Ing Mathematik: Funktionen mehrerer Veränderlicher

Aus Wikibooks

Wechseln zu: Navigation, Suche

Zurück zu Integralrechnung |

Hoch zu Gesamtinhaltsverzeichnis |

Vor zu Mehrfachintegrale

Inhaltsverzeichnis

1 Zylinderkoordinaten
2 Sphärische Koordinaten

[Bearbeiten] Zylinderkoordinaten

[Bearbeiten] Umrechnung von Zylinderkoordinaten in kartesische Koordinaten

Die Zylinderkoordinaten werden durch folgende Gleichungen definiert:

$\begin{matrix}x&=&\rho \cos(\phi) \\ y&=&\rho \sin(\phi) \\ z&=&z \end{matrix}$

[Bearbeiten] Umrechung von kartesischen Koordinaten in Zylinderkoordinaten

Aus den Definitionsgleichungen erhällt man:

$\begin{matrix} \rho=\sqrt{x^2+y^2}&=&\left(x^2+y^2\right)^\frac{1}{2} \\ \phi&=&\mathrm{arctan}(\frac{y}{x}) \end{matrix}$

[Bearbeiten] Ableitungen der Zylinderkoordinaten nach den kartesischen Koordinaten

Leitet man die obigen Gleichungen ab so erhällt man:

$\frac{\partial \rho}{\partial x}=\frac{1}{2}(x^2+y^2)^{-\frac{1}{2}}\cdot 2x =\frac{x}{\rho}= \cos(\phi)$

$\frac{\partial \rho}{\partial y}=\frac{1}{2}(x^2+y^2)^{-\frac{1}{2}}\cdot 2y =\frac{y}{\rho}=\sin(\phi)$

$\frac{\partial \phi}{\partial x}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot -\frac{y}{x^2}=-\frac{y}{\rho^2} =- \frac{\sin(\phi)}{\rho}$

$\frac{\partial \phi}{\partial y}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{1}{x}=\frac{x}{\rho^2} = \frac{\cos(\phi)}{\rho}$

[Bearbeiten] Ableitung einer Funktion in Zylinderkoordinaten nach kartesichen Koordinaten

Will man eine Funktion in Zylinderkoordinaten nach kartesischen Koordinaten ableiten so muss man die (mehrdimensionale) Kettenregel berücksichtigen und erhällt:

$\frac{\partial}{\partial x} f(\rho,\phi,z)=\frac{\partial \rho}{\partial x}\frac{\partial}{\partial \rho} f(r,\phi,z)+\frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi} f(r,\phi,z)= \left(\cos(\phi)\frac{\partial}{\partial \rho}-\frac{\sin(\phi)}{\rho}\frac{\partial}{\partial \phi} \right) f(r,\phi,z)$

$\frac{\partial}{\partial y} f(\rho,\phi,z)=\frac{\partial \rho}{\partial y}\frac{\partial}{\partial \rho} f(r,\phi,z)+\frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi} f(r,\phi,z)= \left(\sin(\phi)\frac{\partial}{\partial \rho}+\frac{\cos(\phi)}{\rho}\frac{\partial}{\partial \phi} \right) f(r,\phi,z)$

[Bearbeiten] Ableitungen der kartesischen Koordinaten nach den Zylinderkoordinaten

$\frac{\partial \mathbf{x}}{\partial \rho}=\begin{pmatrix} \cos(\phi) \\ \sin(\phi) \\ 0\end{pmatrix} \; \frac{\partial \mathbf{x}}{\partial \phi}=\rho \begin{pmatrix} -\sin(\phi) \\ \cos(\phi) \\ 0\end{pmatrix} \; \frac{\partial \mathbf{x}}{\partial z}=\rho \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

$\boldsymbol{\hat{\rho}}= \frac{\frac{\partial \mathbf{x}}{\partial \rho}}{\left|\frac{\partial \mathbf{x}}{\partial \rho}\right|}=\begin{pmatrix} \cos(\phi) \\ \sin(\phi) \\ 0 \end{pmatrix}$

