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<title>Greensche Formeln</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Greensche Formeln</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">In der <a href="Mathematik" title="Mathematik">Mathematik</a>, speziell der <a href="Vektoranalysis" title="Vektoranalysis">Vektoranalysis</a>, sind die beiden greenschen Formeln (manchmal auch greensche Identitäten, greensche Sätze oder Theoreme) spezielle Anwendungen des <a href="Gau%C3%9Fscher_Integralsatz" title="Gaußscher Integralsatz">gaußschen Integralsatzes</a>. Sie sind benannt nach dem Mathematiker <a href="George_Green" title="George Green">George Green</a>. Anwendung finden sie unter anderem in der <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a> bei der Berechnung von Potentialen. Die Formeln sind nicht zu verwechseln mit dem <a href="Satz_von_Green" title="Satz von Green">Satz von Green</a>, bei dem es um ebene Integrale geht.

Im Folgenden sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1caefb347c86337ea7cd0c354acd2294bd7d81d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U\subset \mathbb {R} ^{n}}" loading="lazy"> kompakt mit abschnittweise glattem Rand und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> seien zwei Funktionen auf <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">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> einfach und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> zweifach <a href="Differentialrechnung" title="Differentialrechnung">stetig differenzierbar</a> sei. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"> ist der <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a>.

<div class="mw-heading mw-heading2"><h2 id="Erste_Greensche_Identität">Erste Greensche Identität</h2></div>

</mrow>

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<msup>

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<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{U}(\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} U=\int \limits _{\partial U}\phi {\frac {\partial \psi }{\partial n}}\mathrm {d} S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03b4dca2c1def6a2aa86610a78c14475a89d9fb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.02ex; width:38.187ex; height:7.843ex;" alt="{\displaystyle \int \limits _{U}(\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} U=\int \limits _{\partial U}\phi {\frac {\partial \psi }{\partial n}}\mathrm {d} S}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial \psi }{\partial n}}=\nabla \psi \cdot {\vec {n}}}">

<mfrac>

<mrow>

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<mrow>

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</mfrac>

</mstyle>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial \psi }{\partial n}}=\nabla \psi \cdot {\vec {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/962d53a1d8cbe3955009f606f1c87c787322858f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.459ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\partial \psi }{\partial n}}=\nabla \psi \cdot {\vec {n}}}" loading="lazy"> die Normalenableitung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">, also die <a href="Normalkomponente" title="Normalkomponente">Normalkomponente</a> des Gradienten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> auf dem Rand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial U}">

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

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Diese Identität lässt sich wie folgt beweisen:

</mrow>

</munder>

<mfrac>

<mrow>

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<mrow>

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</mrow>

</mrow>

</mrow>

</munder>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

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</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\int \limits _{\partial U}\phi {\frac {\partial \psi }{\partial n}}\mathrm {d} S=\int \limits _{\partial U}(\phi \,\nabla \psi )\cdot {\vec {n}}\,\mathrm {d} S=\int \limits _{U}\nabla \cdot (\phi \,\nabla \psi )\,\mathrm {d} U=\int \limits _{U}(\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21d12eaa4b04dfc5475b03f902450d3d66410608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:77.176ex; height:7.843ex;" alt="{\displaystyle \;\int \limits _{\partial U}\phi {\frac {\partial \psi }{\partial n}}\mathrm {d} S=\int \limits _{\partial U}(\phi \,\nabla \psi )\cdot {\vec {n}}\,\mathrm {d} S=\int \limits _{U}\nabla \cdot (\phi \,\nabla \psi )\,\mathrm {d} U=\int \limits _{U}(\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} U}" loading="lazy">,</dd></dl>

wobei im zweiten Schritt der <a href="Gau%C3%9Fscher_Integralsatz" title="Gaußscher Integralsatz">gaußsche Integralsatz</a> in der Form

</mrow>

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<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</munder>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\int \limits _{\partial U}{\vec {F}}\cdot {\vec {n}}\;\mathrm {d} S=\int \limits _{U}\nabla \cdot {\vec {F}}\;\mathrm {d} U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a079e843d116840e10fa6fda8a167b0b42d6aaf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.293ex; height:7.509ex;" alt="{\displaystyle \;\int \limits _{\partial U}{\vec {F}}\cdot {\vec {n}}\;\mathrm {d} S=\int \limits _{U}\nabla \cdot {\vec {F}}\;\mathrm {d} U}" loading="lazy"></dd></dl>

benutzt wurde.

