ff1588d761d55f48f61a52249e1dcc24

Micron Document

<!DOCTYPE html>

<title>Laplace-Operator</title>

</head>

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

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der Laplace-Operator ist ein mathematischer <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operator</a>, der zuerst von <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> eingeführt wurde. Es handelt sich um einen linearen <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> innerhalb der <a href="Mehrdimensionale_Analysis" class="mw-redirect" title="Mehrdimensionale Analysis">mehrdimensionalen Analysis</a>. Er wird meist durch das Zeichen <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">, den Großbuchstaben <a href="Delta" title="Delta">Delta</a> des <a href="Griechisches_Alphabet" title="Griechisches Alphabet">griechischen Alphabets</a>, notiert.

Der Laplace-Operator kommt in vielen <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a> vor, die das Verhalten <a href="Feld_(Physik)" title="Feld (Physik)">physikalischer Felder</a> beschreiben. Beispiele sind die <a href="Poisson-Gleichung" title="Poisson-Gleichung">Poisson-Gleichung</a> der <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a>, die <a href="Navier-Stokes-Gleichungen" title="Navier-Stokes-Gleichungen">Navier-Stokes-Gleichungen</a> für Strömungen von Flüssigkeiten oder Gasen und die <a href="Diffusionsgleichung" class="mw-redirect" title="Diffusionsgleichung">Diffusionsgleichung</a> für die <a href="W%C3%A4rmeleitung" title="Wärmeleitung">Wärmeleitung</a>.

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

Der Laplace-Operator ordnet einem zweimal <a href="Differenzierbarkeit" title="Differenzierbarkeit">differenzierbaren</a> <a href="Skalarfeld" title="Skalarfeld">Skalarfeld</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> die <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> seines <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> zu,

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\operatorname {div} \left(\operatorname {grad} \,f\right),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7c769e4439e8c17f2f136b296c7635f46a8fcc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.518ex; height:2.843ex;" alt="{\displaystyle \Delta f=\operatorname {div} \left(\operatorname {grad} \,f\right),}" loading="lazy"></dd></dl>

oder mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> notiert

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\nabla \cdot (\nabla f)=(\nabla \cdot \nabla )f=\nabla ^{2}f.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b84dfc4ac9eda932f4232d408b583ca6bf5182c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.703ex; height:3.176ex;" alt="{\displaystyle \Delta f=\nabla \cdot (\nabla f)=(\nabla \cdot \nabla )f=\nabla ^{2}f.}" loading="lazy"></dd></dl>

Das formale „<a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a>“ des Nabla-Operators mit sich selbst ergibt also den Laplace-Operator. Vor allem im englischsprachigen Raum ist für den Laplace-Operator oft die Schreibweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}" loading="lazy"> zu finden.

Da der Divergenz-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/982dd33ed8ab34888860f2f24827fa15cd52e1eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.167ex; height:2.176ex;" alt="{\displaystyle \operatorname {div} }" loading="lazy"> und der Gradient-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78cb3c8b2bd5fdc5be8f1ce959b75a1a8bac5db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.529ex; height:2.509ex;" alt="{\displaystyle \operatorname {grad} }" loading="lazy"> unabhängig vom gewählten <a href="Koordinatensystem" title="Koordinatensystem">Koordinatensystem</a> sind, ist auch der Laplace-Operator unabhängig vom gewählten Koordinatensystem. Die Darstellung des Laplace-Operators in anderen Koordinatensystemen ergibt sich mit der Kettenregel aus der <a href="Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformation</a>.

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> ergibt sich in <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\sum _{k=1}^{n}{\partial ^{2}f \over \partial x_{k}^{2}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75746adeaf6e7aba567eaa2be0196fc03a6eb0c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.274ex; height:6.843ex;" alt="{\displaystyle \Delta f=\sum _{k=1}^{n}{\partial ^{2}f \over \partial x_{k}^{2}}.}" loading="lazy"></dd></dl>

In einer Dimension reduziert sich der Laplace-Operator somit auf die zweite Ableitung:

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/283f07c0087b7d644c870b5037b42558fe2cc9ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.771ex; height:2.843ex;" alt="{\displaystyle \Delta f=f''}" loading="lazy"></dd></dl>

Der Laplace-Operator einer Funktion kann auch als <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> ihrer <a href="Hesse-Matrix" title="Hesse-Matrix">Hesse-Matrix</a> dargestellt werden:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=\mathrm {Spur} (H(f))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8564d827117352aa75a8d3ac9882fb490c6c8502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.063ex; height:2.843ex;" alt="{\displaystyle \Delta f=\mathrm {Spur} (H(f))}" loading="lazy"></dd></dl>

Der Laplace-Operator kann auch auf <a href="Vektorfeld" title="Vektorfeld">Vektorfelder</a> angewendet werden. Mit dem <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadischen Produkt</a> „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy">“ wird mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

</mstyle>

</mrow>

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

</semantics>

</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">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {v}}:=(\nabla \cdot \nabla ){\vec {v}}=\nabla \cdot (\nabla \otimes {\vec {v}})=\operatorname {div(grad} ({\vec {v}})^{\top })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86d74e80642a8ca4b4c3875bf93aac9932896c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.965ex; height:3.176ex;" alt="{\displaystyle \Delta {\vec {v}}:=(\nabla \cdot \nabla ){\vec {v}}=\nabla \cdot (\nabla \otimes {\vec {v}})=\operatorname {div(grad} ({\vec {v}})^{\top })}" loading="lazy"></dd></dl>

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

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bbd6d9537e60be722c4093bd7c6e3cf655863e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.511ex; height:2.509ex;" alt="{\displaystyle {}^{\top }}" loading="lazy"> steht für <a href="Transponierte_Matrix" title="Transponierte Matrix">Transponierung</a>. In der Literatur findet sich auch ein Divergenz-Operator, der sein Argument gemäß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\widetilde {div}} T=\operatorname {div} (T^{\top })}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {\widetilde {div}} T=\operatorname {div} (T^{\top })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/016b5239ee5670b559a5ef5f79e20e5af07e4e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.684ex; height:3.343ex;" alt="{\displaystyle \operatorname {\widetilde {div}} T=\operatorname {div} (T^{\top })}" loading="lazy"> transponiert. Mit diesem Operator schreibt sich analog zum Skalarfeld:

<mover>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {v}}=\operatorname {{\widetilde {div}}(grad} \,{\vec {v}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13d5c2c3e718d88e0fbda3f59dda6e8f2343cd34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.853ex; height:3.343ex;" alt="{\displaystyle \Delta {\vec {v}}=\operatorname {{\widetilde {div}}(grad} \,{\vec {v}})}" loading="lazy"></dd></dl>

