td><td style="text-align:left;"><code>s\hspace7ex k</code></td></tr><tr><td style="text-align:left;">\htmlClass</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\htmlClass</mtext></mstyle></mrow><annotation encoding="application/x-tex">\htmlClass{foo}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\htmlClass</span></span></span></span></span></td><td style="text-align:left;"><code>\htmlClass{foo}{x}</code> Must enable <code>trust</code> and disable <code>strict</code></td></tr><tr><td style="text-align:left;">\htmlData</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\htmlData</mtext></mstyle></mrow><annotation encoding="application/x-tex">\htmlData{foo=a, bar=b}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\htmlData</span></span></span></span></span></td><td style="text-align:left;"><code>\htmlData{foo=a, bar=b}{x}</code> Must enable <code>trust</code> and disable <code>strict</code></td></tr><tr><td style="text-align:left;">\htmlId</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\htmlId</mtext></mstyle></mrow><annotation encoding="application/x-tex">\htmlId{bar}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\htmlId</span></span></span></span></span></td><td style="text-align:left;"><code>\htmlId{bar}{x}</code> Must enable <code>trust</code> and disable <code>strict</code></td></tr><tr><td style="text-align:left;">\htmlStyle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\htmlStyle</mtext></mstyle></mrow><annotation encoding="application/x-tex">\htmlStyle{color: red;}{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\htmlStyle</span></span></span></span></span></td><td style="text-align:left;"><code>\htmlStyle{color: red;}{x}</code> Must enable <code>trust</code> and disable <code>strict</code></td></tr><tr><td style="text-align:left;">\huge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mi>h</mi><mi>u</mi><mi>g</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\huge huge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8435em;vertical-align:-0.4033em;"></span><span class="mord mathnormal sizing reset-size6 size10">h</span><span class="mord mathnormal sizing reset-size6 size10" style="margin-right:0.03588em;">ug</span><span class="mord mathnormal sizing reset-size6 size10">e</span></span></span></span></td><td style="text-align:left;"><code>\huge huge</code></td></tr><tr><td style="text-align:left;">\Huge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.488em"><mi>H</mi><mi>u</mi><mi>g</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\Huge Huge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1839em;vertical-align:-0.4838em;"></span><span class="mord mathnormal sizing reset-size6 size11" style="margin-right:0.08125em;">H</span><span class="mord mathnormal sizing reset-size6 size11" style="margin-right:0.03588em;">ug</span><span class="mord mathnormal sizing reset-size6 size11">e</span></span></span></span></td><td style="text-align:left;"><code>\Huge Huge</code></td></tr></tbody></table><h2 id="i" tabindex="-1">I <a class="header-anchor" href="#i" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\i</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ı</mtext></mrow><annotation encoding="application/x-tex">\text{\i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">ı</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\i}</code></td></tr><tr><td style="text-align:left;">\idotsint</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\iddots</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1223" target="_blank" rel="noreferrer">Issue #1223</a></td></tr><tr><td style="text-align:left;">\if</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1003" target="_blank" rel="noreferrer">Issue #1003</a></td></tr><tr><td style="text-align:left;">\iff</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>B</mi></mrow><annotation encoding="application/x-tex">A\iff B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:left;"><code>A\iff B</code></td></tr><tr><td style="text-align:left;">\ifmode</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1003" target="_blank" rel="noreferrer">Issue #1003</a></td></tr><tr><td style="text-align:left;">\ifx</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1003" target="_blank" rel="noreferrer">Issue #1003</a></td></tr><tr><td style="text-align:left;">\iiiint</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\iiint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∭</mo></mrow><annotation encoding="application/x-tex">\iiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∭</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\iint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∬</mo></mrow><annotation encoding="application/x-tex">\iint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Im</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\image</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\image</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\imath</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ı</mi></mrow><annotation encoding="application/x-tex">\imath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\impliedby</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mtext>  </mtext><mo>⟸</mo><mtext>  </mtext><mi>Q</mi></mrow><annotation encoding="application/x-tex">P\impliedby Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟸</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span></td><td style="text-align:left;"><code>P\impliedby Q</code></td></tr><tr><td style="text-align:left;">\implies</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>Q</mi></mrow><annotation encoding="application/x-tex">P\implies Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7073em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span></td><td style="text-align:left;"><code>P\implies Q</code></td></tr><tr><td style="text-align:left;">\in</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\includegraphics</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\includegraphics</mtext></mstyle></mrow><annotation encoding="application/x-tex">\includegraphics[height=0.8em, totalheight=0.9em, width=0.9em, alt=KA logo]{https://cdn.kastatic.org/images/apple-touch-icon-57x57-precomposed.new.png}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\includegraphics</span></span></span></span></span></td><td style="text-align:left;"><code>\includegraphics[height=0.8em, totalheight=0.9em, width=0.9em, alt=KA logo]{https://cdn.kastatic.org/images/apple-touch-icon-57x57-precomposed.new.png}</code> Requires <code>trust</code></td></tr><tr><td style="text-align:left;">\inf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\inf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">in<span style="margin-right:0.07778em;">f</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\infin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\infty</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\injlim</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\int</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\int</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\intercal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊺</mo></mrow><annotation encoding="application/x-tex">\intercal</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊺</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\intop</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\intop</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Iota</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">I</mi></mrow><annotation encoding="application/x-tex">\Iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">I</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\iota</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ι</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\isin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\isin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\it</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>a</mi><mi>B</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">{\it AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathit">AaBb</span></span></span></span></span></span></td><td style="text-align:left;"><code>{\it AaBb}</code></td></tr><tr><td style="text-align:left;">\itshape</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="jk" tabindex="-1">JK <a class="header-anchor" href="#jk" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\j</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ȷ</mtext></mrow><annotation encoding="application/x-tex">\text{\j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord latin_fallback">ȷ</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\j}</code></td></tr><tr><td style="text-align:left;">\jmath</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ȷ</mi></mrow><annotation encoding="application/x-tex">\jmath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Join</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\Join</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Kappa</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">K</mi></mrow><annotation encoding="application/x-tex">\Kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">K</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\kappa</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\KaTeX</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>KaTeX</mtext></mrow><annotation encoding="application/x-tex">\KaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">K</span><span class="mspace" style="margin-right:-0.17em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ker</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ker</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ker</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ker</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\kern</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mspace width="-0.25em"></mspace><mi>R</mi></mrow><annotation encoding="application/x-tex">I\kern-2.5pt R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></td><td style="text-align:left;"><code>I\kern-2.5pt R</code></td></tr><tr><td style="text-align:left;">\Ket</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∣</mo><mi>ψ</mi><mo fence="true">⟩</mo></mrow><annotation encoding="application/x-tex">\left|\psi\right\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose delimcenter" style="top:0em;">⟩</span></span></span></span></span></td><td style="text-align:left;"><code>\Ket{\psi}</code></td></tr><tr><td style="text-align:left;">\ket</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded><mi mathvariant="normal">∣</mi><mi>ψ</mi><mo stretchy="false">⟩</mo></mpadded></mrow><annotation encoding="application/x-tex">\mathinner{|{\psi}\rangle}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span><span class="mclose">⟩</span></span></span></span></span></td><td style="text-align:left;"><code>\ket{\psi}</code></td></tr><tr><td style="text-align:left;">\Koppa</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\koppa</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="l" tabindex="-1">L <a class="header-anchor" href="#l" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\L</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\l</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Lambda</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Λ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lambda</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\label</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\land</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\land</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lang</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\lang A\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">A</span><span class="mclose">⟩</span></span></span></span></td><td style="text-align:left;"><code>\lang A\rangle</code></td></tr><tr><td style="text-align:left;">\langle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle A\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">A</span><span class="mclose">⟩</span></span></span></span></td><td style="text-align:left;"><code>\langle A\rangle</code></td></tr><tr><td style="text-align:left;">\Larr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\lArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\larr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\large</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.2em"><mi>l</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\large large</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0667em;vertical-align:-0.2333em;"></span><span class="mord mathnormal sizing reset-size6 size7" style="margin-right:0.01968em;">l</span><span class="mord mathnormal sizing reset-size6 size7">a</span><span class="mord mathnormal sizing reset-size6 size7" style="margin-right:0.02778em;">r</span><span class="mord mathnormal sizing reset-size6 size7" style="margin-right:0.03588em;">g</span><span class="mord mathnormal sizing reset-size6 size7">e</span></span></span></span></td><td style="text-align:left;"><code>\large large</code></td></tr><tr><td style="text-align:left;">\Large</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.44em"><mi>L</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\Large Large</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.264em;vertical-align:-0.28em;"></span><span class="mord mathnormal sizing reset-size6 size8">L</span><span class="mord mathnormal sizing reset-size6 size8">a</span><span class="mord mathnormal sizing reset-size6 size8" style="margin-right:0.02778em;">r</span><span class="mord mathnormal sizing reset-size6 size8" style="margin-right:0.03588em;">g</span><span class="mord mathnormal sizing reset-size6 size8">e</span></span></span></span></td><td style="text-align:left;"><code>\Large Large</code></td></tr><tr><td style="text-align:left;">\LARGE</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.728em"><mi>L</mi><mi>A</mi><mi>R</mi><mi>G</mi><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">\LARGE LARGE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1808em;"></span><span class="mord mathnormal sizing reset-size6 size9">L</span><span class="mord mathnormal sizing reset-size6 size9">A</span><span class="mord mathnormal sizing reset-size6 size9" style="margin-right:0.05764em;">RGE</span></span></span></span></td><td style="text-align:left;"><code>\LARGE LARGE</code></td></tr><tr><td style="text-align:left;">\LaTeX</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>LaTeX</mtext></mrow><annotation encoding="application/x-tex">\LaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">L</span><span class="mspace" style="margin-right:-0.36em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lBrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal" lspace="0em" rspace="0em">⦃</mo></mrow><annotation encoding="application/x-tex">\lBrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen"><span class="mopen">{</span><span class="mspace" style="margin-right:-0.1778em;"></span><span class="mopen">[</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lbrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\lbrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lbrack</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">\lbrack</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lceil</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo></mrow><annotation encoding="application/x-tex">\lceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ldotp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">.</mo></mrow><annotation encoding="application/x-tex">\ldotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em;"></span><span class="mpunct">.</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ldots</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\le</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leadsto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\leadsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\left</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mi>b</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\left\lbrace \dfrac ab \right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.836em;vertical-align:-0.686em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">{</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\left\lbrace \dfrac ab \right.</code></td></tr><tr><td style="text-align:left;">\leftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Leftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\LeftArrow</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Non standard</td></tr><tr><td style="text-align:left;">\leftarrowtail</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftharpoondown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↽</mo></mrow><annotation encoding="application/x-tex">\leftharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↽</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftharpoonup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↼</mo></mrow><annotation encoding="application/x-tex">\leftharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↼</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftleftarrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇇</mo></mrow><annotation encoding="application/x-tex">\leftleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Leftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftrightarrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇆</mo></mrow><annotation encoding="application/x-tex">\leftrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftrightharpoons</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇋</mo></mrow><annotation encoding="application/x-tex">\leftrightharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">⇋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftrightsquigarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↭</mo></mrow><annotation encoding="application/x-tex">\leftrightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">↭</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftroot</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leftthreetimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋋</mo></mrow><annotation encoding="application/x-tex">\leftthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leqalignno</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≦</mo></mrow><annotation encoding="application/x-tex">\leqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≦</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\leqslant</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩽</mo></mrow><annotation encoding="application/x-tex">\leqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩽</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lessapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪅</mo></mrow><annotation encoding="application/x-tex">\lessapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lessdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋖</mo></mrow><annotation encoding="application/x-tex">\lessdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lesseqgtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋚</mo></mrow><annotation encoding="application/x-tex">\lesseqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋚</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lesseqqgtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪋</mo></mrow><annotation encoding="application/x-tex">\lesseqqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lessgtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≶</mo></mrow><annotation encoding="application/x-tex">\lessgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≶</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lesssim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≲</mo></mrow><annotation encoding="application/x-tex">\lesssim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≲</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\let</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lfloor</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo></mrow><annotation encoding="application/x-tex">\lfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">l<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lgroup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟮</mo></mrow><annotation encoding="application/x-tex">\lgroup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.988em;vertical-align:-0.244em;"></span><span class="mopen">⟮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lhd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\lhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊲</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">lim</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\liminf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\liminf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm" style="margin-right:0.07778em;">inf</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\limits</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></munder></mrow><annotation encoding="application/x-tex">\lim\limits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\lim\limits_x</code></td></tr><tr><td style="text-align:left;">\limsup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\limsup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">sup</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ll</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≪</mo></mrow><annotation encoding="application/x-tex">\ll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≪</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\llap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo lspace="0em" rspace="0em">=</mo><mpadded lspace="-1width" width="0px"><mrow><mtext>/</mtext><mtext> </mtext></mrow></mpadded></mrow><annotation encoding="application/x-tex">{=}\llap{/\,}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mrel">=</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord text"><span class="mord textrm">/</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span></span><span class="fix"></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>{=}\llap{/\,}</code></td></tr><tr><td style="text-align:left;">\llbracket</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal" lspace="0em" rspace="0em">⟦</mo></mrow><annotation encoding="application/x-tex">\llbracket</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen"><span class="mopen">[</span><span class="mspace" style="margin-right:-0.1778em;"></span><span class="mopen">[</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\llcorner</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌞</mi></mo></mrow><annotation encoding="application/x-tex">\llcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mopen amsrm">└</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Lleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇚</mo></mrow><annotation encoding="application/x-tex">\Lleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇚</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lll</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\lll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\llless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\llless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lmoustache</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎰</mo></mrow><annotation encoding="application/x-tex">\lmoustache</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.988em;vertical-align:-0.244em;"></span><span class="mopen">⎰</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ln</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ln</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ln</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lnapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪉</mo></mrow><annotation encoding="application/x-tex">\lnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪉</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪇</mo></mrow><annotation encoding="application/x-tex">\lneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lnot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">\lnot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lnsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋦</mo></mrow><annotation encoding="application/x-tex">\lnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋦</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\log</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\long</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Longleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟸</mo></mrow><annotation encoding="application/x-tex">\Longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟸</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\longleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟵</mo></mrow><annotation encoding="application/x-tex">\longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟵</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Longleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟺</mo></mrow><annotation encoding="application/x-tex">\Longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟺</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\longleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟷</mo></mrow><annotation encoding="application/x-tex">\longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟷</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\longmapsto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟼</mo></mrow><annotation encoding="application/x-tex">\longmapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟼</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Longrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟹</mo></mrow><annotation encoding="application/x-tex">\Longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟹</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\longrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\looparrowleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↫</mo></mrow><annotation encoding="application/x-tex">\looparrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\looparrowright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↬</mo></mrow><annotation encoding="application/x-tex">\looparrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lor</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\lor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lower</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lozenge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\lozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lparen</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo></mrow><annotation encoding="application/x-tex">\lparen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Lrarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lrArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\lrArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lrarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lrcorner</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌟</mi></mo></mrow><annotation encoding="application/x-tex">\lrcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mclose amsrm">┘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‘</mi></mrow><annotation encoding="application/x-tex">\lq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">‘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Lsh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↰</mo></mrow><annotation encoding="application/x-tex">\Lsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↰</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">\lt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ltimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋉</mo></mrow><annotation encoding="application/x-tex">\ltimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋉</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lVert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\lVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lvert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">\lvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\lvertneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="m" tabindex="-1">M <a class="header-anchor" href="#m" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\maltese</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✠</mi></mrow><annotation encoding="application/x-tex">\maltese</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mapsto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mathbb</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi></mrow><annotation encoding="application/x-tex">\mathbb{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">AB</span></span></span></span></span></td><td style="text-align:left;"><code>\mathbb{AB}</code><br>KaTeX supports A-Z k</td></tr><tr><td style="text-align:left;">\mathbf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">B</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">123</mn></mrow><annotation encoding="application/x-tex">\mathbf{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\mathbf{AaBb123}</code></td></tr><tr><td style="text-align:left;">\mathbin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false" lspace="0.22em" rspace="0.22em">!</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\mathbin{!}b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin"><span class="mclose">!</span></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\mathbin{!}b</code></td></tr><tr><td style="text-align:left;">\mathcal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">a</mi><mi mathvariant="script">B</mi><mi mathvariant="script">b</mi><mn mathvariant="script">123</mn></mrow><annotation encoding="application/x-tex">\mathcal{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="mord mathnormal">a</span><span class="mord mathcal" style="margin-right:0.03041em;">B</span><span class="mord mathnormal">b</span><span class="mord">123</span></span></span></span></span></td><td style="text-align:left;"><code>\mathcal{AaBb123}</code></td></tr><tr><td style="text-align:left;">\mathchoice</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mrow><mtext> </mtext><mtext> </mtext></mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">a\mathchoice{\,}{\,\,}{\,\,\,}{\,\,\,\,}b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\mathchoice{\,}{\,\,}{\,\,\,}{\,\,\,\,}b</code></td></tr><tr><td style="text-align:left;">\mathclap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mo>∑</mo><mpadded lspace="-0.5width" width="0px"><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></mpadded></munder><msub><mi>x</mi><mi>i</mi></msub></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_{\mathclap{1\le i\le n}} x_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4229em;vertical-align:-1.3728em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="clap mtight"><span class="strut" style="height:0.7955em;vertical-align:-0.136em;"></span><span class="inner mtight"><span><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">n</span></span></span></span><span class="fix"></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3728em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\sum_{\mathclap{1\le i\le n}} x_{i}</code></td></tr><tr><td style="text-align:left;">\mathclose</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mo stretchy="false">(</mo><mi>b</mi><mo lspace="0em" rspace="0em">&gt;</mo><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a + (b\mathclose\gt + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose"><span class="mrel">&gt;</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></td><td style="text-align:left;"><code>a + (b\mathclose\gt + c</code></td></tr><tr><td style="text-align:left;">\mathellipsis</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\mathellipsis</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mathfrak</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">a</mi><mi mathvariant="fraktur">B</mi><mi mathvariant="fraktur">b</mi></mrow><annotation encoding="application/x-tex">\mathfrak{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">AaBb</span></span></span></span></span></td><td style="text-align:left;"><code>\mathfrak{AaBb}</code><br>KaTeX supports A-Za-z</td></tr><tr><td style="text-align:left;">\mathinner</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>b</mi><mpadded><mtext>inside</mtext></mpadded><mi>c</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">ab\mathinner{\text{inside}}cd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord text"><span class="mord">inside</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span></span></span></span></td><td style="text-align:left;"><code>ab\mathinner{\text{inside}}cd</code></td></tr><tr><td style="text-align:left;">\mathit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>a</mi><mi>B</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\mathit{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathit">AaBb</span></span></span></span></span></td><td style="text-align:left;"><code>\mathit{AaBb}</code><br>KaTeX supports A-Za-z</td></tr><tr><td style="text-align:left;">\mathllap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo lspace="0em" rspace="0em">=</mo><mpadded lspace="-1width" width="0px"><mrow><mi mathvariant="normal">/</mi><mtext> </mtext></mrow></mpadded></mrow><annotation encoding="application/x-tex">{=}\mathllap{/\,}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mrel">=</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>{=}\mathllap{/\,}</code></td></tr><tr><td style="text-align:left;">\mathnormal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>a</mi><mi>B</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\mathnormal{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>\mathnormal{AaBb}</code><br>KaTeX supports A-Za-z</td></tr><tr><td style="text-align:left;">\mathop</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo><mo>⋆</mo></mo><mi>a</mi><mi>b</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathop{\star}_a^b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0961em;vertical-align:-0.247em;"></span><span class="mop"><span class="mop">⋆</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\mathop{\star}_a^b</code></td></tr><tr><td style="text-align:left;">\mathopen</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mo lspace="0em" rspace="0em">&lt;</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a + \mathopen\lt b) + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen"><span class="mrel">&lt;</span></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></td><td style="text-align:left;"><code>a + \mathopen\lt b) + c</code></td></tr><tr><td style="text-align:left;">\mathord</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi separator="true">,</mi><mn>234</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>567</mn></mrow><annotation encoding="application/x-tex">1\mathord{,}234{,}567</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">234</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">567</span></span></span></span></td><td style="text-align:left;"><code>1\mathord{,}234{,}567</code></td></tr><tr><td style="text-align:left;">\mathpunct</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo lspace="0em" rspace="0.17em">−</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\mathpunct{-}B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mpunct"><span class="mord">−</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:left;"><code>A\mathpunct{-}B</code></td></tr><tr><td style="text-align:left;">\mathrel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo mathvariant="normal">#</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \mathrel{\#} b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord">#</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a \mathrel{\#} b</code></td></tr><tr><td style="text-align:left;">\mathrlap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="0px"><mrow><mtext> </mtext><mi mathvariant="normal">/</mi></mrow></mpadded><mo lspace="0em" rspace="0em">=</mo></mrow><annotation encoding="application/x-tex">\mathrlap{\,/}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span></span></span><span class="fix"></span></span></span></span><span class="mord"><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"><code>\mathrlap{\,/}{=}</code></td></tr><tr><td style="text-align:left;">\mathring</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\mathring{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\mathring{a}</code></td></tr><tr><td style="text-align:left;">\mathrm</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">B</mi><mi mathvariant="normal">b</mi><mn>123</mn></mrow><annotation encoding="application/x-tex">\mathrm{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\mathrm{AaBb123}</code></td></tr><tr><td style="text-align:left;">\mathscr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">B</mi></mrow><annotation encoding="application/x-tex">\mathscr{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.04087em;">AB</span></span></span></span></span></td><td style="text-align:left;"><code>\mathscr{AaBb123}</code><br>KaTeX supports A-Z</td></tr><tr><td style="text-align:left;">\mathsf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">a</mi><mi mathvariant="sans-serif">B</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">123</mn></mrow><annotation