<div class="mw-heading mw-heading2"><h2 id="Zweite_Greensche_Identität">Zweite Greensche Identität</h2></div>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

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<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{U}(\phi \nabla ^{2}\psi -\psi \nabla ^{2}\phi )\,\mathrm {d} U=\int \limits _{\partial U}\left(\phi {\frac {\partial \psi }{\partial n}}-\psi {\frac {\partial \phi }{\partial n}}\right)\,\mathrm {d} S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9db2f9f4511850c5b48ee0c382c6566c250262c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.02ex; width:47.724ex; height:7.843ex;" alt="{\displaystyle \int \limits _{U}(\phi \nabla ^{2}\psi -\psi \nabla ^{2}\phi )\,\mathrm {d} U=\int \limits _{\partial U}\left(\phi {\frac {\partial \psi }{\partial n}}-\psi {\frac {\partial \phi }{\partial n}}\right)\,\mathrm {d} S}" loading="lazy"></dd></dl>

Die zweite greensche Identität folgt aus der ersten greenschen Identität, wobei nun vorausgesetzt wird, dass auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

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<mrow>

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</mfrac>

</mrow>

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</mstyle>

</mrow>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f9dc0bcaf1230d3c02f9474ac3c0de25df07cb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.02ex; width:38.574ex; height:7.843ex;" alt="{\displaystyle \int \limits _{U}(\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} U=\int \limits _{\partial U}\phi {\frac {\partial \psi }{\partial n}}\,\mathrm {d} S}" loading="lazy">,</dd></dl>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

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<mfrac>

<mrow>

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<mrow>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{U}(\psi \nabla ^{2}\phi +\nabla \psi \cdot \nabla \phi )\,\mathrm {d} U=\int \limits _{\partial U}\psi {\frac {\partial \phi }{\partial n}}\,\mathrm {d} S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9588d3a48e968c0e4301e4111f1f0eb026077e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.02ex; width:38.584ex; height:7.843ex;" alt="{\displaystyle \int \limits _{U}(\psi \nabla ^{2}\phi +\nabla \psi \cdot \nabla \phi )\,\mathrm {d} U=\int \limits _{\partial U}\psi {\frac {\partial \phi }{\partial n}}\,\mathrm {d} S}" loading="lazy"></dd></dl>

Subtrahiert man nun die zweite Gleichung von der ersten Gleichung, so ergibt sich die zweite greensche Identität.

<div class="mw-heading mw-heading2"><h2 id="Anwendungen_in_der_Elektrostatik_(3D)">Anwendungen in der Elektrostatik (3D)</h2></div>

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

Für ein elektrostatisches Potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0ab1c89aaae9805fd0a213f2ffc0dd555f5307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi \!\,}" loading="lazy"> gilt die <a href="Poissongleichung" class="mw-redirect" title="Poissongleichung">Poissongleichung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\phi =-4\pi \rho }">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\phi =-4\pi \rho }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99a2a78246e34558ebbb042138fecc8c78a5db67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.979ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\phi =-4\pi \rho }" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \!\,}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dfe7f773266aa21dd37387252a399dbeebf0f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho \!\,}" loading="lazy"> die elektrische Ladungsdichte ist (<a href="Gau%C3%9Fsches_Einheitensystem" title="Gaußsches Einheitensystem">gaußsches Einheitensystem</a>). Wenn in einem Volumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\!\,}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dcb8c83fc647d606320e0f20ab8ba74f4ba109b.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 Ladungsdichte gegeben ist, und wenn zusätzlich auf dem Rand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1360e3416408f1420e852179eb2552c020599426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.176ex;" alt="{\displaystyle \partial U}" loading="lazy"> die Werte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