Speziell in drei Dimensionen gilt mit dem <a href="Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotationsoperator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rot} }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41e94439acfc407f2b156125b3b0870e28acb1a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.009ex;" alt="{\displaystyle \operatorname {rot} }" loading="lazy">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {v}}=(\nabla \cdot \nabla ){\vec {v}}=\nabla (\nabla \cdot {\vec {v}})-\nabla \times (\nabla \times {\vec {v}})=\operatorname {grad(div} ({\vec {v}}))-\operatorname {rot(rot} ({\vec {v}})),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8683b79c844dcf0a1cb8b4c88ee803a0e30ca91f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.584ex; height:2.843ex;" alt="{\displaystyle \Delta {\vec {v}}=(\nabla \cdot \nabla ){\vec {v}}=\nabla (\nabla \cdot {\vec {v}})-\nabla \times (\nabla \times {\vec {v}})=\operatorname {grad(div} ({\vec {v}}))-\operatorname {rot(rot} ({\vec {v}})),}" loading="lazy"></dd></dl>

was mit der <a href="Gra%C3%9Fmann-Identit%C3%A4t" class="mw-redirect" title="Graßmann-Identität">Graßmann-Identität</a> begründet werden kann. Letztere Formel definiert den sogenannten vektoriellen Laplace-Operator.<a href="#cite_note-1">[1]</a>

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

<div class="mw-heading mw-heading3"><h3 id="In_zwei_Dimensionen">In zwei Dimensionen</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> in <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> ergibt die Anwendung des Laplace-Operators

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd80db750cd1dc7b604643d9491c55cd1a52b88f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.851ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}.}" loading="lazy"></dd></dl>

In <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\varphi )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/deec3051ba5a9c7e5b676df779673dfb5e37a0a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle (r,\varphi )}" loading="lazy"> ergibt sich

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1a7735390f1e42abb70b6bd715f25ea384b05f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.605ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}}" loading="lazy"></dd></dl>

oder

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\,{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c884cf74ee6335e3bae8290a1656c1a48700c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.346ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\,{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="In_drei_Dimensionen">In drei Dimensionen</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> mit drei Variablen ergibt sich in <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}">

</mstyle>

</mrow>

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

</semantics>

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

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33587fedbb79ea8527c34eca0f99b261dd1e4c17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.203ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}" loading="lazy"></dd></dl>

In <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,\varphi ,z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b915f15da61cb0537189d6795ab55d9377537056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.687ex; height:2.843ex;" alt="{\displaystyle (\rho ,\varphi ,z)}" loading="lazy"> ergibt sich

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho \,{\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17097cc03d55b9c0272cf865df33f75e6bf6b559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.551ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho \,{\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}" loading="lazy"></dd></dl>

und in <a href="Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\theta ,\varphi )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0fbf2e667b8c94c169165fc44e0d73e66c901a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.536ex; height:2.843ex;" alt="{\displaystyle (r,\theta ,\varphi )}" loading="lazy">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta \,{\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f2c0f66b81bd49edff15bad190613a1b217b662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.821ex; height:6.343ex;" alt="{\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta \,{\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}.}" loading="lazy"></dd></dl>

Die Ableitungen der Produkte in dieser Darstellung können noch <a href="Kugelkoordinaten#Transformation_des_Laplace-Operators" title="Kugelkoordinaten">entwickelt werden</a>, wobei sich der erste und zweite Term ändern. Der erste (radiale) Term kann in drei äquivalenten Formen geschrieben werden:

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial f}{\partial r}}\right)={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}{\Big (}rf(r){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bae1d5645543da5007a9be60db66ce4d3421514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.07ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}\,{\frac {\partial f}{\partial r}}\right)={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}{\Big (}rf(r){\Big )}}" loading="lazy"></dd></dl>

Entsprechend gilt für den zweiten Term:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta \,{\frac {\partial f}{\partial \theta }}\right)={\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+{\frac {\cot \theta }{r^{2}}}{\frac {\partial f}{\partial \theta }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50c66795928076af9ccb243bb2f9696c5d668de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.101ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta \,{\frac {\partial f}{\partial \theta }}\right)={\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+{\frac {\cot \theta }{r^{2}}}{\frac {\partial f}{\partial \theta }}}" loading="lazy"></dd></dl>

Diese Darstellungen des Laplace-Operators in Zylinder- und Kugelkoordinaten gelten nur für den skalaren Laplace-Operator. Für den Laplace-Operator, der auf vektorwertige Funktionen wirkt, müssen noch weitere Terme berücksichtigt werden, siehe weiter unten den Abschnitt „<a href="#Anwendung_auf_Vektorfelder">Anwendung auf Vektorfelder</a>“.

<div class="mw-heading mw-heading3"><h3 id="In_krummlinigen_Orthogonalkoordinaten">In krummlinigen Orthogonalkoordinaten</h3></div>

In beliebigen <a href="Krummlinige_Koordinaten" title="Krummlinige Koordinaten">krummlinigen Orthogonalkoordinaten</a>, zum Beispiel in <a href="Polarkoordinaten" title="Polarkoordinaten">sphärischen Polarkoordinaten</a>, <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> oder <a href="Elliptische_Koordinaten" title="Elliptische Koordinaten">elliptischen Koordinaten</a> gilt dagegen für den Laplace-Operator die allgemeinere Beziehung

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f={\rm {div\,\,grad\,\,}}f={\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{1}}}\left({\frac {a_{2}a_{3}\,\partial f}{a_{1}\,\partial u_{1}}}\right)+{\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{2}}}\left({\frac {a_{1}a_{3}\,\partial f}{a_{2}\,\partial u_{2}}}\right)+{\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{3}}}\left({\frac {a_{1}a_{2}\,\partial f}{a_{3}\,\partial u_{3}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/049d23bdc0a98ffac3b9bd24b182d7b956d8d9cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:101.206ex; height:6.176ex;" alt="{\displaystyle \Delta f={\rm {div\,\,grad\,\,}}f={\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{1}}}\left({\frac {a_{2}a_{3}\,\partial f}{a_{1}\,\partial u_{1}}}\right)+{\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{2}}}\left({\frac {a_{1}a_{3}\,\partial f}{a_{2}\,\partial u_{2}}}\right)+{\frac {1}{a_{1}a_{2}a_{3}}}\,\,{\frac {\partial }{\partial u_{3}}}\left({\frac {a_{1}a_{2}\,\partial f}{a_{3}\,\partial u_{3}}}\right)}" loading="lazy"></dd></dl>

mit den durch

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\vec {r}}=\sum _{i=1}^{3}\,a_{i}\,{\hat {e}}_{i}(u_{1},u_{2},u_{3})\,\mathrm {d} u_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92e3be5d6388158c52d137d62dafdc8416657a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.089ex; height:7.176ex;" alt="{\displaystyle \mathrm {d} {\vec {r}}=\sum _{i=1}^{3}\,a_{i}\,{\hat {e}}_{i}(u_{1},u_{2},u_{3})\,\mathrm {d} u_{i}}" loading="lazy"></dd>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {grad\,\,}}f=\sum _{i=1}^{3}\,{\frac {\partial f}{a_{i}\,\partial u_{i}}}\,{\hat {e}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b413c2de69ab423c65993eacdc1e13bc33c7efa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.988ex; height:7.176ex;" alt="{\displaystyle {\rm {grad\,\,}}f=\sum _{i=1}^{3}\,{\frac {\partial f}{a_{i}\,\partial u_{i}}}\,{\hat {e}}_{i}}" loading="lazy"></dd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{i}\cdot {\hat {e}}_{k}=\delta _{i,k}={\begin{cases}1&{\text{für }}i=k\\0&{\text{für }}i\neq k\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78e58c80c7fb3baf96207cea199c9e089ff7ac4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.336ex; height:7.509ex;" alt="{\displaystyle {\hat {e}}_{i}\cdot {\hat {e}}_{k}=\delta _{i,k}={\begin{cases}1&{\text{für }}i=k\\0&{\text{für }}i\neq k\end{cases}}}" loading="lazy"></dd></dl>