encoding="application/x-tex">\mathsf{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\mathsf{AaBb123}</code></td></tr><tr><td style="text-align:left;">\mathsterling</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\mathsterling</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mathstrut</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mathtip</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mathtt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">a</mi><mi mathvariant="monospace">B</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">123</mn></mrow><annotation encoding="application/x-tex">\mathtt{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord mathtt">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\mathtt{AaBb123}</code></td></tr><tr><td style="text-align:left;">\matrix</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>{matrix}</code></td></tr><tr><td style="text-align:left;" matrix=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix}a&amp;b\\c&amp;d\end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td></tr><tr><td style="text-align:left;">\max</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\max</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">max</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mbox</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\md</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mdseries</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\measuredangle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∡</mi></mrow><annotation encoding="application/x-tex">\measuredangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">∡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\medspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\medspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\medspace b</code></td></tr><tr><td style="text-align:left;">\mho</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℧</mi></mrow><annotation encoding="application/x-tex">\mho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">℧</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mid</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">R</mi><mo>∣</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x∈ℝ\mid x&gt;0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord amsrm">R</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">}</span></span></span></span></td><td style="text-align:left;"><code>\{x∈ℝ\mid x&gt;0\}</code></td></tr><tr><td style="text-align:left;">\middle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mrow><mo fence="true">(</mo><mi>A</mi><mo fence="true" lspace="0.05em" rspace="0.05em">|</mo><mi>B</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">P\left(A\middle\vert B\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">A</span><span class="delimcenter" style="top:0em;">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></td><td style="text-align:left;"><code>P\left(A\middle\vert B\right)</code></td></tr><tr><td style="text-align:left;">\min</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">min</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mit</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>\mathit</code></td></tr><tr><td style="text-align:left;">\mkern</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace width="1em"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\mkern18mu b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\mkern18mu b</code></td></tr><tr><td style="text-align:left;">\mmlToken</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mod</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mo>≡</mo><mn>5</mn><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>2</mn></mrow><annotation encoding="application/x-tex">3\equiv 5 \mod 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span></span></td><td style="text-align:left;"><code>3\equiv 5 \mod 2</code></td></tr><tr><td style="text-align:left;">\models</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\models</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.999em;vertical-align:-0.249em;"></span><span class="mrel">⊨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\moveleft</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\moveright</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∓</mo></mrow><annotation encoding="application/x-tex">\mp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">∓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mskip</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace width="0.5556em"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\mskip{10mu}b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.5556em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\mskip{10mu}b</code></td></tr><tr><td style="text-align:left;">\mspace</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Mu</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">M</mi></mrow><annotation encoding="application/x-tex">\Mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">M</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\mu</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\multicolumn</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/269" target="_blank" rel="noreferrer">Issue #269</a></td></tr><tr><td style="text-align:left;" multiline=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\multimap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">⊸</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="n" tabindex="-1">N <a class="header-anchor" href="#n" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\N</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nabla</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\natnums</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\natnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\natural</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♮</mi></mrow><annotation encoding="application/x-tex">\natural</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\negmedspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> ⁣</mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\negmedspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:-0.2222em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\negmedspace b</code></td></tr><tr><td style="text-align:left;">\ncong</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≆</mo></mrow><annotation encoding="application/x-tex">\ncong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ne</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nearrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↗</mo></mrow><annotation encoding="application/x-tex">\nearrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↗</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\neg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">\neg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\negthickspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> ⁣</mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\negthickspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:-0.2778em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\negthickspace b</code></td></tr><tr><td style="text-align:left;">\negthinspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> ⁣</mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\negthinspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\negthinspace b</code></td></tr><tr><td style="text-align:left;">\neq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\newcommand</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\newcommand\chk{\checkmark} \chk</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✓</span></span></span></span></td><td style="text-align:left;"><code>\newcommand\chk{\checkmark} \chk</code></td></tr><tr><td style="text-align:left;">\newenvironment</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/37" target="_blank" rel="noreferrer">Issue #37</a></td></tr><tr><td style="text-align:left;">\Newextarrow</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\newline</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace linebreak="newline"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\newline b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\newline b</code></td></tr><tr><td style="text-align:left;">\nexists</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∄</mi></mrow><annotation encoding="application/x-tex">\nexists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">∄</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ngeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≱</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ngeqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ngeqslant</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ngtr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≯</mo></mrow><annotation encoding="application/x-tex">\ngtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ni</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\ni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↚</mo></mrow><annotation encoding="application/x-tex">\nleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↚</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nLeftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇍</mo></mrow><annotation encoding="application/x-tex">\nLeftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇍</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nLeftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇎</mo></mrow><annotation encoding="application/x-tex">\nLeftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇎</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↮</mo></mrow><annotation encoding="application/x-tex">\nleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nleq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≰</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nleqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nleqslant</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≮</mo></mrow><annotation encoding="application/x-tex">\nless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nmid</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∤</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nobreak</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nobreakspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\nobreakspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace nobreak"> </span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\nobreakspace b</code></td></tr><tr><td style="text-align:left;">\noexpand</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nolimits</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\lim\nolimits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\lim\nolimits_x</code></td></tr><tr><td style="text-align:left;">\normalfont</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\normalsize</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1em"><mi>n</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>i</mi><mi>z</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\normalsize normalsize</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal">ma</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">s</span><span class="mord mathnormal">i</span><span class="mord mathnormal">ze</span></span></span></span></td><td style="text-align:left;"><code>\normalsize normalsize</code></td></tr><tr><td style="text-align:left;">\not</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\not =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span></td><td style="text-align:left;"><code>\not =</code></td></tr><tr><td style="text-align:left;">\notag</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\notin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\notni</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∌</mo></mrow><annotation encoding="application/x-tex">\notni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nparallel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∦</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nprec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊀</mo></mrow><annotation encoding="application/x-tex">\nprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊀</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\npreceq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋠</mo></mrow><annotation encoding="application/x-tex">\npreceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nRightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇏</mo></mrow><annotation encoding="application/x-tex">\nRightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇏</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↛</mo></mrow><annotation encoding="application/x-tex">\nrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↛</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nshortmid</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nshortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nshortparallel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nshortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≁</mo></mrow><annotation encoding="application/x-tex">\nsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">≁</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsubseteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsubseteqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsucc</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊁</mo></mrow><annotation encoding="application/x-tex">\nsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊁</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsucceq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋡</mo></mrow><annotation encoding="application/x-tex">\nsucceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋡</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsupseteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊉</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nsupseteqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ntriangleleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋪</mo></mrow><annotation encoding="application/x-tex">\ntriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋪</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ntrianglelefteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋬</mo></mrow><annotation encoding="application/x-tex">\ntrianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ntriangleright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋫</mo></mrow><annotation encoding="application/x-tex">\ntriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ntrianglerighteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋭</mo></mrow><annotation encoding="application/x-tex">\ntrianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋭</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Nu</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi></mrow><annotation encoding="application/x-tex">\Nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">N</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nu</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nVDash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊯</mo></mrow><annotation encoding="application/x-tex">\nVDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nVdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊮</mo></mrow><annotation encoding="application/x-tex">\nVdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nvDash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊭</mo></mrow><annotation encoding="application/x-tex">\nvDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊭</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nvdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊬</mo></mrow><annotation encoding="application/x-tex">\nvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊬</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\nwarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↖</mo></mrow><annotation encoding="application/x-tex">\nwarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↖</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="o" tabindex="-1">O <a class="header-anchor" href="#o" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\O</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ø</mtext></mrow><annotation encoding="application/x-tex">\text{\O}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7806em;vertical-align:-0.0486em;"></span><span class="mord text"><span class="mord">Ø</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\O}</code></td></tr><tr><td style="text-align:left;">\o</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ø</mtext></mrow><annotation encoding="application/x-tex">\text{\o}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.0972em;"></span><span class="mord text"><span class="mord">ø</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\o}</code></td></tr><tr><td style="text-align:left;">\odot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊙</mo></mrow><annotation encoding="application/x-tex">\odot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊙</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\OE</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Œ</mtext></mrow><annotation encoding="application/x-tex">\text{\OE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord latin_fallback">Œ</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\OE}</code></td></tr><tr><td style="text-align:left;">\oe</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>œ</mtext></mrow><annotation encoding="application/x-tex">\text{\oe}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">œ</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\oe}</code></td></tr><tr><td style="text-align:left;">\officialeuro</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\oiiint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∰</mo></mrow><annotation encoding="application/x-tex">\oiiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∭</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:1.304em;"><svg xmlns="http://www.w3.org/2000/svg" width="1.304em" height="0.499em" style="width:1.304em;" viewBox="0 0 1304 499" preserveAspectRatio="xMinYMin"><path d="M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6
+-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z
+m525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0
+85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\oiint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∯</mo></mrow><annotation encoding="application/x-tex">\oiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∬</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:0.957em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.957em" height="0.499em" style="width:0.957em;" viewBox="0 0 957 499" preserveAspectRatio="xMinYMin"><path d="M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6
+-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z
+m368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8
+60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\oint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo></mrow><annotation encoding="application/x-tex">\oint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\oldstyle</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\omega</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Omega</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Omicron</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\Omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">O</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\omicron</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ο</mi></mrow><annotation encoding="application/x-tex">\omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ο</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ominus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊖</mo></mrow><annotation encoding="application/x-tex">\ominus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\operatorname</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">asin</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\operatorname{asin} x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop"><span class="mord mathrm">asin</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></td><td style="text-align:left;">\operatorname{asin} x</td></tr><tr><td style="text-align:left;">\operatorname*</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">asin</mi><mo>⁡</mo></mrow><mi>y</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">\operatorname{asin}\limits_y x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.954em;vertical-align:-0.2861em;"></span><span class="mop"><span class="mop"><span class="mord mathrm">asin</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></td><td style="text-align:left;"><code>\operatorname{asin}\limits_y x</code></td></tr><tr><td style="text-align:left;">\oplus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\or</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\oslash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊘</mo></mrow><annotation encoding="application/x-tex">\oslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\otimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊗</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\over</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>b</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">{a+1 \over b+2}+c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></td><td style="text-align:left;"><code>{a+1 \over b+2}+c</code></td></tr><tr><td style="text-align:left;">\overbrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mover><mrow><mi>x</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>x</mi></mrow><mo stretchy="true">⏞</mo></mover><mrow><mi>n</mi><mtext> times</mtext></mrow></mover></mrow><annotation encoding="application/x-tex">\overbrace{x+⋯+x}^{n\text{ times}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.9822em;vertical-align:-0.0833em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8988em;"><span style="top:-3.2313em;"><span class="pstrut" style="height:3.2313em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
+-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
+ 5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
+c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
+-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span><span style="top:-4.6627em;"><span class="pstrut" style="height:3.2313em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord text mtight"><span class="mord mtight"> times</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overbrace{x+⋯+x}^{n\text{ times}}</code></td></tr><tr><td style="text-align:left;">\overbracket</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\overgroup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏠</mo></mover></mrow><annotation encoding="application/x-tex">\overgroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0253em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 80
+H435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0
+ 435 0h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0
+ 3-1 3-3v-38c-76-158-257-219-435-219H0z"></path></svg></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overgroup{AB}</code></td></tr><tr><td style="text-align:left;">\overleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></mover></mrow><annotation encoding="application/x-tex">\overleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overleftarrow{AB}</code></td></tr><tr><td style="text-align:left;">\overleftharpoon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↼</mo></mover></mrow><annotation encoding="application/x-tex">\overleftharpoon{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overleftharpoon{AB}</code></td></tr><tr><td style="text-align:left;">\overleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></mover></mrow><annotation encoding="application/x-tex">\overleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overleftrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\overline</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a long argument</mtext><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\text{a long argument}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0889em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">a long argument</span></span></span></span><span style="top:-3.8144em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overline{\text{a long argument}}</code></td></tr><tr><td style="text-align:left;">\overlinesegment</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></mover></mrow><annotation encoding="application/x-tex">\overlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overlinesegment{AB}</code></td></tr><tr><td style="text-align:left;">\overparen</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>\overgroup</code></td></tr><tr><td style="text-align:left;">\Overrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⇒</mo></mover></mrow><annotation encoding="application/x-tex">\Overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2433em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2433em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.56em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\Overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\overrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\overrightharpoon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">⇀</mo></mover></mrow><annotation encoding="application/x-tex">\overrightharpoon{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9526em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9526em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overrightharpoon{ac}</code></td></tr><tr><td style="text-align:left;">\overset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\overset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overset{!}{=}</code></td></tr><tr><td style="text-align:left;">\overwithdelims</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\owns</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\owns</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="p" tabindex="-1">P <a class="header-anchor" href="#p" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\P</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>¶</mtext></mrow><annotation encoding="application/x-tex">\text{\P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">¶</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\P}</code></td></tr><tr><td style="text-align:left;">\pagecolor</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax" target="_blank" rel="noreferrer">Deprecated</a></td></tr><tr><td style="text-align:left;">\parallel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\parallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\part</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax" target="_blank" rel="noreferrer">Deprecated</a></td></tr><tr><td style="text-align:left;">\partial</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∂</mi></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="margin-right:0.05556em;">∂</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\perp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\perp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\phantom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="normal">Γ</mi><mrow><mi>i</mi><mphantom><mi>j</mi></mphantom><mi>k</mi></mrow><mrow><mphantom><mi>i</mi></mphantom><mi>j</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Gamma^{\phantom{i}j}_{i\phantom{j}k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.38em;vertical-align:-0.4374em;"></span><span class="mord"><span class="mord">Γ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9426em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;color:transparent;">j</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="color:transparent;">i</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4374em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\Gamma^{\phantom{i}j}_{i\phantom{j}k}</code></td></tr><tr><td style="text-align:left;">\phase</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Phi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\phi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Pi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\pi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" picture=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\pitchfork</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋔</mo></mrow><annotation encoding="application/x-tex">\pitchfork</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⋔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\plim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">plim</mi><mo>⁡</mo></mo></mrow><annotation encoding="application/x-tex">\plim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mop"><span class="mord mathrm">plim</span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\plusmn</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo></mrow><annotation encoding="application/x-tex">\plusmn</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">±</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\pm</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo></mrow><annotation encoding="application/x-tex">\pm</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">±</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\pmatrix</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>{pmatrix}</code></td></tr><tr><td style="text-align:left;" pmatrix=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix}a&amp;b\\c&amp;d\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{pmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{pmatrix}</code></td></tr><tr><td style="text-align:left;">\pmb</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle style="text-shadow:0.02em 0.01em 0.04px;"><mi>μ</mi></mstyle></mrow><annotation encoding="application/x-tex">\pmb{\mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord" style="text-shadow:0.02em 0.01em 0.04px;"><span class="mord mathnormal">μ</span></span></span></span></span></td><td style="text-align:left;"><code>\pmb{\mu}</code></td></tr><tr><td style="text-align:left;">\pmod</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\pmod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.3333em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"><code>x\pmod a</code></td></tr><tr><td style="text-align:left;">\pod</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \pod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"><code>x \pod a</code></td></tr><tr><td style="text-align:left;">\pounds</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\pounds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Pr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Pr</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\Pr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">Pr</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\prec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≺</mo></mrow><annotation encoding="application/x-tex">\prec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≺</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\precapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪷</mo></mrow><annotation encoding="application/x-tex">\precapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪷</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\preccurlyeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≼</mo></mrow><annotation encoding="application/x-tex">\preccurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≼</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\preceq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪯</mo></mrow><annotation encoding="application/x-tex">\preceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\precnapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪹</mo></mrow><annotation encoding="application/x-tex">\precnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪹</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\precneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪵</mo></mrow><annotation encoding="application/x-tex">\precneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪵</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\precnsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋨</mo></mrow><annotation encoding="application/x-tex">\precnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\precsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≾</mo></mrow><annotation encoding="application/x-tex">\precsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≾</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\prime</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">′</mo></mrow><annotation encoding="application/x-tex">\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">′</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\prod</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∏</mo></mrow><annotation encoding="application/x-tex">\prod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\projlim</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\propto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\propto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">∝</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\providecommand</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Hello</mtext></mrow><annotation encoding="application/x-tex">\providecommand\greet{\text{Hello}} \greet</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Hello</span></span></span></span></span></td><td style="text-align:left;"><code>\providecommand\greet{\text{Hello}} \greet</code></td></tr><tr><td style="text-align:left;">\psi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Psi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ψ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\pu</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>123</mn><mtext> </mtext><mfrac><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">J</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">l</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">{123~\mathchoice{\textstyle\frac{\mathrm{kJ}}{\mathrm{mol}}}{\frac{\mathrm{kJ}}{\mathrm{mol}}}{\frac{\mathrm{kJ}}{\mathrm{mol}}}{\frac{\mathrm{kJ}}{\mathrm{mol}}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mord"><span class="mord">123</span><span class="mspace nobreak"> </span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mol</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">kJ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\pu{123 kJ//mol}</code> Requires an <a href="https://github.com/KaTeX/KaTeX/tree/master/contrib/mhchem/" target="_blank" rel="noreferrer">extension</a></td></tr></tbody></table><h2 id="qr" tabindex="-1">QR <a class="header-anchor" href="#qr" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\Q</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>\Bbb{Q}</code></td></tr><tr><td style="text-align:left;">\qquad</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace width="2em"></mspace><mspace width="2em"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\qquad\qquad{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:2em;"></span><span class="mspace" style="margin-right:2em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>a\qquad\qquad{b}</code></td></tr><tr><td style="text-align:left;">\quad</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mspace width="1em"></mspace><mspace width="1em"></mspace><mi>b</mi></mrow><annotation encoding="application/x-tex">a\quad\quad{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span></span></td><td style="text-align:left;"><code>a\quad\quad{b}</code></td></tr><tr><td style="text-align:left;">\R</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\r</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\r{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\r{a}}</code></td></tr><tr><td style="text-align:left;">\raise</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>\raisebox</code></td></tr><tr><td style="text-align:left;">\raisebox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mpadded voffset="2pt"><mstyle scriptlevel="0" displaystyle="false"><mtext>ighe</mtext></mstyle></mpadded><mi>r</mi></mrow><annotation encoding="application/x-tex">h\raisebox{2pt}{ighe}r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8944em;"></span><span class="mord mathnormal">h</span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8944em;"><span style="top:-3.2em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">ighe</span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></td><td style="text-align:left;"><code>h\raisebox{2pt}{ighe}r</code></td></tr><tr><td style="text-align:left;">\rang</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle A\rang</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">A</span><span class="mclose">⟩</span></span></span></span></td><td style="text-align:left;"><code>\langle A\rang</code></td></tr><tr><td style="text-align:left;">\rangle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle A\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal">A</span><span class="mclose">⟩</span></span></span></span></td><td style="text-align:left;"><code>\langle A\rangle</code></td></tr><tr><td style="text-align:left;">\Rarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\rArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rBrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal" lspace="0em" rspace="0em">⦄</mo></mrow><annotation encoding="application/x-tex">\rBrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose"><span class="mclose">]</span><span class="mspace" style="margin-right:-0.1778em;"></span><span class="mclose">}</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rbrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\rbrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">}</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rbrack</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\rbrack</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">]</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" rcases=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\begin{rcases}a&amp;\text{if }b\\c&amp;\text{if }d\end{rcases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">}</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{rcases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{rcases}</code></td></tr><tr><td style="text-align:left;">\rceil</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">\rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">⌉</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Re</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\Re</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\real</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\real</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Reals</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\Reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\reals</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ref</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/350" target="_blank" rel="noreferrer">Issue #350</a></td></tr><tr><td style="text-align:left;">\relax</td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\renewcommand</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ahoy!