English+GermanWikipedia - /page/entry.mu

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<title>Differentialform</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Differentialform</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der Begriff Differentialform (oft auch alternierende Differentialform genannt) geht auf den <a href="Mathematiker" title="Mathematiker">Mathematiker</a> <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Joseph Cartan</a> zurück. Differentialformen sind ein grundlegendes Konzept der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a>. Sie erlauben eine koordinatenunabhängige <a href="Integralrechnung" title="Integralrechnung">Integration</a> auf allgemeinen <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierten</a> <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbaren Mannigfaltigkeiten</a>.

<div class="mw-heading mw-heading2"><h2 id="Kontext">Kontext</h2></div>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">

<ul><li>eine <a href="Offene_Menge" title="Offene Menge">offene Teilmenge</a> des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></li>

<li>oder eine differenzierbare <a href="Untermannigfaltigkeit_des_%E2%84%9Dn" title="Untermannigfaltigkeit des ℝn">Untermannigfaltigkeit des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></a></li>

<li>oder eine <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeit</a>.</li></ul>

In jedem dieser Fälle gibt es

<ul><li>den Begriff der <a href="Differenzierbarkeit" title="Differenzierbarkeit">differenzierbaren</a> Funktion auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e6cb664175067eedff7aed897cc764fb937f537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U;}" loading="lazy"> der Raum der beliebig oft differenzierbaren Funktionen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> werde mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(U)}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4551e29a7341d79592ff93adc25a23fb2744b0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.265ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(U)}" loading="lazy"> bezeichnet;</li>

<li>den Begriff des <a href="Tangentialraum" title="Tangentialraum">Tangentialraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} _{p}U}">

<msub>

</mrow>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {T} _{p}U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1149857ffedc5f89bcc0246f8f061dfd342eb5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.52ex; height:2.843ex;" alt="{\displaystyle \mathrm {T} _{p}U}" loading="lazy"> an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in U;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in U;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fef8d17375e6f1ac9909a6d76a8d167bce5804b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.529ex; height:2.509ex;" alt="{\displaystyle p\in U;}" loading="lazy"></li>

<li>den Begriff der <a href="Richtungsableitung" title="Richtungsableitung">Richtungsableitung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial f}{\partial X}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial f}{\partial X}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08417c4a5288930334311eeca5ab12fa1fb84e80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.168ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\partial f}{\partial X}}}" loading="lazy"> für einen Tangentialvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in \mathrm {T} _{p}U}">

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</mrow>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\in \mathrm {T} _{p}U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad1ae151811b166777334f80b55ec82a9ad24e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.34ex; height:2.843ex;" alt="{\displaystyle X\in \mathrm {T} _{p}U}" loading="lazy"> und eine differenzierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f;}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a952ac5e1012390ade8a8ef01750c49e0c35f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f;}" loading="lazy"></li>

<li>den Begriff des differenzierbaren <a href="Vektorfeld" title="Vektorfeld">Vektorfeldes</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e6cb664175067eedff7aed897cc764fb937f537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U;}" loading="lazy"> der Raum der Vektorfelder auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> sei mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\mathrm {T} U)}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (\mathrm {T} U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57e44715ce92becc8e7febe164f5607074b3605a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.723ex; height:2.843ex;" alt="{\displaystyle \Gamma (\mathrm {T} U)}" loading="lazy"> bezeichnet.</li></ul>

Der <a href="Dualraum" title="Dualraum">Dualraum</a> des Tangentialraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} _{p}U}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {T} _{p}U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1149857ffedc5f89bcc0246f8f061dfd342eb5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.52ex; height:2.843ex;" alt="{\displaystyle \mathrm {T} _{p}U}" loading="lazy"> wird als Kotangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} _{p}^{*}U}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {T} _{p}^{*}U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/860cd1160e8b8e9fc5e6f4e3c3f9a5a65ba02b8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.52ex; height:3.176ex;" alt="{\displaystyle \mathrm {T} _{p}^{*}U}" loading="lazy"> bezeichnet.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Differentialform">Differentialform</h3></div>

Eine Differentialform vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> oder kurz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> ist ein glatter <a href="Schnitt_(Faserb%C3%BCndel)" title="Schnitt (Faserbündel)">Schnitt</a> in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-ten <a href="Gra%C3%9Fmann-Algebra#Äußere_Potenz" title="Graßmann-Algebra">äußeren Potenz</a> des <a href="Kotangentialb%C3%BCndel" class="mw-redirect" title="Kotangentialbündel">Kotangentialbündels</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. In symbolischer Schreibweise bedeutet dies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Gamma (\Lambda ^{k}(T^{*}U))}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Gamma (\Lambda ^{k}(T^{*}U))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f8cc00d68e988a12f917d0b8acb0805511dd04b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.616ex; height:3.176ex;" alt="{\displaystyle \omega \in \Gamma (\Lambda ^{k}(T^{*}U))}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}U}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T^{*}U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ca27329e88eafe267ecee5907b18c27aabf0668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.557ex; height:2.343ex;" alt="{\displaystyle T^{*}U}" loading="lazy"> das Kotangentialbündel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(T^{*}U)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(T^{*}U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ebe7c26e4a8406ca3c051a101be0acef1af78ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.068ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{k}(T^{*}U)}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-te äußere Potenz von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}U}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T^{*}U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ca27329e88eafe267ecee5907b18c27aabf0668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.557ex; height:2.343ex;" alt="{\displaystyle T^{*}U}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\Lambda ^{k}(T^{*}U))}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (\Lambda ^{k}(T^{*}U))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/844266f051fa926e39955fbdc5f50f1447b02dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.33ex; height:3.176ex;" alt="{\displaystyle \Gamma (\Lambda ^{k}(T^{*}U))}" loading="lazy"> somit die Menge der glatten Schnitte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(T^{*}U)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(T^{*}U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ebe7c26e4a8406ca3c051a101be0acef1af78ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.068ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{k}(T^{*}U)}" loading="lazy"> bezeichnet.

Dies bedeutet, dass jedem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66f04c97c84237ad610995f15df5cd7fd0034d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.882ex; height:2.509ex;" alt="{\displaystyle p\in U}" loading="lazy"> eine <a href="Multilinearform#Alternierende_Multilinearformen" title="Multilinearform">alternierende Multilinearform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03e5bac876eaf90d1ef45de136723aa1c2838b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.505ex; height:2.343ex;" alt="{\displaystyle \omega _{p}}" loading="lazy"> auf dem Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}U}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4ac5eea70db932d8db09938c6541c483a56b87c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.199ex; height:2.843ex;" alt="{\displaystyle T_{p}U}" loading="lazy"> zugeordnet wird, und zwar so, dass für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> glatte Vektorfelder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b6ecebf8cf8de4436ef9a6310a4937a5cef6d6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{k}}" loading="lazy"> die Funktion

<msub>

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</msub>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\mapsto \omega _{p}((X_{1})_{p},\ldots ,(X_{k})_{p})\in \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebed975e97647a455acfca0c81d52540b18c9ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:30.613ex; height:3.009ex;" alt="{\displaystyle p\mapsto \omega _{p}((X_{1})_{p},\ldots ,(X_{k})_{p})\in \mathbb {R} }" loading="lazy"></dd></dl>

<a href="Glatte_Funktion" title="Glatte Funktion">glatt</a>, also beliebig oft differenzierbar, ist.

Alternativ dazu kann man eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> als eine alternierende, glatte <a href="Multilinearform" title="Multilinearform">multilineare</a> Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \colon (\Gamma TU)^{k}\to C^{\infty }(U)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \colon (\Gamma TU)^{k}\to C^{\infty }(U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8dda95368d2d52939657ef52f383aee89e1da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.129ex; height:3.176ex;" alt="{\displaystyle \omega \colon (\Gamma TU)^{k}\to C^{\infty }(U)}" loading="lazy"> auffassen. Das bedeutet: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> ordnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> <a href="Vektorfeld" title="Vektorfeld">Vektorfeldern</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b6ecebf8cf8de4436ef9a6310a4937a5cef6d6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{k}}" loading="lazy"> eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (X_{1},\ldots ,X_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega (X_{1},\ldots ,X_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/745d867168f5bce419ad6eac837f612ff8d64ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.425ex; height:2.843ex;" alt="{\displaystyle \omega (X_{1},\ldots ,X_{k})}" loading="lazy"> zu, sodass

<msub>

</mrow>

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</mrow>

</msubsup>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega (X_{1},\ldots ,X'_{i}+X''_{i},\ldots ,X_{k})=\omega (X_{1},\ldots ,X'_{i},\ldots ,X_{k})+\omega (X_{1},\ldots ,X''_{i},\ldots ,X_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b755dd87965c072d13aabe35110c65e1a4d7d884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:79.304ex; height:3.009ex;" alt="{\displaystyle \omega (X_{1},\ldots ,X'_{i}+X''_{i},\ldots ,X_{k})=\omega (X_{1},\ldots ,X'_{i},\ldots ,X_{k})+\omega (X_{1},\ldots ,X''_{i},\ldots ,X_{k})}" loading="lazy"></li>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega (X_{1},\ldots ,f\cdot X_{i},\ldots ,X_{k})=f\cdot \omega (X_{1},\ldots ,X_{i},\ldots ,X_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2cdc44c2cc3becb53dba107e561c6eb1a4c397.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.668ex; height:2.843ex;" alt="{\displaystyle \omega (X_{1},\ldots ,f\cdot X_{i},\ldots ,X_{k})=f\cdot \omega (X_{1},\ldots ,X_{i},\ldots ,X_{k})}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{\infty }(U),1\leq i\leq k}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C^{\infty }(U),1\leq i\leq k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddadf2b89e3c465ff2b2cd4b964bcc0949bd1bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.791ex; height:2.843ex;" alt="{\displaystyle f\in C^{\infty }(U),1\leq i\leq k}" loading="lazy"></li></ul>

und

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega (X_{1},\ldots ,X_{i},\ldots ,X_{j},\ldots ,X_{k})=-\omega (X_{1},\ldots ,X_{j},\ldots ,X_{i},\ldots ,X_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daa184462f759dca8fe6cd82d740b6ee532cd849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:65.585ex; height:3.009ex;" alt="{\displaystyle \omega (X_{1},\ldots ,X_{i},\ldots ,X_{j},\ldots ,X_{k})=-\omega (X_{1},\ldots ,X_{j},\ldots ,X_{i},\ldots ,X_{k})}" loading="lazy"></li></ul>

gilt.

Alternative unter Rückgriff auf <a href="Tensorfeld" title="Tensorfeld">Tensorfelder</a>: Eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form ist ein alternierendes, kovariantes Tensorfeld der Stufe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Raum_der_Differentialformen">Raum der Differentialformen</h3></div>

Die Menge der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Formen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> bildet einen <a href="Vektorraum" title="Vektorraum">Vektorraum</a> und wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{k}(U)}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f853af9f78c1fdfb132da6497dfc8fd718b746b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.359ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(U)}" loading="lazy"> bezeichnet. Weiterhin setzt man

</mrow>

</mrow>

</munderover>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (U)=\bigoplus _{k=1}^{\infty }\Omega ^{k}(U).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bda0ffe0ea0a9db59d536293132ba5951bee418f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.272ex; height:6.843ex;" alt="{\displaystyle \Omega (U)=\bigoplus _{k=1}^{\infty }\Omega ^{k}(U).}" loading="lazy"></dd></dl>

Für endlichdimensionale Mannigfaltigkeiten ist diese Summe endlich, da für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>\dim U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>\dim U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47f7e7dbc7953e6d3a87791c2171ebd6609d2f94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.355ex; height:2.176ex;" alt="{\displaystyle k>\dim U}" loading="lazy"> der Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{k}(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f853af9f78c1fdfb132da6497dfc8fd718b746b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.359ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(U)}" loading="lazy"> der <a href="Nullvektorraum" title="Nullvektorraum">Nullvektorraum</a> ist. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (U)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c053a26918e0562501161dcc3bada8056a141f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.27ex; height:2.843ex;" alt="{\displaystyle \Omega (U)}" loading="lazy"> ist eine <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra</a> mit dem äußeren Produkt als Multiplikation und somit auch wieder ein Vektorraum. Aus <a href="Topologischer_Raum" title="Topologischer Raum">topologischer</a> Sicht ist dieser Raum auch eine <a href="Garbe_(Mathematik)" title="Garbe (Mathematik)">Garbe</a>.