impliziert definierten Größen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i},u_{i},{\hat {e}}_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i},u_{i},{\hat {e}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d68f87bc19e2b8de8d7afd0c737d5614d703546c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.318ex; height:2.509ex;" alt="{\displaystyle a_{i},u_{i},{\hat {e}}_{i}}" loading="lazy">. Dabei haben nicht die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} u_{i}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd2e793d6d6b247449777500fec90a085fb78fe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.422ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} u_{i}}" loading="lazy">, sondern die Größen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} l_{i}:=a_{i}\cdot \mathrm {d} u_{i}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} l_{i}:=a_{i}\cdot \mathrm {d} u_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829ef9466f4aefb0e1581eb204f139c58e6065a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.661ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} l_{i}:=a_{i}\cdot \mathrm {d} u_{i}}" loading="lazy"> die physikalische Dimension einer „Länge“, wobei zu beachten ist, dass die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"> nicht konstant sind, sondern von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69b0c788a124a32684f109737f7cfab7611d6a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2b5855eefa1e5c167320e2fb16e432c4931b166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167275af21aa3ae1ef607cd482e4336fc30845c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{3}}" loading="lazy"> abhängen können.

Für noch allgemeinere Koordinaten gilt die <a href="Laplace-Beltrami-Operator" class="mw-redirect" title="Laplace-Beltrami-Operator">Laplace-Beltrami-Beziehung</a>.

<div class="mw-heading mw-heading3"><h3 id="Anwendung_auf_Vektorfelder">Anwendung auf Vektorfelder</h3></div>

In einem kartesischen Koordinatensystem mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy">-, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">- und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-Koordinaten und Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{x,y,z}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{x,y,z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d7a78bec27761a9465409fca86057c20f218fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.965ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{x,y,z}}" loading="lazy"> gilt:

<mover>

</mover>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {v}}={\frac {\partial ^{2}}{\partial x^{2}}}{\vec {v}}+{\frac {\partial ^{2}}{\partial y^{2}}}{\vec {v}}+{\frac {\partial ^{2}}{\partial z^{2}}}{\vec {v}}=\Delta v_{x}{\hat {e}}_{x}+\Delta v_{y}{\hat {e}}_{y}+\Delta v_{z}{\hat {e}}_{z}={\begin{pmatrix}\Delta v_{x}\\\Delta v_{y}\\\Delta v_{z}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46352497393171450c62478ed03cb9439719b410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:69.067ex; height:9.509ex;" alt="{\displaystyle \Delta {\vec {v}}={\frac {\partial ^{2}}{\partial x^{2}}}{\vec {v}}+{\frac {\partial ^{2}}{\partial y^{2}}}{\vec {v}}+{\frac {\partial ^{2}}{\partial z^{2}}}{\vec {v}}=\Delta v_{x}{\hat {e}}_{x}+\Delta v_{y}{\hat {e}}_{y}+\Delta v_{z}{\hat {e}}_{z}={\begin{pmatrix}\Delta v_{x}\\\Delta v_{y}\\\Delta v_{z}\end{pmatrix}}}" loading="lazy"></dd></dl>

Bei Verwendung von Zylinder- bzw. Kugelkoordinaten ist die Differentiation der Basisvektoren zu beachten. Es ergibt sich in Zylinderkoordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,\varphi ,z)}">

</mstyle>

</mrow>

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

</semantics>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {v}}=\left(\Delta v_{\rho }-{\frac {1}{\rho ^{2}}}v_{\rho }-{\frac {2}{\rho ^{2}}}{\frac {\partial v_{\varphi }}{\partial \varphi }}\right){\hat {e}}_{\rho }+\left(\Delta v_{\varphi }-{\frac {1}{\rho ^{2}}}v_{\varphi }+{\frac {2}{\rho ^{2}}}{\frac {\partial v_{\rho }}{\partial \varphi }}\right){\hat {e}}_{\varphi }+\Delta v_{z}{\hat {e}}_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbf5a9be3cf607524b4c5c529cf4267070c60a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.683ex; height:6.343ex;" alt="{\displaystyle \Delta {\vec {v}}=\left(\Delta v_{\rho }-{\frac {1}{\rho ^{2}}}v_{\rho }-{\frac {2}{\rho ^{2}}}{\frac {\partial v_{\varphi }}{\partial \varphi }}\right){\hat {e}}_{\rho }+\left(\Delta v_{\varphi }-{\frac {1}{\rho ^{2}}}v_{\varphi }+{\frac {2}{\rho ^{2}}}{\frac {\partial v_{\rho }}{\partial \varphi }}\right){\hat {e}}_{\varphi }+\Delta v_{z}{\hat {e}}_{z}}" loading="lazy"></dd></dl>

und in Kugelkoordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\theta ,\varphi )}">

</mstyle>

</mrow>

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

</semantics>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta {\vec {v}}=&\left(\Delta v_{r}-{\frac {2}{r^{2}}}v_{r}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}-{\frac {2}{r^{2}}}{\frac {\partial v_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\tan \theta }}v_{\theta }\right){\hat {e}}_{r}\\&+\left(\Delta v_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial v_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta }\right){\hat {e}}_{\theta }\\&+\left(\Delta v_{\varphi }+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial v_{r}}{\partial \varphi }}-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial v_{\theta }}{\partial \varphi }}\right){\hat {e}}_{\varphi }\,.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e8ed6fba9bd0cf1fb2e9b26a9db199f286eeb5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:65.551ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\Delta {\vec {v}}=&\left(\Delta v_{r}-{\frac {2}{r^{2}}}v_{r}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}-{\frac {2}{r^{2}}}{\frac {\partial v_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\tan \theta }}v_{\theta }\right){\hat {e}}_{r}\\&+\left(\Delta v_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial v_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta }\right){\hat {e}}_{\theta }\\&+\left(\Delta v_{\varphi }+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial v_{r}}{\partial \varphi }}-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial v_{\theta }}{\partial \varphi }}\right){\hat {e}}_{\varphi }\,.\end{aligned}}}" loading="lazy"></dd></dl>

Die zu den Laplace-Ableitungen der Vektorkomponenten hinzu kommenden Terme resultieren aus den Ableitungen der Basisvektoren.<a href="#cite_note-2">[2]</a>

<td>Beweis

</td></tr>

<tr>

<td>In Zylinderkoordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,\varphi ,z)}">

</mstyle>

</mrow>

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

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{\rho }={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}},\quad {\hat {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/051c60fdbea2551de53f4b311768b17a1aa4d6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:50.868ex; height:9.509ex;" alt="{\displaystyle {\hat {e}}_{\rho }={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}},\quad {\hat {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}" loading="lazy">

als orthonormale Basisvektoren genommen. Ihre Ableitungen lauten:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{\rho ,\varphi }={\hat {e}}_{\varphi }\quad {\text{und}}\quad {\hat {e}}_{\varphi ,\varphi }=-{\hat {e}}_{\rho }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63fd5bcfd70c6d89beef93b494cd992f2ae119b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.537ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{\rho ,\varphi }={\hat {e}}_{\varphi }\quad {\text{und}}\quad {\hat {e}}_{\varphi ,\varphi }=-{\hat {e}}_{\rho }}" loading="lazy">