</mstyle>

</mrow>

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

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<dl><dd>Innerhalb 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_/eb734a37dd21ce173a46342d1cc64c92/2dcb8c83fc647d606320e0f20ab8ba74f4ba109b.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"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ({\vec {r}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})}</annotation>

</semantics>

Beweis: Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{1}({\vec {r}})}">

<msub>

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<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle \phi _{1}({\vec {r}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93a36b6c8b6a0f26441832e6d68ab23601eef983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle \phi _{1}({\vec {r}})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{2}({\vec {r}})}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi _{2}({\vec {r}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67b9a53c7db6eb82103c681387e3cea16e5a23a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle \phi _{2}({\vec {r}})}" loading="lazy"> zwei Potentiale, die dieselben Vorgaben über Ladungsdichte und Randwerte erfüllen. Für die Differenzfunktion gilt dann

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<msub>

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</msub>

<mover>

</mover>

</mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi ({\vec {r}}):=\phi _{1}({\vec {r}})-\phi _{2}({\vec {r}})\qquad {\begin{array}{rl}\nabla ^{2}\chi ({\vec {r}})=0&{\vec {r}}\in U\\\chi ({\vec {r}})=0&{\vec {r}}\in \partial U\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0387e92c3d7db7e0217abfadcc505501456daf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.639ex; height:6.509ex;" alt="{\displaystyle \chi ({\vec {r}}):=\phi _{1}({\vec {r}})-\phi _{2}({\vec {r}})\qquad {\begin{array}{rl}\nabla ^{2}\chi ({\vec {r}})=0&{\vec {r}}\in U\\\chi ({\vec {r}})=0&{\vec {r}}\in \partial U\end{array}}}" loading="lazy"></dd></dl>

Setzt man <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_/eb734a37dd21ce173a46342d1cc64c92/294aaa5d04250dacb8ce55e5072fd4738a9e47b1.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"> in der ersten greenschen Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0ab1c89aaae9805fd0a213f2ffc0dd555f5307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi \!\,}" loading="lazy"> und auch für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \!\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a439446e92271c848e5171e877ba69d9015c661e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi \!\,}" loading="lazy"> ein, so folgt

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{U}\nabla \chi \cdot \nabla \chi \,\mathrm {d} U=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eabdbdece18e417c4b1cbee7d15d07360ca39628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.357ex; height:5.676ex;" alt="{\displaystyle \int _{U}\nabla \chi \cdot \nabla \chi \,\mathrm {d} U=0}" loading="lazy"></dd></dl>

Also muss der Gradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \chi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d23bc4172b63770da6f6a37135533513bb299bf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.391ex; height:2.509ex;" alt="{\displaystyle \nabla \chi }" loading="lazy"> überall in <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_/eb734a37dd21ce173a46342d1cc64c92/2dcb8c83fc647d606320e0f20ab8ba74f4ba109b.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"> verschwinden, somit <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_/eb734a37dd21ce173a46342d1cc64c92/20ccded05dc5366a538db649e7cb6648c02896f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.842ex; height:2.009ex;" alt="{\displaystyle \chi \,}" loading="lazy"> konstant sein, und wegen seines Null-Randwerts sogar konstant gleich null sein. Also gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{1}({\vec {r}})=\!\,\phi _{2}({\vec {r}})}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{1}({\vec {r}})=\!\,\phi _{2}({\vec {r}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38bcb4860a3fd5ee9b47b13dcda6545477224bf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.043ex; height:2.843ex;" alt="{\displaystyle \phi _{1}({\vec {r}})=\!\,\phi _{2}({\vec {r}})}" loading="lazy"> innerhalb von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\!\,}">