$\boldsymbol{\hat{\phi}}= \frac{\frac{\partial \mathbf{x}}{\partial \phi}}{\left|\frac{\partial \mathbf{x}}{\partial \phi}\right|}=\begin{pmatrix} -\sin(\phi) \\ \cos(\phi) \\ 0 \end{pmatrix}$

$\boldsymbol{\hat{z}}= \frac{\frac{\partial \mathbf{x}}{\partial z}}{\left|\frac{\partial \mathbf{x}}{\partial z}\right|}=\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

$\nabla f(r,\phi,z)=\left(\boldsymbol{\hat{\rho}} \frac{\partial}{\partial \rho}+\boldsymbol{\hat{\phi}}\frac{1}{\rho}\frac{\partial}{\partial \phi} + \boldsymbol{\hat{z}}\frac{\partial}{\partial z} \right) f(r,\phi,z)$

$\mathbf{A}:=\begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix}=\begin{pmatrix} A_\rho \cos(\phi) -A_\phi \sin(\phi) \\ A_\rho \sin(\phi) + A_\phi \cos(\phi) \\ A_z \end{pmatrix}$

$\frac{\partial }{\partial x} A_x= \left(\frac{x}{\rho}\frac{\partial}{\partial \rho}-\frac{y}{\rho^2}\frac{\partial}{\partial \phi} \right) \left( A_\rho \cos(\phi) -A_\phi \sin(\phi) \right)= \frac{x}{\rho}\frac{\partial}{\partial \rho}A_\rho \cos(\phi)+ \frac{y}{\rho^2}\frac{\partial}{\partial \phi} A_\phi \sin(\phi) - \frac{x}{\rho}\frac{\partial}{\partial \rho} A_\phi \sin(\phi) -\frac{y}{\rho^2}\frac{\partial}{\partial \phi}A_\rho \cos(\phi)$

$\frac{\partial }{\partial x} A_x= \frac{x}{\rho} \cos(\phi) \frac{\partial A_\rho}{\partial \rho} + \frac{y}{\rho^2} \sin(\phi) \frac{\partial A_\phi}{\partial \phi} + \frac{y}{\rho^2} A_\phi \cos(\phi) - \frac{x}{\rho} \sin(\phi) \frac{\partial A_\phi} {\partial \rho} - \frac{y}{\rho^2}\cos(\phi) \frac{\partial A_\rho}{\partial \phi} + \frac{y}{\rho^2} A_\rho \sin(\phi)$

$\frac{\partial }{\partial x} A_x= \cos(\phi)^2 \frac{\partial A_\rho}{\partial \rho} + \frac{1}{\rho} \sin(\phi)^2 \frac{\partial A_\phi}{\partial \phi} + \frac{1}{\rho} A_\phi \cos(\phi)\sin(\phi) - \cos(\phi) \sin(\phi) \frac{\partial A_\phi} {\partial \rho} - \frac{1}{\rho} \sin(\phi) \cos(\phi) \frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} A_\rho \sin(\phi)^2$

$\frac{\partial }{\partial y} A_y= \left(\frac{y}{\rho}\frac{\partial}{\partial \rho}+\frac{x}{\rho^2}\frac{\partial}{\partial \phi} \right) \left( A_\rho \sin(\phi) +A_\phi \cos(\phi) \right)= \frac{y}{\rho}\frac{\partial}{\partial \rho}A_\rho \sin(\phi)+ \frac{x}{\rho^2}\frac{\partial}{\partial \phi} A_\phi \cos(\phi) + \frac{y}{\rho}\frac{\partial}{\partial \rho} A_\phi \cos(\phi) +\frac{x}{\rho^2}\frac{\partial}{\partial \phi}A_\rho \sin(\phi)$

$\frac{\partial }{\partial y} A_y= \frac{y}{\rho} \sin(\phi) \frac{\partial A_\rho}{\partial \rho} - \frac{x}{\rho^2} A_\phi \sin(\phi) + \frac{x}{\rho^2}\cos(\phi) \frac{\partial A_\phi}{\partial \phi} + \frac{y}{\rho}\cos(\phi) \frac{\partial A_\phi }{\partial \rho} + \frac{x}{\rho^2} \sin(\phi) \frac{\partial A_\rho }{\partial \phi}+ \frac{x}{\rho^2} A_\rho \cos(\phi)$