</mtext></mrow><annotation encoding="application/x-tex">\def\hail{Hi!}\renewcommand\hail{\text{Ahoy!}} \hail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Ahoy!</span></span></span></span></span></td><td style="text-align:left;"><code>\def\hail{Hi!}</code><br><code>\renewcommand\hail{\text{Ahoy!}}</code><br><code>\hail</code></td></tr><tr><td style="text-align:left;">\renewenvironment</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\require</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\restriction</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\restriction</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rfloor</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">⌋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rgroup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟯</mo></mrow><annotation encoding="application/x-tex">\rgroup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mclose">⟯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rhd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\rhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊳</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Rho</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">P</mi></mrow><annotation encoding="application/x-tex">\Rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">P</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rho</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\right</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mi>b</mi></mfrac></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left.\dfrac a b\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.836em;vertical-align:-0.686em;"></span><span class="minner"><span class="mopen nulldelimiter"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\left.\dfrac a b\right)</code></td></tr><tr><td style="text-align:left;">\Rightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightarrowtail</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↣</mo></mrow><annotation encoding="application/x-tex">\rightarrowtail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightharpoondown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇁</mo></mrow><annotation encoding="application/x-tex">\rightharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇁</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightharpoonup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇀</mo></mrow><annotation encoding="application/x-tex">\rightharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇀</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightleftarrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇄</mo></mrow><annotation encoding="application/x-tex">\rightleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇄</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightleftharpoons</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇌</mo></mrow><annotation encoding="application/x-tex">\rightleftharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.682em;vertical-align:-0.011em;"></span><span class="mrel">⇌</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightrightarrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇉</mo></mrow><annotation encoding="application/x-tex">\rightrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇉</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightsquigarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rightthreetimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋌</mo></mrow><annotation encoding="application/x-tex">\rightthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋌</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\risingdotseq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≓</mo></mrow><annotation encoding="application/x-tex">\risingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rlap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="0px"><mrow><mtext> </mtext><mtext>/</mtext></mrow></mpadded><mo lspace="0em" rspace="0em">=</mo></mrow><annotation encoding="application/x-tex">\rlap{\,/}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord text"><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord textrm">/</span></span></span></span><span class="fix"></span></span></span></span><span class="mord"><span class="mrel">=</span></span></span></span></span></td><td style="text-align:left;"><code>\rlap{\,/}{=}</code></td></tr><tr><td style="text-align:left;">\rm</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">B</mi><mi mathvariant="normal">b</mi><mn>12</mn></mrow><annotation encoding="application/x-tex">\rm AaBb12</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">AaBb12</span></span></span></span></span></td><td style="text-align:left;"><code>\rm AaBb12</code></td></tr><tr><td style="text-align:left;">\rmoustache</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎱</mo></mrow><annotation encoding="application/x-tex">\rmoustache</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mclose">⎱</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\root</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rotatebox</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/681" target="_blank" rel="noreferrer">Issue #681</a></td></tr><tr><td style="text-align:left;">\rparen</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rparen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\rq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rrbracket</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal" lspace="0em" rspace="0em">⟧</mo></mrow><annotation encoding="application/x-tex">\rrbracket</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose"><span class="mclose">]</span><span class="mspace" style="margin-right:-0.1778em;"></span><span class="mclose">]</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Rrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇛</mo></mrow><annotation encoding="application/x-tex">\Rrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇛</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Rsh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↱</mo></mrow><annotation encoding="application/x-tex">\Rsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↱</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rtimes</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋊</mo></mrow><annotation encoding="application/x-tex">\rtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Rule</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>\rule</code></td></tr><tr><td style="text-align:left;">\rule</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mpadded height="0.6em" voffset="0.6em"><mspace mathbackground="black" width="0.862em" height="0.431em"></mspace></mpadded><mi>x</mi></mrow><annotation encoding="application/x-tex">x\rule[6pt]{2ex}{1ex}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.031em;"></span><span class="mord mathnormal">x</span><span class="mord rule" style="border-right-width:0.862em;border-top-width:0.431em;bottom:0.6em;"></span><span class="mord mathnormal">x</span></span></span></span></td><td style="text-align:left;"><code>x\rule[6pt]{2ex}{1ex}x</code></td></tr><tr><td style="text-align:left;">\rVert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\rVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\rvert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">\rvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">∣</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="s" tabindex="-1">S <a class="header-anchor" href="#s" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\S</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\S}</code></td></tr><tr><td style="text-align:left;">\Sampi</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sampi</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sc</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/471" target="_blank" rel="noreferrer">Issue #471</a></td></tr><tr><td style="text-align:left;">\scalebox</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\scr</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>\mathscr</code></td></tr><tr><td style="text-align:left;">\scriptscriptstyle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mfrac><mi>c</mi><mi>d</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle \frac cd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6748em;vertical-align:-0.266em;"></span><span class="mord mtight sizing reset-size6 size1"><span class="mopen nulldelimiter sizing reset-size1 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8176em;"><span style="top:-2.468em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.387em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.532em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size1 size6"></span></span></span></span></span></td><td style="text-align:left;"><code>\scriptscriptstyle \frac cd</code></td></tr><tr><td style="text-align:left;">\scriptsize</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.7em"><mi>s</mi><mi>c</mi><mi>r</mi><mi>i</mi><mi>p</mi><mi>t</mi><mi>s</mi><mi>i</mi><mi>z</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptsize scriptsize</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5978em;vertical-align:-0.1361em;"></span><span class="mord mathnormal sizing reset-size6 size3" style="margin-right:0.02778em;">scr</span><span class="mord mathnormal sizing reset-size6 size3">i</span><span class="mord mathnormal sizing reset-size6 size3">pt</span><span class="mord mathnormal sizing reset-size6 size3">s</span><span class="mord mathnormal sizing reset-size6 size3">i</span><span class="mord mathnormal sizing reset-size6 size3">ze</span></span></span></span></td><td style="text-align:left;"><code>\scriptsize scriptsize</code></td></tr><tr><td style="text-align:left;">\scriptstyle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>+</mo><mstyle scriptlevel="1" displaystyle="false"><mfrac><mi>c</mi><mi>d</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\frac ab + {\scriptstyle \frac cd}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7249em;vertical-align:-0.2408em;"></span><span class="mord"><span class="mord mtight sizing reset-size6 size3"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\frac ab + {\scriptstyle \frac cd}</code></td></tr><tr><td style="text-align:left;">\sdot</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\sdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\searrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↘</mo></mrow><annotation encoding="application/x-tex">\searrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">sec</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sect</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\sect}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\sect}</code></td></tr><tr><td style="text-align:left;">\setlength</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/687" target="_blank" rel="noreferrer">Issue #687</a></td></tr><tr><td style="text-align:left;">\setminus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\setminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">a</mi><mi mathvariant="sans-serif">B</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">123</mn></mrow><annotation encoding="application/x-tex">\sf AaBb123</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\sf AaBb123</code></td></tr><tr><td style="text-align:left;">\sharp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♯</mi></mrow><annotation encoding="application/x-tex">\sharp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♯</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\shortmid</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\shortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\shortparallel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\shortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\shoveleft</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\shoveright</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sideset</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Sigma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sigma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">∼</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\simeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sin</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sinh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sinh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sinh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sinh</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sixptsize</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.6em"><mi>s</mi><mi>i</mi><mi>x</mi><mi>p</mi><mi>t</mi><mi>s</mi><mi>i</mi><mi>z</mi><mi>e</mi></mstyle></mrow><annotation encoding="application/x-tex">\sixptsize sixptsize</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5124em;vertical-align:-0.1167em;"></span><span class="mord mathnormal sizing reset-size6 size2">s</span><span class="mord mathnormal sizing reset-size6 size2">i</span><span class="mord mathnormal sizing reset-size6 size2">x</span><span class="mord mathnormal sizing reset-size6 size2">pt</span><span class="mord mathnormal sizing reset-size6 size2">s</span><span class="mord mathnormal sizing reset-size6 size2">i</span><span class="mord mathnormal sizing reset-size6 size2">ze</span></span></span></span></td><td style="text-align:left;"><code>\sixptsize sixptsize</code></td></tr><tr><td style="text-align:left;">\sh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sh</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\skew</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\skip</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sl</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\small</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mi>s</mi><mi>m</mi><mi>a</mi><mi>l</mi><mi>l</mi></mstyle></mrow><annotation encoding="application/x-tex">\small small</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;"></span><span class="mord mathnormal sizing reset-size6 size5">s</span><span class="mord mathnormal sizing reset-size6 size5">ma</span><span class="mord mathnormal sizing reset-size6 size5" style="margin-right:0.01968em;">ll</span></span></span></span></td><td style="text-align:left;"><code>\small small</code></td></tr><tr><td style="text-align:left;">\smallfrown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\smallfrown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\smallint</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo largeop="false">∫</mo></mrow><annotation encoding="application/x-tex">\smallint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" smallmatrix=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="1"><mtable rowspacing="0.1em" columnspacing="0.2778em"><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="1" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="1" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable></mstyle></mrow><annotation encoding="application/x-tex">\begin{smallmatrix} a &amp; b \\ c &amp; d \end{smallmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3322em;vertical-align:-0.4161em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9161em;"><span style="top:-3.13em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight sizing reset-size6 size3"><span class="mord mathnormal mtight">a</span></span></span><span style="top:-2.4639em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight sizing reset-size6 size3"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4161em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.1945em;"></span><span class="arraycolsep" style="width:0.1945em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9161em;"><span style="top:-3.13em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight sizing reset-size6 size3"><span class="mord mathnormal mtight">b</span></span></span><span style="top:-2.4639em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight sizing reset-size6 size3"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4161em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{smallmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{smallmatrix}</code></td></tr><tr><td style="text-align:left;">\smallsetminus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\smallsetminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∖</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\smallsmile</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smallsmile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\smash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><msup><mi>x</mi><mpadded height="0px" depth="0px"><mn>2</mn></mpadded></msup><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(x^{\smash{2}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.363em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></td><td style="text-align:left;"><code>\left(x^{\smash{2}}\right)</code></td></tr><tr><td style="text-align:left;">\smile</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\smiley</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sout</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="horizontalstrike"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\sout{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ab</span><span class="mord mathnormal">c</span></span></span><span style="top:-3.2155em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy sout"></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\sout{abc}</code></td></tr><tr><td style="text-align:left;">\Space</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>\space</code></td></tr><tr><td style="text-align:left;">\space</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\space b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace"> </span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\space b</code></td></tr><tr><td style="text-align:left;">\spades</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spades</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\spadesuit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spadesuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sphericalangle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∢</mi></mrow><annotation encoding="application/x-tex">\sphericalangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5576em;vertical-align:-0.0352em;"></span><span class="mord amsrm">∢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" split=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/1345" target="_blank" rel="noreferrer">Issue #1345</a></td></tr><tr><td style="text-align:left;">\sqcap</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊓</mo></mrow><annotation encoding="application/x-tex">\sqcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊓</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sqcup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊔</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\square</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\square</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sqrt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mroot><mi>x</mi><mn>3</mn></mroot></mrow><annotation encoding="application/x-tex">\sqrt[3]{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6586em;"><span style="top:-2.8363em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊑</mo></mrow><annotation encoding="application/x-tex">\sqsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sqsupset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊐</mo></mrow><annotation encoding="application/x-tex">\sqsupset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sqsupseteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊒</mo></mrow><annotation encoding="application/x-tex">\sqsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊒</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ss</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ß</mtext></mrow><annotation encoding="application/x-tex">\text{\ss}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">ß</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\ss}</code></td></tr><tr><td style="text-align:left;">\stackrel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\stackrel{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\stackrel{!}{=}</code></td></tr><tr><td style="text-align:left;">\star</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">⋆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Stigma</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\stigma</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\strut</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\style</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">Non standard</td></tr><tr><td style="text-align:left;">\sub</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\sub</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;" subarray=""></td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sube</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊆</mo></mrow><annotation encoding="application/x-tex">\sube</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Subset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋐</mo></mrow><annotation encoding="application/x-tex">\Subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\subset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\subseteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊆</mo></mrow><annotation encoding="application/x-tex">\subseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\subseteqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫅</mo></mrow><annotation encoding="application/x-tex">\subseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\subsetneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\subsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊊</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\subsetneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\subsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\substack</td><td style="text-align:left;">$$\sum_{\substack{0&lt;i&lt;m\0&lt;j&lt;n}}$$</td><td style="text-align:left;"><code>\sum_{\substack{0&lt;i&lt;m\\0&lt;j&lt;n}}</code></td></tr><tr><td style="text-align:left;">\succ</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≻</mo></mrow><annotation encoding="application/x-tex">\succ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≻</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪸</mo></mrow><annotation encoding="application/x-tex">\succapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪸</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succcurlyeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≽</mo></mrow><annotation encoding="application/x-tex">\succcurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≽</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succeq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪰</mo></mrow><annotation encoding="application/x-tex">\succeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succnapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪺</mo></mrow><annotation encoding="application/x-tex">\succnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪺</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪶</mo></mrow><annotation encoding="application/x-tex">\succneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪶</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succnsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋩</mo></mrow><annotation encoding="application/x-tex">\succnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋩</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\succsim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≿</mo></mrow><annotation encoding="application/x-tex">\succsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≿</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sum</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\sup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">sup</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supe</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊇</mo></mrow><annotation encoding="application/x-tex">\supe</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Supset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋑</mo></mrow><annotation encoding="application/x-tex">\Supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supseteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊇</mo></mrow><annotation encoding="application/x-tex">\supseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊇</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supseteqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫆</mo></mrow><annotation encoding="application/x-tex">\supseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫆</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supsetneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\supsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊋</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\supsetneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\supsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫌</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\surd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">√</mi></mrow><annotation encoding="application/x-tex">\surd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.2em;"></span><span class="mord">√</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\swarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↙</mo></mrow><annotation encoding="application/x-tex">\swarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↙</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="t" tabindex="-1">T <a class="header-anchor" href="#t" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\tag</td><td style="text-align:left;">$$\tag{3.1c} a^2+b^2=c^2$$</td><td style="text-align:left;"><code>\tag{3.1c} a^2+b^2=c^2</code></td></tr><tr><td style="text-align:left;">\tag*</td><td style="text-align:left;">$$\tag*{3.1c} a^2+b^2=c^2$$</td><td style="text-align:left;"><code>\tag*{3.1c} a^2+b^2=c^2</code></td></tr><tr><td style="text-align:left;">\tan</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">tan</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\tanh</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tanh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tanh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">tanh</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Tau</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">T</mi></mrow><annotation encoding="application/x-tex">\Tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">T</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\tau</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\tbinom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\tbinom n k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\tbinom n k</code></td></tr><tr><td style="text-align:left;">\TeX</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>TeX</mtext></mrow><annotation encoding="application/x-tex">\TeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\text</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> yes </mtext><mi mathvariant="normal">&amp;</mi><mtext> no </mtext></mrow><annotation encoding="application/x-tex">\text{ yes }\&amp;\text{ no }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"> yes </span></span><span class="mord">&amp;</span><span class="mord text"><span class="mord"> no </span></span></span></span></span></td><td style="text-align:left;"><code>\text{ yes }\&amp;\text{ no }</code></td></tr><tr><td style="text-align:left;">\textasciitilde</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>~</mtext></mrow><annotation encoding="application/x-tex">\text{\textasciitilde}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord">~</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textasciitilde}</code></td></tr><tr><td style="text-align:left;">\textasciicircum</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>^</mtext></mrow><annotation encoding="application/x-tex">\text{\textasciicircum}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">^</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textasciicircum}</code></td></tr><tr><td style="text-align:left;">\textbackslash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>\</mtext></mrow><annotation encoding="application/x-tex">\text{\textbackslash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">\</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textbackslash}</code></td></tr><tr><td style="text-align:left;">\textbar</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>|</mtext></mrow><annotation encoding="application/x-tex">\text{\textbar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">|</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textbar}</code></td></tr><tr><td style="text-align:left;">\textbardbl</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>∥</mtext></mrow><annotation encoding="application/x-tex">\text{\textbardbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">∥</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textbardbl}</code></td></tr><tr><td style="text-align:left;">\textbf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="bold">AaBb123</mtext></mrow><annotation encoding="application/x-tex">\textbf{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textbf">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\textbf{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textbraceleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>{</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">{</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textbraceleft}</code></td></tr><tr><td style="text-align:left;">\textbraceright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>}</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">}</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textbraceright}</code></td></tr><tr><td style="text-align:left;">\textcircled</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>◯</mo></mover></mrow><annotation encoding="application/x-tex">\text{\textcircled a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textcircled a}</code></td></tr><tr><td style="text-align:left;">\textcolor</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{F=ma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:blue;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:blue;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:blue;">ma</span></span></span></span></td><td style="text-align:left;"><code>\textcolor{blue}{F=ma}</code></td></tr><tr><td style="text-align:left;">\textdagger</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>†</mtext></mrow><annotation encoding="application/x-tex">\text{\textdagger}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">†</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textdagger}</code></td></tr><tr><td style="text-align:left;">\textdaggerdbl</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‡</mtext></mrow><annotation encoding="application/x-tex">\text{\textdaggerdbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">‡</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textdaggerdbl}</code></td></tr><tr><td style="text-align:left;">\textdegree</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>°</mtext></mrow><annotation encoding="application/x-tex">\text{\textdegree}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">°</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textdegree}</code></td></tr><tr><td style="text-align:left;">\textdollar</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>$</mtext></mrow><annotation encoding="application/x-tex">\text{\textdollar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord text"><span class="mord">$</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textdollar}</code></td></tr><tr><td style="text-align:left;">\textellipsis</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\text{\textellipsis}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="mord text"><span class="minner">…</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textellipsis}</code></td></tr><tr><td style="text-align:left;">\textemdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>—</mtext></mrow><annotation encoding="application/x-tex">\text{\textemdash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">—</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textemdash}</code></td></tr><tr><td style="text-align:left;">\textendash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>–</mtext></mrow><annotation encoding="application/x-tex">\text{\textendash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">–</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textendash}</code></td></tr><tr><td style="text-align:left;">\textgreater</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&gt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textgreater}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&gt;</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textgreater}</code></td></tr><tr><td style="text-align:left;">\textit</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="italic">AaBb</mtext></mrow><annotation encoding="application/x-tex">\textit{AaBb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textit">AaBb</span></span></span></span></span></td><td style="text-align:left;"><code>\textit{AaBb}</code></td></tr><tr><td style="text-align:left;">\textless</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&lt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textless}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&lt;</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textless}</code></td></tr><tr><td style="text-align:left;">\textmd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AaBb123</mtext></mrow><annotation encoding="application/x-tex">\textmd{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textmd">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\textmd{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textnormal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AB</mtext></mrow><annotation encoding="application/x-tex">\textnormal{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord textrm">AB</span></span></span></span></span></td><td style="text-align:left;"><code>\textnormal{AB}</code></td></tr><tr><td style="text-align:left;">\textquotedblleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>“</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">“</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textquotedblleft}</code></td></tr><tr><td style="text-align:left;">\textquotedblright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>”</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">”</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textquotedblright}</code></td></tr><tr><td style="text-align:left;">\textquoteleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‘</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">‘</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textquoteleft}</code></td></tr><tr><td style="text-align:left;">\textquoteright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>’</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">’</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textquoteright}</code></td></tr><tr><td style="text-align:left;">\textregistered</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>®</mtext></mrow><annotation encoding="application/x-tex">\text{\textregistered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sizing reset-size6 size3">R</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textregistered}</code></td></tr><tr><td