Man kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03e5bac876eaf90d1ef45de136723aa1c2838b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.505ex; height:2.343ex;" alt="{\displaystyle \omega _{p}}" loading="lazy"> als Element der <a href="Gra%C3%9Fmann-Algebra#Äußere_Potenz" title="Graßmann-Algebra">äußeren Potenz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(T_{p}^{*}U)}">

<msup>

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</msup>

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</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(T_{p}^{*}U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c71daec840f93ab3a7d6b6ef2b4051b65d25b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.068ex; height:3.343ex;" alt="{\displaystyle \Lambda ^{k}(T_{p}^{*}U)}" loading="lazy"> auffassen; infolgedessen definiert das <a href="Gra%C3%9Fmann-Algebra#Äußeres_Produkt" title="Graßmann-Algebra">äußere Produkt</a> (d. h. das Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"> in der <a href="Gra%C3%9Fmann-Algebra#Äußere_Algebra" title="Graßmann-Algebra">äußeren Algebra</a>) Abbildungen

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U),\quad (\omega ,\eta )\mapsto \omega \wedge \eta ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/957f513a4773c4107ba7a8c16ad9ef24994fc0ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.598ex; height:3.176ex;" alt="{\displaystyle \Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U),\quad (\omega ,\eta )\mapsto \omega \wedge \eta ,}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \wedge \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \wedge \eta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60d8de1dff983c3e53066825841da1a3e53be0ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.198ex; height:2.509ex;" alt="{\displaystyle \omega \wedge \eta }" loading="lazy"> durch

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (\omega \wedge \eta )_{p}=\omega _{p}\wedge \eta _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a08ba0bd4a01a9ebde7044219539625bff27f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.467ex; height:3.009ex;" alt="{\displaystyle (\omega \wedge \eta )_{p}=\omega _{p}\wedge \eta _{p}}" loading="lazy"></dd></dl>

punktweise definiert ist.

Dieses Produkt ist graduiert-kommutativ, es gilt

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \wedge \eta =(-1)^{\deg \omega \cdot \deg \eta }\cdot \eta \wedge \omega ;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a34493111b9a597eff027f30468bc5872a43c4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.845ex; height:3.176ex;" alt="{\displaystyle \omega \wedge \eta =(-1)^{\deg \omega \cdot \deg \eta }\cdot \eta \wedge \omega ;}" loading="lazy"></dd></dl>

dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \deg \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bacbaa1688570d4dc6596153cb109908b8bd00d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.32ex; height:2.509ex;" alt="{\displaystyle \deg \omega }" loading="lazy"> den Grad von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b0d8eba2c8829fecf9414b15b1d02c24db3a553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.093ex; height:2.009ex;" alt="{\displaystyle \omega ,}" loading="lazy"> d. h.: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg \omega =k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \deg \omega =k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/495eff5fdce6a3a4f8a991a28a842623f473228f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.63ex; height:2.509ex;" alt="{\displaystyle \deg \omega =k}" loading="lazy">. Demnach ist das Produkt zweier Formen ungeraden Grades antikommutativ und in allen anderen Kombinationen kommutativ.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<ul><li><a href="Glatte_Funktion" title="Glatte Funktion">Glatte Funktionen</a> sind 0-Formen.</li>

<li><a href="Pfaffsche_Form" title="Pfaffsche Form">Pfaffsche Formen</a> sind 1-Formen.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Koordinatendarstellung">Koordinatendarstellung</h2></div>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale differenzierbare Mannigfaltigkeit. Weiter sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86bece9c0272e14e23c6c43e0164f292d329e19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.956ex; height:2.843ex;" alt="{\displaystyle (U,x)}" loading="lazy"> ein <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">lokales Koordinatensystem (eine Karte)</a>. Dann ist

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B=\{\mathrm {d} x_{i_{1}}|_{p}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}|_{p}\ {\big |}\ i_{1}<\ldots <i_{k}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac8bf2db948d6863f4676b219031abc43218c5b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.499ex; height:3.343ex;" alt="{\displaystyle B=\{\mathrm {d} x_{i_{1}}|_{p}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}|_{p}\ {\big |}\ i_{1}<\ldots <i_{k}\}}" loading="lazy"></dd></dl>

eine Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(T_{p}^{*}M).}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(T_{p}^{*}M).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/700d5dc67cf991614dc17dccd479beee318c575d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.374ex; height:3.343ex;" alt="{\displaystyle \Lambda ^{k}(T_{p}^{*}M).}" loading="lazy"> Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x_{i}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec2a183d6fe7cb959456b8fa4368a76262da0966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.422ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} x_{i}}" loading="lazy"> das <a href="Totales_Differential" title="Totales Differential">totale Differential</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-ten Koordinatenfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">. Das heißt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x_{i}|_{p}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x_{i}|_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/834195127e66e18a74f089932bcd4046e364c269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.128ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} x_{i}|_{p}}" loading="lazy"> ist diejenige Linearform auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}(M)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d56850194cc0defe2c7bb2fd8d940835a686bcc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.668ex; height:3.009ex;" alt="{\displaystyle T_{p}(M)}" loading="lazy">, die den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-ten Basisvektor der Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial }{\partial x_{1}}}|_{p},\ldots ,{\frac {\partial }{\partial x_{n}}}|_{p}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial }{\partial x_{1}}}|_{p},\ldots ,{\frac {\partial }{\partial x_{n}}}|_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9b125eb6277eb7b75f52552cd5fa48ccb27fe02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.803ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {\partial }{\partial x_{1}}}|_{p},\ldots ,{\frac {\partial }{\partial x_{n}}}|_{p}}" loading="lazy"> auf 1 und alle anderen auf 0 abbildet.

Jede Differentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega ^{k}(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega ^{k}(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd4450f2d016c637248e8f3e283dc7de6b88e34b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.305ex; height:3.176ex;" alt="{\displaystyle \omega \in \Omega ^{k}(M)}" loading="lazy"> hat auf jeder Karte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86bece9c0272e14e23c6c43e0164f292d329e19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.956ex; height:2.843ex;" alt="{\displaystyle (U,x)}" loading="lazy"> eine eindeutige Darstellung

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega |_{U}=\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,\mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b72dba0a1b6ac5931eb86bf322f327f7457a36f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.684ex; height:5.843ex;" alt="{\displaystyle \omega |_{U}=\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,\mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}" loading="lazy"></dd></dl>

mit geeigneten differenzierbaren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i_{1},\ldots ,i_{k}}.}">

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i_{1},\ldots ,i_{k}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16589def84f223c894c23a8c3fafdc547b8da02e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.775ex; height:2.343ex;" alt="{\displaystyle a_{i_{1},\ldots ,i_{k}}.}" loading="lazy">

Aus der Koordinatendarstellung ergibt sich, dass für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e81682bf174c978e9008ffb557ba4da2cf7478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k>n}" loading="lazy"> die Nullform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2bb95bb72b12e6b52d1e87c670c3f423041bccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.707ex; height:2.176ex;" alt="{\displaystyle \omega =0}" loading="lazy"> die einzige Differentialform ist.

<div class="mw-heading mw-heading2"><h2 id="Äußere_Ableitung">Äußere Ableitung</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="%C3%84u%C3%9Fere_Ableitung" title="Äußere Ableitung">Äußere Ableitung</a></div>

Die äußere Ableitung ist ein Operator, der einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Differentialform eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f9f13644a6be482d7ddb19a6e0c706564773085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}" loading="lazy">-Differentialform zuordnet. Betrachtet man sie auf der Menge der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">-Differentialformen, also auf der Menge der glatten Funktionen, so entspricht die äußere Ableitung der <a href="Differentialrechnung" title="Differentialrechnung">üblichen Ableitung</a> für Funktionen.

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

Die äußere Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \omega }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3147db820f16a7d610d03639f49d726fa1431420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.738ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \omega }" loading="lazy"> einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> wird induktiv mithilfe der <a href="Lie-Ableitung" title="Lie-Ableitung">Lie-Ableitung</a> und der <a href="Cartan-Formel" title="Cartan-Formel">Cartan-Formel</a>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X}=i_{X}\circ \mathrm {d} +\mathrm {d} \circ i_{X}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6343cfcde4c6bbe2c92844584d91f9b47116dabd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.019ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{X}=i_{X}\circ \mathrm {d} +\mathrm {d} \circ i_{X}}" loading="lazy"></dd></dl>

definiert; dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ein Vektorfeld, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce4db2d17b365a7321dbfdb8f8bc512dd911ea54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.236ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{X}}" loading="lazy"> die Lie-Ableitung und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{X}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb2f1b6a552361bf8b27aaa44d209c55fa807585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.435ex; height:2.509ex;" alt="{\displaystyle i_{X}}" loading="lazy"> die Einsetzung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

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<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Ist beispielsweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> eine 1-Form, so ist

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {L}}_{X}\omega )(Y)={\mathcal {L}}_{X}(\omega (Y))-\omega ({\mathcal {L}}_{X}(Y))=X\omega (Y)-\omega ([X,Y])}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e5d19a3a4697c1fbd0fd56cc72b21fee4d5bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.444ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {L}}_{X}\omega )(Y)={\mathcal {L}}_{X}(\omega (Y))-\omega ({\mathcal {L}}_{X}(Y))=X\omega (Y)-\omega ([X,Y])}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle ((\mathrm {d} \circ i_{X})\omega )(Y)=(\mathrm {d} (\omega (X)))(Y)=Y\omega (X),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb79f5b02a3c24d1e13cab0e1410ba362204d76e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.15ex; height:2.843ex;" alt="{\displaystyle ((\mathrm {d} \circ i_{X})\omega )(Y)=(\mathrm {d} (\omega (X)))(Y)=Y\omega (X),}" loading="lazy"></dd></dl>

also

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega (X,Y)=X\omega (Y)-Y\omega (X)-\omega ([X,Y])}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23489c86ec22a50b42c393036dcd838753c47fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.467ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} \omega (X,Y)=X\omega (Y)-Y\omega (X)-\omega ([X,Y])}" loading="lazy"></dd></dl>

für Vektorfelder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">

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<annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy">; dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,Y]}">

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<annotation encoding="application/x-tex">{\displaystyle [X,Y]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94470b44d283fde62130212956058ca6b727da37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle [X,Y]}" loading="lazy"> die <a href="Lie-Ableitung" title="Lie-Ableitung">Lie-Klammer</a>.