$\frac{\partial }{\partial y} A_y= \sin(\phi)^2 \frac{\partial A_\rho}{\partial \rho} - \frac{1}{\rho} A_\phi \cos(\phi)\sin(\phi) + \frac{1}{\rho}\cos(\phi)^2 \frac{\partial A_\phi}{\partial \phi} + \cos(\phi) \sin(\phi) \frac{\partial A_\phi }{\partial \rho} + \frac{1}{\rho} \cos(\phi) \sin(\phi) \frac{\partial A_\rho }{\partial \phi}+ \frac{1}{\rho} A_\rho \cos(\phi)^2$

$\frac{\partial }{\partial x} A_x +\frac{\partial }{\partial y} A_y = \frac{\partial A_\rho}{\partial \rho} + \frac{1}{\rho} \frac{\partial A_\phi}{\partial \phi} + \frac{1}{\rho} A_\rho = \frac{1} {\rho} \frac{\partial}{\partial \rho} \rho A_\rho + \frac{1}{\rho} \frac{\partial A_\phi}{\partial \phi}$

$\nabla \cdot \mathbf{A} = \frac{1} {\rho} \frac{\partial}{\partial \rho} \rho A_\rho + \frac{1}{\rho} \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{ \partial z}$

[Bearbeiten] Sphärische Koordinaten

$\begin{matrix}x&=&r \sin(\Theta) \cos(\phi) \\ y&=&r \sin(\Theta) \sin(\phi) \\ z&=&r \cos(\Theta) \end{matrix}$

$\begin{matrix} r&=&\sqrt{x^2+y^2+z^2}=\left(x^2+y^2+z^2\right)^\frac{1}{2} \\ \phi&=&\mathrm{arctan}(\frac{y}{x}) \\ \Theta&=&\mathrm{arctan}(\frac{\sqrt{x^2+y^2}}{z}) \end{matrix}$

$\frac{\partial r}{\partial x}=\frac{1}{2}(x^2+y^2+z^2)^{-\frac{1}{2}}\cdot 2x =\frac{x}{r}=\sin(\Theta) \cos(\phi)$

$\frac{\partial r}{\partial y}=\frac{y}{r}=\sin(\Theta) \sin(\phi)$

$\frac{\partial r}{\partial z}=\frac{z}{r}=\cos(\Theta)$

$\rho:=\sqrt{x^2+y^2}= r \sin(\Theta)$

$\frac{\partial \phi}{\partial x}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot -\frac{y}{x^2}=-\frac{y}{\rho^2}=-\frac{\sin(\phi)}{\rho} =-\frac{\sin(\phi)}{r \sin(\Theta)}$

$\frac{\partial \phi}{\partial y}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{1}{x}=\frac{x}{\rho^2}=\frac{\cos(\phi)}{\rho}= \frac{\cos(\phi)}{r \sin(\Theta)}$

$\frac{\partial \phi}{\partial z}=0$

$\frac{\partial \Theta}{\partial x}=\frac{1}{1+\frac{x^2+y^2}{z^2}}\cdot \frac{2x}{2 z\sqrt{x^2+y^2}}=\frac{z}{r^2}\cdot \cos(\phi)= \frac{1}{r}\cdot \cos(\Theta) \cos(\phi)$

$\frac{\partial \Theta}{\partial y}=\frac{1}{1+\frac{x^2+y^2}{z^2}}\cdot \frac{2y}{2 z\sqrt{x^2+y^2}}=\frac{z}{r^2}\cdot \sin(\phi)= \frac{1}{r}\cdot \cos(\Theta) \sin(\phi)$

$\frac{\partial \Theta}{\partial z}=\frac{1}{1+\frac{x^2+y^2}{z^2}}\cdot \frac{-\sqrt{x^2+y^2}}{z^2}=-\frac{\rho}{r^2}=-\frac{\sin(\Theta)}{r}$