style="text-align:left;">\textrm</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AaBb123</mtext></mrow><annotation encoding="application/x-tex">\textrm{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textrm">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\textrm{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textsc</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/471" target="_blank" rel="noreferrer">Issue #471</a></td></tr><tr><td style="text-align:left;">\textsf</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="sans-serif">AaBb123</mtext></mrow><annotation encoding="application/x-tex">\textsf{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\textsf{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textsl</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\textsterling</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>£</mtext></mrow><annotation encoding="application/x-tex">\text{\textsterling}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">£</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textsterling}</code></td></tr><tr><td style="text-align:left;">\textstyle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="false"><msubsup><mo>∑</mo><mn>0</mn><mi>n</mi></msubsup></mstyle></mrow><annotation encoding="application/x-tex">\textstyle\sum_0^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\textstyle\sum_0^n</code></td></tr><tr><td style="text-align:left;">\texttip</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\texttt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">AaBb123</mtext></mrow><annotation encoding="application/x-tex">\texttt{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\texttt{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textunderscore</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>_</mtext></mrow><annotation encoding="application/x-tex">\text{\textunderscore}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">_</span></span></span></span></span></td><td style="text-align:left;"><code>\text{\textunderscore}</code></td></tr><tr><td style="text-align:left;">\textup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AaBb123</mtext></mrow><annotation encoding="application/x-tex">\textup{AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textup">AaBb123</span></span></span></span></span></td><td style="text-align:left;"><code>\textup{AaBb123}</code></td></tr><tr><td style="text-align:left;">\textvisiblespace</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\tfrac</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\tfrac ab</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>\tfrac ab</code></td></tr><tr><td style="text-align:left;">\tg</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">t<span style="margin-right:0.01389em;">g</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\th</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>th</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\th</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">th</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\therefore</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∴</mo></mrow><annotation encoding="application/x-tex">\therefore</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∴</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Theta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Θ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\theta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\thetasym</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\thetasym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\thickapprox</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\thickapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel amsrm">≈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\thicksim</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\thicksim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">∼</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\thickspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext>  </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\thickspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\thickspace b</code></td></tr><tr><td style="text-align:left;">\thinspace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mtext> </mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a\thinspace b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span></td><td style="text-align:left;"><code>a\thinspace b</code></td></tr><tr><td style="text-align:left;">\tilde</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>M</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9202em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\tilde M</code></td></tr><tr><td style="text-align:left;">\times</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Tiny</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">see <code>\tiny</code></td></tr><tr><td style="text-align:left;">\tiny</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.5em"><mi>t</mi><mi>i</mi><mi>n</mi><mi>y</mi></mstyle></mrow><annotation encoding="application/x-tex">\tiny tiny</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.427em;vertical-align:-0.0972em;"></span><span class="mord mathnormal sizing reset-size6 size1">t</span><span class="mord mathnormal sizing reset-size6 size1">in</span><span class="mord mathnormal sizing reset-size6 size1" style="margin-right:0.03588em;">y</span></span></span></span></td><td style="text-align:left;"><code>\tiny tiny</code></td></tr><tr><td style="text-align:left;">\to</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\toggle</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\top</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊤</mi></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊤</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\triangle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi></mrow><annotation encoding="application/x-tex">\triangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\triangledown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▽</mi></mrow><annotation encoding="application/x-tex">\triangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▽</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\triangleleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◃</mo></mrow><annotation encoding="application/x-tex">\triangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">◃</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\trianglelefteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\trianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊴</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\triangleq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≜</mo></mrow><annotation encoding="application/x-tex">\triangleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9987em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≜</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\triangleright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▹</mo></mrow><annotation encoding="application/x-tex">\triangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">▹</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\trianglerighteq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\trianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊵</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\tt</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">a</mi><mi mathvariant="monospace">B</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">123</mn></mrow><annotation encoding="application/x-tex">{\tt AaBb123}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord"><span class="mord mathtt">AaBb123</span></span></span></span></span></span></td><td style="text-align:left;"><code>{\tt AaBb123}</code></td></tr><tr><td style="text-align:left;">\twoheadleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↞</mo></mrow><annotation encoding="application/x-tex">\twoheadleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↞</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\twoheadrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↠</mo></mrow><annotation encoding="application/x-tex">\twoheadrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↠</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="u" tabindex="-1">U <a class="header-anchor" href="#u" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\u</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\text{\u{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\u{a}}</code></td></tr><tr><td style="text-align:left;">\Uarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\uArr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\uArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\uarr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\ulcorner</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌜</mi></mo></mrow><annotation encoding="application/x-tex">\ulcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mopen amsrm">┌</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\underbrace</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><munder><mrow><mi>x</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>x</mi></mrow><mo stretchy="true">⏟</mo></munder><mrow><mi>n</mi><mtext> times</mtext></mrow></munder></mrow><annotation encoding="application/x-tex">\underbrace{x+⋯+x}_{n\text{ times}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.9822em;vertical-align:-1.3988em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5833em;"><span style="top:-1.6012em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mord text mtight"><span class="mord mtight"> times</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5833em;"><span class="svg-align" style="top:-2.2687em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
+ 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
+ 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
+-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
+c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
+ 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
+ 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
+-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
+ 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
+-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7313em;"><span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3988em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underbrace{x+⋯+x}_{n\text{ times}}</code></td></tr><tr><td style="text-align:left;">\underbracket</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\undergroup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏡</mo></munder></mrow><annotation encoding="application/x-tex">\undergroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;vertical-align:-0.342em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.658em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 262
+H435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219
+ 435 219h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18
+ 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.342em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\undergroup{AB}</code></td></tr><tr><td style="text-align:left;">\underleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></munder></mrow><annotation encoding="application/x-tex">\underleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underleftarrow{AB}</code></td></tr><tr><td style="text-align:left;">\underleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></munder></mrow><annotation encoding="application/x-tex">\underleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underleftrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\underrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></munder></mrow><annotation encoding="application/x-tex">\underrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\underline</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mtext>a long argument</mtext><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\text{a long argument}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0889em;vertical-align:-0.3944em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.6456em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">a long argument</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3944em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underline{\text{a long argument}}</code></td></tr><tr><td style="text-align:left;">\underlinesegment</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></munder></mrow><annotation encoding="application/x-tex">\underlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underlinesegment{AB}</code></td></tr><tr><td style="text-align:left;">\underparen</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;">See <code>\undergroup</code></td></tr><tr><td style="text-align:left;">\underrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></munder></mrow><annotation encoding="application/x-tex">\underrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underrightarrow{AB}</code></td></tr><tr><td style="text-align:left;">\underset</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><munder><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></munder></mo></mrow><annotation encoding="application/x-tex">\underset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.119em;vertical-align:-0.7521em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3669em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\underset{!}{=}</code></td></tr><tr><td style="text-align:left;">\unicode</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\unlhd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\unlhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊴</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\unrhd</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\unrhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊵</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\up</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Uparrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\uparrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Updownarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇕</mo></mrow><annotation encoding="application/x-tex">\Updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇕</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\updownarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↕</mo></mrow><annotation encoding="application/x-tex">\updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">↕</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\upharpoonleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↿</mo></mrow><annotation encoding="application/x-tex">\upharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↿</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\upharpoonright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\upharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\uplus</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊎</mo></mrow><annotation encoding="application/x-tex">\uplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊎</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\uproot</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\upshape</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Upsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Υ</mi></mrow><annotation encoding="application/x-tex">\Upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Υ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\upsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>υ</mi></mrow><annotation encoding="application/x-tex">\upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">υ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\upuparrows</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇈</mo></mrow><annotation encoding="application/x-tex">\upuparrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇈</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\urcorner</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌝</mi></mo></mrow><annotation encoding="application/x-tex">\urcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mclose amsrm">┐</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\url</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.8em"><mstyle mathcolor="#cc0000"><mtext>\url</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\footnotesize\url{https://katex.org/}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8em;vertical-align:-0.2em;"></span><span class="mord text sizing reset-size6 size4" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\url</span></span></span></span></span></td><td style="text-align:left;"><code>\url{https://katex.org/}</code> Requires <code>trust</code></td></tr><tr><td style="text-align:left;">\utilde</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">~</mo></munder></mrow><annotation encoding="application/x-tex">\utilde{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0893em;vertical-align:-0.406em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.594em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><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
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.406em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\utilde{AB}</code></td></tr></tbody></table><h2 id="v" tabindex="-1">V <a class="header-anchor" href="#v" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\v</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\v{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\text{\v{a}}</code></td></tr><tr><td style="text-align:left;">\varcoppa</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varDelta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\varDelta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Δ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varepsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varGamma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\varGamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Γ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varinjlim</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varkappa</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϰ</mi></mrow><annotation encoding="application/x-tex">\varkappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varLambda</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\varLambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Λ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varliminf</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varlimsup</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varnothing</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\varnothing</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∅</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varOmega</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\varOmega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ω</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varPhi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\varPhi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Φ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varphi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varPi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\varPi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Π</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varpi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϖ</mi></mrow><annotation encoding="application/x-tex">\varpi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">ϖ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varprojlim</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varpropto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\varpropto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∝</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varPsi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\varPsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ψ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varPsi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\varPsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ψ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varrho</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi></mrow><annotation encoding="application/x-tex">\varrho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϱ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varSigma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\varSigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Σ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varsigma</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ς</mi></mrow><annotation encoding="application/x-tex">\varsigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5278em;vertical-align:-0.0972em;"></span><span class="mord mathnormal" style="margin-right:0.07986em;">ς</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varstigma</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varsubsetneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\varsubsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varsubsetneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\varsubsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varsupsetneq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\varsupsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varsupsetneqq</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\varsupsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varTheta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\varTheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Θ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vartheta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\vartheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vartriangle</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\vartriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">△</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vartriangleleft</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\vartriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊲</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vartriangleright</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\vartriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊳</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varUpsilon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Υ</mi></mrow><annotation encoding="application/x-tex">\varUpsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Υ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\varXi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\varXi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ξ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vcentcolon</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mo><mo>:</mo></mo></mo></mrow><annotation encoding="application/x-tex">\vcentcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vcenter</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Vdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊩</mo></mrow><annotation encoding="application/x-tex">\Vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊩</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vDash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\vDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊢</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vdots</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.03em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vec</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>F</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" 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+-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
+H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
+c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\vec{F}</code></td></tr><tr><td style="text-align:left;">\vee</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\veebar</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊻</mo></mrow><annotation encoding="application/x-tex">\veebar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊻</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\verb</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">\frac a b</mtext></mrow><annotation encoding="application/x-tex">\verb!\frac a b!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord text"><span class="mord texttt">\frac a b</span></span></span></span></span></td><td style="text-align:left;"><code>\verb!\frac a b!</code></td></tr><tr><td style="text-align:left;">\Vert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vert</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vfil</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vfill</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\vline</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/269" target="_blank" rel="noreferrer">Issue #269</a></td></tr><tr><td style="text-align:left;" Vmatrix=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∥</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∥</mo></mrow><annotation encoding="application/x-tex">\begin{Vmatrix}a&amp;b\\c&amp;d\end{Vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><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
+M367 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 M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><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
+M367 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 M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{Vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Vmatrix}</code></td></tr><tr><td style="text-align:left;" vmatrix=""></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><annotation encoding="application/x-tex">\begin{vmatrix}a&amp;b\\c&amp;d\end{vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><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"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><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"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{vmatrix}</code></td></tr><tr><td style="text-align:left;">\vphantom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mpadded width="0px"><mphantom><mi>M</mi></mphantom></mpadded><mi>a</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\vphantom{M}a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord rlap"><span class="inner"><span class="mord" style="color:transparent;"><span class="mord mathnormal" style="margin-right:0.10903em;color:transparent;">M</span></span></span><span class="fix"></span></span><span class="mord mathnormal">a</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\overline{\vphantom{M}a}</code></td></tr><tr><td style="text-align:left;">\Vvdash</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊪</mo></mrow><annotation encoding="application/x-tex">\Vvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊪</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="w" tabindex="-1">W <a class="header-anchor" href="#w" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\wedge</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\weierp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\weierp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\widecheck</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\widecheck{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9833em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\widecheck{AB}</code></td></tr><tr><td style="text-align:left;">\widehat</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9833em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\widehat{AB}</code></td></tr><tr><td style="text-align:left;">\wideparen</td><td style="text-align:left;"><span style="color:firebrick;">Not supported</span></td><td style="text-align:left;"><a href="https://github.com/KaTeX/KaTeX/issues/560" target="_blank" rel="noreferrer">Issue #560</a></td></tr><tr><td style="text-align:left;">\widetilde</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">~</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9693em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9693em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><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
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\widetilde{AB}</code></td></tr><tr><td style="text-align:left;">\wp</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\wp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\wr</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≀</mo></mrow><annotation encoding="application/x-tex">\wr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">≀</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table><h2 id="x" tabindex="-1">X <a class="header-anchor" href="#x" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\xcancel</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike downdiagonalstrike"><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\xcancel{ABC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cancel-lap"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord cancel-pad"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.6833em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.6833em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xcancel{ABC}</code></td></tr><tr><td style="text-align:left;">\xdef</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">\def\foo{a}\xdef\bar{\foo}\def\foo{}\bar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></td><td style="text-align:left;"><code>\def\foo{a}\xdef\bar{\foo}\def\foo{}\bar</code></td></tr><tr><td style="text-align:left;">\Xi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ξ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\xi</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\xhookleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↩</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3
+ 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0
+-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21
+ 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xhookleftarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xhookrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↪</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5
+-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3
+-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21
+ 71.5 23h399859zM103 281v-40h399897v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xhookrightarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xLeftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇐</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
+c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
+ 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
+-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
+-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
+m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xLeftarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">←</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xleftarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xleftharpoondown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↽</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333
+ 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5
+ 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667
+-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xleftharpoondown{abc}</code></td></tr><tr><td style="text-align:left;">\xleftharpoonup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↼</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xleftharpoonup{abc}</code></td></tr><tr><td style="text-align:left;">\xLeftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.56em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
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+m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
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+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xLeftrightarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xleftrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xleftrightarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xleftrightharpoons</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇋</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5
+ 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3
+-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7
+-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z
+m0 0v40h400000v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8
+ 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3
+ 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3
+-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z
+m0-194v40h400000v-40zm0 0v40h400000v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xleftrightharpoons{abc}</code></td></tr><tr><td style="text-align:left;">\xlongequal</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">=</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xlongequal{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 50 h400000 v40H0z m0 194h40000v40H0z
+M0 50 h400000 v40H0z m0 194h40000v40H0z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xlongequal{abc}</code></td></tr><tr><td style="text-align:left;">\xmapsto</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↦</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xmapsto{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.5em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V448H0V74H40V241H400000v40z
+M40 281 V448H0V74H40V241H400000v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xmapsto{abc}</code></td></tr><tr><td style="text-align:left;">\xRightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇒</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xRightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xRightarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xrightarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xrightharpoondown</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇁</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8
+ 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5
+-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95
+-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xrightharpoondown{abc}</code></td></tr><tr><td style="text-align:left;">\xrightharpoonup</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇀</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xrightharpoonup{abc}</code></td></tr><tr><td style="text-align:left;">\xrightleftharpoons</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇌</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightleftharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12
+ 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7
+-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0
+v40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11
+-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7
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+m0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xrightleftharpoons{abc}</code></td></tr><tr><td style="text-align:left;">\xtofrom</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇄</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtofrom{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1251em;vertical-align:-0.014em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1111em;"><span style="top:-3.325em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.686em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.528em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMinYMin slice"><path d="M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23
+-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8
+c28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3
+ 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23
+ 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32
+-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142
+-167z M100 147v40h399900v-40zM0 341v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.014em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xtofrom{abc}</code></td></tr><tr><td style="text-align:left;">\xtwoheadleftarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↞</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 167c68 40
+ 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69
+-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3
+-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19
+-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101
+ 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xtwoheadleftarrow{abc}</code></td></tr><tr><td style="text-align:left;">\xtwoheadrightarrow</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↠</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167