Die allgemeine Formel lautet

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\mathrm {d} \omega (X_{0},\ldots ,X_{k})&=&\sum _{i=0}^{k}(-1)^{i}X_{i}\omega (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})+\\[0.5em]&&+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})\,;\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ac03313f0f6cfc0c30d75776386d2018e25f274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:82.502ex; height:8.843ex;" alt="{\displaystyle {\begin{array}{rcl}\mathrm {d} \omega (X_{0},\ldots ,X_{k})&=&\sum _{i=0}^{k}(-1)^{i}X_{i}\omega (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})+\\[0.5em]&&+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})\,;\end{array}}}" loading="lazy"></dd></dl>

dabei bedeutet das Dach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97daf05d8be90237eb4d2e4a46d3733ca829bd1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:2.009ex;" alt="{\displaystyle {\hat {}}}" loading="lazy"> im Zeichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {X}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46da0c7536c39c4ad37652a75cc51fa214685e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:3.176ex;" alt="{\displaystyle {\hat {X}}_{i}}" loading="lazy">, dass das entsprechende Argument wegzulassen ist.

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3></div>

Die äußere Ableitung hat folgende Eigenschaften:

<ul><li>Die äußere Ableitung ist eine <a href="Antiderivation" class="mw-redirect" title="Antiderivation">Antiderivation</a>. Das heißt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a15022657616b297a2c995d1b314a3aa3442c0cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} }" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-linear, und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \Omega ^{k}(U),\beta \in \Omega ^{l}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in \Omega ^{k}(U),\beta \in \Omega ^{l}(U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed50e2b2b750cd0cff9e110ebacb320d425e1f37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.886ex; height:3.176ex;" alt="{\displaystyle \alpha \in \Omega ^{k}(U),\beta \in \Omega ^{l}(U)}" loading="lazy"> gilt die Leibnizregel</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (\alpha \wedge \beta )=\mathrm {d} \alpha \wedge \beta +(-1)^{k}\alpha \wedge \mathrm {d} \beta .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/099e0828c54ee5eb4cd474550443dec1aea5ac02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.348ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} (\alpha \wedge \beta )=\mathrm {d} \alpha \wedge \beta +(-1)^{k}\alpha \wedge \mathrm {d} \beta .}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f\in C^{\infty }(U),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f39c6bfcfc55c499a832f46d2b1e37bcb566cd79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.031ex; height:2.843ex;" alt="{\displaystyle f\in C^{\infty }(U),}" loading="lazy"> dann stimmt die äußere Ableitung mit dem <a href="Totales_Differential" title="Totales Differential">totalen Differential</a> überein.</li>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{2}=\mathrm {d} \circ \mathrm {d} =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60131c77b02035abf45e7703196b1784c09ab15a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.486ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} ^{2}=\mathrm {d} \circ \mathrm {d} =0}" loading="lazy"></li>

<li>Die äußere Ableitung respektiert Einschränkungen. Es sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset V\subset M}">

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<annotation encoding="application/x-tex">{\displaystyle U\subset V\subset M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74a933cfc7c1c434da75458caa2197bb2eb4e9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.209ex; height:2.176ex;" alt="{\displaystyle U\subset V\subset M}" loading="lazy"> offen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \Omega ^{k}(V).}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in \Omega ^{k}(V).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5208037a993d38c782ed89bf87576541323f3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.338ex; height:3.176ex;" alt="{\displaystyle \alpha \in \Omega ^{k}(V).}" loading="lazy"> Dann gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} (\alpha |_{U})=(\mathrm {d} \alpha )|_{U}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (\alpha |_{U})=(\mathrm {d} \alpha )|_{U}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae690e084e78b17b3460bf299a4d56a0232e596f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.203ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} (\alpha |_{U})=(\mathrm {d} \alpha )|_{U}.}" loading="lazy"> Man nennt die äußere Ableitung deshalb auch einen lokalen Operator.</li></ul>

Diese vier Eigenschaften charakterisieren die äußere Ableitung vollständig. Das heißt, man kann aus diesen Eigenschaften die obige Summenformel herleiten. Rechnet man mit der äußeren Ableitung, so bevorzugt man das Rechnen mit den Eigenschaften der Ableitung und vermeidet die obige Formel.

<div class="mw-heading mw-heading3"><h3 id="Koordinatendarstellung_der_äußeren_Ableitung">Koordinatendarstellung der äußeren Ableitung</h3></div>

Die äußere Ableitung einer Differentialform

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<annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}(x)\cdot \mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc7003ba834905da17fcb29dbd879c3c70fcc7ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.975ex; height:5.843ex;" alt="{\displaystyle \omega =\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}(x)\cdot \mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}" loading="lazy"></dd></dl>

in Koordinatendarstellung lautet

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega =\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}\mathrm {d} a_{i_{1},\ldots ,i_{k}}\wedge \mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49e8ba9c0cf3aa03c38ca102a4dc34537fba4696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.325ex; height:5.843ex;" alt="{\displaystyle \mathrm {d} \omega =\sum _{1\leq i_{1}<\ldots <i_{k}\leq n}\mathrm {d} a_{i_{1},\ldots ,i_{k}}\wedge \mathrm {d} x_{i_{1}}\wedge \ldots \wedge \mathrm {d} x_{i_{k}}}" loading="lazy"></dd></dl>

mit den totalen Differentialen der Koeffizientenfunktionen

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} a_{i_{1},\ldots ,i_{k}}=\sum _{j=1}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\mathrm {d} x_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3df6633830a2598d0a73aa567e5176e118b3444.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.076ex; height:7.176ex;" alt="{\displaystyle \mathrm {d} a_{i_{1},\ldots ,i_{k}}=\sum _{j=1}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\mathrm {d} x_{j}}" loading="lazy">.</dd></dl>

Um die dabei entstehenden Ausdrücke wieder durch die Standardbasis auszudrücken, sind die Identitäten

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x_{i}\wedge \mathrm {d} x_{j}=-\mathrm {d} x_{j}\wedge \mathrm {d} x_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d36d0fa0341f63e46125b96d3b1f602c510b1aec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.979ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} x_{i}\wedge \mathrm {d} x_{j}=-\mathrm {d} x_{j}\wedge \mathrm {d} x_{i}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x_{i}\wedge \mathrm {d} x_{i}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bc00fd5e527bb4483dda93cf11a7fb31cf0da49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.687ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} x_{i}\wedge \mathrm {d} x_{i}=0}" loading="lazy"></dd></dl>

wichtig.

<div class="mw-heading mw-heading3"><h3 id="Beispiel">Beispiel</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2,k=1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af153497a1679d2d7b57644734577ccef49e766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.162ex; height:2.509ex;" alt="{\displaystyle n=2,k=1}" loading="lazy"> gilt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} (a_{1}\cdot \mathrm {d} x_{1}+a_{2}\cdot \mathrm {d} x_{2})&=\mathrm {d} a_{1}\wedge \mathrm {d} x_{1}+\mathrm {d} a_{2}\wedge \mathrm {d} x_{2}\\[0.5em]&=\left({\frac {\partial a_{1}}{\partial x_{1}}}\mathrm {d} x_{1}+{\frac {\partial a_{1}}{\partial x_{2}}}\mathrm {d} x_{2}\right)\wedge \mathrm {d} x_{1}+\left({\frac {\partial a_{2}}{\partial x_{1}}}\mathrm {d} x_{1}+{\frac {\partial a_{2}}{\partial x_{2}}}\mathrm {d} x_{2}\right)\wedge \mathrm {d} x_{2}\\[0.5em]&={\frac {\partial a_{1}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{1}+{\frac {\partial a_{1}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{1}+{\frac {\partial a_{2}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}+{\frac {\partial a_{2}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{2}\\[0.5em]&=\left({\frac {\partial a_{2}}{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{2}}}\right)\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04d6cb8b377aad410081e447e865d9f732ae8c66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.65ex; margin-bottom: -0.188ex; width:98.203ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} (a_{1}\cdot \mathrm {d} x_{1}+a_{2}\cdot \mathrm {d} x_{2})&=\mathrm {d} a_{1}\wedge \mathrm {d} x_{1}+\mathrm {d} a_{2}\wedge \mathrm {d} x_{2}\\[0.5em]&=\left({\frac {\partial a_{1}}{\partial x_{1}}}\mathrm {d} x_{1}+{\frac {\partial a_{1}}{\partial x_{2}}}\mathrm {d} x_{2}\right)\wedge \mathrm {d} x_{1}+\left({\frac {\partial a_{2}}{\partial x_{1}}}\mathrm {d} x_{1}+{\frac {\partial a_{2}}{\partial x_{2}}}\mathrm {d} x_{2}\right)\wedge \mathrm {d} x_{2}\\[0.5em]&={\frac {\partial a_{1}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{1}+{\frac {\partial a_{1}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{1}+{\frac {\partial a_{2}}{\partial x_{1}}}\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}+{\frac {\partial a_{2}}{\partial x_{2}}}\cdot \mathrm {d} x_{2}\wedge \mathrm {d} x_{2}\\[0.5em]&=\left({\frac {\partial a_{2}}{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{2}}}\right)\cdot \mathrm {d} x_{1}\wedge \mathrm {d} x_{2}.\end{aligned}}}" loading="lazy"></dd></dl>

Allgemein gilt für die äußere Ableitung einer 1-Form

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \left(\sum _{i=1}^{n}a_{i}\cdot \mathrm {d} x_{i}\right)=\sum _{1\leq i<j\leq n}\left({\frac {\partial a_{j}}{\partial x_{i}}}-{\frac {\partial a_{i}}{\partial x_{j}}}\right)\cdot \mathrm {d} x_{i}\wedge \mathrm {d} x_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03d4a28f6d61d6c14c98046f2582c5b7bf9c2015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.772ex; height:7.676ex;" alt="{\displaystyle \mathrm {d} \left(\sum _{i=1}^{n}a_{i}\cdot \mathrm {d} x_{i}\right)=\sum _{1\leq i<j\leq n}\left({\frac {\partial a_{j}}{\partial x_{i}}}-{\frac {\partial a_{i}}{\partial x_{j}}}\right)\cdot \mathrm {d} x_{i}\wedge \mathrm {d} x_{j}}" loading="lazy"></dd></dl>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">

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<annotation encoding="application/x-tex">{\displaystyle n=3}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy"> bilden also die Koeffizienten der äußeren Ableitung einer 1-Form die <a href="Rotation_(Mathematik)" class="mw-redirect" title="Rotation (Mathematik)">Rotation</a> des aus den Koeffizienten der 1-Form gebildeten Vektors.

<div class="mw-heading mw-heading2"><h2 id="Weitere_Operationen_auf_Differentialformen">Weitere Operationen auf Differentialformen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Inneres_Produkt">Inneres Produkt</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in T(M)}">

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<annotation encoding="application/x-tex">{\displaystyle X\in T(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f876322057c422c95c695a08bcc7227530cb02d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.708ex; height:2.843ex;" alt="{\displaystyle X\in T(M)}" loading="lazy"> ein glattes Vektorfeld. Das innere Produkt ist eine lineare Abbildung

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<annotation encoding="application/x-tex">{\displaystyle i_{X}\colon \Omega ^{k}(M)\to \Omega ^{k-1}(M),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e2421832b4d6528fc983ea850f17fb011f238a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.867ex; height:3.176ex;" alt="{\displaystyle i_{X}\colon \Omega ^{k}(M)\to \Omega ^{k-1}(M),}" loading="lazy"></dd></dl>

die durch

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<annotation encoding="application/x-tex">{\displaystyle \omega \mapsto \omega (X,\cdot ,\ldots ,\cdot )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/519e5c2486e3d1c8a94818ac23a8b4eb48badc3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.801ex; height:2.843ex;" alt="{\displaystyle \omega \mapsto \omega (X,\cdot ,\ldots ,\cdot )}" loading="lazy"></dd></dl>

gegeben ist. Das heißt, das innere Produkt bildet eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> auf eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}">

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<annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69f74fa2adbbab50f6969acb2af719045435461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}" loading="lazy">-Form ab, indem die Form an einem festen Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ausgewertet wird. Diese Abbildung ist ein Analogon der <a href="Tensorverj%C3%BCngung" title="Tensorverjüngung">Tensorverjüngung</a> auf dem Raum der Differentialformen. Deshalb wird diese Operation im Englischen auch manchmal „contraction“ genannt.

