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

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Die Poisson-Gleichung, benannt nach dem französischen <a href="Mathematiker" title="Mathematiker">Mathematiker</a> und <a href="Physiker" title="Physiker">Physiker</a> <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>, ist eine <a href="Elliptische_partielle_Differentialgleichung" title="Elliptische partielle Differentialgleichung">elliptische partielle Differentialgleichung</a> zweiter Ordnung, die als Teil von <a href="Randwertproblem" title="Randwertproblem">Randwertproblemen</a> in weiten Teilen der Physik Anwendung findet.

<div class="mw-heading mw-heading2"><h2 id="Mathematische_Formulierung">Mathematische Formulierung</h2></div>

Die Poisson-Gleichung lautet allgemein

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c1d2f433a6dc29a5230a45ed7609be30ea24cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; width:9.451ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle -\Delta u=f}" loading="lazy"></dd></dl>

Dabei bezeichnet

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.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 \Delta }" loading="lazy"> den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a></li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> die gesuchte Lösung</li>

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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 Funktion. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\equiv 0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3f8c50ad49a75e73abba74c35712a860f5bfa47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f\equiv 0}" loading="lazy">, so wird die Gleichung zur <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a>.</li></ul>

Um die Poisson-Gleichung zu lösen, müssen noch weitere Informationen gegeben sein, z. B. in Form einer <a href="Dirichlet-Randbedingung" title="Dirichlet-Randbedingung">Dirichlet-Randbedingung</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-\Delta u&=f&{\text{in}}&\Omega \\\quad u&=g&{\text{auf}}&\partial \Omega \end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baf0f296f107d3426e57e9c79285058b5ade0f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.431ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}-\Delta u&=f&{\text{in}}&\Omega \\\quad u&=g&{\text{auf}}&\partial \Omega \end{cases}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79774d994aac0be34ef390915fed12cbce816f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{n}}" loading="lazy"> offen und beschränkt.

In diesem Fall konstruiert man eine Lösung mithilfe der <a href="Fundamentall%C3%B6sung" title="Fundamentallösung">Fundamentallösung</a> <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/aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> der Laplace-Gleichung:

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<annotation encoding="application/x-tex">{\displaystyle \Phi (x):={\begin{cases}-{\dfrac {1}{2\pi }}\ln |x|&n=2\\{\dfrac {1}{(n-2)\omega _{n}}}\cdot {\dfrac {1}{|x|^{n-2}}}&n\geq 3\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f49cb3987ba211e3296cc51df95a64d336a38512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:38.523ex; height:11.509ex;" alt="{\displaystyle \Phi (x):={\begin{cases}-{\dfrac {1}{2\pi }}\ln |x|&n=2\\{\dfrac {1}{(n-2)\omega _{n}}}\cdot {\dfrac {1}{|x|^{n-2}}}&n\geq 3\end{cases}}}" loading="lazy"></dd></dl>

Dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{n}={\tfrac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{n}={\tfrac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5412a79bc1b3332e2f6e9f7c86af85b4dedbc837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.542ex; height:6.343ex;" alt="{\displaystyle \omega _{n}={\tfrac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}" loading="lazy"> den Flächeninhalt der <a href="Sph%C3%A4re_(Mathematik)" title="Sphäre (Mathematik)">Einheitssphäre</a> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<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 <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a>.

Durch die <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Phi *f)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7445addbd6b07985f5595727b0c141af37e6f41b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.961ex; height:2.843ex;" alt="{\displaystyle (\Phi *f)}" loading="lazy"> erhält man eine Lösung der Poisson-Gleichung.

Um auch die Randwertbedingung zu erfüllen, kann man die <a href="Greensche_Funktion#Partielle_Differentialgleichungen" title="Greensche Funktion">greensche Funktion</a> verwenden

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<annotation encoding="application/x-tex">{\displaystyle G(x,y):=\Phi (y-x)-\phi ^{x}(y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4e6aafa3be7fac4a13e047eb7557a0c45f9eacf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.076ex; height:2.843ex;" alt="{\displaystyle G(x,y):=\Phi (y-x)-\phi ^{x}(y)}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a4efcc1644682275e82a999e5ffdc7630c54315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.558ex; height:2.676ex;" alt="{\displaystyle \phi ^{x}}" loading="lazy"> ist dabei eine Korrekturfunktion, die