</mstyle>

</mrow>

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

</semantics>

<a href="Nota_bene" class="mw-redirect" title="Nota bene">N.B.</a> Bei dem Beweis wird die Poissongleichung und somit die Ladungsdichte nur innerhalb von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\!\,}">

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Abschirmung_durch_geschlossene_Leiterfläche">Abschirmung durch geschlossene Leiterfläche</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1360e3416408f1420e852179eb2552c020599426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.176ex;" alt="{\displaystyle \partial U}" loading="lazy"> sei eine geschlossene Leiterfläche, so dass das elektrostatische Potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0ab1c89aaae9805fd0a213f2ffc0dd555f5307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi \!\,}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1360e3416408f1420e852179eb2552c020599426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.176ex;" alt="{\displaystyle \partial U}" loading="lazy"> einen konstanten Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}\!\,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89e272523e8fcce7a2235cc0daa805c0dae40f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.44ex; height:2.509ex;" alt="{\displaystyle \phi _{0}\!\,}" loading="lazy"> hat (<a href="%C3%84quipotentialfl%C3%A4che" title="Äquipotentialfläche">Äquipotentialfläche</a>). Zum Beispiel lässt sich

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9865cd83838aca54c50f80f6d3fc8f478add259.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.701ex; height:2.509ex;" alt="{\displaystyle \phi _{0}\!\,=0}" loading="lazy"> physikalisch realisieren, indem die Leiterfläche geerdet wird. Nach dem Eindeutigkeitssatz ist der Potentialverlauf innerhalb von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\!\,}">

</mstyle>

</mrow>

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

</semantics>

</mstyle>

</mrow>

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

</semantics>

Wenn die geschlossene Leiterfläche nicht geerdet ist, dann sind die Randwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0ab1c89aaae9805fd0a213f2ffc0dd555f5307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi \!\,}" loading="lazy"> immer noch konstant, aber mit unbekanntem Wert. Dieser Wert kann davon abhängen, welche Ladungen außerhalb 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_/eb734a37dd21ce173a46342d1cc64c92/2dcb8c83fc647d606320e0f20ab8ba74f4ba109b.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"> vorhanden sind. Der Beweis des Eindeutigkeitssatzes lässt sich dahingehend verallgemeinern, dass die Differenzfunktion <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_/eb734a37dd21ce173a46342d1cc64c92/294aaa5d04250dacb8ce55e5072fd4738a9e47b1.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"> noch konstant in <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_/eb734a37dd21ce173a46342d1cc64c92/2dcb8c83fc647d606320e0f20ab8ba74f4ba109b.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">, aber nicht mehr gleich null ist. Für die elektrische Feldstärke, die durch Ableitungen aus dem Potential gewonnen wird, spielt die Konstante keine Rolle; die elektrische Feldstärke ist also auch ohne Erdung abgeschirmt.

<div class="mw-heading mw-heading3"><h3 id="Symmetrie_der_greenschen_Funktion">Symmetrie der greenschen Funktion</h3></div>

Die <a href="Greensche_Funktion" title="Greensche Funktion">greensche Funktion</a> mit <a href="Dirichlet-Randbedingung" title="Dirichlet-Randbedingung">Dirichlet-Randbedingung</a> und mit vektoriellem Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{1}\in U}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{1}\in U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71f9928eb2661172179378de392e3f8e027c7217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.901ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{1}\in U}" loading="lazy"> ist definiert durch

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{cl}\nabla ^{2}G_{D}({\vec {r}},{\vec {r}}_{1})=\delta ({\vec {r}}-{\vec {r}}_{1})&{\vec {r}}\in U\\G_{D}({\vec {r}},{\vec {r}}_{1})=0&{\vec {r}}\in \partial U\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c6fb085667ba0c8be452b4be4a1e749d7f461d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.289ex; height:6.509ex;" alt="{\displaystyle {\begin{array}{cl}\nabla ^{2}G_{D}({\vec {r}},{\vec {r}}_{1})=\delta ({\vec {r}}-{\vec {r}}_{1})&{\vec {r}}\in U\\G_{D}({\vec {r}},{\vec {r}}_{1})=0&{\vec {r}}\in \partial U\end{array}}}" loading="lazy"></dd></dl>