Das innere Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{X}}">

<msub>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb2f1b6a552361bf8b27aaa44d209c55fa807585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.435ex; height:2.509ex;" alt="{\displaystyle i_{X}}" loading="lazy"> ist eine <a href="Antiderivation" class="mw-redirect" title="Antiderivation">Antiderivation</a>. Das heißt, für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega ^{k}(M)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd4450f2d016c637248e8f3e283dc7de6b88e34b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.305ex; height:3.176ex;" alt="{\displaystyle \omega \in \Omega ^{k}(M)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \in \Omega ^{l}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \nu \in \Omega ^{l}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6982a9f7b8004f2a20577a132f0458c613ad7d61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.725ex; height:3.176ex;" alt="{\displaystyle \nu \in \Omega ^{l}(M)}" loading="lazy"> gilt die <a href="Produktregel" title="Produktregel">Leibnizregel</a>

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<annotation encoding="application/x-tex">{\displaystyle i_{X}(\omega \wedge \nu )=(i_{X}\omega )\wedge \nu +(-1)^{k}\omega \wedge (i_{X}\nu ).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9262451737e9bd69f091b68fc7b81d5da336fe68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.968ex; height:3.176ex;" alt="{\displaystyle i_{X}(\omega \wedge \nu )=(i_{X}\omega )\wedge \nu +(-1)^{k}\omega \wedge (i_{X}\nu ).}" loading="lazy"></dd></dl>

Außerdem gilt für das innere Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{X}\circ i_{X}=0.}">

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<annotation encoding="application/x-tex">{\displaystyle i_{X}\circ i_{X}=0.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e65a0efcd1d6622f96a18c7a6f4737b522c0107f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.972ex; height:2.509ex;" alt="{\displaystyle i_{X}\circ i_{X}=0.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Rücktransport_(Pullback)_von_Differentialformen">Rücktransport (Pullback) von Differentialformen</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="R%C3%BCcktransport" title="Rücktransport">Rücktransport</a></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> eine <a href="Glatte_Abbildung" class="mw-redirect" title="Glatte Abbildung">glatte Abbildung</a> zwischen differenzierbaren <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a>, so ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega ^{k}(N)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eda45e16e1ddb29a9aab85033007b5a4f16d341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.926ex; height:3.176ex;" alt="{\displaystyle \omega \in \Omega ^{k}(N)}" loading="lazy"> die mittels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> zurückgeholte Form

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<annotation encoding="application/x-tex">{\displaystyle (f^{*}\omega )\in \Omega ^{k}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e778b3c21840ed595ee8b6c4c28d20f5036c9da8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.489ex; height:3.176ex;" alt="{\displaystyle (f^{*}\omega )\in \Omega ^{k}(M)}" loading="lazy"> wie folgt definiert:

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<annotation encoding="application/x-tex">{\displaystyle (f^{*}\omega )(X_{1},\ldots ,X_{k}):=\omega (\mathrm {d} f(X_{1}),\ldots ,\mathrm {d} f(X_{k}))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe02566904433ba53b9d39ac634daf435920f5ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.54ex; height:2.843ex;" alt="{\displaystyle (f^{*}\omega )(X_{1},\ldots ,X_{k}):=\omega (\mathrm {d} f(X_{1}),\ldots ,\mathrm {d} f(X_{k}))}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df\colon TM\to TN}">

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<annotation encoding="application/x-tex">{\displaystyle df\colon TM\to TN}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2dc2c2f046f7e98d0a069f7a0ba762f08958fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.921ex; height:2.509ex;" alt="{\displaystyle df\colon TM\to TN}" loading="lazy"> die durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> induzierte Abbildung der <a href="Differentialrechnung" title="Differentialrechnung">Ableitungen</a>, auch „push-forward“ genannt. Das Zurückziehen ist mit der äußeren Ableitung und dem äußeren Produkt verträglich:

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<annotation encoding="application/x-tex">{\displaystyle f^{*}(\mathrm {d} \omega )=\mathrm {d} (f^{*}\omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3664f22fc689fabdbacf585fc5905a74f01ee7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.943ex; height:2.843ex;" alt="{\displaystyle f^{*}(\mathrm {d} \omega )=\mathrm {d} (f^{*}\omega )}" loading="lazy"></li></ul>

<dl><dd>(ausführlicher geschrieben: auf der linken Seite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} =\mathrm {d} ^{(N)}}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} =\mathrm {d} ^{(N)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77f3cc9e881b830a90a62b5457e137879feb2bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.654ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} =\mathrm {d} ^{(N)}}" loading="lazy">, auf der rechten Seite dagegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} =\mathrm {d} ^{(M)}}">

</mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} =\mathrm {d} ^{(M)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bfea6ca930a8019cee1dbbecaf5508d78a84ba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.922ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} =\mathrm {d} ^{(M)}}" loading="lazy">) und</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f^{*}(\omega \wedge \eta )=(f^{*}\omega )\wedge (f^{*}\eta )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0221b4e8ce701ea8c1a8674fb7d26c4edbb1b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.046ex; height:2.843ex;" alt="{\displaystyle f^{*}(\omega \wedge \eta )=(f^{*}\omega )\wedge (f^{*}\eta )}" loading="lazy"></li></ul>

<msup>

<mtext>pback</mtext>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \omega ,\eta \in \Omega ^{\text{pback}}(N)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a909a2cfc4ef1a381397ae449ebb0fda4f41ed71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.521ex; height:3.176ex;" alt="{\displaystyle \omega ,\eta \in \Omega ^{\text{pback}}(N)}" loading="lazy"></dd></dl>

Insbesondere induziert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> eine Abbildung zwischen den <a href="De-Rham-Kohomologie" title="De-Rham-Kohomologie">De-Rham-Kohomologie</a>-Gruppen (siehe unten)

<msup>

<mtext>pback</mtext>

</mrow>

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</mrow>

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f^{\text{pback}}\colon \mathrm {H} _{\text{dR}}^{k}(N)\longrightarrow \mathrm {H} _{\text{dR}}^{k}(M),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4dc38ec0612775e68784394b04724c46aa15ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.899ex; height:3.176ex;" alt="{\displaystyle f^{\text{pback}}\colon \mathrm {H} _{\text{dR}}^{k}(N)\longrightarrow \mathrm {H} _{\text{dR}}^{k}(M),}" loading="lazy"></dd></dl>

wobei die Umkehr der Pfeilrichtung gegenüber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> zu beachten ist („pull-back“, „Kohomologie“ statt „Homologie“).

<div class="mw-heading mw-heading3"><h3 id="Duale_Form_und_Stern-Operator">Duale Form und Stern-Operator</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Hodge-Stern-Operator" title="Hodge-Stern-Operator">Hodge-Stern-Operator</a></div>

Betrachtet werden äußere Formen in einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum, in dem ein inneres Produkt (Metrik) definiert ist, sodass eine orthonormale Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> des Raumes gebildet werden kann. Die zu einer äußeren Form von Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> in diesem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum duale Form ist eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n-k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e9f25b19fdb4fc155f89d4f926e9d9e4e440da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.256ex; height:2.843ex;" alt="{\displaystyle (n-k)}" loading="lazy">-Form

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle *(e_{1}\wedge e_{2}\wedge \ldots \wedge e_{k})=e_{k+1}\wedge e_{k+2}\wedge \ldots \wedge e_{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49c4b3deeefa4b5357040cda1daa71897d5897ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.954ex; height:2.843ex;" alt="{\displaystyle *(e_{1}\wedge e_{2}\wedge \ldots \wedge e_{k})=e_{k+1}\wedge e_{k+2}\wedge \ldots \wedge e_{n}.}" loading="lazy"></dd></dl>

Dabei seien beide Seiten in orientierter Form geschrieben. Formal wird die duale Form durch Anwendung des (Hodge-)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">-Operators bezeichnet. Speziell für Differentialformen im dreidimensionalen euklidischen Raum ergibt sich:

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *\mathrm {d} x=\mathrm {d} y\wedge \mathrm {d} z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b71f471ea9a3a654b65728cbbe61e4262470422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.294ex; height:2.509ex;" alt="{\displaystyle *\mathrm {d} x=\mathrm {d} y\wedge \mathrm {d} z}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *\mathrm {d} y=\mathrm {d} z\wedge \mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7a64eaf35ee1ac48c00789bc0b769caf43cf32d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.294ex; height:2.509ex;" alt="{\displaystyle *\mathrm {d} y=\mathrm {d} z\wedge \mathrm {d} x}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *\mathrm {d} z=\mathrm {d} x\wedge \mathrm {d} y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab571c192c49fc7d21f0f5a575cf62dbd8981886.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.294ex; height:2.509ex;" alt="{\displaystyle *\mathrm {d} z=\mathrm {d} x\wedge \mathrm {d} y}" loading="lazy"></dd></dl>

mit den 1-Formen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx,dy,dz}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle dx,dy,dz}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0df5a109b0087620b32449bd5ea7b4c1fffd9db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.289ex; height:2.509ex;" alt="{\displaystyle dx,dy,dz}" loading="lazy">. Dabei wurde berücksichtigt, dass die orientierte Reihenfolge hier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,z),(z,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y,z),(z,x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7011eb81da6b0f98945dc2f0b3b96c82675a71d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.382ex; height:2.843ex;" alt="{\displaystyle (y,z),(z,x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> ist (zyklische Vertauschungen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22a8c93372e8f8b6e24d523bd5545aed3430baf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" loading="lazy">).

Das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">-Symbol soll die Tatsache unterstreichen, dass damit ein inneres Produkt im Raum der Formen auf einem zugrundeliegenden Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> gegeben ist, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \wedge *\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \wedge *\beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e91397e5902c22772aca7695eeb19a3a2b26a34e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.565ex; height:2.509ex;" alt="{\displaystyle \alpha \wedge *\beta }" loading="lazy"> lässt sich für zwei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Formen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> als Volumenform schreiben und das Integral

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge *\beta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00bd9013cb77f6105fa300c405661cdf18ff5fe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.965ex; height:5.676ex;" alt="{\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge *\beta }" loading="lazy"></dd></dl>

liefert eine reelle Zahl. Der Zusatz dual zeigt an, dass die zweifache Anwendung auf eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form wieder die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form ergibt – bis auf das <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a>, das gesondert betrachtet werden muss. Genauer gilt für eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form in einem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen Raum, dessen Metrik die Signatur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> hat (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/062261785446129907c44618616bc7c8dca7bd5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.16ex; height:2.343ex;" alt="{\displaystyle s=+1}" loading="lazy"> im euklidischen Raum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/999944118796b0e4485e997249775b0d9925772f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.16ex; height:2.343ex;" alt="{\displaystyle s=-1}" loading="lazy"> im Minkowski-Raum):

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {*}{*}\alpha =(-1)^{k(n-k)}s\;\alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88361b92f13f032c84be4aefcc93d40efa8dc0c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.403ex; height:3.343ex;" alt="{\displaystyle {*}{*}\alpha =(-1)^{k(n-k)}s\;\alpha }" loading="lazy"></dd></dl>

Oben wurde gezeigt, wie sich im 3-dimensionalen euklidischen Raum bei äußerer Ableitung einer 1-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> die 2-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\wedge \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\wedge \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/446fbffe28278b0d3d26d8601af15eb88b2b2818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.286ex; height:2.176ex;" alt="{\displaystyle d\wedge \alpha }" loading="lazy"> ergibt mit den Komponenten des Rotations-Vektors der <a href="Vektoranalysis" title="Vektoranalysis">Vektoranalysis</a> als Koeffizienten. Diese 2-Form kann man mit Hilfe des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">-Operators nun auch formal direkt als 1-Form (rot-Vektor) schreiben: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *(d\wedge \alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *(d\wedge \alpha )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79a66e9d9592d13b88fa92442004b3670ab9cb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.258ex; height:2.843ex;" alt="{\displaystyle *(d\wedge \alpha )}" loading="lazy">. Analog wird der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">-Operator zur „Übersetzung“ des oben formulierten Satzes von Stokes in die Vektoranalysis-Form benutzt.