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\Delta \phi ^{x}=0&{\text{in}}\ \Omega \\\phi ^{x}=\Phi (y-x)&{\text{auf}}\ \partial \Omega \end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0019c9c405bea458e894cecc507d5aa8d1fac6b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.03ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\Delta \phi ^{x}=0&{\text{in}}\ \Omega \\\phi ^{x}=\Phi (y-x)&{\text{auf}}\ \partial \Omega \end{cases}}}" loading="lazy"></dd></dl></dd></dl>

erfüllt. Sie ist im Allgemeinen von <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_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.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 \Omega }" loading="lazy"> abhängig und nur für einfache Gebiete leicht zu finden.

Kennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x,y)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3197a4895281863ac09cb8c7c4371d21e93d11f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.155ex; height:2.843ex;" alt="{\displaystyle G(x,y)}" loading="lazy">, so ist eine Lösung des Randwertproblems von oben gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle u(x)=-\int _{\partial \Omega }g(y){\frac {\partial G(x,y)}{\partial n}}\mathrm {d} \sigma (y)+\int _{\Omega }f(y)G(x,y)\mathrm {d} y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a763596f2f170017caccdac93570e123a584984d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.555ex; height:6.176ex;" alt="{\displaystyle u(x)=-\int _{\partial \Omega }g(y){\frac {\partial G(x,y)}{\partial n}}\mathrm {d} \sigma (y)+\int _{\Omega }f(y)G(x,y)\mathrm {d} y}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> das <a href="Oberfl%C3%A4chenma%C3%9F" class="mw-redirect" title="Oberflächenmaß">Oberflächenmaß</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16feddaad462c2a1c9efdaeee062a0484a023fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }" loading="lazy"> bezeichne.

Die Lösung kann man auch mithilfe des Perron-Verfahrens oder mittels <a href="Variationsrechnung" title="Variationsrechnung">Variationsrechnung</a> finden.

<div class="mw-heading mw-heading2"><h2 id="Anwendungen_in_der_Physik">Anwendungen in der Physik</h2></div>

Der Poisson-Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \Phi (\mathbf {r} )=f(\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta \Phi (\mathbf {r} )=f(\mathbf {r} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08b084144964fcc0684715b4d50f967cd693de5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:13.814ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \Delta \Phi (\mathbf {r} )=f(\mathbf {r} )}" loading="lazy"> genügen beispielsweise das <a href="Coulombpotential" class="mw-redirect" title="Coulombpotential">elektrostatische Potential</a> und das <a href="Gravitationspotential" class="mw-redirect" title="Gravitationspotential">Gravitationspotential</a>, jeweils mit Formelzeichen <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/aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy">. Dabei ist die Funktion <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"> proportional zur elektrischen <a href="Ladungsdichte" title="Ladungsdichte">Ladungsdichte</a> bzw. zur <a href="Massendichte" class="mw-redirect" title="Massendichte">Massendichte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {r} ).}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} ).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e0da000b6b21e69b4a7715f17c0c05ff780d012.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.76ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} ).}" loading="lazy">

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {r} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb64b30ae67dec8ef9cb06c1d3537f00b9a7efed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.19ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {r} )}" loading="lazy"> überall bekannt, so ist die allgemeine Lösung der Poisson-Gleichung, die für große Abstände gegen Null geht, das Integral<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{4\,\pi }}\int \mathrm {d} ^{3}r'\,{\frac {f(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/405351a63c617c03af1c3db8c4bb1e9a1beff439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.508ex; height:6.509ex;" alt="{\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{4\,\pi }}\int \mathrm {d} ^{3}r'\,{\frac {f(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}}" loading="lazy">.</dd></dl>

In Worten: jede <a href="Ladung_(Physik)" title="Ladung (Physik)">Ladung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{3}r'\,f(\mathbf {r} ')}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{3}r'\,f(\mathbf {r} ')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ea0b29cb19d70d91366978657a6257b92b92c83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.342ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} ^{3}r'\,f(\mathbf {r} ')}" loading="lazy"> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} '}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1317b633a9366fab48e4b85ddec1cd1c0a8c31b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} '}" loading="lazy"> im

kleinen Gebiet der Größe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{3}r'}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{3}r'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bef930d406517d2df33110da85bb5f6d6c2c1b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.08ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} ^{3}r'}" loading="lazy"> trägt additiv bei zum 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/aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> mit ihrem elektrostatischen oder Gravitationspotential:

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} ^{3}r'\,f(\mathbf {r} ')}{4\,\pi \,|\mathbf {r} -\mathbf {r} '|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c713a3321338e54535602c4d0a77b5f77431e2db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.936ex; height:6.843ex;" alt="{\displaystyle -{\frac {\mathrm {d} ^{3}r'\,f(\mathbf {r} ')}{4\,\pi \,|\mathbf {r} -\mathbf {r} '|}}}" loading="lazy"></dd></dl></dd></dl>

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

Da das <a href="Elektrostatisches_Feld" class="mw-redirect" title="Elektrostatisches Feld">elektrostatische Feld</a> ein <a href="Konservative_Kraft" title="Konservative Kraft">konservatives Feld</a> ist, kann es über den <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}\Phi }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04e4f80de82c2c9eed28d700628ce3b46d42e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}\Phi }" loading="lazy"> eines <a href="Potential_(Physik)" title="Potential (Physik)">Potentials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0aae204acbbf8529f6aebad36b0968d6eb4d8658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.589ex; height:2.843ex;" alt="{\displaystyle \Phi (\mathbf {r} )}" loading="lazy"> ausgedrückt werden:

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )=-{\vec {\nabla }}\Phi (\mathbf {r} ).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c2ad1be88334be5907f289e61e9ad08d6069e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.747ex; height:3.343ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )=-{\vec {\nabla }}\Phi (\mathbf {r} ).}" loading="lazy"></dd></dl>

Mit Anwendung der <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> ergibt sich

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\cdot \mathbf {E} (\mathbf {r} )=-\Delta \Phi (\mathbf {r} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd8ec13d3695d9e24dd776f79d5be6993ca40ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.715ex; height:3.343ex;" alt="{\displaystyle {\vec {\nabla }}\cdot \mathbf {E} (\mathbf {r} )=-\Delta \Phi (\mathbf {r} )}" loading="lazy"></dd></dl>

mit dem <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.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 \Delta }" loading="lazy">.

Gemäß der ersten <a href="Maxwell-Gleichungen" title="Maxwell-Gleichungen">Maxwellgleichung</a> gilt jedoch auch

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f4595ece0f13ece3791e58fe097c1a6d8453e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.331ex; height:5.676ex;" alt="{\displaystyle {\vec {\nabla }}\cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon }}}" loading="lazy"></dd></dl>

mit

<ul><li>der <a href="Ladungsdichte" title="Ladungsdichte">Ladungsdichte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77f477411625125978c0a18946bdfae2c1f13bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}" loading="lazy"></li>

<li>der <a href="Permittivit%C3%A4t" title="Permittivität">Permittivität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\varepsilon _{\mathrm {r} }\cdot \varepsilon _{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon _{\mathrm {r} }\cdot \varepsilon _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5981eb78d62a83a780ffe137c49ec7b9ba68cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.959ex; height:2.009ex;" alt="{\displaystyle \varepsilon =\varepsilon _{\mathrm {r} }\cdot \varepsilon _{0}}" loading="lazy">.</li></ul>

Damit folgt für die Poisson-Gleichung des elektrischen Feldes

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \Phi (\mathbf {r} )=-{\frac {\rho (\mathbf {r} )}{\varepsilon }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a7629a539e556d5596c6fc37619141579169b11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -1.838ex; width:16.381ex; height:5.676ex;" aria-hidden="true" alt="{\displaystyle \Delta \Phi (\mathbf {r} )=-{\frac {\rho (\mathbf {r} )}{\varepsilon }}}" loading="lazy"></dd></dl>

Der Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {r} )=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e4134ae2b6c8b5de205eeaad70897f533bb0eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )=0}" loading="lazy"> für jeden Ort im betrachteten Gebiet wird als <a href="Laplace-Gleichung#Bedeutung_in_der_Physik" title="Laplace-Gleichung">Laplace-Gleichung der Elektrostatik</a> bezeichnet.

