Mathematik-Online-Kurs: Vektoranalysis-Differentiation-Koordinatentransformation-Differentialoperatoren in Kugelkoordinaten

$\displaystyle x=r\cos\varphi\sin\vartheta,\quad y=r\sin\varphi\sin\vartheta,\quad z=r\cos\vartheta$

$\displaystyle U(x,y,z)=\Phi(r,\vartheta,\varphi)$

$\displaystyle \vec{F}(x,y,z)= F_x\vec{e}_x+F_y\vec{e}_y+F_z\vec{e}_z = \Psi_r \... ...{e}_\vartheta + \Psi_\varphi \vec{e}_\varphi = \vec{\Psi}(r,\vartheta,\varphi)$

$\displaystyle \operatorname{grad} U$	$\displaystyle = \partial_r \Phi \vec{e}_r + \frac{1}{r}\partial_\vartheta \Phi ... ..._\vartheta + \frac{1}{r\sin\vartheta} \partial_\varphi \Phi \vec{e}_\varphi \,,$
$\displaystyle \Delta U$	$\displaystyle = \frac{1}{r^2}\partial_r \left(r^2\partial_r \Phi\right) + \frac... ...rtheta} \partial_\vartheta \left(\sin\vartheta\partial_\vartheta \Phi\right)\,,$
$\displaystyle \operatorname{div} \vec{F}$	$\displaystyle = \frac{1}{r^2} \partial_r \left(r^2\Psi_r\right) + \frac{1}{r\si... ...1}{r\sin\vartheta}\partial_\vartheta\left(\sin\vartheta\Psi_\vartheta\right)\,,$
$\displaystyle \operatorname{rot}\vec{F}$	$\displaystyle = \frac{1}{r\sin\vartheta} \left(\partial_\vartheta(\sin\vartheta\Psi_\varphi)-\partial_\varphi \Psi_\vartheta\right)\vec{e}_r$
	$\displaystyle +\frac{1}{r\sin\vartheta} \left(\partial_\varphi \Psi_r - \sin\va... ...\partial_r (r\Psi_\vartheta)-\partial_\vartheta\Psi_r\right)\vec{e}_\varphi \,.$

$\displaystyle \operatorname{grad}U = \partial_r\Phi \vec{e}_r$

$\displaystyle \Delta U = \frac{1}{r^2}\partial_r\left(r^2\partial_r \Phi \right) = \Phi'' + \frac{2}{r}\Phi' \,.$

$\begin{displaymath} \operatorname{grad}U = sr^{s-1}\vec{e}_r = s(x^2+y^2+z^2)^{s/2-1}\,\left( \begin{array}{c} x\\ y\\ z\\ \end{array}\right) \end{displaymath}$

$\displaystyle \Delta U = s(s+1) r^{s-2} \,.$

$\displaystyle \vec{F}(x,y,z)= \psi(r)\vec{e}_r$

$\displaystyle \operatorname{div} \vec{F} = \frac{1}{r^2}\partial_r\left(r^2\partial_r \psi\right) = \psi' + \frac{2}{r} \psi \,.$

$\displaystyle \operatorname{div}\vec{F}= (s+2)r^{s-1} \,.$

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	Mathematik-Online-Kurs: Vektoranalysis - Differentiation - Koordinatentransformation
	Differentialoperatoren in Kugelkoordinaten