<div class="mw-heading mw-heading2"><h2 id="De-Rham-Kohomologie">De-Rham-Kohomologie</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="De-Rham-Kohomologie" title="De-Rham-Kohomologie">De-Rham-Kohomologie</a></div>

Aus der <a href="Graduierung_(Algebra)" title="Graduierung (Algebra)">graduierten Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (U)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c053a26918e0562501161dcc3bada8056a141f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.27ex; height:2.843ex;" alt="{\displaystyle \Omega (U)}" loading="lazy"> kann zusammen mit der äußeren Ableitung ein <a href="Kokettenkomplex" class="mw-redirect" title="Kokettenkomplex">Kokettenkomplex</a> konstruiert werden. Aus diesem wird dann mit den üblichen Methoden der <a href="Homologische_Algebra" title="Homologische Algebra">homologischen Algebra</a> eine <a href="Kohomologie" title="Kohomologie">Kohomologie</a> definiert. <a href="Georges_de_Rham" title="Georges de Rham">Georges de Rham</a> konnte zeigen, dass diese nach ihm benannte Kohomologietheorie mit der <a href="Singul%C3%A4re_Kohomologie" title="Singuläre Kohomologie">singulären Kohomologie</a> übereinstimmt. Um die De-Rham-Kohomologie zu definieren, werden zuerst die Begriffe der exakten und der geschlossenen Differentialform definiert:

<div class="mw-heading mw-heading3"><h3 id="Exakte_und_geschlossene_Formen">Exakte und geschlossene Formen</h3></div>

Eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> heißt geschlossen, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \omega =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94beef0188c62c22491e78300a0b90442f9cd40f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.999ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \omega =0}" loading="lazy"> gilt; sie heißt exakt, wenn es eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69f74fa2adbbab50f6969acb2af719045435461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> gibt, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\mathrm {d} \eta }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\mathrm {d} \eta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8f3fdfe8f90b0a10ed3b430a79024c0334f92e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.006ex; height:2.676ex;" alt="{\displaystyle \omega =\mathrm {d} \eta }" loading="lazy"> gilt. Aufgrund der Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathrm {d} \eta =0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathrm {d} \eta =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d456a0281c540b3b209d988db06729e3201d5ed8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.015ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} \mathrm {d} \eta =0}" loading="lazy"> ist jede exakte Form geschlossen. Man beachte, dass Geschlossenheit im Gegensatz zu Exaktheit eine lokale Eigenschaft ist: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V_{\alpha }\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{V_{\alpha }\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eaa6bdef85db640af1f0f1ec8adf29607172b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.964ex; height:2.843ex;" alt="{\displaystyle \{V_{\alpha }\}}" loading="lazy"> eine <a href="Offene_%C3%9Cberdeckung" class="mw-redirect" title="Offene Überdeckung">offene Überdeckung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c21c569e498e0197798e3428b2bc6f25c0a457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" loading="lazy"> so ist eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> genau dann geschlossen, wenn die Einschränkung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\alpha }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/582d40a9ff663187250948b07bb66456162c2042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle V_{\alpha }}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> geschlossen ist.

<div class="mw-heading mw-heading3"><h3 id="Die_De-Rham-Kohomologiegruppen">Die De-Rham-Kohomologiegruppen</h3></div>

Der <a href="Faktorraum" title="Faktorraum">Faktorraum</a>

<dl><dd>(Menge aller geschlossenen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Formen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big /}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\big /}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f255915283628a9801ed610c5e614d33d11af8fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.344ex; height:3.176ex;" alt="{\displaystyle {\big /}}" loading="lazy"> (Menge aller exakten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Formen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">)</dd></dl>

heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-te <a href="De-Rham-Kohomologie" title="De-Rham-Kohomologie">De-Rham-Kohomologiegruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U).}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/679acf4d6586d1ca5632096fe27fb447da579fe5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.338ex; height:3.176ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U).}" loading="lazy"> Sie enthält Informationen über die globale topologische Struktur von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Das_Lemma_von_Poincaré">Das Lemma von Poincaré</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Poincar%C3%A9-Lemma" title="Poincaré-Lemma">Poincaré-Lemma</a></div>

Das Lemma von Poincaré besagt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)=0}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39f35a0b23a7f538d436f49c7735be1fa39d23e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.952ex; height:3.176ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)=0}" loading="lazy"> gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy"> und <a href="Sterngebiet" title="Sterngebiet">Sterngebiete</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Allgemeiner gilt die Aussage dieses Lemmas für <a href="Zusammenziehbar" class="mw-redirect" title="Zusammenziehbar">zusammenziehbare</a> offene Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ef548febfc9981762740107858be9e3a5576c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}" loading="lazy"> Der Beweis ist konstruktiv, d. h., es werden explizite Beispiele konstruiert, was für Anwendungen sehr wichtig ist. Man beachte, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f106c470ba5083ec37b5bc1eaa6c8c80b998e145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.691ex; height:3.343ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)}" loading="lazy"> aus den <a href="Lokal_konstante_Funktion" title="Lokal konstante Funktion">lokal konstanten Funktionen</a> besteht, da es per definitionem keine exakten 0-Formen gibt. Es ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)\not =0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)\not =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5522f25e1f2edba071184f0706c64590eef7800b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.952ex; height:3.343ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{0}(U)\not =0}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\neq \varnothing .}">

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<annotation encoding="application/x-tex">{\displaystyle U\neq \varnothing .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d77f050de6c9fe46e95b739b6afbe616e8e98fb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.336ex; height:2.676ex;" alt="{\displaystyle U\neq \varnothing .}" loading="lazy">

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> geschlossen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\mathrm {d} \eta '}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =\mathrm {d} \eta '}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84e8aacc563fdcf46ab08dca4c57f30424227cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.419ex; height:3.009ex;" alt="{\displaystyle \eta =\mathrm {d} \eta '}" loading="lazy"> exakt, so folgt

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<annotation encoding="application/x-tex">{\displaystyle \omega \wedge \eta =\omega \wedge \mathrm {d} \eta '=(-1)^{\deg \omega }\mathrm {d} (\omega \wedge \eta ')\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34fdf69f4eb0c53b7a492430e3a207e5fd40ca9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.484ex; height:3.176ex;" alt="{\displaystyle \omega \wedge \eta =\omega \wedge \mathrm {d} \eta '=(-1)^{\deg \omega }\mathrm {d} (\omega \wedge \eta ')\,.}" loading="lazy"></dd></dl>

Entsprechendes gilt, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> exakt und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> geschlossen ist. Damit gibt es induzierte Abbildungen

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)\times H_{\mathrm {dR} }^{m}(U)\longrightarrow \mathrm {H} _{\mathrm {dR} }^{k+m}(U).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66a1f45fcb08b3b32eb684009be89506ff8035ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.298ex; height:3.343ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)\times H_{\mathrm {dR} }^{m}(U)\longrightarrow \mathrm {H} _{\mathrm {dR} }^{k+m}(U).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ein_Beispiel_aus_der_Elektrodynamik">Ein Beispiel aus der Elektrodynamik</h3></div>

In der <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> impliziert das <a href="Poincar%C3%A9-Lemma" title="Poincaré-Lemma">Lemma von Poincaré</a>, dass zu jedem Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}},{\vec {B}}}">

<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}},{\vec {B}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8916cfaee999c32881516cfa5f4ec4618f9879b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.574ex; height:3.176ex;" alt="{\displaystyle {\vec {E}},{\vec {B}}}" loading="lazy"> elektromagnetischer Felder, die zu einer zweistufigen alternierenden Differentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> in einem vierdimensionalen <a href="Minkowskiraum" class="mw-redirect" title="Minkowskiraum">Minkowskiraum</a> zusammengefasst werden können, eine einstufige Vektorpotentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bea2a33c6a245bb75f86e94c6025677d2e8851b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.093ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} }" loading="lazy"> existiert, ein sogenanntes „Viererpotential“, siehe auch <a href="Vierervektor" title="Vierervektor">Vierervektor</a>. Auch Strom- und Ladungsdichten können zu einem Vierervektor bzw. zu einer entsprechenden 3-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca5d874dbb32b8b33e83ca521f592808387a486e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:0.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} }" loading="lazy"> zusammengefasst werden.

Die relativistischen <a href="Maxwell-Gleichung" class="mw-redirect" title="Maxwell-Gleichung">Maxwell-Gleichungen</a> der Elektrodynamik auf einer vierdimensionalen Raum-Zeit-Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> (mit Metrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\alpha ,\beta }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{\alpha ,\beta }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9538f08bae20ad2427e26bac8308e19d161c0a23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.792ex; height:2.343ex;" alt="{\displaystyle g_{\alpha ,\beta }}" loading="lazy"> und Determinante der Metrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, wobei hier natürlich die Signatur eines <a href="Minkowski-Raum" title="Minkowski-Raum">Minkowski-Raumes</a> vorliegt, etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {diag} (+1,-1,-1,-1)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {diag} (+1,-1,-1,-1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb566caea0f8e0770c2d088706287e9c94025976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.058ex; height:2.843ex;" alt="{\displaystyle \mathrm {diag} (+1,-1,-1,-1)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0,1,2,3,}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =0,1,2,3,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe3a0dabcee75bbec6ae9912bf6800775bd7b923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.985ex; height:2.509ex;" alt="{\displaystyle \alpha =0,1,2,3,}" loading="lazy"> entsprechend der Definition des <a href="Linienelement" class="mw-redirect" title="Linienelement">Linienelements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5496d5dfa7c75b160324a51bc702b3c7abfdb717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.383ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} s}" loading="lazy">) lauten beispielsweise unter Verwendung dieser Symbolik:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {F} =2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/912ac8af8029d8c3dc955116f4f493aae4142203.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.003ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} \mathbf {F} =2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0}" loading="lazy"></dd></dl>

(die sogenannte <a href="Bianchi-Identit%C3%A4t" class="mw-redirect" title="Bianchi-Identität">Bianchi-Identität</a>) und

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (*\mathbf {F} )={F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }=\mathbf {j} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f985153c370efa589195b43704a20a8e13496468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:46.308ex; height:3.676ex;" alt="{\displaystyle \mathrm {d} (*\mathbf {F} )={F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }=\mathbf {j} }" loading="lazy"></dd></dl>

mit dem elektromagnetischen Feldtensor ausgedrückt als 2-Form

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa5678ca34e431be5a1e99475e49ec957055d2a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.982ex; height:3.343ex;" alt="{\displaystyle \mathbf {F} =F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\,,}" loading="lazy"></dd></dl>

z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1,2}=B_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{1,2}=B_{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3aaa31b9e2847b344cdcc08963856639bb057e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.692ex; height:2.843ex;" alt="{\displaystyle F_{1,2}=B_{z}}" loading="lazy"> mit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-Komponente des Vektors der magnetischen Induktion, und mit dem Strom (geschrieben als 3-Form)