$\frac{\partial \mathbf{x}}{\partial r}=\begin{pmatrix} \sin(\Theta) \cos(\phi) \\ \sin(\Theta) \sin(\phi) \\ \cos(\Theta) \end{pmatrix}$

$\frac{\partial \mathbf{x}}{\partial \phi}=\begin{pmatrix} - r \sin(\Theta) \sin(\phi) \\ r \sin(\Theta) \cos(\phi) \\ 0 \end{pmatrix}$

$\frac{\partial \mathbf{x}}{\partial \Theta}= \begin{pmatrix} r \cos(\Theta) \cos(\phi) \\ r \cos(\Theta) \sin(\phi) \\ -r \sin(\Theta) \end{pmatrix}$

$\boldsymbol{\hat{r}}=\frac{\frac{\partial \mathbf{x}}{\partial r}}{|\frac{\partial \mathbf{x}}{\partial r}|}=\begin{pmatrix} \sin(\Theta) \cos(\phi) \\ \sin(\Theta) \sin(\phi) \\ \cos(\Theta) \end{pmatrix}$

$\boldsymbol{\hat{\phi}}=\frac{\frac{\partial \mathbf{x}}{\partial \phi}}{|\frac{\partial \mathbf{x}}{\partial \phi}|}=\begin{pmatrix} - \sin(\phi) \\ \cos(\phi) \\ 0 \end{pmatrix}$

$\boldsymbol{\hat{\Theta}}=\frac{\frac{\partial \mathbf{x}}{\partial \Theta}}{|\frac{\partial \mathbf{x}}{\partial \Theta}|}= \begin{pmatrix} \cos(\Theta) \cos(\phi) \\ \cos(\Theta) \sin(\phi) \\ - \sin(\Theta) \end{pmatrix}$

$\frac{\partial }{\partial x} f(r,\Theta,\phi) = \left( \frac{\partial r}{\partial x}\frac{\partial }{\partial r}+ \frac{\partial \phi}{\partial x}\frac{\partial }{\partial \phi}+ \frac{\partial \Theta}{\partial x}\frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)= \left( \sin(\Theta) \cos(\phi)\frac{\partial }{\partial r}- \frac{\sin(\phi)}{r \sin(\Theta)}\frac{\partial }{\partial \phi}+ \frac{1}{r}\cdot \cos(\Theta) \cos(\phi)\frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)$

$\frac{\partial }{\partial y} f(r,\Theta,\phi) = \left( \frac{\partial r}{\partial y}\frac{\partial }{\partial r}+ \frac{\partial \phi}{\partial y}\frac{\partial }{\partial \phi}+ \frac{\partial \Theta}{\partial y}\frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)= \left( \sin(\Theta) \sin(\phi)\frac{\partial }{\partial r}+ \frac{\cos(\phi)}{r \sin(\Theta)}\frac{\partial }{\partial \phi}+ \frac{1}{r}\cdot \cos(\Theta) \sin(\phi)\frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)$

$\frac{\partial }{\partial z} f(r,\Theta,\phi) = \left( \frac{\partial r}{\partial z}\frac{\partial }{\partial r}+ \frac{\partial \phi}{\partial z}\frac{\partial }{\partial \phi}+ \frac{\partial \Theta}{\partial z}\frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)= \left( \cos(\Theta)\frac{\partial }{\partial r}- \frac{1}{r}\cdot \sin(\Theta) \frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)$

$\nabla f(r,\Theta,\phi) = \boldsymbol{\hat{r}} \frac{\partial }{\partial r} + \frac{1}{r \sin(\Theta)}\boldsymbol{\hat{\phi}} \frac{\partial }{\partial \phi}+ \frac{1}{r} \boldsymbol{\hat{\Theta}} \frac{\partial }{\partial \Theta}$

$\mathbf{A}(r,\Theta,\phi)=\begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix}=A_r\boldsymbol{\hat{r}} +A_\Theta\boldsymbol{\hat{\Theta}}+A_\phi\boldsymbol{\hat{\phi}} =\begin{pmatrix} A_r\sin(\Theta) \cos(\phi)+A_\Theta \cos(\Theta) \cos(\phi)-A_\phi \sin(\phi) \\ A_r\sin(\Theta) \sin(\phi)+A_\Theta\cos(\Theta) \sin(\phi) + A_\phi \cos(\phi)\\ A_r\cos(\Theta) - A_\Theta\sin(\Theta) \end{pmatrix}$