+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"></path></svg></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\xtwoheadrightarrow{abc}</code></td></tr></tbody></table><h2 id="yz" tabindex="-1">YZ <a class="header-anchor" href="#yz" aria-hidden="true">#</a></h2><table><thead><tr><th style="text-align:left;">Symbol/Function</th><th style="text-align:left;">Rendered</th><th style="text-align:left;">Source or Comment</th></tr></thead><tbody><tr><td style="text-align:left;">\yen</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¥</mi></mrow><annotation encoding="application/x-tex">\yen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm" style="margin-right:0.025em;">¥</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Z</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex">\Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">Z</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\Zeta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Z</mi></mrow><annotation encoding="application/x-tex">\Zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">Z</span></span></span></span></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;">\zeta</td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07378em;">ζ</span></span></span></span></td><td style="text-align:left;"></td></tr></tbody></table></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/application/markdown-it-katex/support-table.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/application/markdown-it-katex/support-function" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>2: KaTeX supported functions</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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-import{_ as t,o as l,c as e,d as n,a as s,b as a}from"./app.fb7dd446.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":[{"level":2,"title":"12-5-1: Learning Objectives","slug":"_12-5-1-learning-objectives","link":"#_12-5-1-learning-objectives","children":[]},{"level":2,"title":"12-5-2: Molecularity and the Rate-Determining Step","slug":"_12-5-2-molecularity-and-the-rate-determining-step","link":"#_12-5-2-molecularity-and-the-rate-determining-step","children":[{"level":3,"title":"15-2-1: Two-Step Mechanism","slug":"_15-2-1-two-step-mechanism","link":"#_15-2-1-two-step-mechanism","children":[]}]},{"level":2,"title":"12-5-3: Using Molecularity to Describe a Rate Law","slug":"_12-5-3-using-molecularity-to-describe-a-rate-law","link":"#_12-5-3-using-molecularity-to-describe-a-rate-law","children":[]},{"level":2,"title":"12-5-4: Identifying the Rate-Determining Step","slug":"_12-5-4-identifying-the-rate-determining-step","link":"#_12-5-4-identifying-the-rate-determining-step","children":[{"level":3,"title":"12-5-4-1: Example-A Reaction with an Intermediate","slug":"_12-5-4-1-example-a-reaction-with-an-intermediate","link":"#_12-5-4-1-example-a-reaction-with-an-intermediate","children":[]}]}],"relativePath":"academic/chemistry/notes/12-5.md","lastUpdated":1677742530000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n('<h1 id="_12-5-reaction-mechanism" tabindex="-1">12-5: Reaction Mechanism <a class="header-anchor" href="#_12-5-reaction-mechanism" aria-hidden="true">#</a></h1><h2 id="_12-5-1-learning-objectives" tabindex="-1">12-5-1: Learning Objectives <a class="header-anchor" href="#_12-5-1-learning-objectives" aria-hidden="true">#</a></h2><div class="tip custom-block"><p class="custom-block-title">Learning Objectives</p><p>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.</p></div><p>In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and 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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-hidden":"true"},"#")],-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 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In this case, further experiments are needed to distinguish between them. 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+import{_ as t,o as l,c as e,d as n,a as s,b as a}from"./app.0679696f.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":[{"level":2,"title":"12-5-1: Learning Objectives","slug":"_12-5-1-learning-objectives","link":"#_12-5-1-learning-objectives","children":[]},{"level":2,"title":"12-5-2: Molecularity and the Rate-Determining Step","slug":"_12-5-2-molecularity-and-the-rate-determining-step","link":"#_12-5-2-molecularity-and-the-rate-determining-step","children":[{"level":3,"title":"15-2-1: Two-Step Mechanism","slug":"_15-2-1-two-step-mechanism","link":"#_15-2-1-two-step-mechanism","children":[]}]},{"level":2,"title":"12-5-3: Using Molecularity to Describe a Rate Law","slug":"_12-5-3-using-molecularity-to-describe-a-rate-law","link":"#_12-5-3-using-molecularity-to-describe-a-rate-law","children":[]},{"level":2,"title":"12-5-4: Identifying the Rate-Determining Step","slug":"_12-5-4-identifying-the-rate-determining-step","link":"#_12-5-4-identifying-the-rate-determining-step","children":[{"level":3,"title":"12-5-4-1: Example-A Reaction with an Intermediate","slug":"_12-5-4-1-example-a-reaction-with-an-intermediate","link":"#_12-5-4-1-example-a-reaction-with-an-intermediate","children":[]}]}],"relativePath":"academic/chemistry/notes/12-5.md","lastUpdated":1677768301000}'),m={name:"academic/chemistry/notes/12-5.md"},i=n('<h1 id="_12-5-reaction-mechanism" tabindex="-1">12-5: Reaction Mechanism <a class="header-anchor" href="#_12-5-reaction-mechanism" aria-hidden="true">#</a></h1><h2 id="_12-5-1-learning-objectives" tabindex="-1">12-5-1: Learning Objectives <a class="header-anchor" href="#_12-5-1-learning-objectives" aria-hidden="true">#</a></h2><div class="tip custom-block"><p class="custom-block-title">Learning Objectives</p><p>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.</p></div><p>In an internal combustion engine, for example, isooctane reacts with oxygen to give carbon dioxide and 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\\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 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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-hidden":"true"},"#")],-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 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similarity index 99%
rename from assets/academic_chemistry_notes_12-5.md.33bb053e.lean.js
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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-hidden":"true"},"#")],-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 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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. 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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-hidden":"true"},"#")],-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. 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diff --git a/assets/academic_chemistry_problems_02-20.md.36b990c9.js b/assets/academic_chemistry_problems_02-20.md.93196f40.js
similarity index 99%
rename from assets/academic_chemistry_problems_02-20.md.36b990c9.js
rename to assets/academic_chemistry_problems_02-20.md.93196f40.js
index c4d1cbe0..7b88bb17 100644
--- a/assets/academic_chemistry_problems_02-20.md.36b990c9.js
+++ b/assets/academic_chemistry_problems_02-20.md.93196f40.js
@@ -1 +1 @@
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diff --git a/assets/academic_chemistry_problems_02-20.md.36b990c9.lean.js b/assets/academic_chemistry_problems_02-20.md.93196f40.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_02-20.md.36b990c9.lean.js
rename to assets/academic_chemistry_problems_02-20.md.93196f40.lean.js
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diff --git a/assets/academic_chemistry_problems_03-02-1.md.d06dd648.js b/assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-1.md.d06dd648.js
rename to assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.js
index 9a72cdb6..687eecd6 100644
--- a/assets/academic_chemistry_problems_03-02-1.md.d06dd648.js
+++ b/assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.js
@@ -1 +1 @@
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diff --git a/assets/academic_chemistry_problems_03-02-1.md.d06dd648.lean.js b/assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-1.md.d06dd648.lean.js
rename to assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.lean.js
index 9a72cdb6..687eecd6 100644
--- a/assets/academic_chemistry_problems_03-02-1.md.d06dd648.lean.js
+++ b/assets/academic_chemistry_problems_03-02-1.md.c5dfe88a.lean.js
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diff --git a/assets/academic_chemistry_problems_03-02-2.md.7eeed216.js b/assets/academic_chemistry_problems_03-02-2.md.47d45d00.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-2.md.7eeed216.js
rename to assets/academic_chemistry_problems_03-02-2.md.47d45d00.js
index 43c07b84..8107d1e3 100644
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diff --git a/assets/academic_chemistry_problems_03-02-2.md.7eeed216.lean.js b/assets/academic_chemistry_problems_03-02-2.md.47d45d00.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-2.md.7eeed216.lean.js
rename to assets/academic_chemistry_problems_03-02-2.md.47d45d00.lean.js
index 43c07b84..8107d1e3 100644
--- a/assets/academic_chemistry_problems_03-02-2.md.7eeed216.lean.js
+++ b/assets/academic_chemistry_problems_03-02-2.md.47d45d00.lean.js
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diff --git a/assets/academic_chemistry_problems_03-02-3.md.f8561ad7.js b/assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-3.md.f8561ad7.js
rename to assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.js
index a42752f3..95b1735a 100644
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+++ b/assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.js
@@ -1 +1 @@
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")])])]),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 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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 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"),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('<ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It increases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> It decreases by a factor of 2.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It decreases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It does not change.</li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2>',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,i,c,o,h,p,d,g];function x(w,y,k,f,b,v){return e(),l("div",null,u)}const X=t(m,[["render",x]]);export{_ as __pageData,X as default};
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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 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"),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('<ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It increases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> It decreases by a factor of 2.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It decreases by a factor of 4.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> It does not change.</li></ul><h2 id="solution" tabindex="-1">Solution <a class="header-anchor" href="#solution" aria-hidden="true">#</a></h2>',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,i,c,o,h,p,d,g];function x(w,y,k,f,b,v){return e(),l("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.f8561ad7.lean.js b/assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.lean.js
similarity index 99%
rename from assets/academic_chemistry_problems_03-02-3.md.f8561ad7.lean.js
rename to assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.lean.js
index 741cd756..4c8c9fe6 100644
--- a/assets/academic_chemistry_problems_03-02-3.md.f8561ad7.lean.js
+++ b/assets/academic_chemistry_problems_03-02-3.md.cbe8dadf.lean.js
@@ -1 +1 @@
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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 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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 . 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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 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"),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. 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-import{_ as e,o as t,c as o,d as a}from"./app.fb7dd446.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":[{"level":2,"title":"Major Patterns of Organization","slug":"major-patterns-of-organization","link":"#major-patterns-of-organization","children":[]},{"level":2,"title":"General to Specific or Specific to General","slug":"general-to-specific-or-specific-to-general","link":"#general-to-specific-or-specific-to-general","children":[]},{"level":2,"title":"Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End?","slug":"does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end","link":"#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end","children":[]},{"level":2,"title":"Are There Other Ways to Think about Organizing My Ideas?","slug":"are-there-other-ways-to-think-about-organizing-my-ideas","link":"#are-there-other-ways-to-think-about-organizing-my-ideas","children":[]},{"level":2,"title":"Methods of Development","slug":"methods-of-development","link":"#methods-of-development","children":[{"level":3,"title":"Cause and Effect (or Effect and Cause)","slug":"cause-and-effect-or-effect-and-cause","link":"#cause-and-effect-or-effect-and-cause","children":[]},{"level":3,"title":"Problem-Solution","slug":"problem-solution","link":"#problem-solution","children":[]},{"level":3,"title":"Chronology or Narrative","slug":"chronology-or-narrative","link":"#chronology-or-narrative","children":[]},{"level":3,"title":"Comparison and Contrast","slug":"comparison-and-contrast","link":"#comparison-and-contrast","children":[]}]}],"relativePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1677742530000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a('<h1 id="patterns-of-organization-and-methods-of-development" tabindex="-1">Patterns of Organization and Methods of Development <a class="header-anchor" href="#patterns-of-organization-and-methods-of-development" aria-hidden="true">#</a></h1><blockquote><p>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.</p></blockquote><h2 id="major-patterns-of-organization" tabindex="-1">Major Patterns of Organization <a class="header-anchor" href="#major-patterns-of-organization" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/19875203514_543f7bff51_z-300x201.jpg" alt="A fruit pie."></p><p>Read the following sentences:</p><ul><li>Now take the pie out of the oven and let it cool on the stovetop.</li><li>Mix the dry ingredients with the liquid ingredients.</li><li>Set the pie crust aside while you make the filling.</li></ul><p>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.</p><p>First, let’s take a look at a couple of ways to think about organization.</p><h2 id="general-to-specific-or-specific-to-general" tabindex="-1">General to Specific or Specific to General <a class="header-anchor" href="#general-to-specific-or-specific-to-general" aria-hidden="true">#</a></h2><p>It might be useful to think about organizing your topic like a triangle:</p><p><img src="https://r2.toshiki.dev/image/Patterns_of_Organizaiton_Triangles.png" alt="Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General."></p><p>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.</p><p>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:</p><p>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.</p><p>An essay that uses the specific-to-general structure might start like this:</p><p>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.</p><p>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.</p><h2 id="does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" tabindex="-1">Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? <a class="header-anchor" href="#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" aria-hidden="true">#</a></h2><p>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.</p><h2 id="are-there-other-ways-to-think-about-organizing-my-ideas" tabindex="-1">Are There Other Ways to Think about Organizing My Ideas? <a class="header-anchor" href="#are-there-other-ways-to-think-about-organizing-my-ideas" aria-hidden="true">#</a></h2><p>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:</p><ul><li>Most important information first (consider what you want readers to focus on first)</li><li>Chronological order (the order in time that events take place)</li><li>Compare and contrast (ideas are organized together because of their relationship to each other)</li></ul><p>The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.</p><p>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.<br> Topics:</p><ul><li>Facing fears</li><li>Safety in sports</li><li>Community policing</li><li>Educating prisoners</li><li>Sex education</li></ul><h2 id="methods-of-development" tabindex="-1">Methods of Development <a class="header-anchor" href="#methods-of-development" aria-hidden="true">#</a></h2><p>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.</p><p>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.</p><p>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.</p><h3 id="cause-and-effect-or-effect-and-cause" tabindex="-1">Cause and Effect (or Effect and Cause) <a class="header-anchor" href="#cause-and-effect-or-effect-and-cause" aria-hidden="true">#</a></h3><p>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 <em>cause</em>, <em>effect</em>, and <em>result</em>, to cue the reader about your about the relationships that you’re establishing.</p><h3 id="problem-solution" tabindex="-1">Problem-Solution <a class="header-anchor" href="#problem-solution" aria-hidden="true">#</a></h3><p>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:</p><ul><li>Describe the problem, followed by the solution.</li><li>Propose the solution first and then describe the problems that motivated it.</li><li>Or a problem may be followed by several solutions, one of which is selected as the best.</li></ul><p>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 <em>problem</em> and <em>solution</em> to signal these sections of your paper for your reader.</p><h3 id="chronology-or-narrative" tabindex="-1">Chronology or Narrative <a class="header-anchor" href="#chronology-or-narrative" aria-hidden="true">#</a></h3><p>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 <em>then</em>, <em>next</em>, and <em>finally</em> to make the parts of the chronology clear.</p><h3 id="comparison-and-contrast" tabindex="-1">Comparison and Contrast <a class="header-anchor" href="#comparison-and-contrast" aria-hidden="true">#</a></h3><p>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 <em>by contrast</em>, <em>in comparison</em>, <em>while</em>, <em>some</em>, and <em>others</em>.</p><div class="tip custom-block"><p class="custom-block-title">Summary</p><p>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.</p></div><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li><p>This chapter was adapted from “<a href="https://openoregon.pressbooks.pub/wrd/chapter/patterns-of-organization-and-methods-of-development/" target="_blank" rel="noreferrer">Patterns of Organization and Methods of Development</a>” in <em>The Word on College Reading and Writing</em> by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>. Adapted by Allison Kilgannon.</p></li><li><p>“<a href="https://flic.kr/p/whiByE" target="_blank" rel="noreferrer">Peach and lavender pie</a>” by <a href="https://www.flickr.com/photos/heatherjoan/" target="_blank" rel="noreferrer">Heather Joan</a> is licensed under a <a href="https://creativecommons.org/licenses/by-nc-nd/2.0/" target="_blank" rel="noreferrer">CC BY-NC-ND 2.0 Licence</a>.</p></li><li><p>“General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>.</p></li><li><p>Kilgannon, Allison. “Patterns of Organization and Methods of Development.” <a href="http://Opentextbc.ca" target="_blank" rel="noreferrer">Opentextbc.ca</a>, 20 Aug. 2021, <a href="http://opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development" target="_blank" rel="noreferrer">opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These four methods of development</a>.</p></li></ul></div>',41),r=[n];function s(l,h,d,c,p,u){return t(),o("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
+import{_ as e,o as t,c as o,d as a}from"./app.0679696f.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":[{"level":2,"title":"Major Patterns of Organization","slug":"major-patterns-of-organization","link":"#major-patterns-of-organization","children":[]},{"level":2,"title":"General to Specific or Specific to General","slug":"general-to-specific-or-specific-to-general","link":"#general-to-specific-or-specific-to-general","children":[]},{"level":2,"title":"Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End?","slug":"does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end","link":"#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end","children":[]},{"level":2,"title":"Are There Other Ways to Think about Organizing My Ideas?","slug":"are-there-other-ways-to-think-about-organizing-my-ideas","link":"#are-there-other-ways-to-think-about-organizing-my-ideas","children":[]},{"level":2,"title":"Methods of Development","slug":"methods-of-development","link":"#methods-of-development","children":[{"level":3,"title":"Cause and Effect (or Effect and Cause)","slug":"cause-and-effect-or-effect-and-cause","link":"#cause-and-effect-or-effect-and-cause","children":[]},{"level":3,"title":"Problem-Solution","slug":"problem-solution","link":"#problem-solution","children":[]},{"level":3,"title":"Chronology or Narrative","slug":"chronology-or-narrative","link":"#chronology-or-narrative","children":[]},{"level":3,"title":"Comparison and Contrast","slug":"comparison-and-contrast","link":"#comparison-and-contrast","children":[]}]}],"relativePath":"academic/literature/writing/methods-of-development.md","lastUpdated":1677768301000}'),i={name:"academic/literature/writing/methods-of-development.md"},n=a('<h1 id="patterns-of-organization-and-methods-of-development" tabindex="-1">Patterns of Organization and Methods of Development <a class="header-anchor" href="#patterns-of-organization-and-methods-of-development" aria-hidden="true">#</a></h1><blockquote><p>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.</p></blockquote><h2 id="major-patterns-of-organization" tabindex="-1">Major Patterns of Organization <a class="header-anchor" href="#major-patterns-of-organization" aria-hidden="true">#</a></h2><p><img src="https://r2.toshiki.dev/image/19875203514_543f7bff51_z-300x201.jpg" alt="A fruit pie."></p><p>Read the following sentences:</p><ul><li>Now take the pie out of the oven and let it cool on the stovetop.</li><li>Mix the dry ingredients with the liquid ingredients.</li><li>Set the pie crust aside while you make the filling.</li></ul><p>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.</p><p>First, let’s take a look at a couple of ways to think about organization.</p><h2 id="general-to-specific-or-specific-to-general" tabindex="-1">General to Specific or Specific to General <a class="header-anchor" href="#general-to-specific-or-specific-to-general" aria-hidden="true">#</a></h2><p>It might be useful to think about organizing your topic like a triangle:</p><p><img src="https://r2.toshiki.dev/image/Patterns_of_Organizaiton_Triangles.png" alt="Two triangles. The first is an inverted pyramid for General to Specific, the second is a pyramid for Specific to General."></p><p>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.</p><p>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:</p><p>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.</p><p>An essay that uses the specific-to-general structure might start like this:</p><p>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.</p><p>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.</p><h2 id="does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" tabindex="-1">Does the Triangle Mean the Essay Keeps Getting More Specific or More Broad until the Very End? <a class="header-anchor" href="#does-the-triangle-mean-the-essay-keeps-getting-more-specific-or-more-broad-until-the-very-end" aria-hidden="true">#</a></h2><p>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.</p><h2 id="are-there-other-ways-to-think-about-organizing-my-ideas" tabindex="-1">Are There Other Ways to Think about Organizing My Ideas? <a class="header-anchor" href="#are-there-other-ways-to-think-about-organizing-my-ideas" aria-hidden="true">#</a></h2><p>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:</p><ul><li>Most important information first (consider what you want readers to focus on first)</li><li>Chronological order (the order in time that events take place)</li><li>Compare and contrast (ideas are organized together because of their relationship to each other)</li></ul><p>The section on Methods of Development, below, offers more detail about some of these organizational patterns, along with some others.</p><p>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.<br> Topics:</p><ul><li>Facing fears</li><li>Safety in sports</li><li>Community policing</li><li>Educating prisoners</li><li>Sex education</li></ul><h2 id="methods-of-development" tabindex="-1">Methods of Development <a class="header-anchor" href="#methods-of-development" aria-hidden="true">#</a></h2><p>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.</p><p>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.</p><p>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.</p><h3 id="cause-and-effect-or-effect-and-cause" tabindex="-1">Cause and Effect (or Effect and Cause) <a class="header-anchor" href="#cause-and-effect-or-effect-and-cause" aria-hidden="true">#</a></h3><p>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 <em>cause</em>, <em>effect</em>, and <em>result</em>, to cue the reader about your about the relationships that you’re establishing.</p><h3 id="problem-solution" tabindex="-1">Problem-Solution <a class="header-anchor" href="#problem-solution" aria-hidden="true">#</a></h3><p>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:</p><ul><li>Describe the problem, followed by the solution.</li><li>Propose the solution first and then describe the problems that motivated it.</li><li>Or a problem may be followed by several solutions, one of which is selected as the best.</li></ul><p>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 <em>problem</em> and <em>solution</em> to signal these sections of your paper for your reader.</p><h3 id="chronology-or-narrative" tabindex="-1">Chronology or Narrative <a class="header-anchor" href="#chronology-or-narrative" aria-hidden="true">#</a></h3><p>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 <em>then</em>, <em>next</em>, and <em>finally</em> to make the parts of the chronology clear.</p><h3 id="comparison-and-contrast" tabindex="-1">Comparison and Contrast <a class="header-anchor" href="#comparison-and-contrast" aria-hidden="true">#</a></h3><p>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 <em>by contrast</em>, <em>in comparison</em>, <em>while</em>, <em>some</em>, and <em>others</em>.</p><div class="tip custom-block"><p class="custom-block-title">Summary</p><p>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.</p></div><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li><p>This chapter was adapted from “<a href="https://openoregon.pressbooks.pub/wrd/chapter/patterns-of-organization-and-methods-of-development/" target="_blank" rel="noreferrer">Patterns of Organization and Methods of Development</a>” in <em>The Word on College Reading and Writing</em> by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, which is licensed under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>. Adapted by Allison Kilgannon.</p></li><li><p>“<a href="https://flic.kr/p/whiByE" target="_blank" rel="noreferrer">Peach and lavender pie</a>” by <a href="https://www.flickr.com/photos/heatherjoan/" target="_blank" rel="noreferrer">Heather Joan</a> is licensed under a <a href="https://creativecommons.org/licenses/by-nc-nd/2.0/" target="_blank" rel="noreferrer">CC BY-NC-ND 2.0 Licence</a>.</p></li><li><p>“General to Specific vs. Specific to General Triangles” by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear is under a <a href="https://creativecommons.org/licenses/by-nc/4.0/" target="_blank" rel="noreferrer">CC BY-NC 4.0 Licence</a>.</p></li><li><p>Kilgannon, Allison. “Patterns of Organization and Methods of Development.” <a href="http://Opentextbc.ca" target="_blank" rel="noreferrer">Opentextbc.ca</a>, 20 Aug. 2021, <a href="http://opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These%20four%20methods%20of%20development" target="_blank" rel="noreferrer">opentextbc.ca/advancedenglish/chapter/patterns-of-organization-and-methods-of-development/#:~:text=These four methods of development</a>.</p></li></ul></div>',41),r=[n];function s(l,h,d,c,p,u){return t(),o("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
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楕円の性質","slug":"_12-7-楕円の性質","link":"#_12-7-楕円の性質","children":[]},{"level":3,"title":"12.8: 円の中心","slug":"_12-8-円の中心","link":"#_12-8-円の中心","children":[]},{"level":3,"title":"12.9: nge-Lenz ベクトル","slug":"_12-9-nge-lenz-ベクトル","link":"#_12-9-nge-lenz-ベクトル","children":[]}]}],"relativePath":"academic/physics/ipho-formulas-jpn/12.md","lastUpdated":1677742530000}'),p={name:"academic/physics/ipho-formulas-jpn/12.md"},c=l('<h1 id="formulas-for-ipho-日本語版-section-12" tabindex="-1">Formulas for IPhO 日本語版: Section 12 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-12" aria-hidden="true">#</a></h1><h2 id="_12-kepler-の法則" tabindex="-1">12: Kepler の法則 <a class="header-anchor" href="#_12-kepler-の法則" aria-hidden="true">#</a></h2><h3 id="_12-1-f-u" tabindex="-1">12.1: F &amp; U <a class="header-anchor" href="#_12-1-f-u" 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diff --git a/assets/academic_physics_ipho-formulas-jpn_12.md.cb5336d0.lean.js b/assets/academic_physics_ipho-formulas-jpn_12.md.6107078a.lean.js
similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_12.md.cb5336d0.lean.js
rename to assets/academic_physics_ipho-formulas-jpn_12.md.6107078a.lean.js
index d9debd5c..8b17a78b 100644
--- a/assets/academic_physics_ipho-formulas-jpn_12.md.cb5336d0.lean.js
+++ b/assets/academic_physics_ipho-formulas-jpn_12.md.6107078a.lean.js
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diff --git a/assets/academic_physics_ipho-formulas-jpn_3.md.91380de9.js b/assets/academic_physics_ipho-formulas-jpn_3.md.85923133.js
similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_3.md.91380de9.js
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tabindex="-1">3.3: 曲線運動 <a class="header-anchor" href="#_3-3-曲線運動" aria-hidden="true">#</a></h3><ol start="3"><li>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.</li></ol><h3 id="_3-4-剛体の運動" tabindex="-1">3.4: 剛体の運動 <a class="header-anchor" href="#_3-4-剛体の運動" aria-hidden="true">#</a></h3>',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 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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 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\\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 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使える.</li></ol><h3 id="_3-8-ベクトル" tabindex="-1">3.8: ベクトル <a class="header-anchor" href="#_3-8-ベクトル" aria-hidden="true">#</a></h3><ol start="8"><li>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.</li></ol>',4);function z(k,f,M,A,B,L){const t=p("Badge");return m(),e("div",null,[c,o,h,g,d,u,y,v,s("h3",b,[a("3.6: 弾道問題 "),i(t,{type:"tip",text:"supplemental"}),a(),x]),_,w])}const S=n(r,[["render",z]]);export{C as __pageData,S as default};
diff --git a/assets/academic_physics_ipho-formulas-jpn_3.md.91380de9.lean.js b/assets/academic_physics_ipho-formulas-jpn_3.md.85923133.lean.js
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rename from assets/academic_physics_ipho-formulas-jpn_3.md.91380de9.lean.js
rename to assets/academic_physics_ipho-formulas-jpn_3.md.85923133.lean.js
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--- a/assets/academic_physics_ipho-formulas-jpn_3.md.91380de9.lean.js
+++ b/assets/academic_physics_ipho-formulas-jpn_3.md.85923133.lean.js
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diff --git a/assets/academic_physics_ipho-formulas-jpn_4.md.4dcbe948.js b/assets/academic_physics_ipho-formulas-jpn_4.md.2c63d1ff.js
similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_4.md.4dcbe948.js
rename to assets/academic_physics_ipho-formulas-jpn_4.md.2c63d1ff.js
index 1f84b92c..1bea48c8 100644
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@@ -1,4 +1,4 @@
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similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_4.md.4dcbe948.lean.js
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diff --git a/assets/academic_physics_ipho-formulas-jpn_5.md.0b1b59d9.lean.js b/assets/academic_physics_ipho-formulas-jpn_5.md.39f9b67a.lean.js
similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_5.md.0b1b59d9.lean.js
rename to assets/academic_physics_ipho-formulas-jpn_5.md.39f9b67a.lean.js
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+++ b/assets/academic_physics_ipho-formulas-jpn_5.md.39f9b67a.lean.js
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similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_6.md.2e4aa53e.lean.js
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+++ b/assets/academic_physics_ipho-formulas-jpn_6.md.14b89910.lean.js
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rename from assets/academic_physics_ipho-formulas-jpn_7.md.d55993e1.js
rename to assets/academic_physics_ipho-formulas-jpn_7.md.6e7beac8.js
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rename from assets/academic_physics_ipho-formulas-jpn_8.md.ce924806.js
rename to assets/academic_physics_ipho-formulas-jpn_8.md.17d2156e.js
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diff --git a/assets/academic_physics_ipho-formulas-jpn_9.md.f7359776.js b/assets/academic_physics_ipho-formulas-jpn_9.md.75a8812f.js
similarity index 99%
rename from assets/academic_physics_ipho-formulas-jpn_9.md.f7359776.js
rename to assets/academic_physics_ipho-formulas-jpn_9.md.75a8812f.js
index f86bbfc6..4efb9c06 100644
--- a/assets/academic_physics_ipho-formulas-jpn_9.md.f7359776.js
+++ b/assets/academic_physics_ipho-formulas-jpn_9.md.75a8812f.js
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diff --git a/assets/academic_physics_ipho-formulas-jpn_9.md.f7359776.lean.js b/assets/academic_physics_ipho-formulas-jpn_9.md.75a8812f.lean.js
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@@ -1 +1 @@
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diff --git a/assets/academic_vocabulary_2023_02_2023-02-27.md.7aa70be9.js b/assets/academic_vocabulary_2023_02_2023-02-27.md.6e53f878.js
similarity index 96%
rename from assets/academic_vocabulary_2023_02_2023-02-27.md.7aa70be9.js
rename to assets/academic_vocabulary_2023_02_2023-02-27.md.6e53f878.js
index 928ebd29..59945907 100644
--- a/assets/academic_vocabulary_2023_02_2023-02-27.md.7aa70be9.js
+++ b/assets/academic_vocabulary_2023_02_2023-02-27.md.6e53f878.js
@@ -1 +1 @@
-import{_ as t,o as e,c as r,d as o}from"./app.fb7dd446.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","lastUpdated":1677742530000}'),a={name:"academic/vocabulary/2023/02/2023-02-27.md"},d=o('<h1 id="_2023-2-27-vocabulary" tabindex="-1">2023-2-27: Vocabulary <a class="header-anchor" href="#_2023-2-27-vocabulary" aria-hidden="true">#</a></h1><blockquote><p>This table of vocabularies are from &quot;Sadlier Vocabulary Workshop-Level G (Unit 4)&quot; with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.</p></blockquote><table><thead><tr><th>Vocabulary</th><th>Definition</th></tr></thead><tbody><tr><td>Atrophy</td><td>(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away</td></tr><tr><td>Bastion</td><td>A fortified place, stronghold</td></tr><tr><td>Concord</td><td>A state of agreement, harmony, unanimity; a treaty, pact, covenant</td></tr><tr><td>Consummate</td><td>(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection</td></tr><tr><td>Disarray</td><td>(n.) disorder, confusion; (v.) to throw into disorder</td></tr><tr><td>Exigency</td><td>Urgency, pressure; urgent demand, pressing need; an emergency</td></tr><tr><td>Flotsam</td><td>Floating debris; homeless, impoverished people</td></tr><tr><td>Frenetic</td><td>Frenzied, highly agitated</td></tr><tr><td>Glean</td><td>To gather bit by bit; to gather small quantities of grain left in a field by the reapers.</td></tr><tr><td>Grouse</td><td>(n.) A type of game bird; a complaint; (v.) to complain, grumble</td></tr><tr><td>Incarcerate</td><td>To imprison, confine, jail</td></tr><tr><td>Incumbent</td><td>(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of</td></tr><tr><td>Jocular</td><td>Humorous, jesting, jolly, joking</td></tr><tr><td>Ludicrous</td><td>Ridiculous, laughable, absurd</td></tr><tr><td>Mordant</td><td>Biting or caustic in thought, manner, or style; sharply or bitterly harsh.</td></tr><tr><td>Nettle</td><td>(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely</td></tr><tr><td>Pecuniary</td><td>Consisting of or measured in money; of or related to money</td></tr><tr><td>Pusillanimous</td><td>Contemptibly cowardly or mean-spirited</td></tr><tr><td>Recumbent</td><td>In a reclining position, lying down, in the posture of one sleeping or resting.</td></tr><tr><td>Stratagem</td><td>A scheme to outwit or deceive an opponent or to gain and end.</td></tr></tbody></table><div class="tip custom-block"><p class="custom-block-title">Reference</p><ul><li>&quot;Sadlier Vocabulary Workshop- Level G (Unit 4).&quot; kensiezoe. Quizlet, n.d. <a href="https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/" target="_blank" rel="noreferrer">https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/</a></li></ul></div>',4),i=[d];function n(s,l,c,u,p,h){return e(),r("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default};