Bis auf einen Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4\pi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa0a7f74ab3340b3c6e11bedfff6fd38d806981a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.303ex; height:2.343ex;" alt="{\displaystyle -4\pi }" loading="lazy"> entspricht das der Poissongleichung für ein Potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ({\vec {r}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11af2cbe34341b0013f28e6e16c94b8d4b7c7603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \phi ({\vec {r}})}" loading="lazy">, das von einer <a href="Punktladung" title="Punktladung">Punktladung</a> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3846ad020fcaf0f97523dc407977b9f4f3e87e8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{1}}" loading="lazy"> erzeugt wird, und das auf der geerdeten Oberfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1360e3416408f1420e852179eb2552c020599426.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.176ex;" alt="{\displaystyle \partial U}" loading="lazy"> den Randwert 0 hat. Die Existenz einer solchen Funktion ist physikalisch klar, und wegen des Eindeutigkeitssatzes ist sie eindeutig bestimmt. Obwohl die Rollen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"> (Messpunkt) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{D}({\vec {r}}_{2},{\vec {r}}_{1})=G_{D}({\vec {r}}_{1},{\vec {r}}_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aae5506a79f9e6c686375af5d9e8034512479404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.733ex; height:2.843ex;" alt="{\displaystyle G_{D}({\vec {r}}_{2},{\vec {r}}_{1})=G_{D}({\vec {r}}_{1},{\vec {r}}_{2})}" loading="lazy"></dd></dl>

Beweis: Setzt man in der zweiten greenschen Formel

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})=G_{D}({\vec {r}},{\vec {r}}_{1})\qquad \psi ({\vec {r}})=G_{D}({\vec {r}},{\vec {r}}_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/353cb89439f6208bbe16a0e29265f4ce6e272ba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.332ex; height:2.843ex;" alt="{\displaystyle \phi ({\vec {r}})=G_{D}({\vec {r}},{\vec {r}}_{1})\qquad \psi ({\vec {r}})=G_{D}({\vec {r}},{\vec {r}}_{2})}" loading="lazy"></dd></dl>

so erhält man auf der linken Seite Integrale mit Delta-Funktionen, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{D}({\vec {r}}_{2},{\vec {r}}_{1})-G_{D}({\vec {r}}_{1},{\vec {r}}_{2})}">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{D}({\vec {r}}_{2},{\vec {r}}_{1})-G_{D}({\vec {r}}_{1},{\vec {r}}_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/054197ccfb87dbe315a60a23436cdbcd27f450b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.475ex; height:2.843ex;" alt="{\displaystyle G_{D}({\vec {r}}_{2},{\vec {r}}_{1})-G_{D}({\vec {r}}_{1},{\vec {r}}_{2})}" loading="lazy"> ergeben. Auf der rechten Seite verschwinden die Integranden wegen der Randwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{D}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f3f76e984daf24cd9a54cfe1d27f8e460efce18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.42ex; height:2.509ex;" alt="{\displaystyle G_{D}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Potential_ausgedrückt_durch_Ladungsdichte_und_Randwerte">Potential ausgedrückt durch Ladungsdichte und Randwerte</h3></div>

Verwendet man in der zweiten greenschen Formel als Integrationsvariable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{1}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3846ad020fcaf0f97523dc407977b9f4f3e87e8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{1}}" loading="lazy"> und lässt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \!\,}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef0ab1c89aaae9805fd0a213f2ffc0dd555f5307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi \!\,}" loading="lazy"> das elektrostatische Potential sein, so erhält man mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ({\vec {r}}_{1})=G_{D}({\vec {r}},{\vec {r}}_{1})}">