Hier wie im Folgenden bedeutet ein Index nach einem Komma eine Ableitung nach der angegebenen Koordinate, beispielsweise

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{\rho ,\varphi }:={\frac {\partial }{\partial \varphi }}{\hat {e}}_{\rho }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dddda8abfc0bc4c6ba9d9a12d0366b5bd03ea703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.346ex; height:6.009ex;" alt="{\displaystyle {\hat {e}}_{\rho ,\varphi }:={\frac {\partial }{\partial \varphi }}{\hat {e}}_{\rho }.}" loading="lazy">

Die Anwendung des Laplace-Operators

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3cb70168041c7b20aa243e5721542c393772498.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.48ex; height:6.343ex;" alt="{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}" loading="lazy">

auf ein Vektorfeld ergibt:

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left({\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\=&{\frac {\partial ^{2}}{\partial \rho ^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})+{\frac {\partial ^{2}}{\partial z^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\=&v_{\rho ,\rho \rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho \rho }{\hat {e}}_{\varphi }+v_{z,\rho \rho }{\hat {e}}_{z}+{\frac {1}{\rho }}(v_{\rho ,\rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho }{\hat {e}}_{\varphi }+v_{z,\rho }{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}{\frac {\partial }{\partial \varphi }}(v_{\rho ,\varphi }{\hat {e}}_{\rho }+v_{\rho }{\hat {e}}_{\varphi }+v_{\varphi ,\varphi }{\hat {e}}_{\varphi }-v_{\varphi }{\hat {e}}_{\rho }+v_{z,\varphi }{\hat {e}}_{z})\\&+v_{\rho ,zz}{\hat {e}}_{\rho }+v_{\varphi ,zz}{\hat {e}}_{\varphi }+v_{z,zz}{\hat {e}}_{z}\\=&v_{\rho ,\rho \rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho \rho }{\hat {e}}_{\varphi }+v_{z,\rho \rho }{\hat {e}}_{z}+{\frac {1}{\rho }}(v_{\rho ,\rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho }{\hat {e}}_{\varphi }+v_{z,\rho }{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}(v_{\rho ,\varphi \varphi }{\hat {e}}_{\rho }+2v_{\rho ,\varphi }{\hat {e}}_{\varphi }-v_{\rho }{\hat {e}}_{\rho }+v_{\varphi ,\varphi \varphi }{\hat {e}}_{\varphi }-2v_{\varphi ,\varphi }{\hat {e}}_{\rho }-v_{\varphi }{\hat {e}}_{\varphi }+v_{z,\varphi \varphi }{\hat {e}}_{z})\\&+v_{\rho ,zz}{\hat {e}}_{\rho }+v_{\varphi ,zz}{\hat {e}}_{\varphi }+v_{z,zz}{\hat {e}}_{z}\\=&+\left(\Delta v_{\rho }-{\frac {1}{\rho ^{2}}}v_{\rho }-{\frac {2}{\rho ^{2}}}v_{\varphi ,\varphi }\right){\hat {e}}_{\rho }+\left(\Delta v_{\varphi }-{\frac {1}{\rho ^{2}}}v_{\varphi }+{\frac {2}{\rho ^{2}}}v_{\rho ,\varphi }\right){\hat {e}}_{\varphi }+\Delta v_{z}{\hat {e}}_{z},\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/849fc62a2cdfcb15fe4547088e009971fe983881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -27.416ex; margin-bottom: -0.256ex; width:75.276ex; height:56.509ex;" alt="{\displaystyle {\begin{aligned}&\left({\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\=&{\frac {\partial ^{2}}{\partial \rho ^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})+{\frac {\partial ^{2}}{\partial z^{2}}}(v_{\rho }{\hat {e}}_{\rho }+v_{\varphi }{\hat {e}}_{\varphi }+v_{z}{\hat {e}}_{z})\\=&v_{\rho ,\rho \rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho \rho }{\hat {e}}_{\varphi }+v_{z,\rho \rho }{\hat {e}}_{z}+{\frac {1}{\rho }}(v_{\rho ,\rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho }{\hat {e}}_{\varphi }+v_{z,\rho }{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}{\frac {\partial }{\partial \varphi }}(v_{\rho ,\varphi }{\hat {e}}_{\rho }+v_{\rho }{\hat {e}}_{\varphi }+v_{\varphi ,\varphi }{\hat {e}}_{\varphi }-v_{\varphi }{\hat {e}}_{\rho }+v_{z,\varphi }{\hat {e}}_{z})\\&+v_{\rho ,zz}{\hat {e}}_{\rho }+v_{\varphi ,zz}{\hat {e}}_{\varphi }+v_{z,zz}{\hat {e}}_{z}\\=&v_{\rho ,\rho \rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho \rho }{\hat {e}}_{\varphi }+v_{z,\rho \rho }{\hat {e}}_{z}+{\frac {1}{\rho }}(v_{\rho ,\rho }{\hat {e}}_{\rho }+v_{\varphi ,\rho }{\hat {e}}_{\varphi }+v_{z,\rho }{\hat {e}}_{z})\\&+{\frac {1}{\rho ^{2}}}(v_{\rho ,\varphi \varphi }{\hat {e}}_{\rho }+2v_{\rho ,\varphi }{\hat {e}}_{\varphi }-v_{\rho }{\hat {e}}_{\rho }+v_{\varphi ,\varphi \varphi }{\hat {e}}_{\varphi }-2v_{\varphi ,\varphi }{\hat {e}}_{\rho }-v_{\varphi }{\hat {e}}_{\varphi }+v_{z,\varphi \varphi }{\hat {e}}_{z})\\&+v_{\rho ,zz}{\hat {e}}_{\rho }+v_{\varphi ,zz}{\hat {e}}_{\varphi }+v_{z,zz}{\hat {e}}_{z}\\=&+\left(\Delta v_{\rho }-{\frac {1}{\rho ^{2}}}v_{\rho }-{\frac {2}{\rho ^{2}}}v_{\varphi ,\varphi }\right){\hat {e}}_{\rho }+\left(\Delta v_{\varphi }-{\frac {1}{\rho ^{2}}}v_{\varphi }+{\frac {2}{\rho ^{2}}}v_{\rho ,\varphi }\right){\hat {e}}_{\varphi }+\Delta v_{z}{\hat {e}}_{z},\end{aligned}}}" loading="lazy">

also die im Text angegebene Formel.

