Formelsammlung Physik: Nabla-Operator

Dies ist eine Liste von einigen Formeln der Vorlage:W im Zusammenhang mit gebräuchlichen Vorlage:Wen. Dabei bezeichnen $\widehat{𝒙},\widehat{𝒚},\widehat{𝒛},\hat{\mathit{\rho }},\hat{\mathit{\varphi }},\hat{\mathit{\theta }},\widehat{𝒓}$ die Vorlage:Wen in den jeweiligen Koordinatenrichtungen; $\mathrm{atan2}(.)$ ist der Vorlage:W; $f$, $g$ sind Vorlage:We und $𝐀$, $𝐁$, $𝐂$ sind Vorlage:Wen.

Tabelle mit Vorlage:W in Zylinder und Kugelkoordinaten

Operation	Vorlage:W $(x,y,z)$	Vorlage:W $(\rho ,\varphi ,z)$	Vorlage:W $(r,\theta ,\varphi )$
Definition der Koordinaten		$[\begin{array}{ccc}x& =& \rho \cos \varphi \\ y& =& \rho \sin \varphi \\ z& =& z\end{array}$	$[\begin{array}{ccc}x& =& r\sin \theta \cos \varphi \\ y& =& r\sin \theta \sin \varphi \\ z& =& r\cos \theta \end{array}$
		$[\begin{array}{ccc}\rho & =& \sqrt{x^2+y^2}\\ \varphi & =& \mathrm{atan}(y/x)\\ z& =& z\end{array}$	$[\begin{array}{ccc}r& =& \sqrt{x^2+y^2+z^2}\\ \theta & =& \arccos (z/r)\\ \varphi & =& \mathrm{atan}(y/x)\end{array}$
$𝐀$	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\widehat{𝒛}$	$A_r\widehat{𝒓}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$
$\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\widehat{𝒛}$	$\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$
$\mathrm{\nabla }\cdot 𝐀$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$
$\mathrm{\nabla }\times 𝐀$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\widehat{𝐱}& +\\ (\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\widehat{𝐲}& +\\ (\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\widehat{𝐳}& \end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}& +\\ (\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}& +\\ \frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\widehat{𝒛}& \end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })\widehat{𝒓}& +\\ \frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}(r{A}_{\varphi }))\hat{\mathit{\theta }}& +\\ \frac{1}{r}(\frac{\partial }{\partial r}(r{A}_{\theta })-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}& \end{array}$
$\mathrm{\Delta }f={\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$
$\mathrm{\Delta }𝐀={\mathrm{\nabla }}^2𝐀$	$\mathrm{\Delta }{A}_x\widehat{𝐱}+\mathrm{\Delta }{A}_y\widehat{𝐲}+\mathrm{\Delta }{A}_z\widehat{𝐳}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\rho }}& +\\ (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{\varphi }}& +\\ (\mathrm{\Delta }{A}_z)\widehat{𝒛}& \end{array}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_r-\frac{2A_r}{r^2}-\frac{2A_{\theta }\cos \theta }{r^2\sin \theta }-\frac{2}{r^2}\frac{\partial A_{\theta }}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi })\widehat{𝒓}& +\\ (\mathrm{\Delta }{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\theta }}& +\\ (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{r^2{\sin }^2\theta }+\frac{2}{r^2{\sin }^2\theta }\frac{\partial A_r}{\partial \varphi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{\varphi }}& \end{array}$