$\begin{matrix} \frac{\partial A_x}{\partial x} &=& \left( \sin(\Theta) \cos(\phi)\frac{\partial }{\partial r}- \frac{\sin(\phi)}{r \sin(\Theta)}\frac{\partial }{\partial \phi}+ \frac{1}{r}\cdot \cos(\Theta) \cos(\phi)\frac{\partial }{\partial \Theta} \right)\left(A_r\sin(\Theta) \cos(\phi)+A_\Theta \cos(\Theta) \cos(\phi)-A_\phi \sin(\phi) \right) \\ &=& \sin^2(\Theta) \cos^2(\phi)\frac{\partial A_r}{\partial r}+ \sin(\Theta) \cos(\Theta)\cos^2(\phi)\frac{\partial A_\Theta}{\partial r}- \sin(\Theta) \cos(\phi)\sin(\phi)\frac{\partial A_\phi}{\partial r} \\ & & - \frac{\sin(\phi)\cos(\phi)}{r} \frac{\partial A_r}{\partial \phi}+A_r\frac{\sin^2(\phi)}{r}- \frac{\sin(\phi)\cos(\phi)\cos(\Theta)}{r \sin(\Theta)} \frac{\partial A_\Theta}{\partial \phi} + A_\Theta \frac{\sin^2(\phi)\cos(\Theta)}{r \sin(\Theta)} + \frac{\sin^2(\phi)}{r \sin(\Theta)} \frac{\partial A_\phi}{\partial \phi} + A_\phi \frac{\sin(\phi)\cos(\phi)}{r \sin(\Theta)} \\ & & + \frac{1}{r}\cos(\Theta)\sin(\Theta) \cos^2(\phi) \frac{\partial A_r}{\partial \Theta} + \frac{1}{r} A_r \cos^2(\Theta) \cos^2(\phi)\\ & & + \frac{1}{r}\cos^2(\Theta) \cos^2(\phi) \frac{\partial A_\Theta}{\partial \Theta} - \frac{1}{r} A_\Theta \cos(\Theta)\sin(\Theta) \cos^2(\phi) - \frac{1}{r} \cos(\Theta)\cos(\phi)\sin(\phi) \frac{\partial A_\phi}{\partial \Theta} \end{matrix}$

$\begin{matrix} \frac{\partial A_y}{\partial y} &=& \left( \sin(\Theta) \sin(\phi)\frac{\partial }{\partial r}+ \frac{\cos(\phi)}{r \sin(\Theta)}\frac{\partial }{\partial \phi}+ \frac{1}{r}\cdot \cos(\Theta) \sin(\phi)\frac{\partial }{\partial \Theta} \right)\left(A_r\sin(\Theta) \sin(\phi)+A_\Theta \cos(\Theta) \sin(\phi)+A_\phi \cos(\phi) \right) \\ &=& \sin^2(\Theta) \sin^2(\phi)\frac{\partial A_r}{\partial r}+ \sin(\Theta) \cos(\Theta)\sin^2(\phi)\frac{\partial A_\Theta}{\partial r}+ \sin(\Theta) \cos(\phi)\sin(\phi)\frac{\partial A_\phi}{\partial r} \\ & & + \frac{\sin(\phi)\cos(\phi)}{r} \frac{\partial A_r}{\partial \phi} + A_r\frac{\cos^2(\phi)}{r}+ \frac{\sin(\phi)\cos(\phi)\cos(\Theta)}{r \sin(\Theta)} \frac{\partial A_\Theta}{\partial \phi} + A_\Theta \frac{\cos^2(\phi)\cos(\Theta)}{r \sin(\Theta)} + \frac{\cos^2(\phi)}{r \sin(\Theta)} \frac{\partial A_\phi}{\partial \phi} - A_\phi \frac{\sin(\phi)\cos(\phi)}{r \sin(\Theta)} \\ & & + \frac{1}{r}\cos(\Theta)\sin(\Theta) \sin^2(\phi) \frac{\partial A_r}{\partial \Theta} + \frac{1}{r} A_r \cos^2(\Theta) \sin^2(\phi)\\ & & + \frac{1}{r}\cos^2(\Theta) \sin^2(\phi) \frac{\partial A_\Theta}{\partial \Theta} - \frac{1}{r} A_\Theta \cos(\Theta)\sin(\Theta) \sin^2(\phi) + \frac{1}{r} \cos(\Theta)\sin(\phi)\cos(\phi) \frac{\partial A_\phi}{\partial \Theta} \end{matrix}$