+import{_ as t,o as e,c as r,d as o}from"./app.0679696f.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","lastUpdated":1677768301000}'),a={name:"academic/vocabulary/2023/02/2023-02-27.md"},d=o('<h1 id="_2023-2-27-vocabulary" tabindex="-1">2023-2-27: Vocabulary <a class="header-anchor" href="#_2023-2-27-vocabulary" aria-hidden="true">#</a></h1><blockquote><p>This table of vocabularies are from &quot;Sadlier Vocabulary Workshop-Level G (Unit 4)&quot; with their corresponding textbook definitions and part of speeches (from some of them) made into a collection.</p></blockquote><table><thead><tr><th>Vocabulary</th><th>Definition</th></tr></thead><tbody><tr><td>Atrophy</td><td>(n.) the wasting away of a body organ or tissue; any progressive decline or failure; (v.) to waste away</td></tr><tr><td>Bastion</td><td>A fortified place, stronghold</td></tr><tr><td>Concord</td><td>A state of agreement, harmony, unanimity; a treaty, pact, covenant</td></tr><tr><td>Consummate</td><td>(adj.) complete or perfect in the highest degree; (v.) to bring to a state of completion or perfection</td></tr><tr><td>Disarray</td><td>(n.) disorder, confusion; (v.) to throw into disorder</td></tr><tr><td>Exigency</td><td>Urgency, pressure; urgent demand, pressing need; an emergency</td></tr><tr><td>Flotsam</td><td>Floating debris; homeless, impoverished people</td></tr><tr><td>Frenetic</td><td>Frenzied, highly agitated</td></tr><tr><td>Glean</td><td>To gather bit by bit; to gather small quantities of grain left in a field by the reapers.</td></tr><tr><td>Grouse</td><td>(n.) A type of game bird; a complaint; (v.) to complain, grumble</td></tr><tr><td>Incarcerate</td><td>To imprison, confine, jail</td></tr><tr><td>Incumbent</td><td>(adj.) obligatory, required; (n.) one who holds a specific office at the time spoken of</td></tr><tr><td>Jocular</td><td>Humorous, jesting, jolly, joking</td></tr><tr><td>Ludicrous</td><td>Ridiculous, laughable, absurd</td></tr><tr><td>Mordant</td><td>Biting or caustic in thought, manner, or style; sharply or bitterly harsh.</td></tr><tr><td>Nettle</td><td>(n.) a prickly or stinging plant; (v.) to arouse displeasure, impatience, or anger; to vex or irritate severely</td></tr><tr><td>Pecuniary</td><td>Consisting of or measured in money; of or related to money</td></tr><tr><td>Pusillanimous</td><td>Contemptibly cowardly or mean-spirited</td></tr><tr><td>Recumbent</td><td>In a reclining position, lying down, in the posture of one sleeping or resting.</td></tr><tr><td>Stratagem</td><td>A scheme to outwit or deceive an opponent or to gain and end.</td></tr></tbody></table><div class="tip custom-block"><p class="custom-block-title">Reference</p><ul><li>&quot;Sadlier Vocabulary Workshop- Level G (Unit 4).&quot; kensiezoe. Quizlet, n.d. <a href="https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/" target="_blank" rel="noreferrer">https://quizlet.com/cn/732875555/sadlier-vocabulary-workshop-level-g-unit-4-flash-cards/</a></li></ul></div>',4),i=[d];function n(s,l,c,u,p,h){return e(),r("div",null,i)}const b=t(a,[["render",n]]);export{y as __pageData,b as default};
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similarity index 77%
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similarity index 79%
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diff --git a/assets/application_markdown-it-katex_index.md.c8d7aacf.js b/assets/application_markdown-it-katex_index.md.c8d7aacf.js
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@@ -0,0 +1,84 @@
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+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">npm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">install</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">markdown-it</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">--save</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>First install package with your preferred package manager (npm, yarn, pnpm), or include javascript before the closing <code>&lt;/body&gt;</code> for markdown-it-katex&#39;s core utils to be loaded for the static page.</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-PCRGl" id="tab-IuKsIbL" checked="checked"><label for="tab-IuKsIbL">npm</label><input type="radio" name="group-PCRGl" id="tab-xE2mBll"><label for="tab-xE2mBll">yarn</label><input type="radio" name="group-PCRGl" id="tab-DF2UJDw"><label for="tab-DF2UJDw">pnpm</label><input type="radio" name="group-PCRGl" id="tab-MoaQtA2"><label for="tab-MoaQtA2">cdn</label></div><div class="blocks"><div class="language-sh active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">npm</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">install</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">-D</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">npm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">install</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">-D</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-sh line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">yarn</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">add</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">--dev</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">yarn</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">add</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">--dev</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-sh line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">sh</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">pnpm</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">add</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">-D</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">pnpm</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">add</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">-D</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">@andatoshiki/markdown-it-katex</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#676E95;font-style:italic;">&lt;!-- your other body contents ... --&gt;</span></span>
+<span class="line"><span style="color:#89DDFF;">    &lt;</span><span style="color:#F07178;">script</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">src</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;">&gt;&lt;/</span><span style="color:#F07178;">script</span><span style="color:#89DDFF;">&gt;</span></span>
+<span class="line"><span style="color:#89DDFF;">&lt;/</span><span style="color:#F07178;">body</span><span style="color:#89DDFF;">&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#7F848E;font-style:italic;">&lt;!-- your other body contents ... --&gt;</span></span>
+<span class="line"><span style="color:#ABB2BF;">    &lt;</span><span style="color:#E06C75;">script</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">src</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://unpkg.com/@andatoshiki/markdown-it-katex@0.0.3/markdown-it-katex.min.js&quot;</span><span style="color:#ABB2BF;">&gt;&lt;/</span><span style="color:#E06C75;">script</span><span style="color:#ABB2BF;">&gt;</span></span>
+<span class="line"><span style="color:#ABB2BF;">&lt;/</span><span style="color:#E06C75;">body</span><span style="color:#ABB2BF;">&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br></div></div></div></div><p>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 <code>&lt;head&gt;</code> tag or import it into a currently linked CSS stylesheets to enable styles for KaTeX globally as follows,</p><p>Or, you could clone or download the entire repository source of <a href="https://github.com/katex/katex" target="_blank" rel="noreferrer">KaTeX</a> 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.</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-_3GVv" id="tab-o_wQppj" checked="checked"><label for="tab-o_wQppj">jsdelivr</label><input type="radio" name="group-_3GVv" id="tab-tSEOCGG"><label for="tab-tSEOCGG">jsd-gcore</label><input type="radio" name="group-_3GVv" id="tab-7G28YIT"><label for="tab-7G28YIT">jsd-fastly</label><input type="radio" name="group-_3GVv" id="tab-NnkVzJj"><label for="tab-NnkVzJj">cdnjs</label><input type="radio" name="group-_3GVv" id="tab-kno0ksX"><label for="tab-kno0ksX">unpkg</label><input type="radio" name="group-_3GVv" id="tab-v7NtdeU"><label for="tab-v7NtdeU">scss or css</label></div><div class="blocks"><div class="language-html active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdn.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://gcore.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://fastly.jsdelivr.net/npm/katex@0.16/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.0/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://unpkg.com/katex@0.16.0/dist/katex.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://unpkg.com/katex@0.16.0/dist/katex.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><div class="language-scss line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">scss</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#676E95;font-style:italic;">// ... your other styles</span></span>
+<span class="line"><span style="color:#89DDFF;font-style:italic;">@import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css</span><span style="color:#89DDFF;">&#39;</span><span style="color:#89DDFF;">;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#7F848E;font-style:italic;">// ... your other styles</span></span>
+<span class="line"><span style="color:#C678DD;">@import</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css&#39;</span><span style="color:#ABB2BF;">;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br></div></div></div></div><p>If you are using the default markdown-it parser, I personally recommend that you use the GitHub markdown CSS (<a href="https://github.com/sindresorhus/github-markdown-css/" target="_blank" rel="noreferrer">github-markdown-css</a>) for styling your HTML output with a similar style replica of GitHub&#39;s markdown styling to your familiarity.</p><div class="language-html line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">html</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&lt;</span><span style="color:#F07178;">link</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">rel</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">stylesheet</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> </span><span style="color:#C792EA;">href</span><span style="color:#89DDFF;">=</span><span style="color:#89DDFF;">&quot;</span><span style="color:#C3E88D;">https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css</span><span style="color:#89DDFF;">&quot;</span><span style="color:#89DDFF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">&lt;</span><span style="color:#E06C75;">link</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">rel</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;stylesheet&quot;</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">href</span><span style="color:#ABB2BF;">=</span><span style="color:#98C379;">&quot;https://cdn.jsdelivr.net/npm/github-markdown-css@5.2.0/github-markdown.min.css&quot;</span><span style="color:#ABB2BF;"> /&gt;</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>This forked project maintained by <a href="https://github.com/andatoshiki" target="_blank" rel="noreferrer">Anda Toshiki</a> comes with the update of KaTeX components with higher style version support (this documentation uses KaTeX version <code>16.0</code>, without hassels), later versions may works, but no guarantees are given by the developers, if you are not obsessed with the latest released version, <code>16.0</code> may fits your need for loading KaTeX on a personal blog site or small educational sites; yet it should work fully functionally.</p><div class="danger custom-block"><p class="custom-block-title">Warning</p><p>Since this project is a fork of the <a href="https://www.npmjs.com/package/markdown-it-katex" target="_blank" rel="noreferrer">original</a> markdown-it-katex project that hasn&#39;t been receiving any active updates of its code source for years, the latest katex style version supported is somewhere around <code>ver. 0.9.0</code> which clearly is outdated and results in broken styles with overflowing and other potential bug presents.</p></div><h2 id="_2-usage" tabindex="-1">2: Usage <a class="header-anchor" href="#_2-usage" aria-hidden="true">#</a></h2><p>RTo render equations, you need to include the <code>markdown-it-katex</code> plugin in the <code>markdown-it</code> components in your JavaScript or TypeScript file as follows,</p><div class="vp-code-group"><div class="tabs"><input type="radio" name="group-UAUng" id="tab-Dzumkds" checked="checked"><label for="tab-Dzumkds">js</label><input type="radio" name="group-UAUng" id="tab-pwVvLRi"><label for="tab-pwVvLRi">ts</label></div><div class="blocks"><div class="language-js active line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#C792EA;">var</span><span style="color:#A6ACCD;"> md </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">require</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">markdown-it</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)()</span><span style="color:#89DDFF;">,</span></span>
+<span class="line"><span style="color:#A6ACCD;">    mk </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">require</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">andatoshiki/markdown-it-katex</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#676E95;font-style:italic;">// double backslash is required for javascript strings, but not html input</span></span>
+<span class="line"><span style="color:#C792EA;">var</span><span style="color:#A6ACCD;"> result </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">render</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"># Math Rulez! </span><span style="color:#A6ACCD;">\\n</span><span style="color:#C3E88D;">  $</span><span style="color:#A6ACCD;">\\\\</span><span style="color:#C3E88D;">sqrt{3x-1}+(1+x)^2$</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#C678DD;">var</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">md</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">require</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;markdown-it&#39;</span><span style="color:#ABB2BF;">)(),</span></span>
+<span class="line"><span style="color:#ABB2BF;">    </span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">require</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;andatoshiki/markdown-it-katex&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#7F848E;font-style:italic;">// double backslash is required for javascript strings, but not html input</span></span>
+<span class="line"><span style="color:#C678DD;">var</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">result</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">render</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;# Math Rulez! </span><span style="color:#56B6C2;">\\n</span><span style="color:#98C379;">  $</span><span style="color:#56B6C2;">\\\\</span><span style="color:#98C379;">sqrt{3x-1}+(1+x)^2$&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br></div></div><div class="language-ts line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">ts</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;font-style:italic;">import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">*</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;font-style:italic;">as</span><span style="color:#A6ACCD;"> mk </span><span style="color:#89DDFF;font-style:italic;">from</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">markdown-it-katex</span><span style="color:#89DDFF;">&#39;</span></span>
+<span class="line"><span style="color:#89DDFF;font-style:italic;">import</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">*</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;font-style:italic;">as</span><span style="color:#A6ACCD;"> MarkdownIt </span><span style="color:#89DDFF;font-style:italic;">from</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">@andatoshiki/markdown-it</span><span style="color:#89DDFF;">&#39;</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#C792EA;">const</span><span style="color:#A6ACCD;"> md </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">new</span><span style="color:#A6ACCD;"> </span><span style="color:#82AAFF;">MarkdownIt</span><span style="color:#A6ACCD;">()</span></span>
+<span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#676E95;font-style:italic;">// double backslash is not required for TypeScript strings or template literals</span></span>
+<span class="line"><span style="color:#C792EA;">const</span><span style="color:#A6ACCD;"> result </span><span style="color:#89DDFF;">=</span><span style="color:#A6ACCD;"> md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">render</span><span style="color:#A6ACCD;">(</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"># Math Rulez! </span><span style="color:#A6ACCD;">\\n</span><span style="color:#C3E88D;">  $</span><span style="color:#A6ACCD;">\\\\</span><span style="color:#C3E88D;">sqrt{3x-1}+(1+x)^2$</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#C678DD;">import</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">*</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">as</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">from</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;markdown-it-katex&#39;</span></span>
+<span class="line"><span style="color:#C678DD;">import</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">*</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">as</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">MarkdownIt</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">from</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&#39;@andatoshiki/markdown-it&#39;</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#C678DD;">const</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#C678DD;">new</span><span style="color:#ABB2BF;"> </span><span style="color:#61AFEF;">MarkdownIt</span><span style="color:#ABB2BF;">()</span></span>
+<span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span>
+<span class="line"><span style="color:#7F848E;font-style:italic;">// double backslash is not required for TypeScript strings or template literals</span></span>
+<span class="line"><span style="color:#C678DD;">const</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">result</span><span style="color:#ABB2BF;"> </span><span style="color:#56B6C2;">=</span><span style="color:#ABB2BF;"> </span><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">render</span><span style="color:#ABB2BF;">(</span><span style="color:#98C379;">&#39;# Math Rulez! </span><span style="color:#56B6C2;">\\n</span><span style="color:#98C379;">  $</span><span style="color:#56B6C2;">\\\\</span><span style="color:#98C379;">sqrt{3x-1}+(1+x)^2$&#39;</span><span style="color:#ABB2BF;">)</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br><span class="line-number">5</span><br><span class="line-number">6</span><br><span class="line-number">7</span><br><span class="line-number">8</span><br></div></div></div></div><h3 id="_3-configuration" tabindex="-1">3: Configuration <a class="header-anchor" href="#_3-configuration" aria-hidden="true">#</a></h3><p>The following list of variable are the customizable components of <code>markdown-it-katex</code>, adjust to your needs,</p><ul><li><code>katex</code>: You can change KaTeX version by passing the instance.</li><li><code>blockClass</code>: Class added to KaTeX block.</li><li>Apply any other <a href="https://katex.org/docs/options.html" target="_blank" rel="noreferrer">KaTeX options</a> if needed as a regard to the docs</li></ul><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#A6ACCD;">md</span><span style="color:#89DDFF;">.</span><span style="color:#82AAFF;">use</span><span style="color:#A6ACCD;">(mk</span><span style="color:#89DDFF;">,</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">{</span><span style="color:#A6ACCD;"> </span><span style="color:#F07178;">blockClass</span><span style="color:#89DDFF;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">math-block</span><span style="color:#89DDFF;">&#39;</span><span style="color:#89DDFF;">,</span><span style="color:#A6ACCD;"> </span><span style="color:#F07178;">errorColor</span><span style="color:#89DDFF;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;"> #cc0000</span><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">}</span><span style="color:#A6ACCD;">)</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">md</span><span style="color:#ABB2BF;">.</span><span style="color:#61AFEF;">use</span><span style="color:#ABB2BF;">(</span><span style="color:#E06C75;">mk</span><span style="color:#ABB2BF;">, { </span><span style="color:#E06C75;">blockClass</span><span style="color:#ABB2BF;">: </span><span style="color:#98C379;">&#39;math-block&#39;</span><span style="color:#ABB2BF;">, </span><span style="color:#E06C75;">errorColor</span><span style="color:#ABB2BF;">: </span><span style="color:#98C379;">&#39; #cc0000&#39;</span><span style="color:#ABB2BF;"> })</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><h2 id="_4-examples" tabindex="-1">4: Examples <a class="header-anchor" href="#_4-examples" aria-hidden="true">#</a></h2><h3 id="_4-1-inline-math" tabindex="-1">4.1: Inline math <a class="header-anchor" href="#_4-1-inline-math" aria-hidden="true">#</a></h3><p>To render your LaTeX equations inline, enclose them with a single dollar sign <code>$</code> on each side of the equation as follows,</p><div class="language-tex line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">tex</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">$</span><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">sqrt</span><span style="color:#89DDFF;">{</span><span style="color:#F78C6C;">3</span><span style="color:#FFCB6B;">x</span><span style="color:#89DDFF;">-</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">}+(</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">+</span><span style="color:#FFCB6B;">x</span><span style="color:#89DDFF;">)^</span><span style="color:#F78C6C;">2</span><span style="color:#89DDFF;">$</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">$</span><span style="color:#D19A66;">\\sqrt</span><span style="color:#E5C07B;">{</span><span style="color:#D19A66;">3</span><span style="color:#E5C07B;">x-</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">}+(</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">+x)^</span><span style="color:#D19A66;">2</span><span style="color:#E5C07B;">$</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div>`,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
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+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=n(`<h3 id="_4-2-blocked-math" tabindex="-1">4.2: Blocked math <a class="header-anchor" href="#_4-2-blocked-math" aria-hidden="true">#</a></h3><p>To render blocks, use double dollar sign (<code>$$</code>). This mode utilizes larger symbols and centers the result as a block displayed as a <code>&lt;div&gt;</code> block in HTML output, as follows,</p><div class="language-tex line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">tex</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">$$</span></span>
+<span class="line"><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{\\</span><span style="color:#FFCB6B;">partial</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\\</span><span style="color:#FFCB6B;">partial </span><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\\left(\\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{</span><span style="color:#FFCB6B;">y</span><span style="color:#89DDFF;">^{\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}\\right)</span></span>
+<span class="line"><span style="color:#FFCB6B;">= </span><span style="color:#89DDFF;">\\left(\\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{</span><span style="color:#FFCB6B;">y</span><span style="color:#89DDFF;">^{\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">}\\right)</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\\left\\{(\\</span><span style="color:#FFCB6B;">log y</span><span style="color:#89DDFF;">)^</span><span style="color:#FFCB6B;">r </span><span style="color:#89DDFF;">+</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">sum</span><span style="color:#89DDFF;">_{</span><span style="color:#FFCB6B;">i=</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">}^</span><span style="color:#FFCB6B;">r </span><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">frac </span><span style="color:#89DDFF;">{(-</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">)^</span><span style="color:#FFCB6B;"> Ir </span><span style="color:#89DDFF;">\\</span><span style="color:#FFCB6B;">cdots </span><span style="color:#89DDFF;">(</span><span style="color:#FFCB6B;">r</span><span style="color:#89DDFF;">-</span><span style="color:#FFCB6B;">i</span><span style="color:#89DDFF;">+</span><span style="color:#F78C6C;">1</span><span style="color:#89DDFF;">)</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">(\\</span><span style="color:#FFCB6B;">log y</span><span style="color:#89DDFF;">)^{</span><span style="color:#FFCB6B;">ri</span><span style="color:#89DDFF;">}}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">{\\</span><span style="color:#FFCB6B;">omega</span><span style="color:#89DDFF;">^</span><span style="color:#FFCB6B;">i</span><span style="color:#89DDFF;">}</span><span style="color:#FFCB6B;"> </span><span style="color:#89DDFF;">\\right\\}</span></span>
+<span class="line"><span style="color:#89DDFF;">$$</span></span>
+<span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">$$</span></span>
+<span class="line"><span style="color:#D19A66;">\\frac</span><span style="color:#E5C07B;"> {</span><span style="color:#D19A66;">\\partial</span><span style="color:#E5C07B;">^r} {</span><span style="color:#D19A66;">\\partial</span><span style="color:#E5C07B;"> </span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">^r} \\left(</span><span style="color:#D19A66;">\\frac</span><span style="color:#E5C07B;"> {y^{</span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">}} {</span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">}\\right)</span></span>
+<span class="line"><span style="color:#E5C07B;">= \\left(</span><span style="color:#D19A66;">\\frac</span><span style="color:#E5C07B;"> {y^{</span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">}} {</span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">}\\right) \\left\\{(</span><span style="color:#D19A66;">\\log</span><span style="color:#E5C07B;"> y)^r + </span><span style="color:#D19A66;">\\sum</span><span style="color:#E5C07B;">_{i=</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">}^r </span><span style="color:#D19A66;">\\frac</span><span style="color:#E5C07B;"> {(-</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">)^ Ir </span><span style="color:#D19A66;">\\cdots</span><span style="color:#E5C07B;"> (r-i+</span><span style="color:#D19A66;">1</span><span style="color:#E5C07B;">) (</span><span style="color:#D19A66;">\\log</span><span style="color:#E5C07B;"> y)^{ri}} {</span><span style="color:#D19A66;">\\omega</span><span style="color:#E5C07B;">^i} \\right\\}</span></span>
+<span class="line"><span style="color:#E5C07B;">$$</span></span>
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br><span class="line-number">3</span><br><span class="line-number">4</span><br></div></div><p>the above block equation renders the LaTeX equation as a block with output as 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new file mode 100644
index 00000000..33f35b64
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new file mode 100644
index 00000000..66ae3328
--- /dev/null
+++ b/assets/application_markdown-it-katex_support-function.md.3fce9fc3.lean.js
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new file mode 100644
index 00000000..5f9c5930
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+++ b/assets/application_markdown-it-katex_support-table.md.86956241.lean.js
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diff --git a/assets/getting-started.md.b2a2d774.js b/assets/getting-started.md.43139067.js
similarity index 98%
rename from assets/getting-started.md.b2a2d774.js
rename to assets/getting-started.md.43139067.js
index 95b1fe75..a9704d52 100644
--- a/assets/getting-started.md.b2a2d774.js
+++ b/assets/getting-started.md.43139067.js
@@ -1,4 +1,4 @@
-import{_ as s,o as a,c as n,d as o}from"./app.fb7dd446.js";const h=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,"showFooter":true},"headers":[{"level":2,"title":"Issue","slug":"issue","link":"#issue","children":[]},{"level":2,"title":"Solution","slug":"solution","link":"#solution","children":[{"level":3,"title":"Solution 1","slug":"solution-1","link":"#solution-1","children":[]},{"level":3,"title":"Solution 2","slug":"solution-2","link":"#solution-2","children":[]}]}],"relativePath":"getting-started.md","lastUpdated":1677742530000}'),e={name:"getting-started.md"},l=o(`<h1 id="git-push-results-in-authentication-failed" tabindex="-1">Git push results in &quot;Authentication Failed” <a class="header-anchor" href="#git-push-results-in-authentication-failed" aria-hidden="true">#</a></h1><h2 id="issue" tabindex="-1">Issue <a class="header-anchor" href="#issue" aria-hidden="true">#</a></h2><div class="language-bash line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">bash</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">git</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">push</span></span>
+import{_ as s,o as a,c as n,d as o}from"./app.0679696f.js";const h=JSON.parse('{"title":"Git push results in \\"Authentication Failed”","description":"","frontmatter":{"title":"Git push results in \\"Authentication Failed”","author":"Anda Toshiki","date":"2023/2/21/ 10:54","editLink":true,"lastUpdated":true,"showFooter":true},"headers":[{"level":2,"title":"Issue","slug":"issue","link":"#issue","children":[]},{"level":2,"title":"Solution","slug":"solution","link":"#solution","children":[{"level":3,"title":"Solution 1","slug":"solution-1","link":"#solution-1","children":[]},{"level":3,"title":"Solution 2","slug":"solution-2","link":"#solution-2","children":[]}]}],"relativePath":"getting-started.md","lastUpdated":1677768301000}'),e={name:"getting-started.md"},l=o(`<h1 id="git-push-results-in-authentication-failed" tabindex="-1">Git push results in &quot;Authentication Failed” <a class="header-anchor" href="#git-push-results-in-authentication-failed" aria-hidden="true">#</a></h1><h2 id="issue" tabindex="-1">Issue <a class="header-anchor" href="#issue" aria-hidden="true">#</a></h2><div class="language-bash line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">bash</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">git</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">push</span></span>
 <span class="line"><span style="color:#FFCB6B;">Username</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">for</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://github.com</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">\${</span><span style="color:#A6ACCD;">{ GITHUB_USERNAME </span><span style="color:#89DDFF;">}</span><span style="color:#C3E88D;">}</span></span>
 <span class="line"><span style="color:#FFCB6B;">Password</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">for</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://andatoshiki@github.com</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">\${</span><span style="color:#A6ACCD;">{ GITHUB_PASSWORD </span><span style="color:#89DDFF;">}</span><span style="color:#C3E88D;">}</span><span style="color:#A6ACCD;"> </span><span style="color:#676E95;font-style:italic;"># or GitHub personal access token</span></span>
 <span class="line"><span style="color:#FFCB6B;">remote:</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">Invalid</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">username</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">or</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">password.</span></span>
diff --git a/assets/getting-started.md.b2a2d774.lean.js b/assets/getting-started.md.43139067.lean.js
similarity index 85%
rename from assets/getting-started.md.b2a2d774.lean.js
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index e988b1a9..4fb8aa13 100644
--- a/assets/getting-started.md.b2a2d774.lean.js
+++ b/assets/getting-started.md.43139067.lean.js
@@ -1 +1 @@
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diff --git a/assets/index.md.3ca7fb00.js b/assets/index.md.784aa176.js
similarity index 95%
rename from assets/index.md.3ca7fb00.js
rename to assets/index.md.784aa176.js
index ba1c27b1..40729ebe 100644
--- a/assets/index.md.3ca7fb00.js
+++ b/assets/index.md.784aa176.js
@@ -1 +1 @@
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similarity index 95%
rename from assets/index.md.3ca7fb00.lean.js
rename to assets/index.md.784aa176.lean.js
index ba1c27b1..40729ebe 100644
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diff --git a/assets/javascript_notes_1_1-1.md.b34917d9.js b/assets/javascript_notes_1_1-1.md.59faf06a.js
similarity index 99%
rename from assets/javascript_notes_1_1-1.md.b34917d9.js
rename to assets/javascript_notes_1_1-1.md.59faf06a.js
index 948b64ce..28c7a6f4 100644
--- a/assets/javascript_notes_1_1-1.md.b34917d9.js
+++ b/assets/javascript_notes_1_1-1.md.59faf06a.js
@@ -1,4 +1,4 @@
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+import{_ as t,o as l,c as o,a as s,b as e,e as i,d as a,r as p}from"./app.0679696f.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":[{"level":2,"title":"1-1-1: Numbers","slug":"_1-1-1-numbers","link":"#_1-1-1-numbers","children":[]},{"level":2,"title":"1-1-2: Arithmetic","slug":"_1-1-2-arithmetic","link":"#_1-1-2-arithmetic","children":[]},{"level":2,"title":"1-1-3: Strings","slug":"_1-1-3-strings","link":"#_1-1-3-strings","children":[{"level":3,"title":"1-1-3-FR: Further Reading","slug":"_1-1-3-fr-further-reading","link":"#_1-1-3-fr-further-reading","children":[]}]}],"relativePath":"javascript/notes/1/1-1.md","lastUpdated":1677768301000}'),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-hidden":"true"},"#")],-1),d={id:"_1-1-1-numbers",tabindex:"-1"},h=s("a",{class:"header-anchor",href:"#_1-1-1-numbers","aria-hidden":"true"},"#",-1),u=a(`<p>Values of the <em>number</em> type are numeric values, normally written as follows,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#F78C6C;">144</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#D19A66;">144</span></span>
 <span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><ul><li><p>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,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#F78C6C;">0100000001100010000000000000000000000000000000000000000000000000</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#D19A66;">0100000001100010000000000000000000000000000000000000000000000000</span></span>
diff --git a/assets/javascript_notes_1_1-1.md.b34917d9.lean.js b/assets/javascript_notes_1_1-1.md.59faf06a.lean.js
similarity index 98%
rename from assets/javascript_notes_1_1-1.md.b34917d9.lean.js
rename to assets/javascript_notes_1_1-1.md.59faf06a.lean.js
index c1957dc7..af1e6459 100644
--- a/assets/javascript_notes_1_1-1.md.b34917d9.lean.js
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diff --git a/assets/javascript_notes_1_1-2.md.ed1a45b0.js b/assets/javascript_notes_1_1-2.md.96b2e994.js
similarity index 65%
rename from assets/javascript_notes_1_1-2.md.ed1a45b0.js
rename to assets/javascript_notes_1_1-2.md.96b2e994.js
index 1414649c..8311e48f 100644
--- a/assets/javascript_notes_1_1-2.md.ed1a45b0.js
+++ b/assets/javascript_notes_1_1-2.md.96b2e994.js
@@ -1 +1 @@
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+import{_ as t,o as e,c as a}from"./app.0679696f.js";const l=JSON.parse('{"title":"1-2","description":"","frontmatter":{"title":"1-2"},"headers":[],"relativePath":"javascript/notes/1/1-2.md","lastUpdated":1677768301000}'),s={name:"javascript/notes/1/1-2.md"};function r(o,c,n,p,i,_){return e(),a("div")}const m=t(s,[["render",r]]);export{l as __pageData,m as default};
diff --git a/assets/javascript_notes_1_1-2.md.ed1a45b0.lean.js b/assets/javascript_notes_1_1-2.md.96b2e994.lean.js
similarity index 65%
rename from assets/javascript_notes_1_1-2.md.ed1a45b0.lean.js
rename to assets/javascript_notes_1_1-2.md.96b2e994.lean.js
index 1414649c..8311e48f 100644
--- a/assets/javascript_notes_1_1-2.md.ed1a45b0.lean.js
+++ b/assets/javascript_notes_1_1-2.md.96b2e994.lean.js
@@ -1 +1 @@
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diff --git a/assets/roadmap.md.40ca8366.js b/assets/roadmap.md.418e09e0.js
similarity index 97%
rename from assets/roadmap.md.40ca8366.js
rename to assets/roadmap.md.418e09e0.js
index a59a4cee..2e31f94c 100644
--- a/assets/roadmap.md.40ca8366.js
+++ b/assets/roadmap.md.418e09e0.js
@@ -1 +1 @@
