Laplaceoperator in Kugelkoordinaten:

\[\vec{\nabla}^{2}F\left(\vec{r}\right)=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\,\frac{\partial F\left(\vec{r}\right)}{\partial r}\right)+\frac{1}{r^{2}\sin\vartheta}\frac{\partial}{\partial\vartheta}\left(\sin\vartheta\,\frac{\partial F\left(\vec{r}\right)}{\partial\vartheta}\right)+\frac{1}{r^{2}\sin^{2}\vartheta}\frac{\partial^{2}F\left(\vec{r}\right)}{\partial\phi^{2}} \]

Der Radialteil kann auch so geschrieben werden:

\[\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\,\frac{\partial F\left(\vec{r}\right)}{\partial r}\right)=\frac{\partial^{2}F\left(\vec{r}\right)}{\partial r^{2}}+\frac{2}{r}\frac{\partial F\left(\vec{r}\right)}{\partial r}=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}\Big(r^{2}F\left(\vec{r}\right)\Big) \]

Laplaceoperator in Zylinderkoordinaten:

\[\vec{\nabla}^{2}F\left(\vec{r}\right)=\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\,\frac{\partial F\left(\vec{r}\right)}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}F\left(\vec{r}\right)}{\partial\phi^{2}}+\frac{\partial^{2}F\left(\vec{r}\right)}{\partial z^{2}} \]

Herleitung:

Kugelkoordinaten:

\[\left(x,y,z\right)\rightarrow\left(r,\theta,\phi\right)\]

\[x=r\sin\theta\cos\phi \]


\[y=r\sin\theta\sin\phi \]


\[z=r\cos\theta \]


\[r=\sqrt{x^{2}+y^{2}+z^{2}} \]


\[\theta=\arccos\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right) \]


\[\phi=\arctan\left(\frac{y}{x}\right) \]


Ableitungen mit Kettenregel:

\[\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial x}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial x}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial y}=\frac{\partial r}{\partial y}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial y}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial y}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial z}=\frac{\partial r}{\partial z}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial z}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial z}\frac{\partial}{\partial\phi} \]


Partielle Ableitungen:

\[\frac{\partial r}{\partial x}=\frac{x}{r}=\sin\theta\cos\phi \]


\[\frac{\partial\theta}{\partial x}=\frac{\cos\theta\cos\phi}{r} \frac{\partial\phi}{\partial x}=-\frac{\sin\phi}{r\sin\theta} \]


\[\frac{\partial r}{\partial y}=\frac{y}{r}=\sin\theta\sin\phi \]


\[\frac{\partial\theta}{\partial y}=\frac{\cos\theta\sin\phi}{r} \frac{\partial\phi}{\partial y}=\frac{\cos\phi}{r\sin\theta} \]


\[\frac{\partial r}{\partial z}=\frac{z}{r}=\cos\theta \]


\[\frac{\partial\theta}{\partial z}=-\frac{\sin\theta}{r} \]


\[\frac{\partial\phi}{\partial z}=0 \]


F├╝hrt zu Ableitungen:

\[\frac{\partial}{\partial x}=\sin\theta\cos\phi\frac{\partial}{\partial r}+\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial\theta}-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial\phi} \]

\[\frac{\partial}{\partial y}=\sin\theta\sin\phi\frac{\partial}{\partial r}+\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial\theta}+\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial z}=\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta} \]


Laplace-Operator:

\[\vec{\nabla}^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}\sin^{2}\theta}\left[\sin\theta\frac{\partial}{\partial}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{\partial^{2}}{\partial\phi^{2}}\right]\]