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} =j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52af7b55ecaa04b22a4bb426e2bcdccd2f0032c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.164ex; width:34.873ex; height:3.676ex;" alt="{\displaystyle \mathbf {j} =j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }.}" loading="lazy"></dd></dl>

Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\alpha \beta \gamma \delta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{\alpha \beta \gamma \delta }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/670a6d080598763bddf814dd66f911eeecdce000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.804ex; height:2.343ex;" alt="{\displaystyle \epsilon _{\alpha \beta \gamma \delta }}" loading="lazy"> das Antisymmetrisierungssymbol (<a href="Levi-Civita-Symbol" title="Levi-Civita-Symbol">Levi-Civita-Symbol</a>) und das Semikolon steht für die <a href="Kovariante_Ableitung" class="mw-redirect" title="Kovariante Ableitung">kovariante Ableitung</a>. Wie üblich wird über doppelt vorkommende Indizes summiert (<a href="Einsteinsche_Summenkonvention" title="Einsteinsche Summenkonvention">Einsteinsche Summenkonvention</a>) und es werden natürliche Einheiten verwendet (Lichtgeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> ersetzt durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">). Durch Anwendung des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">-Operators kann man den zweiten Satz der vier Maxwellgleichungen auch alternativ mit einer 1-Form für den Strom schreiben. Aus den Maxwellgleichungen sieht man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\mathbf {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *\mathbf {F} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6341c8751ed60863f6bc4c3e100e60ce3b394e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.176ex;" alt="{\displaystyle *\mathbf {F} }" loading="lazy"> in der Elektrodynamik ganz unterschiedlichen Gleichungen gehorchen, die Dualität also keine Symmetrie dieser Theorie ist. Das liegt daran, dass die Dualität elektrische und magnetische Felder vertauscht, in der Elektrodynamik aber keine magnetischen Monopole bekannt sind. Die freien Maxwellgleichungen, die sich für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} \equiv 0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} \equiv 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e38d30607159e6d3baa7957907db81df4aa51e34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:5.241ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} \equiv 0}" loading="lazy"> ergeben, haben dagegen duale Symmetrie.

Die Potentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> ist nur bis auf einen additiven Zusatz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \chi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c79a701eb500fbcdc72eaca200c5f0f8ba3b4ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.748ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \chi }" loading="lazy"> eindeutig: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} +\mathrm {d} \chi }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +\mathrm {d} \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51c49a81256b1964b60d4de1b2db9e11b533d2b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.608ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} +\mathrm {d} \chi }" loading="lazy"> ergeben dasselbe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy">, mit einer Eichform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy">, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} (\mathrm {d} \chi )\equiv 0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (\mathrm {d} \chi )\equiv 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8d9317ce16a2e83f25e917576ee5891fe0dfd6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.11ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} (\mathrm {d} \chi )\equiv 0}" loading="lazy"> erfüllt, aber ansonsten willkürlich ist. Man kann diese zusätzliche sogenannte Eichfreiheit benutzen, um punktweise zusätzliche Nebenbedingungen zu erfüllen. In der Elektrodynamik fordert man beispielsweise, dass für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> überall die zusätzliche sogenannte Lorenz-Bedingung (<a href="Lorenz-Eichung" title="Lorenz-Eichung">Lorenz-Eichung</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} (*\mathbf {A} )=0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} (*\mathbf {A} )=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a5750c78f803b725663a5e0f2a7933fc004d672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.545ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} (*\mathbf {A} )=0}" loading="lazy"> gelten soll, in den vier Komponenten lautet diese Bedingung einfach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\nu }A^{\nu }=0}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\nu }A^{\nu }=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45836379151c25048069f646f258e9703d322ffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.445ex; height:2.676ex;" alt="{\displaystyle \partial _{\nu }A^{\nu }=0}" loading="lazy">. Durch diese „Eichfixierung“ ergibt sich schließlich als eindeutige Lösung aller vier Maxwell-Gleichungen das sogenannte „retardierte Potential“:

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

<msup>

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</msup>

<msup>

</msup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A^{\nu }(\mathbf {r} ,t)=\int \,{\frac {j^{\nu }(\mathbf {r'} ,t-{\frac {|\mathbf {r} -\mathbf {r'} |}{c}})}{4\pi |\mathbf {r} -\mathbf {r'} |}}\,\mathrm {d} ^{3}r'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45b695942222daf6c5a783a674920aaba9a9cadc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.492ex; height:7.676ex;" alt="{\displaystyle A^{\nu }(\mathbf {r} ,t)=\int \,{\frac {j^{\nu }(\mathbf {r'} ,t-{\frac {|\mathbf {r} -\mathbf {r'} |}{c}})}{4\pi |\mathbf {r} -\mathbf {r'} |}}\,\mathrm {d} ^{3}r'}" loading="lazy"></dd></dl>

Beim Übergang zum Dualen ist zu beachten, dass man es nicht mit dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4abb9b9dab94f7b25a4210364f0f9032704bfb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}" loading="lazy">, sondern mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {M} ^{4}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {M} ^{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89df94817b783aa643c441e2d09b25af1ae621db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.248ex; height:2.676ex;" alt="{\displaystyle \mathbb {M} ^{4}}" loading="lazy"> zu tun hat, der eine andere Metrik, nämlich die <a href="Minkowski-Metrik" class="mw-redirect" title="Minkowski-Metrik">Minkowski-Metrik</a>, trägt. Das bei <a href="Lorentztransformation" class="mw-redirect" title="Lorentztransformation">Lorentztransformationen</a> invariante Linienelement ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s^{2}=-c^{2}\mathrm {d} \tau ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}-c^{2}\mathrm {d} t^{2}=-\mathrm {d} x_{\nu }\mathrm {d} x^{\nu }}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=-c^{2}\mathrm {d} \tau ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}-c^{2}\mathrm {d} t^{2}=-\mathrm {d} x_{\nu }\mathrm {d} x^{\nu }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80631c3396452613d7d076dea1f83a2bf44512da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.855ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} s^{2}=-c^{2}\mathrm {d} \tau ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}-c^{2}\mathrm {d} t^{2}=-\mathrm {d} x_{\nu }\mathrm {d} x^{\nu }}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \tau }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \tau }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a18897d8edf14fe5f4b48c7582d5d21f14bdd53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \tau }" loading="lazy"> das Differential der <a href="Eigenzeit" class="mw-redirect" title="Eigenzeit">Eigenzeit</a> ist und die Summenkonvention verwendet wurde. Ko- und kontravariante <a href="Vierervektor" title="Vierervektor">Vierervektorkomponenten</a> unterscheiden sich nun. Zwar ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\mathrm {d} x_{0}\equiv +\mathrm {d} x^{0}=c\,\mathrm {d} t}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\mathrm {d} x_{0}\equiv +\mathrm {d} x^{0}=c\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c20ff10e354842c0ec4a04fbcc368e1616a60fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.692ex; height:3.009ex;" alt="{\displaystyle +\mathrm {d} x_{0}\equiv +\mathrm {d} x^{0}=c\,\mathrm {d} t}" loading="lazy">, aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {d} x_{1}\equiv +\mathrm {d} x^{1}=\mathrm {d} x}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\mathrm {d} x_{1}\equiv +\mathrm {d} x^{1}=\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/088c1c2676a66ab6f13358fee6e29bf9d4ff96f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.788ex; height:3.009ex;" alt="{\displaystyle -\mathrm {d} x_{1}\equiv +\mathrm {d} x^{1}=\mathrm {d} x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {d} x_{2}\equiv +\mathrm {d} x^{2}=\mathrm {d} y}">

</mrow>

<msub>

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</msub>

</mrow>

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\mathrm {d} x_{2}\equiv +\mathrm {d} x^{2}=\mathrm {d} y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9063270cfdd6fc349647ae43effa1201b3b794c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.614ex; height:3.009ex;" alt="{\displaystyle -\mathrm {d} x_{2}\equiv +\mathrm {d} x^{2}=\mathrm {d} y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {d} x_{3}\equiv +\mathrm {d} x^{3}=\mathrm {d} z}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\mathrm {d} x_{3}\equiv +\mathrm {d} x^{3}=\mathrm {d} z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/489f3c0e328b5459821ce81f040dc1f0b021d867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.547ex; height:3.009ex;" alt="{\displaystyle -\mathrm {d} x_{3}\equiv +\mathrm {d} x^{3}=\mathrm {d} z}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Integrationstheorie">Integrationstheorie</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Orientierung">Orientierung</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">Orientierung (Mathematik)</a></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\dim U,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=\dim U,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fccee74e580a79ed96ed0aacb47c6494f97fcb20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.185ex; height:2.509ex;" alt="{\displaystyle n=\dim U,}" loading="lazy"> so heißt eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Form auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c21c569e498e0197798e3428b2bc6f25c0a457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" loading="lazy"> die in keinem Punkt verschwindet, eine <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">Orientierung</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> zusammen mit einer derartigen Form heißt orientiert. Eine Orientierung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> definiert Orientierungen der Tangential- und Kotangentialräume: Eine Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{1},\ldots ,\eta _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta _{1},\ldots ,\eta _{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/072db1172e053b27f4fee59cf5ba754c4b366da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.762ex; height:2.176ex;" alt="{\displaystyle \eta _{1},\ldots ,\eta _{n}}" loading="lazy"> des Kotangentialraums in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> sei positiv orientiert, wenn

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{p}=a\cdot \eta _{1}\wedge \ldots \wedge \eta _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63f71107be09f35efaaa69a8fb854a4823ad1cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.026ex; height:2.676ex;" alt="{\displaystyle \alpha _{p}=a\cdot \eta _{1}\wedge \ldots \wedge \eta _{n}}" loading="lazy"></dd></dl>

mit einer positiven Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> gilt. Eine Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n}}">

<msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac794f5521dcce89913085a6d566e7cdb615dbb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}" loading="lazy"> des Tangentialraums in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> sei positiv orientiert, wenn

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha (X_{1},\ldots ,X_{n})>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3275e26bd32ff3715cd2d2a4268594018994c7ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.857ex; height:2.843ex;" alt="{\displaystyle \alpha (X_{1},\ldots ,X_{n})>0}" loading="lazy"></dd></dl>

gilt.

Zwei Orientierungen heißen äquivalent, wenn sie sich nur um einen überall positiven Faktor unterscheiden; diese Bedingung ist äquivalent dazu, dass sie auf jedem Tangential- oder Kotangentialraum dieselbe Orientierung definieren.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> <a href="Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">zusammenhängend</a>, so gibt es entweder gar keine oder genau zwei Äquivalenzklassen.

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> heißt orientierbar, wenn eine Orientierung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> existiert.