-import{_ as e,o as t,c as s,d as i}from"./app.fb7dd446.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[{"level":2,"title":"Homepage","slug":"homepage","link":"#homepage","children":[]},{"level":2,"title":"Footer","slug":"footer","link":"#footer","children":[]},{"level":2,"title":"Configs","slug":"configs","link":"#configs","children":[]}],"relativePath":"roadmap.md","lastUpdated":1677742530000}'),a={name:"roadmap.md"},l=i('<h1 id="theme" tabindex="-1">Theme <a class="header-anchor" href="#theme" aria-hidden="true">#</a></h1><h2 id="homepage" tabindex="-1">Homepage <a class="header-anchor" href="#homepage" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><p><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Finish three hero action blocks with keywords and its corresponded descriptions.</p></li><li class="task-list-item"><p><input class="task-list-item-checkbox" disabled="" type="checkbox"> Finish my learning platform section using SVGs of different online academies from simple icons.</p><ul><li><p>Descriptions</p></li><li><p>Icon in SVG format</p></li><li><p>Platform name</p></li><li><p>Linkable SVG icon/button</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Khan academy</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> GitHub</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Libretext</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> OpenStax</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> BCcampus</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikibooks</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikipedia</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Bookboon</li><li>[ ]</li></ul></li></ul></li></ul><h2 id="footer" tabindex="-1">Footer <a class="header-anchor" href="#footer" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Footer copyright section new-lining and other additional information designs <ul><li>Refer to footer of <a href="https://status.toshiki.dev" target="_blank" rel="noreferrer">https://status.toshiki.dev</a></li></ul></li></ul><h2 id="configs" tabindex="-1">Configs <a class="header-anchor" href="#configs" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Header configs including different properties and meta configs.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete navigation configuration. <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Navigation divider and text title, line breakings. <ul><li>Refer to: <a href="https://github.com/maomao1996/mm-notes/blob/master/docs/.vitepress/configs/nav.ts" target="_blank" rel="noreferrer">mm-notes: <code>nav.ts</code></a>.</li></ul></li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete sidebar configs for different directories.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Vitepress sidebar generation plugin for huge bundled clips article directory.</li></ul><h1 id="articles" tabindex="-1">Articles <a class="header-anchor" href="#articles" aria-hidden="true">#</a></h1><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <code>&lt;head&gt;</code> tag documentation or guide by <a href="https://github.com/joshbuchea" target="_blank" rel="noreferrer">joshbuchea</a>/<a href="https://github.com/joshbuchea/HEAD" target="_blank" rel="noreferrer">HEAD</a>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Write index page for every directory, <strong>no Chat GPT allowed</strong>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Chemistry review notes.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> JavaScript 0-100 notes per chapter. <ul><li>Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)</li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Outliers novel clips</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Migrate clipped articles and archives from Notion to Vitepress.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Fresh copy of &quot;How To Ask Questions The Smart Way&quot; by Eric Steven Raymond <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Reading notes</li><li><a href="http://www.catb.org/~esr/faqs/smart-questions.html" target="_blank" rel="noreferrer">http://www.catb.org/~esr/faqs/smart-questions.html</a></li></ul></li></ul>',9),c=[l];function o(n,r,d,h,k,p){return t(),s("div",null,c)}const u=e(a,[["render",o]]);export{m as __pageData,u as default};
+import{_ as e,o as t,c as s,d as i}from"./app.0679696f.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[{"level":2,"title":"Homepage","slug":"homepage","link":"#homepage","children":[]},{"level":2,"title":"Footer","slug":"footer","link":"#footer","children":[]},{"level":2,"title":"Configs","slug":"configs","link":"#configs","children":[]}],"relativePath":"roadmap.md","lastUpdated":1677768301000}'),a={name:"roadmap.md"},l=i('<h1 id="theme" tabindex="-1">Theme <a class="header-anchor" href="#theme" aria-hidden="true">#</a></h1><h2 id="homepage" tabindex="-1">Homepage <a class="header-anchor" href="#homepage" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><p><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Finish three hero action blocks with keywords and its corresponded descriptions.</p></li><li class="task-list-item"><p><input class="task-list-item-checkbox" disabled="" type="checkbox"> Finish my learning platform section using SVGs of different online academies from simple icons.</p><ul><li><p>Descriptions</p></li><li><p>Icon in SVG format</p></li><li><p>Platform name</p></li><li><p>Linkable SVG icon/button</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Khan academy</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> GitHub</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Libretext</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> OpenStax</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> BCcampus</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikibooks</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikipedia</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Bookboon</li><li>[ ]</li></ul></li></ul></li></ul><h2 id="footer" tabindex="-1">Footer <a class="header-anchor" href="#footer" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Footer copyright section new-lining and other additional information designs <ul><li>Refer to footer of <a href="https://status.toshiki.dev" target="_blank" rel="noreferrer">https://status.toshiki.dev</a></li></ul></li></ul><h2 id="configs" tabindex="-1">Configs <a class="header-anchor" href="#configs" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Header configs including different properties and meta configs.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete navigation configuration. <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Navigation divider and text title, line breakings. <ul><li>Refer to: <a href="https://github.com/maomao1996/mm-notes/blob/master/docs/.vitepress/configs/nav.ts" target="_blank" rel="noreferrer">mm-notes: <code>nav.ts</code></a>.</li></ul></li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete sidebar configs for different directories.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Vitepress sidebar generation plugin for huge bundled clips article directory.</li></ul><h1 id="articles" tabindex="-1">Articles <a class="header-anchor" href="#articles" aria-hidden="true">#</a></h1><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <code>&lt;head&gt;</code> tag documentation or guide by <a href="https://github.com/joshbuchea" target="_blank" rel="noreferrer">joshbuchea</a>/<a href="https://github.com/joshbuchea/HEAD" target="_blank" rel="noreferrer">HEAD</a>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Write index page for every directory, <strong>no Chat GPT allowed</strong>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Chemistry review notes.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> JavaScript 0-100 notes per chapter. <ul><li>Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)</li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Outliers novel clips</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Migrate clipped articles and archives from Notion to Vitepress.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Fresh copy of &quot;How To Ask Questions The Smart Way&quot; by Eric Steven Raymond <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Reading notes</li><li><a href="http://www.catb.org/~esr/faqs/smart-questions.html" target="_blank" rel="noreferrer">http://www.catb.org/~esr/faqs/smart-questions.html</a></li></ul></li></ul>',9),c=[l];function o(n,r,d,h,k,p){return t(),s("div",null,c)}const u=e(a,[["render",o]]);export{m as __pageData,u as default};
diff --git a/assets/roadmap.md.40ca8366.lean.js b/assets/roadmap.md.418e09e0.lean.js
similarity index 78%
rename from assets/roadmap.md.40ca8366.lean.js
rename to assets/roadmap.md.418e09e0.lean.js
index 6d642e3a..f8c841d0 100644
--- a/assets/roadmap.md.40ca8366.lean.js
+++ b/assets/roadmap.md.418e09e0.lean.js
@@ -1 +1 @@
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+import{_ as e,o as t,c as s,d as i}from"./app.0679696f.js";const m=JSON.parse('{"title":"Theme","description":"","frontmatter":{},"headers":[{"level":2,"title":"Homepage","slug":"homepage","link":"#homepage","children":[]},{"level":2,"title":"Footer","slug":"footer","link":"#footer","children":[]},{"level":2,"title":"Configs","slug":"configs","link":"#configs","children":[]}],"relativePath":"roadmap.md","lastUpdated":1677768301000}'),a={name:"roadmap.md"},l=i("",9),c=[l];function o(n,r,d,h,k,p){return t(),s("div",null,c)}const u=e(a,[["render",o]]);export{m as __pageData,u as default};
diff --git a/assets/save_reading_index.md.280f8fa3.js b/assets/save_reading_index.md.d6ac8f3e.js
similarity index 77%
rename from assets/save_reading_index.md.280f8fa3.js
rename to assets/save_reading_index.md.d6ac8f3e.js
index 24062e68..c83d103d 100644
--- a/assets/save_reading_index.md.280f8fa3.js
+++ b/assets/save_reading_index.md.d6ac8f3e.js
@@ -1 +1 @@
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+import{_ as t,o as a,c as o,a as e,b as n}from"./app.0679696f.js";const x=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","lastUpdated":1677768301000}'),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-hidden":"true"},"#")],-1),r=[d];function c(i,_,l,m,p,h){return a(),o("div",null,r)}const g=t(s,[["render",c]]);export{x as __pageData,g as default};
diff --git a/assets/save_reading_index.md.280f8fa3.lean.js b/assets/save_reading_index.md.d6ac8f3e.lean.js
similarity index 77%
rename from assets/save_reading_index.md.280f8fa3.lean.js
rename to assets/save_reading_index.md.d6ac8f3e.lean.js
index 24062e68..c83d103d 100644
--- a/assets/save_reading_index.md.280f8fa3.lean.js
+++ b/assets/save_reading_index.md.d6ac8f3e.lean.js
@@ -1 +1 @@
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+import{_ as t,o as a,c as o,a as e,b as n}from"./app.0679696f.js";const x=JSON.parse('{"title":"Welcome to Reading Notes","description":"","frontmatter":{},"headers":[],"relativePath":"save/reading/index.md","lastUpdated":1677768301000}'),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-hidden":"true"},"#")],-1),r=[d];function c(i,_,l,m,p,h){return a(),o("div",null,r)}const g=t(s,[["render",c]]);export{x as __pageData,g as default};
diff --git a/assets/save_reading_outliers_1.md.82be654c.js b/assets/save_reading_outliers_1.md.380a0117.js
similarity index 98%
rename from assets/save_reading_outliers_1.md.82be654c.js
rename to assets/save_reading_outliers_1.md.380a0117.js
index 4eeb67d2..2b29deeb 100644
--- a/assets/save_reading_outliers_1.md.82be654c.js
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-import{_ as e,o as t,c as a,d as o}from"./app.fb7dd446.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":[{"level":2,"title":"Summary: Introduction: The Roseto Mystery","slug":"summary-introduction-the-roseto-mystery","link":"#summary-introduction-the-roseto-mystery","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]}]},{"level":2,"title":"Summary: Chapter One: The Matthew Effect","slug":"summary-chapter-one-the-matthew-effect","link":"#summary-chapter-one-the-matthew-effect","children":[{"level":3,"title":"Section 1.","slug":"section-1-1","link":"#section-1-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2-1","link":"#section-2-1","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]},{"level":3,"title":"Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect","slug":"analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect","link":"#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect","children":[]}]}],"relativePath":"save/reading/outliers/1.md","lastUpdated":1677742530000}'),s={name:"save/reading/outliers/1.md"},n=o('<h1 id="introduction-chapter-1-the-roseto-mystery" tabindex="-1">Introduction &amp; Chapter 1: The Roseto Mystery <a class="header-anchor" href="#introduction-chapter-1-the-roseto-mystery" aria-hidden="true">#</a></h1><h2 id="summary-introduction-the-roseto-mystery" tabindex="-1">Summary: Introduction: The Roseto Mystery <a class="header-anchor" href="#summary-introduction-the-roseto-mystery" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>outlier</em>.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>Outliers</em> to do for our understanding of success what Wolf did for our understanding of health.</p><h2 id="summary-chapter-one-the-matthew-effect" tabindex="-1">Summary: Chapter One: The Matthew Effect <a class="header-anchor" href="#summary-chapter-one-the-matthew-effect" aria-hidden="true">#</a></h2><h3 id="section-1-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2-1" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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:</p><p>“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.”</p><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h3 id="analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" tabindex="-1">Analysis: Introduction: The Roseto Mystery &amp; Chapter 1: The Matthew Effect <a class="header-anchor" href="#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" aria-hidden="true">#</a></h3><p>Gladwell&#39;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.</p><p>Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that &quot;success leads to more success.&quot; 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&#39;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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Introduction &amp; Chapter 1 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section1/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section1/</a>.</li></ul></div>',28),i=[n];function r(h,l,d,c,u,m){return t(),a("div",null,i)}const p=e(s,[["render",r]]);export{y as __pageData,p as default};
+import{_ as e,o as t,c as a,d as o}from"./app.0679696f.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":[{"level":2,"title":"Summary: Introduction: The Roseto Mystery","slug":"summary-introduction-the-roseto-mystery","link":"#summary-introduction-the-roseto-mystery","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]}]},{"level":2,"title":"Summary: Chapter One: The Matthew Effect","slug":"summary-chapter-one-the-matthew-effect","link":"#summary-chapter-one-the-matthew-effect","children":[{"level":3,"title":"Section 1.","slug":"section-1-1","link":"#section-1-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2-1","link":"#section-2-1","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]},{"level":3,"title":"Analysis: Introduction: The Roseto Mystery & Chapter 1: The Matthew Effect","slug":"analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect","link":"#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect","children":[]}]}],"relativePath":"save/reading/outliers/1.md","lastUpdated":1677768301000}'),s={name:"save/reading/outliers/1.md"},n=o('<h1 id="introduction-chapter-1-the-roseto-mystery" tabindex="-1">Introduction &amp; Chapter 1: The Roseto Mystery <a class="header-anchor" href="#introduction-chapter-1-the-roseto-mystery" aria-hidden="true">#</a></h1><h2 id="summary-introduction-the-roseto-mystery" tabindex="-1">Summary: Introduction: The Roseto Mystery <a class="header-anchor" href="#summary-introduction-the-roseto-mystery" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>outlier</em>.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>Outliers</em> to do for our understanding of success what Wolf did for our understanding of health.</p><h2 id="summary-chapter-one-the-matthew-effect" tabindex="-1">Summary: Chapter One: The Matthew Effect <a class="header-anchor" href="#summary-chapter-one-the-matthew-effect" aria-hidden="true">#</a></h2><h3 id="section-1-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2-1" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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:</p><p>“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.”</p><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h3 id="analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" tabindex="-1">Analysis: Introduction: The Roseto Mystery &amp; Chapter 1: The Matthew Effect <a class="header-anchor" href="#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" aria-hidden="true">#</a></h3><p>Gladwell&#39;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.</p><p>Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that &quot;success leads to more success.&quot; 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&#39;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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Introduction &amp; Chapter 1 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section1/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section1/</a>.</li></ul></div>',28),i=[n];function r(h,l,d,c,u,m){return t(),a("div",null,i)}const p=e(s,[["render",r]]);export{y as __pageData,p as default};
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similarity index 98%
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-import{_ as e,o as t,c as a,d as o}from"./app.fb7dd446.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":[{"level":2,"title":"Summary: Chapter Two: The 10,000-Hour Rule","slug":"summary-chapter-two-the-10-000-hour-rule","link":"#summary-chapter-two-the-10-000-hour-rule","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Two: The 10,000-Hour Rule","slug":"analysis-chapter-two-the-10-000-hour-rule","link":"#analysis-chapter-two-the-10-000-hour-rule","children":[]}],"relativePath":"save/reading/outliers/2.md","lastUpdated":1677742530000}'),i={name:"save/reading/outliers/2.md"},n=o('<h1 id="chapter-two-the-10-000-hour-rule" tabindex="-1">Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h1><h2 id="summary-chapter-two-the-10-000-hour-rule" tabindex="-1">Summary: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#summary-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><p>Bill Joy and Bill Gates were also born at just the right time. In 1975, <em>Popular Electronics</em> 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.</p><h2 id="analysis-chapter-two-the-10-000-hour-rule" tabindex="-1">Analysis: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#analysis-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><p>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&#39;s inborn abilities.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 2 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section2/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section2/</a>. Accessed 27 Feb. 2023.</li></ul></div>',20),r=[n];function s(h,l,c,d,u,m){return t(),a("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
+import{_ as e,o as t,c as a,d as o}from"./app.0679696f.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":[{"level":2,"title":"Summary: Chapter Two: The 10,000-Hour Rule","slug":"summary-chapter-two-the-10-000-hour-rule","link":"#summary-chapter-two-the-10-000-hour-rule","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Two: The 10,000-Hour Rule","slug":"analysis-chapter-two-the-10-000-hour-rule","link":"#analysis-chapter-two-the-10-000-hour-rule","children":[]}],"relativePath":"save/reading/outliers/2.md","lastUpdated":1677768301000}'),i={name:"save/reading/outliers/2.md"},n=o('<h1 id="chapter-two-the-10-000-hour-rule" tabindex="-1">Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h1><h2 id="summary-chapter-two-the-10-000-hour-rule" tabindex="-1">Summary: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#summary-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><p>Bill Joy and Bill Gates were also born at just the right time. In 1975, <em>Popular Electronics</em> 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.</p><h2 id="analysis-chapter-two-the-10-000-hour-rule" tabindex="-1">Analysis: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#analysis-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><p>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&#39;s inborn abilities.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 2 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section2/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section2/</a>. Accessed 27 Feb. 2023.</li></ul></div>',20),r=[n];function s(h,l,c,d,u,m){return t(),a("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.debc70ec.lean.js b/assets/save_reading_outliers_2.md.204111a8.lean.js
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-import{_ as e,o as t,c as a,d as o}from"./app.fb7dd446.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":[{"level":2,"title":"Summary: Chapter Two: The 10,000-Hour Rule","slug":"summary-chapter-two-the-10-000-hour-rule","link":"#summary-chapter-two-the-10-000-hour-rule","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Two: The 10,000-Hour Rule","slug":"analysis-chapter-two-the-10-000-hour-rule","link":"#analysis-chapter-two-the-10-000-hour-rule","children":[]}],"relativePath":"save/reading/outliers/2.md","lastUpdated":1677742530000}'),i={name:"save/reading/outliers/2.md"},n=o("",20),r=[n];function s(h,l,c,d,u,m){return t(),a("div",null,r)}const g=e(i,[["render",s]]);export{f as __pageData,g as default};
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diff --git a/assets/save_reading_outliers_3.md.e81a2b54.js b/assets/save_reading_outliers_3.md.96561c21.js
similarity index 98%
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rename to assets/save_reading_outliers_3.md.96561c21.js
index fb9dcb68..211e0c71 100644
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-import{_ as e,o as t,c as a,d as s}from"./app.fb7dd446.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":[{"level":2,"title":"Summary: Chapter Three: The Trouble with Geniuses, Part 1","slug":"summary-chapter-three-the-trouble-with-geniuses-part-1","link":"#summary-chapter-three-the-trouble-with-geniuses-part-1","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]}]},{"level":2,"title":"Analysis: Chapter Three: The Trouble with Geniuses, Part 1","slug":"analysis-chapter-three-the-trouble-with-geniuses-part-1","link":"#analysis-chapter-three-the-trouble-with-geniuses-part-1","children":[]}],"relativePath":"save/reading/outliers/3.md","lastUpdated":1677742530000}'),i={name:"save/reading/outliers/3.md"},n=s('<h1 id="chapter-3-the-trouble-with-geniuses-part-1" tabindex="-1">Chapter 3: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#chapter-3-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h1><h2 id="summary-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Summary: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#summary-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>Chris Langan appeared on the television trivia show <em>1 vs. 100</em> 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 <em>Principia Mathematica</em> as a teenager. On <em>1 vs. 100</em>, Langan cashed out with winnings of $250,000.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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 <em>enough</em>.</p><p>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 <em>enough</em> to win a Nobel.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><p>Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.</p><h2 id="analysis-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Analysis: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#analysis-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 3 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section3/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section3/</a>. Accessed 27 Feb. 2023.</li></ul></div>',19),o=[n];function r(l,h,c,d,u,m){return t(),a("div",null,o)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
+import{_ as e,o as t,c as a,d as s}from"./app.0679696f.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":[{"level":2,"title":"Summary: Chapter Three: The Trouble with Geniuses, Part 1","slug":"summary-chapter-three-the-trouble-with-geniuses-part-1","link":"#summary-chapter-three-the-trouble-with-geniuses-part-1","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]}]},{"level":2,"title":"Analysis: Chapter Three: The Trouble with Geniuses, Part 1","slug":"analysis-chapter-three-the-trouble-with-geniuses-part-1","link":"#analysis-chapter-three-the-trouble-with-geniuses-part-1","children":[]}],"relativePath":"save/reading/outliers/3.md","lastUpdated":1677768301000}'),i={name:"save/reading/outliers/3.md"},n=s('<h1 id="chapter-3-the-trouble-with-geniuses-part-1" tabindex="-1">Chapter 3: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#chapter-3-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h1><h2 id="summary-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Summary: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#summary-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>Chris Langan appeared on the television trivia show <em>1 vs. 100</em> 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 <em>Principia Mathematica</em> as a teenager. On <em>1 vs. 100</em>, Langan cashed out with winnings of $250,000.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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 <em>enough</em>.</p><p>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 <em>enough</em> to win a Nobel.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><p>Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.</p><h2 id="analysis-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Analysis: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#analysis-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 3 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section3/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section3/</a>. Accessed 27 Feb. 2023.</li></ul></div>',19),o=[n];function r(l,h,c,d,u,m){return t(),a("div",null,o)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
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-import{_ as e,o as t,c as a,d as s}from"./app.fb7dd446.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":[{"level":2,"title":"Summary: Chapter Three: The Trouble with Geniuses, Part 1","slug":"summary-chapter-three-the-trouble-with-geniuses-part-1","link":"#summary-chapter-three-the-trouble-with-geniuses-part-1","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]}]},{"level":2,"title":"Analysis: Chapter Three: The Trouble with Geniuses, Part 1","slug":"analysis-chapter-three-the-trouble-with-geniuses-part-1","link":"#analysis-chapter-three-the-trouble-with-geniuses-part-1","children":[]}],"relativePath":"save/reading/outliers/3.md","lastUpdated":1677742530000}'),i={name:"save/reading/outliers/3.md"},n=s("",19),o=[n];function r(l,h,c,d,u,m){return t(),a("div",null,o)}const p=e(i,[["render",r]]);export{f as __pageData,p as default};
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diff --git a/assets/save_reading_outliers_4.md.b134b0ec.js b/assets/save_reading_outliers_4.md.01bb2e6f.js
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-import{_ as e,o as t,c as a,d as n}from"./app.fb7dd446.js";const g=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":[{"level":2,"title":"Summary: Chapter Four: The Trouble with Geniuses, Part 2","slug":"summary-chapter-four-the-trouble-with-geniuses-part-2","link":"#summary-chapter-four-the-trouble-with-geniuses-part-2","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Four: The Trouble with Geniuses, Part 2","slug":"analysis-chapter-four-the-trouble-with-geniuses-part-2","link":"#analysis-chapter-four-the-trouble-with-geniuses-part-2","children":[]}],"relativePath":"save/reading/outliers/4.md","lastUpdated":1677742530000}'),i={name:"save/reading/outliers/4.md"},s=n('<h1 id="chapter-4-the-trouble-with-geniuses-part-2" tabindex="-1">Chapter 4: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#chapter-4-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h1><h2 id="summary-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Summary: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#summary-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.</p><ul><li><strong>Affluent</strong>: Having an abundant supply of money or possessions of value.</li></ul><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h2 id="analysis-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Analysis: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#analysis-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><p><strong>Family background plays a crucial role in an individual’s success.</strong> While a genius like Chris Langan should thrive on a college campus, he drops out. <strong>Langan doesn’t struggle with academics, but with paperwork, procedures, and environment</strong>. 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, <strong>his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success.</strong> The author delves deeper into the <strong>irony</strong> 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. <strong>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.</strong> <strong>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</strong> Without a family background that encourages success, intelligence is not enough.</p><div class="danger custom-block"><p class="custom-block-title">Contrast &amp; Comparison</p><p>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</p></div><p>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.</p><p>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&#39;s consideration of the Termites&#39; 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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 4 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section4/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section4/</a>.</li></ul></div>',23),o=[s];function r(l,h,c,d,u,p){return t(),a("div",null,o)}const f=e(i,[["render",r]]);export{g as __pageData,f as default};
+import{_ as e,o as t,c as a,d as n}from"./app.0679696f.js";const g=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":[{"level":2,"title":"Summary: Chapter Four: The Trouble with Geniuses, Part 2","slug":"summary-chapter-four-the-trouble-with-geniuses-part-2","link":"#summary-chapter-four-the-trouble-with-geniuses-part-2","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Four: The Trouble with Geniuses, Part 2","slug":"analysis-chapter-four-the-trouble-with-geniuses-part-2","link":"#analysis-chapter-four-the-trouble-with-geniuses-part-2","children":[]}],"relativePath":"save/reading/outliers/4.md","lastUpdated":1677768301000}'),i={name:"save/reading/outliers/4.md"},s=n('<h1 id="chapter-4-the-trouble-with-geniuses-part-2" tabindex="-1">Chapter 4: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#chapter-4-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h1><h2 id="summary-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Summary: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#summary-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.</p><ul><li><strong>Affluent</strong>: Having an abundant supply of money or possessions of value.</li></ul><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h2 id="analysis-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Analysis: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#analysis-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><p><strong>Family background plays a crucial role in an individual’s success.</strong> While a genius like Chris Langan should thrive on a college campus, he drops out. <strong>Langan doesn’t struggle with academics, but with paperwork, procedures, and environment</strong>. 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, <strong>his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success.</strong> The author delves deeper into the <strong>irony</strong> 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. <strong>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.</strong> <strong>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</strong> Without a family background that encourages success, intelligence is not enough.</p><div class="danger custom-block"><p class="custom-block-title">Contrast &amp; Comparison</p><p>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</p></div><p>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.</p><p>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&#39;s consideration of the Termites&#39; 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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 4 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section4/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section4/</a>.</li></ul></div>',23),o=[s];function r(l,h,c,d,u,p){return t(),a("div",null,o)}const f=e(i,[["render",r]]);export{g as __pageData,f as default};
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+import{_ as e,o as t,c as a,d as n}from"./app.0679696f.js";const g=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":[{"level":2,"title":"Summary: Chapter Four: The Trouble with Geniuses, Part 2","slug":"summary-chapter-four-the-trouble-with-geniuses-part-2","link":"#summary-chapter-four-the-trouble-with-geniuses-part-2","children":[{"level":3,"title":"Section 1.","slug":"section-1","link":"#section-1","children":[]},{"level":3,"title":"Section 2.","slug":"section-2","link":"#section-2","children":[]},{"level":3,"title":"Section 3.","slug":"section-3","link":"#section-3","children":[]},{"level":3,"title":"Section 4.","slug":"section-4","link":"#section-4","children":[]},{"level":3,"title":"Section 5.","slug":"section-5","link":"#section-5","children":[]},{"level":3,"title":"Section 6.","slug":"section-6","link":"#section-6","children":[]}]},{"level":2,"title":"Analysis: Chapter Four: The Trouble with Geniuses, Part 2","slug":"analysis-chapter-four-the-trouble-with-geniuses-part-2","link":"#analysis-chapter-four-the-trouble-with-geniuses-part-2","children":[]}],"relativePath":"save/reading/outliers/4.md","lastUpdated":1677768301000}'),i={name:"save/reading/outliers/4.md"},s=n("",23),o=[s];function r(l,h,c,d,u,p){return t(),a("div",null,o)}const f=e(i,[["render",r]]);export{g as __pageData,f as default};
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class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section 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class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _getting-started" data-v-c5936a1e><div><h1 id="git-push-results-in-authentication-failed" tabindex="-1">Git push results in &quot;Authentication Failed” <a class="header-anchor" href="#git-push-results-in-authentication-failed" aria-hidden="true">#</a></h1><h2 id="issue" tabindex="-1">Issue <a class="header-anchor" href="#issue" aria-hidden="true">#</a></h2><div class="language-bash line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">bash</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">git</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">push</span></span>
 <span class="line"><span style="color:#FFCB6B;">Username</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">for</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://github.com</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">${</span><span style="color:#A6ACCD;">{ GITHUB_USERNAME </span><span style="color:#89DDFF;">}</span><span style="color:#C3E88D;">}</span></span>
 <span class="line"><span style="color:#FFCB6B;">Password</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">for</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">https://andatoshiki@github.com</span><span style="color:#89DDFF;">&#39;</span><span style="color:#C3E88D;">:</span><span style="color:#A6ACCD;"> </span><span style="color:#89DDFF;">${</span><span style="color:#A6ACCD;">{ GITHUB_PASSWORD </span><span style="color:#89DDFF;">}</span><span style="color:#C3E88D;">}</span><span style="color:#A6ACCD;"> </span><span style="color:#676E95;font-style:italic;"># or GitHub personal access token</span></span>
 <span class="line"><span style="color:#FFCB6B;">remote:</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">Invalid</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">username</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">or</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">password.</span></span>
@@ -44,10 +44,10 @@
 <span class="line"><span style="color:#FFCB6B;">$</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">git</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">remote</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">set-url</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">origin</span><span style="color:#A6ACCD;"> </span><span style="color:#C3E88D;">git@github.com:</span><span style="color:#A6ACCD;">$USERNAME</span><span style="color:#C3E88D;">/</span><span style="color:#A6ACCD;">$REPONAME</span><span style="color:#C3E88D;">.git</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">git</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">remote</span><span style="color:#ABB2BF;"> </span><span style="color:#D19A66;">-v</span></span>
 <span class="line"><span style="color:#ABB2BF;">$ </span><span style="color:#98C379;">git</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">remote</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">set-url</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">origin</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">git@github.com:</span><span style="color:#E06C75;">$USERNAME</span><span style="color:#98C379;">/</span><span style="color:#E06C75;">$REPONAME</span><span style="color:#98C379;">.git</span></span>