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<annotation encoding="application/x-tex">{\displaystyle \psi ({\vec {r}}_{1})=G_{D}({\vec {r}},{\vec {r}}_{1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83a2e26d2c99d9d5c1e4e88339a7956e28476337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.462ex; height:2.843ex;" alt="{\displaystyle \psi ({\vec {r}}_{1})=G_{D}({\vec {r}},{\vec {r}}_{1})}" loading="lazy"> und mit Hilfe der Symmetrie von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{D}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})=-4\pi \int _{U}G_{D}({\vec {r}},{\vec {r}}_{1})\rho ({\vec {r}}_{1})dU_{1}+\int _{\partial U}\phi ({\vec {r}}_{1}){\frac {\partial G_{D}({\vec {r}},{\vec {r}}_{1})}{\partial n_{1}}}dS_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe7b6e2f8756f59cb2e8b9cbc2cd225aad2cd7e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.317ex; height:6.176ex;" alt="{\displaystyle \phi ({\vec {r}})=-4\pi \int _{U}G_{D}({\vec {r}},{\vec {r}}_{1})\rho ({\vec {r}}_{1})dU_{1}+\int _{\partial U}\phi ({\vec {r}}_{1}){\frac {\partial G_{D}({\vec {r}},{\vec {r}}_{1})}{\partial n_{1}}}dS_{1}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Integralgleichung_für_das_Potential">Integralgleichung für das Potential</h3></div>

Unter Anwendung der oben gezeigten greenschen Formeln lassen sich Ausdrücke für das <a href="Elektrostatik#Potential_und_Spannung" title="Elektrostatik">elektrostatische Potential</a> einer Ladungsverteilung herleiten. Dabei sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\vec {r}}\,')}">

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<annotation encoding="application/x-tex">{\displaystyle \rho ({\vec {r}}\,')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a204fe3b89063e875b2025738c9fedd837f1265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.306ex; height:3.009ex;" alt="{\displaystyle \rho ({\vec {r}}\,')}" loading="lazy"> die <a href="Ladungsdichte" title="Ladungsdichte">Ladungsdichte</a> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}\,'}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68dbd76b6b7b5188666cbdd895cb7a5805897b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.295ex; height:2.509ex;" alt="{\displaystyle {\vec {r}}\,'}" loading="lazy">. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ({\vec {r}})}">

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<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11af2cbe34341b0013f28e6e16c94b8d4b7c7603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \phi ({\vec {r}})}" loading="lazy"> werde das Potenzial am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"> bezeichnet. Gesucht ist die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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Wir setzen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ({\vec {r}}\,')={\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi ({\vec {r}}\,')={\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4889838088e14d18bfed8f25a8764215f751263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.204ex; height:6.009ex;" alt="{\displaystyle \psi ({\vec {r}}\,')={\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}}" loading="lazy">. Es gilt dann:

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<annotation encoding="application/x-tex">{\displaystyle \Delta '\psi ({\vec {r}}\,')=\Delta '{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}=-4\pi \delta ({\vec {r}}-{\vec {r}}\,')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fe214e049d05fbb3c7864e2de12c31245924298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.063ex; height:6.009ex;" alt="{\displaystyle \Delta '\psi ({\vec {r}}\,')=\Delta '{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}=-4\pi \delta ({\vec {r}}-{\vec {r}}\,')}" loading="lazy">,

<dl><dd><ul><li>wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\nabla ^{2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30822e8684827364a4f67d1125a277cd3328ba42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:2.676ex;" alt="{\displaystyle \Delta =\nabla ^{2}}" loading="lazy"> der <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> ist,</li>

<li>der Strich anzeigt, dass dieser Operator auf die gestrichene Variable wirkt</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> die <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a> ist.</li></ul></dd>

<dd>Diese Identität ist also im Sinne von <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">distributionellen Ableitungen</a> zu verstehen.</dd></dl></li>