$\begin{matrix}\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}&=& \sin^2(\Theta)\frac{\partial A_r}{\partial r}+ \sin(\Theta) \cos(\Theta) \frac{ \partial A_\Theta}{\partial r}+ \frac{A_r}{r}+ \frac{A_\Theta \cos(\Theta)}{r \sin (\Theta)} + \frac{1}{r \sin(\Theta)}\frac{\partial A_\phi}{\partial \phi}\\ & & + \frac{1}{r} \sin(\Theta) \cos(\Theta)\frac{\partial A_r}{\partial \Theta}+ \frac{1}{r} \cos^2(\Theta) A_r+\frac{1}{r} \cos^2(\Theta) \frac{\partial A_\Theta}{\partial \Theta}- \frac{1}{r}\sin(\Theta) \cos(\Theta) A_\Theta \end{matrix}$

$\begin{matrix} \frac{\partial A_z}{\partial z}&=& \left( \cos(\Theta)\frac{\partial }{\partial r}- \frac{1}{r}\cdot \sin(\Theta) \frac{\partial }{\partial \Theta} \right) \left( A_r\cos(\Theta) - A_\Theta\sin(\Theta) \right) \\ &=& \cos^2{\Theta}\frac{\partial A_r}{\partial r}- \sin(\Theta) \cos(\Theta)\frac{\partial A_\Theta}{\partial r} -\frac{1}{r} \sin(\Theta) \cos(\Theta) \frac{\partial A_r}{\partial \Theta } + \frac{1}{r} A_r \sin^2(\Theta)+\frac{1}{r} A_\Theta \sin(\Theta) \cos(\Theta)+ \frac{1}{r} \sin^2(\Theta) \frac{\partial A_\Theta}{\partial \Theta} \end{matrix}$

$\nabla \cdot \mathbf{A}=\frac{\partial A_r}{\partial r} + \frac{A_r}{r} + \frac{A_\Theta \cos(\Theta)}{r \sin (\Theta)} +\frac{1}{r \sin(\Theta)}\frac{\partial A_\phi}{\partial \phi}+\frac{A_r}{r} + \frac{1}{r}\frac{\partial A_\Theta}{\partial \Theta}= \frac{1}{r^2}\frac{\partial r^2 A_r}{\partial r}+ \frac{1}{r \sin(\Theta)}\frac{\partial}{\partial \Theta} A_\Theta \sin(\Theta) +\frac{1}{r \sin(\Theta)}\frac{\partial A_\phi}{\partial \phi}$

$\Delta f(r,\Theta,\phi) = \nabla \cdot (\nabla f(r,\Theta,\phi))=\nabla \cdot \left(\boldsymbol{\hat{r}} \frac{\partial }{\partial r} + \frac{1}{r \sin(\Theta)}\boldsymbol{\hat{\phi}} \frac{\partial }{\partial \phi}+ \frac{1}{r} \boldsymbol{\hat{\Theta}} \frac{\partial }{\partial \Theta} \right) f(r,\Theta,\phi)$

$\Delta f(r,\Theta,\phi) = \left( \frac{1}{r^2}\frac{\partial }{\partial r} r^2 \frac{\partial }{\partial r} + \frac{1}{r^2 \sin(\Theta)}\frac{\partial}{\partial \Theta} \sin(\Theta) \frac{\partial}{\partial \Theta} +\frac{1}{r^2 \sin(\Theta)^2}\frac{\partial^2}{\partial \phi^2} \right) f(r,\Theta,\phi)$