-<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br></div></div><h3 id="solution-2" tabindex="-1">Solution 2 <a class="header-anchor" href="#solution-2" aria-hidden="true">#</a></h3><p>GitHub username &amp; password/access token mismatched due to prior temporary Git config setup → Reset Git global username &amp; email config or in current repository</p><p><strong>To set your global username/email configuration:</strong></p><ol><li>Open the command line.</li><li>Set your username: <code>git config --global user.name &quot;FIRST_NAME LAST_NAME&quot;</code></li><li>Set your email address: <code>git config --global user.email &quot;MY_NAME@example.com&quot;</code></li></ol><p><strong>To set repository-specific username/email configuration:</strong></p><ol><li>From the command line, change into the repository directory.</li><li>Set your username: <code>git config user.name &quot;FIRST_NAME LAST_NAME&quot;</code></li><li>Set your email address: <code>git config user.email &quot;MY_NAME@example.com&quot;</code></li><li>Verify your configuration by displaying your configuration file: <code>cat .git/config</code></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/getting-started.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br><span class="line-number">2</span><br></div></div><h3 id="solution-2" tabindex="-1">Solution 2 <a class="header-anchor" href="#solution-2" aria-hidden="true">#</a></h3><p>GitHub username &amp; password/access token mismatched due to prior temporary Git config setup → Reset Git global username &amp; email config or in current repository</p><p><strong>To set your global username/email configuration:</strong></p><ol><li>Open the command line.</li><li>Set your username: <code>git config --global user.name &quot;FIRST_NAME LAST_NAME&quot;</code></li><li>Set your email address: <code>git config --global user.email &quot;MY_NAME@example.com&quot;</code></li></ol><p><strong>To set repository-specific username/email configuration:</strong></p><ol><li>From the command line, change into the repository directory.</li><li>Set your username: <code>git config user.name &quot;FIRST_NAME LAST_NAME&quot;</code></li><li>Set your email address: <code>git config user.email &quot;MY_NAME@example.com&quot;</code></li><li>Verify your configuration by displaying your configuration file: <code>cat .git/config</code></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/getting-started.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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However, this time is never enough to learn everything. That's why recording knowledge for review is essential.</p><!----></article><!--]--><!----></span></div><div class="grid-3 item" data-v-6816157f><span class="VPLink VPFeature" data-v-6816157f data-v-1ef28411 data-v-30c06bd3><!--[--><article class="box" data-v-1ef28411><div class="icon" data-v-1ef28411>☕</div><h2 class="title" data-v-1ef28411>Carpe Diem</h2><p class="details" data-v-1ef28411>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.</p><!----></article><!--]--><!----></span></div><div class="grid-3 item" data-v-6816157f><span class="VPLink VPFeature" data-v-6816157f data-v-1ef28411 data-v-30c06bd3><!--[--><article class="box" data-v-1ef28411><div class="icon" data-v-1ef28411>💡</div><h2 class="title" data-v-1ef28411>Epiphania</h2><p class="details" data-v-1ef28411>An epiphany, derived from the Latin word "epiphania," is a moment of sudden and brilliant realization or insight. 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1 12 4.92zm3.094 13.627a9.119 9.119 0 0 1-3.103.549 8.972 8.972 0 0 1-3.076-.55 8.493 8.493 0 0 1-5.486-7.987v-.857c4.646.017 8.074 3.823 8.074 8.51v.198h.926v-.197c0-4.688 3.445-8.51 8.056-8.51.026.29.043.582.086.856a8.502 8.502 0 0 1-5.477 7.988z"/></svg></a><!--]--></div></div></div></div><!----></article></div><div class="item" data-v-04685dce><article class="VPTeamMembersItem small" data-v-04685dce data-v-1739ab5f><div class="profile" data-v-1739ab5f><figure class="avatar" data-v-1739ab5f><img class="avatar-img" src="https://avatars.githubusercontent.com/u/5837277?v=4" alt="Knut Sveidqvist" data-v-1739ab5f></figure><div class="data" data-v-1739ab5f><h1 class="name" data-v-1739ab5f>Knut Sveidqvist</h1><p class="affiliation" data-v-1739ab5f><span class="title" data-v-1739ab5f>Creator</span><!----><!----></p><!----><div class="links" data-v-1739ab5f><div class="VPSocialLinks" data-v-1739ab5f data-v-f6988cfb><!--[--><a class="VPSocialLink" href="https://github.com/knsv" 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data-v-b07783ac><!--[--><!--]--><div class="VPHero has-image VPHomeHero" data-v-b07783ac data-v-45916f1d><div class="container" data-v-45916f1d><div class="main" data-v-45916f1d><h1 class="name" data-v-45916f1d><span class="clip" data-v-45916f1d>Toshiki&#39;s Notebook</span></h1><p class="text" data-v-45916f1d>Research &amp; Produce</p><p class="tagline" data-v-45916f1d>👨‍💻 Eternal &amp; digital knowledge base for content craetion and notes management.</p><div class="actions" data-v-45916f1d><!--[--><div class="action" data-v-45916f1d><a class="VPButton medium brand" href="/getting-started" data-v-45916f1d data-v-a7c4128c>👉 Start Reading</a></div><div class="action" data-v-45916f1d><a class="VPButton medium alt" href="https://github.com/andatohsiki/toshiki-notebook" target="_blank" rel="noreferrer" data-v-45916f1d data-v-a7c4128c>🗒️ View on GitHub</a></div><!--]--></div></div><div class="image" data-v-45916f1d><div class="image-container" data-v-45916f1d><div class="image-bg" 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data-v-c5936a1e><div style="position:relative;" class="vp-doc _javascript_notes_1_1-1" data-v-c5936a1e><div><h1 id="_1-1-values" tabindex="-1">1-1: Values <a class="header-anchor" href="#_1-1-values" aria-hidden="true">#</a></h1><h2 id="_1-1-1-numbers" tabindex="-1">1-1-1: Numbers <span class="VPBadge danger" data-v-350d3852><!--[-->must know<!--]--></span> <a class="header-anchor" href="#_1-1-1-numbers" aria-hidden="true">#</a></h2><p>Values of the <em>number</em> type are numeric values, normally written as follows,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#F78C6C;">144</span></span>
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id="_1-1-1-numbers" tabindex="-1">1-1-1: Numbers <span class="VPBadge danger" data-v-350d3852><!--[-->must know<!--]--></span> <a class="header-anchor" href="#_1-1-1-numbers" aria-hidden="true">#</a></h2><p>Values of the <em>number</em> type are numeric values, normally written as follows,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#F78C6C;">144</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#D19A66;">144</span></span>
 <span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><ul><li><p>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,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#F78C6C;">0100000001100010000000000000000000000000000000000000000000000000</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#D19A66;">0100000001100010000000000000000000000000000000000000000000000000</span></span>
@@ -66,10 +66,10 @@
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#E5C07B;">A</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">newline</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">character</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">is</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">written</span><span style="color:#ABB2BF;"> </span><span style="color:#E06C75;">like</span><span style="color:#ABB2BF;"> </span><span style="color:#98C379;">&quot;</span><span style="color:#56B6C2;">\n</span><span style="color:#98C379;">&quot;</span><span style="color:#ABB2BF;">.</span></span>
 <span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>the backslash character (<code>\</code>) is used as an escape character to indicate that the following character(s) should be treated specially. In this case, the <code>&quot;\n&quot;</code> sequence is an escape sequence that represents a newline character. The backslash before the <code>n</code> character tells the JavaScript interpreter that it should treat the <code>n</code> as a special character and not just as the letter &quot;n&quot;.</p><p>However, if we want to include an <strong>actual backslash character</strong> (<code>\</code>), we will have to <strong>escape it</strong> by using to two backslashes <code>\\</code>, or else with one single slash the JavaScript interpreter will still determines it as an special operator, Thus <code>\\</code> will print <code>\</code> on output.</p><ul><li><p><strong>But what about the quotation marks used within the quotation marks that wraps the string? Wouldn&#39;t JavaScript interpreter throw error?</strong></p><ul><li><p>In the given JavaScript string, the inner set of double quotation marks (<code>&quot;\\n&quot;</code>) is escaped using a backslash (<code>\</code>) character. This tells the JavaScript interpreter to treat the inner double quotation marks as a regular character instead of a string delimiter.</p><p>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,</p><div class="language-js line-numbers-mode"><button title="Copy Code" class="copy"></button><span class="lang">js</span><pre class="shiki material-theme-palenight vp-code-dark" tabindex="0"><code><span class="line"><span style="color:#89DDFF;">&#39;</span><span style="color:#A6ACCD;">\n</span><span style="color:#89DDFF;">&#39;</span></span>
 <span class="line"></span></code></pre><pre class="shiki one-dark-pro vp-code-light" tabindex="0"><code><span class="line"><span style="color:#98C379;">&#39;</span><span style="color:#56B6C2;">\n</span><span style="color:#98C379;">&#39;</span></span>
-<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>The outer set of double quotation marks in the original string delimit the entire string and will be displayed as regular quotation marks.</p></li></ul></li></ul><blockquote><p><strong>Source</strong>: ChatGPT CA (Code Analysis)</p></blockquote></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/javascript/notes/1/1-1.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
+<span class="line"></span></code></pre><div class="line-numbers-wrapper" aria-hidden="true"><span class="line-number">1</span><br></div></div><p>The outer set of double quotation marks in the original string delimit the entire string and will be displayed as regular quotation marks.</p></li></ul></li></ul><blockquote><p><strong>Source</strong>: ChatGPT CA (Code Analysis)</p></blockquote></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/javascript/notes/1/1-1.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><!----></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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platform section using SVGs of different online academies from simple icons.</p><ul><li><p>Descriptions</p></li><li><p>Icon in SVG format</p></li><li><p>Platform name</p></li><li><p>Linkable SVG icon/button</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Khan academy</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> GitHub</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Libretext</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> OpenStax</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> BCcampus</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikibooks</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikipedia</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Bookboon</li><li>[ ]</li></ul></li></ul></li></ul><h2 id="footer" tabindex="-1">Footer <a class="header-anchor" href="#footer" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Footer copyright section new-lining and other additional information designs <ul><li>Refer to footer of <a href="https://status.toshiki.dev" target="_blank" rel="noreferrer">https://status.toshiki.dev</a></li></ul></li></ul><h2 id="configs" tabindex="-1">Configs <a class="header-anchor" href="#configs" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Header configs including different properties and meta configs.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete navigation configuration. <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Navigation divider and text title, line breakings. <ul><li>Refer to: <a href="https://github.com/maomao1996/mm-notes/blob/master/docs/.vitepress/configs/nav.ts" target="_blank" rel="noreferrer">mm-notes: <code>nav.ts</code></a>.</li></ul></li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete sidebar configs for different directories.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Vitepress sidebar generation plugin for huge bundled clips article directory.</li></ul><h1 id="articles" tabindex="-1">Articles <a class="header-anchor" href="#articles" aria-hidden="true">#</a></h1><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <code>&lt;head&gt;</code> tag documentation or guide by <a href="https://github.com/joshbuchea" target="_blank" rel="noreferrer">joshbuchea</a>/<a href="https://github.com/joshbuchea/HEAD" target="_blank" rel="noreferrer">HEAD</a>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Write index page for every directory, <strong>no Chat GPT allowed</strong>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Chemistry review notes.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> JavaScript 0-100 notes per chapter. <ul><li>Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)</li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Outliers novel clips</li><li class="task-list-item"><input class="task-list-item-checkbox" 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class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section 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class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _roadmap" data-v-c5936a1e><div><h1 id="theme" tabindex="-1">Theme <a class="header-anchor" href="#theme" aria-hidden="true">#</a></h1><h2 id="homepage" tabindex="-1">Homepage <a class="header-anchor" href="#homepage" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><p><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Finish three hero action blocks with keywords and its corresponded descriptions.</p></li><li class="task-list-item"><p><input class="task-list-item-checkbox" disabled="" type="checkbox"> Finish my learning platform section using SVGs of different online academies from simple icons.</p><ul><li><p>Descriptions</p></li><li><p>Icon in SVG format</p></li><li><p>Platform name</p></li><li><p>Linkable SVG icon/button</p><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" checked="" disabled="" type="checkbox"> Khan academy</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> GitHub</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Libretext</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> OpenStax</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> BCcampus</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikibooks</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Wikipedia</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Bookboon</li><li>[ ]</li></ul></li></ul></li></ul><h2 id="footer" tabindex="-1">Footer <a class="header-anchor" href="#footer" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Footer copyright section new-lining and other additional information designs <ul><li>Refer to footer of <a href="https://status.toshiki.dev" target="_blank" rel="noreferrer">https://status.toshiki.dev</a></li></ul></li></ul><h2 id="configs" tabindex="-1">Configs <a class="header-anchor" href="#configs" aria-hidden="true">#</a></h2><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Header configs including different properties and meta configs.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete navigation configuration. <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Navigation divider and text title, line breakings. <ul><li>Refer to: <a href="https://github.com/maomao1996/mm-notes/blob/master/docs/.vitepress/configs/nav.ts" target="_blank" rel="noreferrer">mm-notes: <code>nav.ts</code></a>.</li></ul></li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Complete sidebar configs for different directories.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Vitepress sidebar generation plugin for huge bundled clips article directory.</li></ul><h1 id="articles" tabindex="-1">Articles <a class="header-anchor" href="#articles" aria-hidden="true">#</a></h1><ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> <code>&lt;head&gt;</code> tag documentation or guide by <a href="https://github.com/joshbuchea" target="_blank" rel="noreferrer">joshbuchea</a>/<a href="https://github.com/joshbuchea/HEAD" target="_blank" rel="noreferrer">HEAD</a>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Write index page for every directory, <strong>no Chat GPT allowed</strong>.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Chemistry review notes.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> JavaScript 0-100 notes per chapter. <ul><li>Marijn Haverbeke - Eloquent JavaScript_ A Modern Introduction to Programming (2011, No Starch Press)</li></ul></li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Outliers novel clips</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Migrate clipped articles and archives from Notion to Vitepress.</li><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Fresh copy of &quot;How To Ask Questions The Smart Way&quot; by Eric Steven Raymond <ul class="contains-task-list"><li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"> Reading notes</li><li><a href="http://www.catb.org/~esr/faqs/smart-questions.html" target="_blank" rel="noreferrer">http://www.catb.org/~esr/faqs/smart-questions.html</a></li></ul></li></ul></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/roadmap.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path 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fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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collapsible collapsed" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>Outliers</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Introduction & Chapter 1: The Roseto Mystery</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 2: The 10,000-Hour Rule</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a 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data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e 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tabindex="-1">Introduction &amp; Chapter 1: The Roseto Mystery <a class="header-anchor" href="#introduction-chapter-1-the-roseto-mystery" aria-hidden="true">#</a></h1><h2 id="summary-introduction-the-roseto-mystery" tabindex="-1">Summary: Introduction: The Roseto Mystery <a class="header-anchor" href="#summary-introduction-the-roseto-mystery" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>outlier</em>.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>Outliers</em> to do for our understanding of success what Wolf did for our understanding of health.</p><h2 id="summary-chapter-one-the-matthew-effect" tabindex="-1">Summary: Chapter One: The Matthew Effect <a class="header-anchor" href="#summary-chapter-one-the-matthew-effect" aria-hidden="true">#</a></h2><h3 id="section-1-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2-1" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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:</p><p>“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.”</p><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h3 id="analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" tabindex="-1">Analysis: Introduction: The Roseto Mystery &amp; Chapter 1: The Matthew Effect <a class="header-anchor" href="#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" aria-hidden="true">#</a></h3><p>Gladwell&#39;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.</p><p>Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that &quot;success leads to more success.&quot; 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&#39;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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Introduction &amp; Chapter 1 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section1/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section1/</a>.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/1.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><!----></div><div class="pager" data-v-e033cd21><a class="pager-link next" href="/save/reading/outliers/2" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Chapter 2: The 10,000-Hour Rule</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-b05232f3>Introduction & Chapter 1: The Roseto Mystery</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 2: The 10,000-Hour Rule</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" 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aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _save_reading_outliers_1" data-v-c5936a1e><div><h1 id="introduction-chapter-1-the-roseto-mystery" tabindex="-1">Introduction &amp; Chapter 1: The Roseto Mystery <a class="header-anchor" href="#introduction-chapter-1-the-roseto-mystery" aria-hidden="true">#</a></h1><h2 id="summary-introduction-the-roseto-mystery" tabindex="-1">Summary: Introduction: The Roseto Mystery <a class="header-anchor" href="#summary-introduction-the-roseto-mystery" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>outlier</em>.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><p>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 <em>Outliers</em> to do for our understanding of success what Wolf did for our understanding of health.</p><h2 id="summary-chapter-one-the-matthew-effect" tabindex="-1">Summary: Chapter One: The Matthew Effect <a class="header-anchor" href="#summary-chapter-one-the-matthew-effect" aria-hidden="true">#</a></h2><h3 id="section-1-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2-1" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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:</p><p>“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.”</p><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h3 id="analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" tabindex="-1">Analysis: Introduction: The Roseto Mystery &amp; Chapter 1: The Matthew Effect <a class="header-anchor" href="#analysis-introduction-the-roseto-mystery-chapter-1-the-matthew-effect" aria-hidden="true">#</a></h3><p>Gladwell&#39;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.</p><p>Advantages result in more advantages. Gladwell reinforces this idea with the “Matthew Effect,” which states that &quot;success leads to more success.&quot; 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&#39;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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Introduction &amp; Chapter 1 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section1/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section1/</a>.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/1.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><!----></div><div class="pager" data-v-e033cd21><a class="pager-link next" href="/save/reading/outliers/2" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Chapter 2: The 10,000-Hour Rule</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e 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10,000-Hour Rule <a class="header-anchor" href="#chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h1><h2 id="summary-chapter-two-the-10-000-hour-rule" tabindex="-1">Summary: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#summary-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><p>Bill Joy and Bill Gates were also born at just the right time. In 1975, <em>Popular Electronics</em> 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.</p><h2 id="analysis-chapter-two-the-10-000-hour-rule" tabindex="-1">Analysis: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#analysis-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><p>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&#39;s inborn abilities.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 2 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section2/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section2/</a>. 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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>Outliers</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Introduction & Chapter 1: The Roseto Mystery</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 2: The 10,000-Hour Rule</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a 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data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _save_reading_outliers_2" data-v-c5936a1e><div><h1 id="chapter-two-the-10-000-hour-rule" tabindex="-1">Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h1><h2 id="summary-chapter-two-the-10-000-hour-rule" tabindex="-1">Summary: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#summary-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><p>Bill Joy and Bill Gates were also born at just the right time. In 1975, <em>Popular Electronics</em> 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.</p><h2 id="analysis-chapter-two-the-10-000-hour-rule" tabindex="-1">Analysis: Chapter Two: The 10,000-Hour Rule <a class="header-anchor" href="#analysis-chapter-two-the-10-000-hour-rule" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><p>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&#39;s inborn abilities.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 2 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section2/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section2/</a>. Accessed 27 Feb. 2023.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/2.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/save/reading/outliers/1" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Introduction & Chapter 1: The Roseto Mystery</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/save/reading/outliers/3" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Chapter 3: The Trouble with Geniuses, Part 1</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e 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3: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#chapter-3-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h1><h2 id="summary-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Summary: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#summary-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>Chris Langan appeared on the television trivia show <em>1 vs. 100</em> 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 <em>Principia Mathematica</em> as a teenager. On <em>1 vs. 100</em>, Langan cashed out with winnings of $250,000.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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 <em>enough</em>.</p><p>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 <em>enough</em> to win a Nobel.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><p>Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.</p><h2 id="analysis-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Analysis: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#analysis-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 3 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section3/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section3/</a>. Accessed 27 Feb. 2023.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/save/reading/outliers/2" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Chapter 2: The 10,000-Hour Rule</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/save/reading/outliers/4" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Chapter 4: The Trouble with Geniuses, Part 2</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>Outliers</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Introduction & Chapter 1: The Roseto Mystery</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 2: The 10,000-Hour Rule</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _save_reading_outliers_3" data-v-c5936a1e><div><h1 id="chapter-3-the-trouble-with-geniuses-part-1" tabindex="-1">Chapter 3: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#chapter-3-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h1><h2 id="summary-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Summary: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#summary-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>Chris Langan appeared on the television trivia show <em>1 vs. 100</em> 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 <em>Principia Mathematica</em> as a teenager. On <em>1 vs. 100</em>, Langan cashed out with winnings of $250,000.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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 <em>enough</em>.</p><p>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 <em>enough</em> to win a Nobel.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><p>Gladwell states that to understand Chris Langan’s prospects of becoming a “true outlier,” one must learn more about him.</p><h2 id="analysis-chapter-three-the-trouble-with-geniuses-part-1" tabindex="-1">Analysis: Chapter Three: The Trouble with Geniuses, Part 1 <a class="header-anchor" href="#analysis-chapter-three-the-trouble-with-geniuses-part-1" aria-hidden="true">#</a></h2><p>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&#39;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&#39;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.</p><p>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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 3 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section3/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section3/</a>. Accessed 27 Feb. 2023.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/3.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/save/reading/outliers/2" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Chapter 2: The 10,000-Hour Rule</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/save/reading/outliers/4" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>Chapter 4: The Trouble with Geniuses, Part 2</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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4: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#chapter-4-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h1><h2 id="summary-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Summary: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#summary-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.</p><ul><li><strong>Affluent</strong>: Having an abundant supply of money or possessions of value.</li></ul><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h2 id="analysis-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Analysis: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#analysis-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><p><strong>Family background plays a crucial role in an individual’s success.</strong> While a genius like Chris Langan should thrive on a college campus, he drops out. <strong>Langan doesn’t struggle with academics, but with paperwork, procedures, and environment</strong>. 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, <strong>his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success.</strong> The author delves deeper into the <strong>irony</strong> 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. <strong>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.</strong> <strong>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</strong> Without a family background that encourages success, intelligence is not enough.</p><div class="danger custom-block"><p class="custom-block-title">Contrast &amp; Comparison</p><p>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</p></div><p>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.</p><p>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&#39;s consideration of the Termites&#39; 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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 4 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section4/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section4/</a>.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/4.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T07:35:30.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/save/reading/outliers/3" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Chapter 3: The Trouble with Geniuses, Part 1</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>Outliers</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Introduction & Chapter 1: The Roseto Mystery</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 2: The 10,000-Hour Rule</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 3: The Trouble with Geniuses, Part 1</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/save/reading/outliers/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>Chapter 4: The Trouble with Geniuses, Part 2</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _save_reading_outliers_4" data-v-c5936a1e><div><h1 id="chapter-4-the-trouble-with-geniuses-part-2" tabindex="-1">Chapter 4: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#chapter-4-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h1><h2 id="summary-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Summary: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#summary-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><h3 id="section-1" tabindex="-1">Section 1. <a class="header-anchor" href="#section-1" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-2" tabindex="-1">Section 2. <a class="header-anchor" href="#section-2" aria-hidden="true">#</a></h3><p>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.</p><h3 id="section-3" tabindex="-1">Section 3. <a class="header-anchor" href="#section-3" aria-hidden="true">#</a></h3><p>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.</p><p>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.</p><h3 id="section-4" tabindex="-1">Section 4. <a class="header-anchor" href="#section-4" aria-hidden="true">#</a></h3><p>Oppenheimer was raised by affluent parents who sent him to a progressive private school and took an active interest in his hobbies.</p><ul><li><strong>Affluent</strong>: Having an abundant supply of money or possessions of value.</li></ul><p>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.</p><h3 id="section-5" tabindex="-1">Section 5. <a class="header-anchor" href="#section-5" aria-hidden="true">#</a></h3><p>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.”</p><h3 id="section-6" tabindex="-1">Section 6. <a class="header-anchor" href="#section-6" aria-hidden="true">#</a></h3><p>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.</p><h2 id="analysis-chapter-four-the-trouble-with-geniuses-part-2" tabindex="-1">Analysis: Chapter Four: The Trouble with Geniuses, Part 2 <a class="header-anchor" href="#analysis-chapter-four-the-trouble-with-geniuses-part-2" aria-hidden="true">#</a></h2><p><strong>Family background plays a crucial role in an individual’s success.</strong> While a genius like Chris Langan should thrive on a college campus, he drops out. <strong>Langan doesn’t struggle with academics, but with paperwork, procedures, and environment</strong>. 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, <strong>his story also perfectly proves Gladwell’s point that brainpower alone does not lead to success.</strong> The author delves deeper into the <strong>irony</strong> 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. <strong>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.</strong> <strong>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</strong> Without a family background that encourages success, intelligence is not enough.</p><div class="danger custom-block"><p class="custom-block-title">Contrast &amp; Comparison</p><p>Oppenheimer’s family background gave him the confidence to deal with authority while Langan’s upbringing created a distrust of authority.</p></div><p>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.</p><p>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&#39;s consideration of the Termites&#39; 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.</p><div class="info custom-block"><p class="custom-block-title">Reference</p><ul><li>“Outliers Chapter 4 Summary &amp; Analysis.” SparkNotes, <a href="http://www.sparknotes.com/lit/outliers/section4/" target="_blank" rel="noreferrer">www.sparknotes.com/lit/outliers/section4/</a>.</li></ul></div></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/save/reading/outliers/4.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-02T14:45:01.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/save/reading/outliers/3" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>Chapter 3: The Trouble with Geniuses, Part 1</span></a></div><div class="has-prev pager" data-v-e033cd21><!----></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and ☕ by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code> <i class="fa fa-clock fa-xs fa-beat"></i></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a> 
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