<div class="mw-heading mw-heading3"><h3 id="Integration_von_Differentialformen">Integration von Differentialformen</h3></div>

Es sei wieder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\dim U,}">

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<annotation encoding="application/x-tex">{\displaystyle n=\dim U,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fccee74e580a79ed96ed0aacb47c6494f97fcb20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.185ex; height:2.509ex;" alt="{\displaystyle n=\dim U,}" loading="lazy"> und wir nehmen an, auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> sei eine Orientierung gewählt. Dann gibt es ein kanonisches Integral

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{U}\omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a7db5390e6d5c38ba2aff3f3c1ba65b26706bbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.618ex; height:5.676ex;" alt="{\displaystyle \int _{U}\omega }" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Formen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega .}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \omega .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2243e2ce8b3865a78e46f7e554deb7b3aaa09490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.093ex; height:1.676ex;" alt="{\displaystyle \omega .}" loading="lazy"> Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> eine offene Teilmenge des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

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</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">, sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/737e02a5fbf8bc31d443c91025339f9fd1de1065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}" loading="lazy"> die Standardkoordinatenfunktionen im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> und ist

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \omega =f\,\,\mathrm {d} x_{1}\wedge \ldots \wedge \mathrm {d} x_{n},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14786badea83dbd1e11193b4f4ddcdc3fc29ae0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.649ex; height:2.509ex;" alt="{\displaystyle \omega =f\,\,\mathrm {d} x_{1}\wedge \ldots \wedge \mathrm {d} x_{n},}" loading="lazy"></dd></dl>

so gilt:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{U}\omega =\int _{U}f(x_{1},\dots ,x_{n})\,\,\mathrm {d} x_{1}\ldots \mathrm {d} x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c506ad6cc36d3fabf2b983fafbb21533daf0bd55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.876ex; height:5.676ex;" alt="{\displaystyle \int _{U}\omega =\int _{U}f(x_{1},\dots ,x_{n})\,\,\mathrm {d} x_{1}\ldots \mathrm {d} x_{n}}" loading="lazy"></dd></dl>

Das Integral auf der rechten Seite ist das gewöhnliche <a href="Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-Integral</a> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ef548febfc9981762740107858be9e3a5576c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}" loading="lazy">

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale orientierte Mannigfaltigkeit, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset M}">

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<annotation encoding="application/x-tex">{\displaystyle U\subset M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edd7b08ced14b1958599bba595488f7c314e59da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle U\subset M}" loading="lazy"> offen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon U\to \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \colon U\to \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0293e3348e2d861c663852071c01df8140a8d95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.713ex; height:2.676ex;" alt="{\displaystyle \phi \colon U\to \mathbb {R} ^{n}}" loading="lazy"> eine Karte, so definiert man

<msub>

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \int _{U}\omega =\int _{\phi (U)}(\phi ^{-1})^{*}\omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f46a5126341071db68453e58b9a1e58465915b1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.788ex; height:6.009ex;" alt="{\displaystyle \int _{U}\omega =\int _{\phi (U)}(\phi ^{-1})^{*}\omega }" loading="lazy"></dd></dl>

als Integral der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> über ein Kartengebiet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Die Differentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> wird also mit der Parametrisierung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{-1}\colon \phi (U)\to U}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ^{-1}\colon \phi (U)\to U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f066d2ac889cbb52b4073403210636f0286e1f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.126ex; height:3.176ex;" alt="{\displaystyle \phi ^{-1}\colon \phi (U)\to U}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> auf die offene Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (U)\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (U)\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cde985581367c93ef9f7200c688e5d88a1fc40cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.972ex; height:2.843ex;" alt="{\displaystyle \phi (U)\subset \mathbb {R} ^{n}}" loading="lazy"> zurückgeholt und dann nach obiger Definition integriert.

Aus dem <a href="Transformationssatz" title="Transformationssatz">Transformationssatz</a> folgt, dass diese Definition invariant gegenüber Koordinatenwechsel ist.

Ist allgemeiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> eine <a href="Messbare_Menge" class="mw-redirect" title="Messbare Menge">messbare</a> Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, so definiert man

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{B}\omega =\int _{U}\chi _{B}\omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47a7fd5d62a054e882b1c0e06d140ded3826216d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.256ex; height:5.676ex;" alt="{\displaystyle \int _{B}\omega =\int _{U}\chi _{B}\omega }" loading="lazy"></dd></dl>

mit der <a href="Indikatorfunktion" title="Indikatorfunktion">charakteristischen Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{B}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{B}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccc5d247c0324706579cf85da51d154eb46a6ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.935ex; height:2.009ex;" alt="{\displaystyle \chi _{B}}" loading="lazy">, d. h., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> wird außerhalb von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> null gesetzt.

Zur Definition des Integrals über ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> kann eine Zerlegung

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=\bigcup _{j=1}^{\infty }M_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ea6c1bed376035a02e94776f471ea31222085ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:11.87ex; height:7.176ex;" alt="{\displaystyle M=\bigcup _{j=1}^{\infty }M_{j}}" loading="lazy"></dd></dl>

in abzählbar viele <a href="Paarweise_disjunkt" class="mw-redirect" title="Paarweise disjunkt">paarweise disjunkte</a> messbare Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0238f61c7810b475df799bfc63a067cb5e517b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.843ex;" alt="{\displaystyle M_{j}}" loading="lazy"> gewählt werden, sodass jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0238f61c7810b475df799bfc63a067cb5e517b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.843ex;" alt="{\displaystyle M_{j}}" loading="lazy"> ganz in einem Kartengebiet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d85201a34558a919f8cc0e193492932eedfc6ce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.497ex; height:2.843ex;" alt="{\displaystyle U_{j}}" loading="lazy"> enthalten ist. Damit setzt man

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{M}\omega =\sum _{j=1}^{\infty }\int _{M_{j}}\omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a09ed07a01b3f9f7974f478d966e5d67624065d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.591ex; height:7.176ex;" alt="{\displaystyle \int _{M}\omega =\sum _{j=1}^{\infty }\int _{M_{j}}\omega }" loading="lazy">.</dd></dl>

Für Integrale von Differentialformen gilt der folgende Transformationssatz:

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> ein orientierungserhaltender <a href="Diffeomorphismus" title="Diffeomorphismus">Diffeomorphismus</a>, dann gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega ^{n}(N)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega ^{n}(N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/438137040e1f4ebf46f4b82b8570265150b2635f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.056ex; height:2.843ex;" alt="{\displaystyle \omega \in \Omega ^{n}(N)}" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{N}\omega =\int _{M}f^{*}\omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9114f7b6c4eecf53feadb338ec6c0c2e021aad4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.375ex; height:5.676ex;" alt="{\displaystyle \int _{N}\omega =\int _{M}f^{*}\omega }" loading="lazy"></dd></dl>

mit der auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> zurückgeholten Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\omega }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}\omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b107985e69cf356a4a3c3aa9ff6935cb5986ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.821ex; height:2.676ex;" alt="{\displaystyle f^{*}\omega }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Satz_von_Stokes">Satz von Stokes</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Satz_von_Stokes" title="Satz von Stokes">Satz von Stokes</a></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine kompakte orientierte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale differenzierbare <a href="Mannigfaltigkeit_mit_Rand" title="Mannigfaltigkeit mit Rand">Mannigfaltigkeit mit Rand</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8456299e2b600f44f5aa08920be090af1b35e013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="{\displaystyle \partial M}" loading="lazy"> und versieht man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8456299e2b600f44f5aa08920be090af1b35e013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="{\displaystyle \partial M}" loading="lazy"> mit der induzierten Orientierung, so gilt für jede <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df88c6333caaf6471cf277f24b802ff9931b133e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{M}\mathrm {d} \omega =\int _{\partial M}\omega .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d544e784b04c889e8a3afbb7a1ed3ea5efe1f554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.139ex; height:5.676ex;" alt="{\displaystyle \int _{M}\mathrm {d} \omega =\int _{\partial M}\omega .}" loading="lazy"></dd></dl>

Dieser Satz ist eine weitreichende Verallgemeinerung des <a href="Fundamentalsatz_der_Analysis" title="Fundamentalsatz der Analysis">Hauptsatzes der Differential- und Integralrechnung</a>. Er enthält als Spezialfälle den <a href="Gau%C3%9Fscher_Integralsatz" title="Gaußscher Integralsatz">gaußschen Integralsatz</a> und den <a href="Satz_von_Stokes#Klassischer_Integralsatz_von_Stokes" title="Satz von Stokes">klassischen Integralsatz von Stokes</a>.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> <a href="Geschlossene_Mannigfaltigkeit" title="Geschlossene Mannigfaltigkeit">geschlossen</a>, das heißt, gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial M=\emptyset ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial M=\emptyset ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42177fe84210803b26b7f2fe68005d515829407a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.668ex; height:2.676ex;" alt="{\displaystyle \partial M=\emptyset ,}" loading="lazy"> so folgt für jede exakte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\text{exakt}},}">

<msup>

<mtext>exakt</mtext>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ^{\text{exakt}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af10c0563d44f1c412852b6aa78c5bdaa0285270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.253ex; height:3.009ex;" alt="{\displaystyle \omega ^{\text{exakt}},}" loading="lazy"> d. h. für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \equiv \omega ^{\text{exakt}}=\mathrm {d} \varphi ,}">

<msup>

<mtext>exakt</mtext>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \equiv \omega ^{\text{exakt}}=\mathrm {d} \varphi ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2de69a8bf01f697f6f4182f94cec72cf24f52f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.708ex; height:3.176ex;" alt="{\displaystyle \omega \equiv \omega ^{\text{exakt}}=\mathrm {d} \varphi ,}" loading="lazy"> die Beziehung

<msub>

</mrow>

</msub>

<msup>

<mtext>exakt</mtext>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{M}\omega ^{\text{exakt}}=\int _{M}\mathrm {d} \varphi =\int _{\partial M=\emptyset }\varphi =0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e3ba55818f27a61ddb1df0698df4e3e1d04528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.992ex; height:5.843ex;" alt="{\displaystyle \int _{M}\omega ^{\text{exakt}}=\int _{M}\mathrm {d} \varphi =\int _{\partial M=\emptyset }\varphi =0.}" loading="lazy"></dd></dl>

Zur Verdeutlichung der genannten Eigenschaft von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> benutzt man oft die Formulierung mit einem Kreis-Integral:

<msub>

</mrow>

</msub>

<msup>

<mtext>exakt</mtext>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \oint _{M}\omega ^{\text{exakt}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45da789820f3d4d2d36a93af9c01d1e8004f714a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.505ex; height:5.676ex;" alt="{\displaystyle \oint _{M}\omega ^{\text{exakt}}=0}" loading="lazy"></dd></dl>

Das Integral liefert eine Abbildung

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{\mathrm {dR} }^{n}(M)\to \mathbb {R} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0da9e009af2cc1eb6ec994fc69e61bc860f3302b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.289ex; height:2.843ex;" alt="{\displaystyle \mathrm {H} _{\mathrm {dR} }^{n}(M)\to \mathbb {R} .}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> <a href="Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">zusammenhängend</a>, so ist diese Abbildung ein <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>. Man kommt damit zur <a href="De-Rham-Kohomologie" title="De-Rham-Kohomologie">De-Rham-Kohomologie</a> zurück (s. o.).

<div class="mw-heading mw-heading2"><h2 id="Rechenbeispiele">Rechenbeispiele</h2></div>

Auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> mit den <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22a8c93372e8f8b6e24d523bd5545aed3430baf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" loading="lazy"> seien die 1-Form

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =z^{2}\,\mathrm {d} x+2y\,\mathrm {d} y+xz\,\mathrm {d} z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/639543c3bf75c29fec00df63289fa45d9b2ce459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.718ex; height:3.009ex;" alt="{\displaystyle \omega =z^{2}\,\mathrm {d} x+2y\,\mathrm {d} y+xz\,\mathrm {d} z}" loading="lazy"></dd></dl>

und die 2-Form

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu =z\,\mathrm {d} z\wedge \mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a524f6317e9014e2f4022af3677753f281f8366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.391ex; height:2.176ex;" alt="{\displaystyle \nu =z\,\mathrm {d} z\wedge \mathrm {d} x}" loading="lazy"></dd></dl>

gegeben.