<div class="mw-heading mw-heading3"><h3 id="Elektrodynamik_stationärer_Ströme">Elektrodynamik stationärer Ströme</h3></div>

Als Beispiel wird hier der Emitter einer Silizium-<a href="Solarzelle" title="Solarzelle">Solarzelle</a> betrachtet, der in guter Näherung als rein zweidimensional beschrieben werden kann. Der Emitter befinde sich in der x-y-Ebene, die z-Achse zeige in die Basis hinein. Die laterale Flächenstromdichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} (x,y)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2abd306f2206ac9cea4d0721e32b1e575056f12e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.164ex; width:6.309ex; height:2.843ex;" alt="{\displaystyle \mathbf {j} (x,y)}" loading="lazy"> im Emitter hängt von der am Emitter auftretenden z-Komponente der (Volumen-)Stromdichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{z}(x,y,z=0)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{z}(x,y,z=0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94bf4e22d41cb9ff5c167786cd2d137b01c959d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.003ex; height:2.843ex;" alt="{\displaystyle J_{z}(x,y,z=0)}" loading="lazy"> der Basis ab, was durch die Kontinuitätsgleichung in der Form

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}_{2}\cdot \mathbf {j} (x,y)=-J_{z}(x,y,z=0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e281180041cf6ceaa18705a14d810872400d4f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.724ex; height:3.343ex;" alt="{\displaystyle {\vec {\nabla }}_{2}\cdot \mathbf {j} (x,y)=-J_{z}(x,y,z=0)}" loading="lazy"></dd></dl>

beschrieben werden kann (mit dem zweidimensionalen <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}_{2}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29ec1bb0625079fac1582b07e84d94cb1fcc22cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle {\vec {\nabla }}_{2}}" loading="lazy">). Die Flächenstromdichte hängt über das <a href="Ohmsches_Gesetz#Lokale_Betrachtungsweise" title="Ohmsches Gesetz">lokale ohmsche Gesetz</a> mit dem lateralen elektrischen Feld im Emitter zusammen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} (x,y)=R_{\Box }^{-1}\mathbf {E} (x,y)}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} (x,y)=R_{\Box }^{-1}\mathbf {E} (x,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cbc502c9ea71a6bfc3c2a081353b44898f30ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.164ex; width:20.589ex; height:3.343ex;" alt="{\displaystyle \mathbf {j} (x,y)=R_{\Box }^{-1}\mathbf {E} (x,y)}" loading="lazy">; hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\Box }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ea3fff17b9b8208b36f229ce7b7a8cd045fde40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.509ex;" alt="{\displaystyle R_{\Box }}" loading="lazy"> der als homogen angenommene spezifische <a href="Fl%C3%A4chenwiderstand" title="Flächenwiderstand">Flächenwiderstand</a> des Emitters. Schreibt man (wie im Abschnitt zur Elektrostatik diskutiert) das elektrische Feld als Gradient des elektrischen Potentials, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (x,y)=-{\vec {\nabla }}_{2}\Phi (x,y)}">

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (x,y)=-{\vec {\nabla }}_{2}\Phi (x,y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/563085f4d9285d145065b413ceab7e3ed85148fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.989ex; height:3.343ex;" alt="{\displaystyle \mathbf {E} (x,y)=-{\vec {\nabla }}_{2}\Phi (x,y)}" loading="lazy">, so erhält man für die Potentialverteilung im Emitter eine Poisson-Gleichung in der Form

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{2}\Phi (x,y)=R_{\Box }J_{z}(x,y,z=0).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e1961c0298995d8eaaa748b2651d7ca5888d2f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:31.02ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \Delta _{2}\Phi (x,y)=R_{\Box }J_{z}(x,y,z=0).}" loading="lazy"></dd></dl>

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

Ebenso wie das elektrostatische Feld

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\int \rho _{\mathrm {el} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-{\vec {\nabla }}\Phi _{\mathrm {el} }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6087e165febb9b4226b2743c439340b09b6247ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.607ex; height:6.676ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\int \rho _{\mathrm {el} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-{\vec {\nabla }}\Phi _{\mathrm {el} }(\mathbf {r} )}" loading="lazy">,

ist auch das <a href="Gravitationsbeschleunigung" class="mw-redirect" title="Gravitationsbeschleunigung">Gravitationsfeld</a> g ein konservatives Feld:

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} (\mathbf {r} )=-G\int \rho _{\mathrm {m} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-{\vec {\nabla }}\Phi _{\mathrm {m} }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c21b817207438acd849ced1db8dbe0dc7882e74f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.771ex; height:6.676ex;" alt="{\displaystyle \mathbf {g} (\mathbf {r} )=-G\int \rho _{\mathrm {m} }\left(\mathbf {r} '\right){\frac {\mathbf {r} -\mathbf {r} '}{\left|\mathbf {r} -\mathbf {r} '\right|^{3}}}\;dx'\,dy'\,dz'=-{\vec {\nabla }}\Phi _{\mathrm {m} }(\mathbf {r} )}" loading="lazy">.</dd></dl>

Dabei ist

<ul><li>G die <a href="Gravitationskonstante" title="Gravitationskonstante">Gravitationskonstante</a></li>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {m} }(\mathbf {r} ')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc780f1e029c09cb3d3af5c0b0ec9ba0470d4700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.399ex; height:3.009ex;" alt="{\displaystyle \rho _{\mathrm {m} }(\mathbf {r} ')}" loading="lazy"> die Massendichte.</li></ul>

Da nur die Ladungen durch Massen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e73b22db97f1e7c1ada8304e8d3e5ff63e98672a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.414ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon }}}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fb3535547044874239a93b1e765f0f71f7e610d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.635ex; height:2.343ex;" alt="{\displaystyle -G}" loading="lazy"> ersetzt werden, gilt analog zur ersten Maxwellgleichung

<mover>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\cdot \mathbf {g} =-4\pi G\rho _{\mathrm {m} }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7e0f2e8bf57285c5304a41eb5ed62af77c13e83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.894ex; height:3.343ex;" alt="{\displaystyle {\vec {\nabla }}\cdot \mathbf {g} =-4\pi G\rho _{\mathrm {m} }(\mathbf {r} )}" loading="lazy">.</dd></dl>

Damit ergibt sich die Poisson-Gleichung der Gravitation zu

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\cdot \mathbf {g} =-{\vec {\nabla }}\cdot ({\vec {\nabla }}\Phi _{\mathrm {m} }(\mathbf {r} ))=-4\pi G\rho _{\mathrm {m} }(\mathbf {r} )\Leftrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43d67eb6dffb5860b60184eae25ca4627331aa07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.32ex; height:3.343ex;" alt="{\displaystyle {\vec {\nabla }}\cdot \mathbf {g} =-{\vec {\nabla }}\cdot ({\vec {\nabla }}\Phi _{\mathrm {m} }(\mathbf {r} ))=-4\pi G\rho _{\mathrm {m} }(\mathbf {r} )\Leftrightarrow }" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \Phi _{\mathrm {m} }(\mathbf {r} )=4\pi G\rho _{\mathrm {m} }(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfed0ed3d9cb9f740e4d7e447474f9ba7250be6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:21.261ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \Delta \Phi _{\mathrm {m} }(\mathbf {r} )=4\pi G\rho _{\mathrm {m} }(\mathbf {r} )}" loading="lazy">.</dd></dl>

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

<ul><li><a href="Richard_Courant" title="Richard Courant">Richard Courant</a>, <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>: Methoden der mathematischen Physik. Band 1. Springer, Berlin u. a. 1924 (= Die Grundlehren der mathematischen Wissenschaften, 12). 4. Auflage, ebenda 1993, ISBN 3-540-56796-8.</li>

<li><a href="Lawrence_C._Evans" title="Lawrence C. Evans">Lawrence C. Evans</a>: Partial Differential Equations. American Mathematical Society, Providence RI 1998, ISBN 0-8218-0772-2 (= Graduate studies in mathematics 19).</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Wolfgang Nolting: <cite style="font-style:italic">Grundkurs theoretische Physik</cite>. [Online-ausg. der] 8. [gedr.] Auflage. 3. Elektrodynamik. Springer, Berlin, ISBN 978-3-540-71252-7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Poisson-Gleichung&rft.au=Wolfgang+Nolting&rft.btitle=Grundkurs+theoretische+Physik&rft.edition=%5BOnline-ausg.+der%5D+8.+%5Bgedr.%5D&rft.genre=book&rft.isbn=9783540712527&rft.place=Berlin&rft.pub=Springer&rft.volume=3.+Elektrodynamik." style="display:none">

</li>

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