Mathematik-Online-Lexikon: Erläuterung zu Transformation von Differentialoperatoren

$\displaystyle \vec{e}_\xi = \frac{1}{\alpha}\left(\begin{array}{c} \partial_\xi... ...c} \partial_\zeta x \\ \partial_\zeta y \\ \partial_\zeta z\end{array} \right)$

$\displaystyle \vec{F}(x,y,z) = F_x\vec{e}_x+F_y\vec{e}_y+F_z\vec{e}_z = \Psi_\x... ... \Psi_\eta \vec{e}_\eta + \Psi_\zeta \vec{e}_\zeta= \vec{\Psi}(\xi,\eta,\zeta)$

$\displaystyle \operatorname{grad} U$	$\displaystyle = \frac{1}{\alpha} \partial_\xi \Phi \vec{e}_\xi + \frac{1}{\beta... ...l_\eta \Phi \vec{e}_\eta + \frac{1}{\gamma} \partial_\zeta\Phi \vec{e}_\zeta\,,$
$\displaystyle \operatorname{div} \vec{F}$	$\displaystyle = \frac{1}{\alpha\beta\gamma} \left( \partial_\xi (\beta\gamma\Ps... ...eta (\gamma\alpha\Psi_\eta) + \partial_\zeta (\alpha\beta\Psi_\zeta) \right)\,,$
$\displaystyle \operatorname{rot} \vec{F}$	$\displaystyle = \frac{1}{\beta\gamma} \left(\partial_\eta (\gamma\Psi_\zeta) - ... ...l_\zeta (\alpha\Psi_\xi) - \partial_\xi (\gamma\Psi_\zeta) \right) \vec{e}_\eta$
	$\displaystyle \qquad + \frac{1}{\alpha\beta} \left( \partial_\xi (\beta\Psi_\eta) - \partial_\eta (\alpha\Psi_\xi) \right) \vec{e}_\zeta \,.$

$\displaystyle \Delta U = \frac{1}{\alpha\beta\gamma} \left( \partial_\xi\left(\... ...ial_\zeta\left(\frac{\alpha\beta}{\gamma}\partial_\zeta\Phi\right) \right) \,.$

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	Mathematik-Online-Lexikon: Erläuterung zu
	Transformation von Differentialoperatoren