</td></tr>

<tr>

<td>In Kugelkoordinaten können die Basisvektoren

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{r}={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}},\qquad {\hat {e}}_{\theta }={\begin{pmatrix}\cos \theta \cos \varphi \\\cos \theta \sin \varphi \\-\sin \theta \end{pmatrix}},\qquad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/357f3467c184c23b27d7eb7bc85e1ad57e9352ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:68.958ex; height:9.509ex;" alt="{\displaystyle {\hat {e}}_{r}={\begin{pmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{pmatrix}},\qquad {\hat {e}}_{\theta }={\begin{pmatrix}\cos \theta \cos \varphi \\\cos \theta \sin \varphi \\-\sin \theta \end{pmatrix}},\qquad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}}" loading="lazy">

verwendet werden. Diese Vektoren haben die Ableitungen

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {e}}_{r,\theta }=&{\begin{pmatrix}\cos \theta \cos \varphi \\\cos \theta \sin \varphi \\-\sin \theta \end{pmatrix}}={\hat {e}}_{\theta }\,,\quad {\hat {e}}_{r,\varphi }={\begin{pmatrix}-\sin \theta \sin \varphi \\\sin \theta \cos \varphi \\0\end{pmatrix}}=\sin \theta {\hat {e}}_{\varphi }\\{\hat {e}}_{\theta ,\theta }=&{\begin{pmatrix}-\sin \theta \cos \varphi \\-\sin \theta \sin \varphi \\-\cos \theta \end{pmatrix}}=-{\hat {e}}_{r}\,,\quad {\hat {e}}_{\theta ,\varphi }={\begin{pmatrix}-\cos \theta \sin \varphi \\\cos \theta \cos \varphi \\0\end{pmatrix}}=\cos \theta {\hat {e}}_{\varphi }\\{\hat {e}}_{\varphi ,\varphi }=&{\begin{pmatrix}-\cos \varphi \\-\sin \varphi \\0\end{pmatrix}}={\hat {e}}_{z}\times {\hat {e}}_{\varphi }=-\sin \theta {\hat {e}}_{r}-\cos \theta {\hat {e}}_{\theta }\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1cbff6a386e55aacd5e5770e64f27d01de44595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:68.967ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {e}}_{r,\theta }=&{\begin{pmatrix}\cos \theta \cos \varphi \\\cos \theta \sin \varphi \\-\sin \theta \end{pmatrix}}={\hat {e}}_{\theta }\,,\quad {\hat {e}}_{r,\varphi }={\begin{pmatrix}-\sin \theta \sin \varphi \\\sin \theta \cos \varphi \\0\end{pmatrix}}=\sin \theta {\hat {e}}_{\varphi }\\{\hat {e}}_{\theta ,\theta }=&{\begin{pmatrix}-\sin \theta \cos \varphi \\-\sin \theta \sin \varphi \\-\cos \theta \end{pmatrix}}=-{\hat {e}}_{r}\,,\quad {\hat {e}}_{\theta ,\varphi }={\begin{pmatrix}-\cos \theta \sin \varphi \\\cos \theta \cos \varphi \\0\end{pmatrix}}=\cos \theta {\hat {e}}_{\varphi }\\{\hat {e}}_{\varphi ,\varphi }=&{\begin{pmatrix}-\cos \varphi \\-\sin \varphi \\0\end{pmatrix}}={\hat {e}}_{z}\times {\hat {e}}_{\varphi }=-\sin \theta {\hat {e}}_{r}-\cos \theta {\hat {e}}_{\theta }\end{aligned}}}" loading="lazy">

Anwendung des Laplace-Operators

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4fee04ac3e668ad0f765499480c34c111be7a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.942ex; height:6.343ex;" alt="{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}}" loading="lazy">

auf ein Vektorfeld ergibt:

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)\cdot (v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&{\frac {\partial ^{2}}{\partial r^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {2}{r}}{\frac {\partial }{\partial r}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial }{\partial \varphi }}(v_{r,\varphi }{\hat {e}}_{r}+\sin \theta v_{r}{\hat {e}}_{\varphi }+v_{\theta ,\varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta }{\hat {e}}_{\varphi }+v_{\varphi ,\varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi }{\hat {e}}_{\theta })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}(v_{r,\theta \theta }{\hat {e}}_{r}+v_{r,\theta }{\hat {e}}_{\theta }+v_{r,\theta }{\hat {e}}_{\theta }-v_{r}{\hat {e}}_{r}+v_{\theta ,\theta \theta }{\hat {e}}_{\theta }-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta }{\hat {e}}_{\theta }+v_{\varphi ,\theta \theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}(v_{r,\varphi \varphi }{\hat {e}}_{r}+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }-\sin ^{2}\theta v_{r}{\hat {e}}_{r}-\sin \theta \cos \theta v_{r}{\hat {e}}_{\theta }\\&+v_{\theta ,\varphi \varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }-\sin \theta \cos \theta v_{\theta }{\hat {e}}_{r}-\cos ^{2}\theta v_{\theta }{\hat {e}}_{\theta }\\&+v_{\varphi ,\varphi \varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\sin ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi }\\&-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\cos ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi })\\=&{\Bigl (}v_{r,rr}+{\frac {2}{r}}v_{r,r}+{\frac {1}{r^{2}}}v_{r,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{r,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{r,\varphi \varphi }\\&\qquad -{\frac {1}{r^{2}}}v_{r}-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}\tan \theta }}v_{\theta }-{\frac {1}{r^{2}}}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{\theta }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{r}\\&+{\Bigl (}v_{\theta ,rr}+{\frac {2}{r}}v_{\theta ,r}+{\frac {1}{r^{2}}}v_{\theta ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\theta ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}}}v_{\theta }+{\frac {1}{r^{2}\tan \theta }}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{r}-{\frac {\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{\theta }\\&+{\Bigl (}v_{\varphi ,rr}+{\frac {2}{r}}v_{\varphi ,r}+{\frac {1}{r^{2}}}v_{\varphi ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\varphi ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {\sin ^{2}\theta +\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\varphi }{\Bigr )}{\hat {e}}_{\varphi }\\=&\left(\Delta v_{r}-{\frac {2}{r^{2}}}v_{r}-{\frac {2}{r^{2}}}v_{\theta ,\theta }-{\frac {2}{r^{2}\tan \theta }}v_{\theta }-{\frac {2}{r^{2}\sin \theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{r}\\&+\left(\Delta v_{\theta }+{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{\theta }\\&+\left(\Delta v_{\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi }+{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }\right){\hat {e}}_{\varphi }\end{aligned}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bd51896bf76920babb5e02607ac6efc0cb61403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -64.338ex; width:100.243ex; height:129.843ex;" alt="{\displaystyle {\begin{aligned}&\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)\cdot (v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&{\frac {\partial ^{2}}{\partial r^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {2}{r}}{\frac {\partial }{\partial r}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial }{\partial \varphi }}(v_{r,\varphi }{\hat {e}}_{r}+\sin \theta v_{r}{\hat {e}}_{\varphi }+v_{\theta ,\varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta }{\hat {e}}_{\varphi }+v_{\varphi ,\varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi }{\hat {e}}_{\theta })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}(v_{r,\theta \theta }{\hat {e}}_{r}+v_{r,\theta }{\hat {e}}_{\theta }+v_{r,\theta }{\hat {e}}_{\theta }-v_{r}{\hat {e}}_{r}+v_{\theta ,\theta \theta }{\hat {e}}_{\theta }-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta }{\hat {e}}_{\theta }+v_{\varphi ,\theta \theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}(v_{r,\varphi \varphi }{\hat {e}}_{r}+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }-\sin ^{2}\theta v_{r}{\hat {e}}_{r}-\sin \theta \cos \theta v_{r}{\hat {e}}_{\theta }\\&+v_{\theta ,\varphi \varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }-\sin \theta \cos \theta v_{\theta }{\hat {e}}_{r}-\cos ^{2}\theta v_{\theta }{\hat {e}}_{\theta }\\&+v_{\varphi ,\varphi \varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\sin ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi }\\&-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\cos ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi })\\=&{\Bigl (}v_{r,rr}+{\frac {2}{r}}v_{r,r}+{\frac {1}{r^{2}}}v_{r,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{r,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{r,\varphi \varphi }\\&\qquad -{\frac {1}{r^{2}}}v_{r}-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}\tan \theta }}v_{\theta }-{\frac {1}{r^{2}}}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{\theta }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{r}\\&+{\Bigl (}v_{\theta ,rr}+{\frac {2}{r}}v_{\theta ,r}+{\frac {1}{r^{2}}}v_{\theta ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\theta ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}}}v_{\theta }+{\frac {1}{r^{2}\tan \theta }}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{r}-{\frac {\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{\theta }\\&+{\Bigl (}v_{\varphi ,rr}+{\frac {2}{r}}v_{\varphi ,r}+{\frac {1}{r^{2}}}v_{\varphi ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\varphi ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {\sin ^{2}\theta +\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\varphi }{\Bigr )}{\hat {e}}_{\varphi }\\=&\left(\Delta v_{r}-{\frac {2}{r^{2}}}v_{r}-{\frac {2}{r^{2}}}v_{\theta ,\theta }-{\frac {2}{r^{2}\tan \theta }}v_{\theta }-{\frac {2}{r^{2}\sin \theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{r}\\&+\left(\Delta v_{\theta }+{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{\theta }\\&+\left(\Delta v_{\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi }+{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }\right){\hat {e}}_{\varphi }\end{aligned}},}" loading="lazy">