infinitesimale Verschiebung	$\mathrm{d}𝐥=\mathrm{d}x\widehat{𝐱}+\mathrm{d}y\widehat{𝐲}+\mathrm{d}z\widehat{𝐳}$	$\mathrm{d}𝐥=\mathrm{d}\rho \hat{\mathit{\rho }}+\rho \mathrm{d}\varphi \hat{\mathit{\varphi }}+\mathrm{d}z\widehat{𝒛}$	$\mathrm{d}𝐥=\mathrm{d}r\widehat{𝐫}+r\mathrm{d}\theta \hat{\mathit{\theta }}+r\sin \theta \mathrm{d}\varphi \hat{\mathit{\varphi }}$
infinitesimales Flächenelement	$\begin{array}{cc}\mathrm{d}𝐀=& \mathrm{d}y\mathrm{d}z\widehat{𝐱}+\\ & \mathrm{d}x\mathrm{d}z\widehat{𝐲}+\\ & \mathrm{d}x\mathrm{d}y\widehat{𝐳}\end{array}$	$\begin{array}{cc}\mathrm{d}𝐀=& \rho \mathrm{d}\varphi \mathrm{d}z\hat{\mathit{\rho }}+\\ & \mathrm{d}\rho \mathrm{d}z\hat{\mathit{\varphi }}+\\ & \rho \mathrm{d}\rho \mathrm{d}\varphi \widehat{𝐳}\end{array}$	$\begin{array}{cc}\mathrm{d}𝐀=& r^2\sin \theta \mathrm{d}\theta \mathrm{d}\varphi \widehat{𝐫}+\\ & r\sin \theta \mathrm{d}r\mathrm{d}\varphi \hat{\mathit{\theta }}+\\ & r\mathrm{d}r\mathrm{d}\theta \hat{\mathit{\varphi }}\end{array}$
infinitesimales Volumenelement	$\mathrm{d}V=\mathrm{d}x\mathrm{d}y\mathrm{d}z$	$\mathrm{d}V=\rho \mathrm{d}\rho \mathrm{d}\varphi \mathrm{d}z$	$\mathrm{d}V=r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\varphi$
Nichttriviale Rechenregeln: $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f$ (Vorlage:W) $\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\times (\mathrm{\nabla }f)=0$ $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{r}\mathrm{o}\mathrm{t}𝐀=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$ $\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{t}𝐀=\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-\mathrm{\Delta }𝐀$ $\mathrm{\Delta }(fg)=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$ $\mathrm{\nabla }\cdot (f𝐀)=f\mathrm{\nabla }\cdot 𝐀+𝐀\cdot \mathrm{\nabla }f$ $\mathrm{\nabla }\times f𝐀=f\mathrm{\nabla }\times 𝐀-𝐀\times \mathrm{\nabla }f$ Vorlage:Anker$\mathrm{\nabla }(𝐀\cdot 𝐁)=(𝐀\cdot \mathrm{\nabla })𝐁+(𝐁\cdot \mathrm{\nabla })𝐀+𝐀\times (\mathrm{\nabla }\times 𝐁)+𝐁\times (\mathrm{\nabla }\times 𝐀),$ woraus mit $𝐀=𝐁=𝐯$ unmittelbar die für die Vorlage:W wichtige Weber-Transformation folgt: $(𝐯\cdot \mathrm{\nabla })𝐯=\mathrm{\nabla }\frac{{𝐯}^2}{2}-𝐯\times (\mathrm{\nabla }\times 𝐯)$ $𝐀\times (\mathrm{\nabla }\times 𝐂)={\mathrm{\nabla }}_{𝐂}(𝐀\cdot 𝐂)-(𝐀\cdot \mathrm{\nabla })𝐂=(\mathrm{\nabla }𝐂)\cdot 𝐀-(𝐀\cdot \mathrm{\nabla })𝐂$ $\mathrm{\nabla }\times (𝐀\times 𝐁)=𝐀(\mathrm{\nabla }\cdot 𝐁)-𝐁(\mathrm{\nabla }\cdot 𝐀)+(𝐁\cdot \mathrm{\nabla })𝐀-(𝐀\cdot \mathrm{\nabla })𝐁$ $\mathrm{\nabla }\cdot (𝐀\times 𝐁)=𝐁\cdot (\mathrm{\nabla }\times 𝐀)-𝐀\cdot (\mathrm{\nabla }\times 𝐁)$