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<annotation encoding="application/x-tex">{\displaystyle \Delta '\phi ({\vec {r}}\,')=-4\pi \rho ({\vec {r}}\,')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83acafee65bccaf7d8a094bd7010e33de8659270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.817ex; height:3.009ex;" alt="{\displaystyle \Delta '\phi ({\vec {r}}\,')=-4\pi \rho ({\vec {r}}\,')}" loading="lazy"> mit der Ladungsverteilung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}\,'}">

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Setzen wir beides in die zweite greensche Identität ein, erhalten wir auf der linken Seite:

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{V}(\phi ({\vec {r}}\,')(-4\pi \delta ({\vec {r}}-{\vec {r}}\,'))-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}(-4\pi \rho ({\vec {r}}\,')))\,\mathrm {d} V'=-4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\,\mathrm {d} V'+4\pi \int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ccf44b793f61fe142c6198bf27cf6560fcdae22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.035ex; width:102.312ex; height:7.843ex;" alt="{\displaystyle \int \limits _{V}(\phi ({\vec {r}}\,')(-4\pi \delta ({\vec {r}}-{\vec {r}}\,'))-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}(-4\pi \rho ({\vec {r}}\,')))\,\mathrm {d} V'=-4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\,\mathrm {d} V'+4\pi \int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'}" loading="lazy">.</dd></dl>

Die rechte Seite der Identität ist:

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617bf8fc6fab88e23c511d1dc55c6501b4efcb8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.053ex; width:49.497ex; height:7.676ex;" alt="{\displaystyle \int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}" loading="lazy">.</dd></dl>

Als Identität geschrieben:

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<annotation encoding="application/x-tex">{\displaystyle -4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\,\mathrm {d} V'+4\pi \int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'=\int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e43a4fce2323b9b0745e538f37850e72558128c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:99.873ex; height:7.843ex;" alt="{\displaystyle -4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\,\mathrm {d} V'+4\pi \int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'=\int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}" loading="lazy">.</dd></dl>

Innerhalb des Volumens gilt an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"> wegen der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

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

</semantics>

</mrow>

</munder>

<mover>

</mover>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\mathrm {d} V'=-4\pi \phi ({\vec {r}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c525943d25b8b7c3d5a4089b20a5886bb078b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.689ex; height:7.343ex;" alt="{\displaystyle -4\pi \int \limits _{V}\phi ({\vec {r}}\,')\delta ({\vec {r}}-{\vec {r}}\,')\mathrm {d} V'=-4\pi \phi ({\vec {r}})}" loading="lazy"></dd></dl>

Damit können wir schließlich obige Identität nach dem Potential auflösen und erhalten:

<mover>

</mover>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ({\vec {r}})=\int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f7b4ae1c45a47865195db15ea4e900cf5cb1de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:78.869ex; height:7.843ex;" alt="{\displaystyle \phi ({\vec {r}})=\int \limits _{V}{\frac {\rho ({\vec {r}}\,')}{|{\vec {r}}-{\vec {r}}\,'|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\int \limits _{F}\left(\phi ({\vec {r}}\,'){\frac {\partial }{\partial n'}}{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}-{\frac {1}{|{\vec {r}}-{\vec {r}}\,'|}}{\frac {\partial }{\partial n'}}\phi ({\vec {r}}\,')\right)\mathrm {d} F'}" loading="lazy">.</dd></dl>

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

<ul><li>John David Jackson: Klassische Elektrodynamik. Walter de Gruyter, Berlin 2006, ISBN 3-11-018970-4</li>

<li>Walter Greiner: Theoretische Physik Band 3 – Klassische Elektrodynamik. Verlag Harri Deutsch, Frankfurt am Main, Thun ISBN 3-8171-1184-3</li>

<li><a href="Otto_Forster" title="Otto Forster">Otto Forster</a>: Analysis 3. Integralrechnung im Rn mit Anwendungen. 3. Aufl. Vieweg-Verlag, 1996, ISBN 3-528-27252-X</li></ul></div><div><div class="zim-footer">

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