also dasselbe Ergebnis wie im Text angegeben.

</td></tr></tbody></table>

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

Der Laplace-Operator ist ein <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a>, das heißt: Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> und <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/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> zweimal differenzierbare Funktionen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> Konstanten, so gilt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (a\cdot f+b\cdot g)=a\cdot (\Delta f)+b\cdot (\Delta g).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ad59aee9d789d4a95dc31a60aa4ca78fc048a27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.622ex; height:2.843ex;" alt="{\displaystyle \Delta (a\cdot f+b\cdot g)=a\cdot (\Delta f)+b\cdot (\Delta g).}" loading="lazy"></dd></dl>

Wie für andere lineare Differentialoperatoren auch, gilt für den Laplace-Operator eine <a href="Produktregel#Höhere_partielle_Ableitungen" title="Produktregel">verallgemeinerte Produktregel</a>. Diese lautet

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (fg)=f\Delta g+2\langle \nabla f,\nabla g\rangle +g\Delta f,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/433fea165becb38b29409a9cd30c3b6ef0ad2c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.499ex; height:2.843ex;" alt="{\displaystyle \Delta (fg)=f\Delta g+2\langle \nabla f,\nabla g\rangle +g\Delta f,}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon U\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g\colon U\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f23e84f1ccb55b5a0a92a7b84c4b027eb0244817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.537ex; height:2.509ex;" alt="{\displaystyle f,g\colon U\to \mathbb {R} }" loading="lazy"> zwei zweimal stetig differenzierbare Funktionen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<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"> sind und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> das euklidische <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a> ist.<a href="#cite_note-3">[3]</a>

Der Laplace-Operator ist <a href="Rotationssymmetrie" class="mw-redirect" title="Rotationssymmetrie">drehsymmetrisch</a>, das heißt: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> eine zweimal differenzierbare Funktion und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> eine <a href="Rotationsmatrix" class="mw-redirect" title="Rotationsmatrix">Drehung</a>, so gilt

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\Delta f\right)\circ R=\Delta \left(f\circ R\right),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/413d1b13dacb1857599358c78090a949de9620d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.484ex; height:2.843ex;" alt="{\displaystyle \left(\Delta f\right)\circ R=\Delta \left(f\circ R\right),}" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy">“ für die <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Verkettung</a> von Abbildungen steht.

Das <a href="Hauptsymbol" class="mw-redirect" title="Hauptsymbol">Hauptsymbol</a> des Laplace-Operators ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\|\xi \|^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\|\xi \|^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/099224b0bf5716bfded02b5e6b3b8932acc8b506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.217ex; height:3.176ex;" alt="{\displaystyle -\|\xi \|^{2}}" loading="lazy">. Er ist also ein <a href="Elliptischer_Differentialoperator" class="mw-redirect" title="Elliptischer Differentialoperator">elliptischer Differentialoperator</a> zweiter Ordnung. Daraus folgt, dass er ein <a href="Fredholm-Operator" title="Fredholm-Operator">Fredholm-Operator</a> ist und mittels des <a href="Satz_von_Atkinson" class="mw-redirect" title="Satz von Atkinson">Satzes von Atkinson</a> folgt, dass er modulo eines <a href="Kompakter_Operator" title="Kompakter Operator">kompakten Operators</a> rechts- und linksinvertierbar ist.

Der Laplace-Operator

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\Delta \colon {\mathcal {S}}(\mathbb {R} ^{n})\rightarrow L^{2}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0463b8609ff3eed1fd0a800121823e43830cd291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.933ex; height:3.176ex;" alt="{\displaystyle -\Delta \colon {\mathcal {S}}(\mathbb {R} ^{n})\rightarrow L^{2}(\mathbb {R} ^{n})}" loading="lazy"></dd></dl>

auf dem <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a> ist <a href="Wesentlich_selbstadjungierter_Operator" class="mw-redirect" title="Wesentlich selbstadjungierter Operator">wesentlich selbstadjungiert</a>. Er hat daher einen <a href="Abgeschlossener_Operator" title="Abgeschlossener Operator">Abschluss</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\Delta \colon H^{2}(\mathbb {R} ^{n})\rightarrow L^{2}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72085516c0c3a43eb1c67c72eefffc88804f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.598ex; height:3.176ex;" alt="{\displaystyle -\Delta \colon H^{2}(\mathbb {R} ^{n})\rightarrow L^{2}(\mathbb {R} ^{n})}" loading="lazy"></dd></dl>

zu einem <a href="Selbstadjungierter_Operator" title="Selbstadjungierter Operator">selbstadjungierten Operator</a> auf dem <a href="Sobolev-Raum" title="Sobolev-Raum">Sobolev-Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(\mathbb {R} ^{n})\subset L^{2}(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{2}(\mathbb {R} ^{n})\subset L^{2}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f778a36fc585102e44b925d5feb144c472d8253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.305ex; height:3.176ex;" alt="{\displaystyle H^{2}(\mathbb {R} ^{n})\subset L^{2}(\mathbb {R} ^{n})}" loading="lazy">.<a href="#cite_note-4">[4]</a> Dieser Operator ist zudem nichtnegativ, sein <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> befindet sich also auf der nichtnegativen reellen Achse, das heißt:

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (-\Delta )\subset \mathbb {R} _{0}^{+}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38e6dad2d58539b0b3f1636276e745e89b90ce79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.17ex; height:3.176ex;" alt="{\displaystyle \sigma (-\Delta )\subset \mathbb {R} _{0}^{+}}" loading="lazy"></dd></dl>

Die <a href="Eigenwertgleichung" class="mw-redirect" title="Eigenwertgleichung">Eigenwertgleichung</a>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bab09d583c7c9205cda2d53fc81644b16c9a5bdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.755ex; height:2.509ex;" alt="{\displaystyle -\Delta f=\lambda f}" loading="lazy"></dd></dl>

des Laplace-Operators wird <a href="Helmholtz-Gleichung" title="Helmholtz-Gleichung">Helmholtz-Gleichung</a> genannt. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</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"> ein <a href="Beschr%C3%A4nkte_Menge" title="Beschränkte Menge">beschränktes</a> Gebiet und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{2}(\Omega )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{0}^{2}(\Omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b233b6a5eac076a77bd8e8b23d93f46f17551a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle H_{0}^{2}(\Omega )}" loading="lazy"> der Sobolev-Raum mit den Randwerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.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=0}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">

</mstyle>

</mrow>

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

</semantics>

</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">, dann bilden die <a href="Eigenfunktion" class="mw-redirect" title="Eigenfunktion">Eigenfunktionen</a> des Laplace-Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\Delta \colon H_{0}^{2}(\Omega )\rightarrow L^{2}(\Omega )}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\Delta \colon H_{0}^{2}(\Omega )\rightarrow L^{2}(\Omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f9565f35bbd7826d1bb1fb8221adc73cb221553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.162ex; height:3.343ex;" alt="{\displaystyle -\Delta \colon H_{0}^{2}(\Omega )\rightarrow L^{2}(\Omega )}" loading="lazy"> ein <a href="Vollst%C3%A4ndiges_Orthonormalsystem" class="mw-redirect" title="Vollständiges Orthonormalsystem">vollständiges Orthonormalsystem</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(\Omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2352f79f73ea92b82f762f072e41bb4a4cef2395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega )}" loading="lazy"> und sein Spektrum besteht aus einem rein <a href="Diskretes_Spektrum" class="mw-redirect" title="Diskretes Spektrum">diskreten</a>, reellen <a href="Punktspektrum" class="mw-redirect" title="Punktspektrum">Punktspektrum</a>, das nur in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> einen <a href="H%C3%A4ufungspunkt" title="Häufungspunkt">Häufungspunkt</a> haben kann. Dies folgt aus dem Spektralsatz für selbstadjungierte elliptische Differentialoperatoren.<a href="#cite_note-5">[5]</a>

Anschaulich gibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f(p)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/429a79a2aec6765e33a02b85985f974b6ba0ccae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.193ex; height:2.843ex;" alt="{\displaystyle \Delta f(p)}" loading="lazy"> für eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> an einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> an, wie sich der Mittelwert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> über konzentrische Kugelschalen um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> mit wachsendem Kugelradius gegenüber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(p)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a2be54931c84179e944e716af5bf95657cbce1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.257ex; height:2.843ex;" alt="{\displaystyle f(p)}" loading="lazy"> verändert.

<div class="mw-heading mw-heading2"><h2 id="Poisson-_und_Laplace-Gleichung">Poisson- und Laplace-Gleichung</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Poisson-Gleichung" title="Poisson-Gleichung">Poisson-Gleichung</a> und <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a></div>

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

Der Laplace-Operator tritt in einer Reihe wichtiger Differentialgleichungen auf. Die <a href="Homogene_Gleichung" class="mw-redirect" title="Homogene Gleichung">homogene</a> Differentialgleichung

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae202f10d6fab9eea1024f2b824210a99d3d64d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.717ex; height:2.676ex;" alt="{\displaystyle \Delta \varphi =0}" loading="lazy"></dd></dl>

wird Laplace-Gleichung genannt und zweimal stetig differenzierbare Lösungen dieser Gleichung heißen <a href="Harmonische_Funktion" title="Harmonische Funktion">harmonische Funktionen</a>. Die entsprechende <a href="Inhomogene_Gleichung" class="mw-redirect" title="Inhomogene Gleichung">inhomogene Gleichung</a>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f78b7e5915cbba73986bd67d0e836dea9bce532a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.833ex; height:2.676ex;" alt="{\displaystyle \Delta \varphi =f}" loading="lazy"></dd></dl>

heißt Poisson-Gleichung.

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

Die <a href="Fundamentall%C3%B6sung" title="Fundamentallösung">Fundamentallösung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })}">

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dda1a0cfd16f3b290caeacade1b582b74cd3e16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.401ex; height:3.176ex;" alt="{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })}" loading="lazy"> des Laplace-Operators erfüllt die <a href="Poisson-Gleichung" title="Poisson-Gleichung">Poisson-Gleichung</a>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \,G({\vec {x}},{\vec {x}}^{\,\prime })=\delta ({\vec {x}}-{\vec {x}}^{\,\prime })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77fcf263d2138b33af063aea0c737ca43e76e471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.252ex; height:3.176ex;" alt="{\displaystyle \Delta \,G({\vec {x}},{\vec {x}}^{\,\prime })=\delta ({\vec {x}}-{\vec {x}}^{\,\prime })}" loading="lazy"></dd></dl>

mit der <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</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/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"> auf der rechten Seite. Diese Funktion ist von der Anzahl der Raumdimensionen abhängig.

Im Dreidimensionalen lautet sie:

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })=-{\frac {1}{4\pi \|{\vec {x}}-{\vec {x}}^{\,\prime }\|}}+F({\vec {x}},{\vec {x}}^{\,\prime })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6392ee0320bb88f6a2d9cee4f2a0eb648938fbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.69ex; height:6.176ex;" alt="{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })=-{\frac {1}{4\pi \|{\vec {x}}-{\vec {x}}^{\,\prime }\|}}+F({\vec {x}},{\vec {x}}^{\,\prime })}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \,F({\vec {x}},{\vec {x}}^{\,\prime })=0}">

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \,F({\vec {x}},{\vec {x}}^{\,\prime })=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56c50eb12eda5d3114551a29b2518454e6cc2141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.899ex; height:3.176ex;" alt="{\displaystyle \Delta \,F({\vec {x}},{\vec {x}}^{\,\prime })=0}" loading="lazy"></dd></dl>

Diese Fundamentallösung wird in der <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> als Hilfsmittel zur Lösung von <a href="Randwertproblem" title="Randwertproblem">Randwertproblemen</a> benötigt.