Für das äußere Produkt gilt:

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munder>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \wedge \nu =z^{3}\,\underbrace {\mathrm {d} x\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,0}+2yz\,\underbrace {\mathrm {d} y\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z}+xz^{2}\,\underbrace {\mathrm {d} z\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,0}=2yz\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/527ac7cc47b652b981d88415a320806f785eb81b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:81.689ex; height:6.843ex;" alt="{\displaystyle \omega \wedge \nu =z^{3}\,\underbrace {\mathrm {d} x\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,0}+2yz\,\underbrace {\mathrm {d} y\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z}+xz^{2}\,\underbrace {\mathrm {d} z\wedge \mathrm {d} z\wedge \mathrm {d} x} _{=\,0}=2yz\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z}" loading="lazy"></dd></dl>

Die äußere Ableitung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> ergibt

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega =2z\,\mathrm {d} z\wedge \mathrm {d} x+2\,\mathrm {d} y\wedge \mathrm {d} y+(z\,\mathrm {d} x+x\,\mathrm {d} z)\wedge \mathrm {d} z=2z\,\mathrm {d} z\wedge \mathrm {d} x-z\,\mathrm {d} z\wedge \mathrm {d} x=z\,\mathrm {d} z\wedge \mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a827667db404da145c52c20f9a38059293ea98de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.959ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} \omega =2z\,\mathrm {d} z\wedge \mathrm {d} x+2\,\mathrm {d} y\wedge \mathrm {d} y+(z\,\mathrm {d} x+x\,\mathrm {d} z)\wedge \mathrm {d} z=2z\,\mathrm {d} z\wedge \mathrm {d} x-z\,\mathrm {d} z\wedge \mathrm {d} x=z\,\mathrm {d} z\wedge \mathrm {d} x}" loading="lazy">,</dd></dl>

also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \omega =\nu }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \omega =\nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2498306e2873921db8182753ae528c4b0b8ad1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.069ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \omega =\nu }" loading="lazy">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> exakt und folglich geschlossen, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \nu =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \nu =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2dfacfbdfb150df360fb2b668b0d1eadacad545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.786ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \nu =0}" loading="lazy">. Das lässt sich auch durch direkte Rechnung überprüfen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \nu =\mathrm {d} z\wedge \mathrm {d} z\wedge \mathrm {d} x=0}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \nu =\mathrm {d} z\wedge \mathrm {d} z\wedge \mathrm {d} x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85d8f0ae227f30fa998c89c889e58a8251e1066c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.433ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \nu =\mathrm {d} z\wedge \mathrm {d} z\wedge \mathrm {d} x=0}" loading="lazy">.

Sei weiter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\colon [0,1]\to \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\colon [0,1]\to \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df13fe84b1c39999f3dd19c35cabe2e17bea1041.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.04ex; height:3.176ex;" alt="{\displaystyle c\colon [0,1]\to \mathbb {R} ^{3}}" loading="lazy"> gegeben durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(t)=(t^{2},2t,1)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(t)=(t^{2},2t,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84c41f58121df5445d297784b24d867bad35d082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.69ex; height:3.176ex;" alt="{\displaystyle c(t)=(t^{2},2t,1)}" loading="lazy">, dann folgt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=t^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=t^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e09aad4237419eead99c52a3e0fa5da28228e529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.322ex; height:2.676ex;" alt="{\displaystyle x=t^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=2t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47dbc75fbb595b3afa6c62ae1719755e07417c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.256ex; height:2.509ex;" alt="{\displaystyle y=2t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/078535cde78d90bfa1d9fbb2446204593a921d57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x=2t\,\mathrm {d} t}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x=2t\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4879e05cf9169dc63d0d1214bb9d61ca1402f7e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.242ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x=2t\,\mathrm {d} t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} y=2\,\mathrm {d} t}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} y=2\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20fbd1a5ce3995ccbde145052d4adced83df9b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.228ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} y=2\,\mathrm {d} t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} z=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} z=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef212999fcfccdda70d0d647325a54cb87f1f6dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.642ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} z=0}" loading="lazy"> für die auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"> zurückgeholte Form:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{*}\omega =2t\,\mathrm {d} t+8t\,\mathrm {d} t=10t\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/183a3d4fdeb61c85ce73e27297c0fdb7de2532dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.271ex; height:2.509ex;" alt="{\displaystyle c^{*}\omega =2t\,\mathrm {d} t+8t\,\mathrm {d} t=10t\,\mathrm {d} t}" loading="lazy"></dd></dl>

Für das Integral von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> über die durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> gegebene Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =c([0,1])}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma =c([0,1])}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5f8c8d8fd83a20438d8dcd4950531edd382e138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.02ex; height:2.843ex;" alt="{\displaystyle \Gamma =c([0,1])}" loading="lazy"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> ergibt sich somit

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\Gamma }\omega =\int _{[0,1]}c^{*}\omega =\int _{0}^{1}10t\,\mathrm {d} t=5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/888dc9660062f9e321e65f3afca234418bea2ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.867ex; height:6.509ex;" alt="{\displaystyle \int _{\Gamma }\omega =\int _{[0,1]}c^{*}\omega =\int _{0}^{1}10t\,\mathrm {d} t=5}" loading="lazy">.</dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}=1\}}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{2}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4fe8bd67e7ba169b19dbffe816b882458ebbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.842ex; height:3.343ex;" alt="{\displaystyle S^{2}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}=1\}}" loading="lazy"> die <a href="Einheitssph%C3%A4re" class="mw-redirect" title="Einheitssphäre">Einheitssphäre</a> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b6401d5d0155afb1406770d1eb80badce4e08ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}" loading="lazy"> der Rand der <a href="Einheitskugel" title="Einheitskugel">Einheitskugel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{3}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}<1\}}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle B^{3}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}<1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52731adf20e613d90297208114640ea6207bd3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.085ex; height:3.343ex;" alt="{\displaystyle B^{3}=\{\mathbf {x} \in \mathbb {R} ^{3}\ {\big |}\ \|\mathbf {x} \|_{2}<1\}}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}=\partial B^{3}}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{2}=\partial B^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36be6c626616b495a0144de6b9f43ecf5b02e38f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.811ex; height:2.676ex;" alt="{\displaystyle S^{2}=\partial B^{3}}" loading="lazy">. Nach dem Satz von Stokes gilt also wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \nu =0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \nu =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2dfacfbdfb150df360fb2b668b0d1eadacad545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.786ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \nu =0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \int _{S^{2}}\nu =\int _{B^{3}}\mathrm {d} \nu =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/532a294644bd9ebcaade7871230a3ec5f8900dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.926ex; height:5.676ex;" alt="{\displaystyle \int _{S^{2}}\nu =\int _{B^{3}}\mathrm {d} \nu =0}" loading="lazy">.</dd></dl>

Die 3-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \wedge \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \omega \wedge \nu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ac4ce657a639dd0d994d54b1eca0f6c9298576b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.261ex; height:2.009ex;" alt="{\displaystyle \omega \wedge \nu }" loading="lazy"> kann beispielsweise über den Einheitswürfel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=[0,1]^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle W=[0,1]^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9149f274e23362ba7d6ef68debd748dcc43eaacc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:3.176ex;" alt="{\displaystyle W=[0,1]^{3}}" loading="lazy"> integriert werden. Ihr Integral stimmt mit dem Lebesgue-Integral der Koeffizientenfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)\mapsto 2yz}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (x,y,z)\mapsto 2yz}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47f32100175b095229400555380b6db7b26dd3a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.471ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\mapsto 2yz}" loading="lazy"> überein:

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</mrow>

</mrow>

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</mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \int _{W}\omega \wedge \nu =\int _{W}2yz\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}2yz\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z=2\cdot \int _{0}^{1}y\,\mathrm {d} y\cdot \int _{0}^{1}z\,\mathrm {d} z={\frac {1}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40b242b390ab9ded11787e72d8fb05bee43027d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:87.243ex; height:6.176ex;" alt="{\displaystyle \int _{W}\omega \wedge \nu =\int _{W}2yz\,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}2yz\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z=2\cdot \int _{0}^{1}y\,\mathrm {d} y\cdot \int _{0}^{1}z\,\mathrm {d} z={\frac {1}{2}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Komplexe_Differentialformen">Komplexe Differentialformen</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Komplexe_Differentialform" title="Komplexe Differentialform">Komplexe Differentialform</a></div>

In der Theorie der komplexen Differentialformen wird der hier eingeführte Kalkül auf <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexe Mannigfaltigkeiten</a> übertragen. Dies funktioniert größtenteils analog zur Definition der hier beschriebenen Formen. Jedoch werden hier analog zu den komplexen Zahlen die Räume der komplexen Differentialformen in zwei Räume (reeller) Differentialformen

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{r}(M)\cong \Omega ^{r,r}(M)=\Omega ^{r}(M)\oplus i\Omega ^{r}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69bcc9a32f81031472a5f8b4bb83c5908a14d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.31ex; height:2.843ex;" alt="{\displaystyle {\mathcal {E}}^{r}(M)\cong \Omega ^{r,r}(M)=\Omega ^{r}(M)\oplus i\Omega ^{r}(M)}" loading="lazy"></dd></dl>

zerlegt. Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{p,q}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{p,q}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7e0a1970d3d159455fa85889c94fd85d830534c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.951ex; height:2.343ex;" alt="{\displaystyle \Omega ^{p,q}}" loading="lazy"> heißt dann der Raum der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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<annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9769c58523b9b639866a2d48e657d9c26911143a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}" loading="lazy">-Formen. Auf diesen Räumen kann man analog zur äußeren Ableitung zwei neue Ableitungen definieren. Diese werden <a href="Dolbeault-Operator" class="mw-redirect" title="Dolbeault-Operator">Dolbeault-</a> und Dolbeault-Quer-Operator genannt, und analog zur De-Rham-Kohomologie kann man mit Hilfe des Dolbeault-Quer-Operators wieder eine <a href="Kohomologie" title="Kohomologie">Kohomologie</a> bilden. Diese heißt <a href="Dolbeault-Kohomologie" title="Dolbeault-Kohomologie">Dolbeault-Kohomologie</a>.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Vektorwertige_Differentialformen" title="Vektorwertige Differentialformen">Vektorwertige Differentialformen</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Herbert Amann, <a href="Joachim_Escher_(Mathematiker)" title="Joachim Escher (Mathematiker)">Joachim Escher</a>: Analysis III. 2. Auflage. Birkhäuser, Basel 2008, ISBN 978-3-7643-8883-6, Kapitel XI und XII.</li>

<li><a href="Henri_Cartan" title="Henri Cartan">Henri Cartan</a>: Differentialformen. Bibliographisches Institut, Mannheim 1974, ISBN 3-411-01443-1.</li>

<li><a href="Klaus_J%C3%A4nich" title="Klaus Jänich">Klaus Jänich</a>: Vektoranalysis. 5. Auflage. Springer, Berlin / Heidelberg 2005, ISBN 978-3-540-27338-7.</li>

<li>Shigeyuki Morita: Geometry of differential forms. American Mathematical Society, 2001, ISBN 0821810456.</li>

<li><a href="Harley_Flanders" title="Harley Flanders">Harley Flanders</a>: Differential forms with applications to the physical sciences. Academic Press, 1963.</li>

<li><a href="Harold_Edwards_(Mathematiker)" title="Harold Edwards (Mathematiker)">Harold Edwards</a>: Advanced Calculus – a differential forms approach. Birkhäuser, 1994 (zuerst 1969).</li>

<li><a href="Steven_H._Weintraub" class="mw-redirect" title="Steven H. Weintraub">Steven H. Weintraub</a>: Differential Forms – a complement to vector calculus. Academic Press, 1997.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li>Gunnar Fløystad: <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201504/rnoti-p364.pdf">The Exterior Algebra and Central Notions in Mathematics</a> (PDF; 182 kB). Notices AMS, Band 62, 2015, Nr. 4.</li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/differential+form">Differential Form</a> in nLab.</li></ul></div><div><div class="zim-footer">

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