Im Zweidimensionalen lautet sie:

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })={\frac {\ln(\|{\vec {x}}-{\vec {x}}^{\,\prime }\|)}{2\pi }}+F({\vec {x}},{\vec {x}}^{\,\prime })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe602ae7993c826b26beae20e7ba3eab454ebfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.136ex; height:5.843ex;" alt="{\displaystyle G({\vec {x}},{\vec {x}}^{\,\prime })={\frac {\ln(\|{\vec {x}}-{\vec {x}}^{\,\prime }\|)}{2\pi }}+F({\vec {x}},{\vec {x}}^{\,\prime })}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \,F({\vec {x}},{\vec {x}}^{\,\prime })=0}">

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \,F({\vec {x}},{\vec {x}}^{\,\prime })=0}</annotation>

</semantics>

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

<div class="mw-heading mw-heading3"><h3 id="D’Alembert-Operator">D’Alembert-Operator</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="D%E2%80%99Alembert-Operator" title="D’Alembert-Operator">D’Alembert-Operator</a></div>

Der Laplace-Operator ergibt zusammen mit der zweiten <a href="Zeitableitung" title="Zeitableitung">Zeitableitung</a> den D’Alembert-Operator:

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \square ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\Delta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a4ae977638ec6190b9ea0271f173e3b03764bfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.628ex; height:6.009ex;" alt="{\displaystyle \square ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\Delta }" loading="lazy"></dd></dl>

Dieser Operator kann als eine Verallgemeinerung des Laplace-Operators <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"> auf den <a href="Minkowski-Raum" title="Minkowski-Raum">Minkowski-Raum</a> betrachtet werden.

<div class="mw-heading mw-heading3"><h3 id="Verallgemeinerter_Laplace-Operator">Verallgemeinerter Laplace-Operator</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Verallgemeinerter_Laplace-Operator" title="Verallgemeinerter Laplace-Operator">Verallgemeinerter Laplace-Operator</a></div>

Für den Laplace-Operator, der ursprünglich stets als Operator des euklidischen Raumes verstanden wurde, gab es mit der Formulierung der <a href="Riemannsche_Geometrie" title="Riemannsche Geometrie">riemannschen Geometrie</a> die Möglichkeit der Verallgemeinerung auf <a href="Regul%C3%A4re_Fl%C3%A4che" title="Reguläre Fläche">gekrümmte Flächen</a> und <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">riemannsche</a> beziehungsweise <a href="Semi-Riemannsche_Mannigfaltigkeit" class="mw-redirect" title="Semi-Riemannsche Mannigfaltigkeit">pseudo-riemannsche Mannigfaltigkeiten</a>. Dieser allgemeinere Operator wird als verallgemeinerter Laplace-Operator bezeichnet.

<div class="mw-heading mw-heading2"><h2 id="Diskreter_Laplace-Operator">Diskreter Laplace-Operator</h2></div>

Auf eine diskrete Eingangsfunktion gn bzw. gnm wird der Laplace-Operator über eine <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> angewendet. Dabei kann man folgende einfache Faltungsmasken verwenden:

<dl><dd>1D-Filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {D}}_{x}^{2}\;={\begin{bmatrix}1&-2&1\end{bmatrix}}}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad {\vec {D}}_{x}^{2}\;={\begin{bmatrix}1&-2&1\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceae795bde9f7f2dffe46b3d45755406563c136e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.149ex; height:3.843ex;" alt="{\displaystyle \quad {\vec {D}}_{x}^{2}\;={\begin{bmatrix}1&-2&1\end{bmatrix}}}" loading="lazy"></dd>

<dd>2D-Filter: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1e2e4e275c4bdc95f53b86aa9c4904c67392a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.253ex; height:9.176ex;" alt="{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}}" loading="lazy"></dd></dl>

Für zwei Dimensionen gibt es noch alternative Varianten, die zusätzlich auch diagonale Kanten berücksichtigen, beispielsweise:

<dl><dd>2D-Filter: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6ef879da092b4b5c8daad9fe1cd1286cd88fcc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:23.253ex; height:9.176ex;" alt="{\displaystyle \quad \mathbf {D} _{xy}^{2}={\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}}" loading="lazy"></dd></dl>

Diese Faltungsmasken erhält man durch die Diskretisierung der Differenzenquotienten. Dabei entspricht der Laplace-Operator einer gewichteten Summe über den Wert an benachbarten Punkten. Die <a href="Kantendetektion" title="Kantendetektion">Kantendetektion</a> in der <a href="Bildverarbeitung" title="Bildverarbeitung">Bildverarbeitung</a> (siehe <a href="Laplace-Filter" title="Laplace-Filter">Laplace-Filter</a>) ist ein mögliches Anwendungsgebiet diskreter Laplace-Operatoren. Dort taucht eine Kante als Nulldurchgang der zweiten Ableitung des Signals auf. Auch bei der Diskretisierung von Differentialgleichungen oder in der Graphentheorie werden diskrete Laplace-Operatoren genutzt.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Biharmonische_Gleichung" class="mw-redirect" title="Biharmonische Gleichung">Biharmonische Gleichung</a></li>

<li><a href="Fraktionaler_Laplace-Operator" title="Fraktionaler Laplace-Operator">Fraktionaler Laplace-Operator</a></li></ul>

Anwendungen

<ul><li><a href="Potentialstr%C3%B6mung" title="Potentialströmung">Potentialströmung</a></li>

<li><a href="Airysche_Spannungsfunktion" title="Airysche Spannungsfunktion">Airysche Spannungsfunktion</a></li>

<li><a href="Navier-Cauchy-Gleichungen" title="Navier-Cauchy-Gleichungen">Navier-Cauchy-Gleichungen</a></li></ul>

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

<ul><li>Bronstein, Semendjajew, Musiol, Mühlig: Taschenbuch der Mathematik. Harri Deutsch, 1999, 4. Auflage, ISBN 3-8171-2004-4.</li>

<li><a href="Otto_Forster" title="Otto Forster">Otto Forster</a>: Analysis. Band 3: Maß- und Integrationstheorie, Integralsätze im Rn und Anwendungen, 8. verbesserte Auflage. Springer Spektrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5.</li>

<li>Russell Merris: Laplacian matrices of graphs: a survey. In: Linear Algebra and its Applications. 197–198, 143–176 (1994). ISSN 0024-3795</li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1193">Wie „krümme“ ich Nabla und Delta?</a> Herleitung des Nablaoperators für orthonormal krummlinige Koordinaten. Auf: <a href="Matroids_Matheplanet" title="Matroids Matheplanet">matheplanet.com</a>.</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> <a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/VectorLaplacian.html">Vector Laplacian.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> M. Bestehorn: <cite style="font-style:italic">Hydrodynamik und Strukturbildung</cite>. Springer, 2006, ISBN 978-3-540-33796-6, S. 378.<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:Laplace-Operator&rft.au=M.+Bestehorn&rft.btitle=Hydrodynamik+und+Strukturbildung&rft.date=2006&rft.genre=book&rft.isbn=9783540337966&rft.pages=378&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a href="Otto_Forster" title="Otto Forster">Otto Forster</a>: Analysis 2. Differentialrechnung im Rn. Gewöhnliche Differentialgleichungen. Vieweg-Verlag, 7. Aufl. 2006, ISBN 3-528-47231-6, S. 61.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="Dirk_Werner_(Mathematiker)" title="Dirk Werner (Mathematiker)">Dirk Werner</a>: Funktionalanalysis. 6., korrigierte Auflage, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6, S. 349.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a href="Lawrence_C._Evans" title="Lawrence C. Evans">Lawrence Craig Evans</a>: Partial Differential Equations. American Mathematical Society, Providence 2002, ISBN 0-8218-0772-2, S. 334–335.

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-10-27" href="https://de.wikipedia.org/wiki/?title=Laplace-Operator&oldid=260995640">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>