Formelsammlung Physik: Nabla-Operator

Dies ist eine Liste von einigen Formeln der Vektoranalysis im Zusammenhang mit gebräuchlichen Koordinatensystemen. Dabei bezeichnen ${\boldsymbol {\hat {x}}},{\boldsymbol {\hat {y}}},{\boldsymbol {\hat {z}}},{\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {r}}}$ die Einheitsvektoren in den jeweiligen Koordinatenrichtungen; $\operatorname {atan2} (.)$ ist der Arkustangens mit zwei Argumenten; $f$, $g$ sind Skalare und $\mathbf {A}$, $\mathbf {B}$, $\mathbf {C}$ sind Vektoren.

Tabelle mit Nabla-Operator in Zylinder und Kugelkoordinaten

Operation	Kartesische Koordinaten $(x,y,z)$	Zylinderkoordinaten $(\rho ,\phi ,z)$	Kugelkoordinaten $(r,\theta ,\phi )$
Definition der Koordinaten		$\left[{\begin{matrix}x&=&\rho \cos \phi \\y&=&\rho \sin \phi \\z&=&z\end{matrix}}\right.$	$\left[{\begin{matrix}x&=&r\sin \theta \cos \phi \\y&=&r\sin \theta \sin \phi \\z&=&r\cos \theta \end{matrix}}\right.$
		$\left[{\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\operatorname {atan} (y/x)\\z&=&z\end{matrix}}\right.$	$\left[{\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arccos(z/r)\\\phi &=&\operatorname {atan} (y/x)\end{matrix}}\right.$
$\mathbf {A}$	$A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$
$\nabla f$	${\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}$	${\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}$
$\nabla \cdot \mathbf {A}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial (\rho A_{\rho }) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial (r^{2}A_{r}) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }(A_{\theta }\sin \theta )+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$
$\nabla \times \mathbf {A}$	${\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right)\mathbf {\hat {x}} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right)\mathbf {\hat {y}} &+\\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right)\mathbf {\hat {z}} &\ \end{matrix}}$	${\begin{matrix}\left({1 \over \rho }{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}\right){\boldsymbol {\hat {\rho }}}&+\\\left({\partial A_{\rho } \over \partial z}-{\partial A_{z} \over \partial \rho }\right){\boldsymbol {\hat {\phi }}}&+\\{1 \over \rho }\left({\partial (\rho A_{\phi }) \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}{1 \over r\sin \theta }\left({\partial \over \partial \theta }(A_{\phi }\sin \theta )-{\partial A_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {r}}}&+\\{1 \over r}\left({1 \over \sin \theta }{\partial A_{r} \over \partial \phi }-{\partial \over \partial r}(rA_{\phi })\right){\boldsymbol {\hat {\theta }}}&+\\{1 \over r}\left({\partial \over \partial r}(rA_{\theta })-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {\hat {\phi }}}&\ \end{matrix}}$
$\Delta f=\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$
$\Delta \mathbf {A} =\nabla ^{2}\mathbf {A}$	$\Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}}$	${\begin{matrix}(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi }){\boldsymbol {\hat {\rho }}}&+\\(\Delta A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi }){\boldsymbol {\hat {\phi }}}&+\\(\Delta A_{z}){\boldsymbol {\hat {z}}}&\ \end{matrix}}$	${\begin{matrix}(\Delta A_{r}-{2A_{r} \over r^{2}}-{2A_{\theta }\cos \theta \over r^{2}\sin \theta }-{2 \over r^{2}}{\partial A_{\theta } \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\phi } \over \partial \phi }){\boldsymbol {\hat {r}}}&+\\(\Delta A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\phi } \over \partial \phi }){\boldsymbol {\hat {\theta }}}&+\\(\Delta A_{\phi }-{A_{\phi } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin ^{2}\theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \phi }){\boldsymbol {\hat {\phi }}}&\end{matrix}}$
infinitesimale Verschiebung	$\mathrm {d} \mathbf {l} =\mathrm {d} x\,\mathbf {\hat {x}} +\mathrm {d} y\,\mathbf {\hat {y}} +\mathrm {d} z\,\mathbf {\hat {z}}$	$\mathrm {d} \mathbf {l} =\mathrm {d} \rho \,{\boldsymbol {\hat {\rho }}}+\rho \,\mathrm {d} \phi \,{\boldsymbol {\hat {\phi }}}+\mathrm {d} z\,{\boldsymbol {\hat {z}}}$	$\mathrm {d} \mathbf {l} =\mathrm {d} r\,\mathbf {\hat {r}} +r\,\mathrm {d} \theta \,{\boldsymbol {\hat {\theta }}}+r\sin \theta \,\mathrm {d} \phi \,{\boldsymbol {\hat {\phi }}}$
infinitesimales Flächenelement	${\begin{matrix}\mathrm {d} \mathbf {A} =&\mathrm {d} y\mathrm {d} z\,\mathbf {\hat {x}} +\\&\mathrm {d} x\mathrm {d} z\,\mathbf {\hat {y}} +\\&\mathrm {d} x\mathrm {d} y\,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathrm {d} \mathbf {A} =&\rho \,\mathrm {d} \phi \mathrm {d} z\,{\boldsymbol {\hat {\rho }}}+\\&\mathrm {d} \rho \mathrm {d} z\,{\boldsymbol {\hat {\phi }}}+\\&\rho \,\mathrm {d} \rho \mathrm {d} \phi \,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathrm {d} \mathbf {A} =&r^{2}\sin \theta \,\mathrm {d} \theta \mathrm {d} \phi \,\mathbf {\hat {r}} +\\&r\sin \theta \,\mathrm {d} r\mathrm {d} \phi \,{\boldsymbol {\hat {\theta }}}+\\&r\,\mathrm {d} r\mathrm {d} \theta \,{\boldsymbol {\hat {\phi }}}\end{matrix}}$
infinitesimales Volumenelement	$\mathrm {d} V=\mathrm {d} x\mathrm {d} y\mathrm {d} z\,$	$\mathrm {d} V=\rho \,\mathrm {d} \rho \mathrm {d} \phi \mathrm {d} z\,$	$\mathrm {d} V=r^{2}\sin \theta \,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \phi \,$
Nichttriviale Rechenregeln: $\operatorname {div\ grad\ } f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ (Laplace-Operator) $\operatorname {rot\ grad\ } f=\nabla \times (\nabla f)=0$ $\operatorname {div\ rot\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {rot\ rot\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\Delta \mathbf {A}$ $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$ $\nabla \cdot (f\mathbf {A} )=f\nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla f$ $\nabla \times f\mathbf {A} =f\nabla \times \mathbf {A} -\mathbf {A} \times \nabla f$ $\nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} ),$ woraus mit $\mathbf {A} =\mathbf {B} =\mathbf {v}$ unmittelbar die für die Strömungslehre wichtige Weber-Transformation folgt: $(\mathbf {v} \cdot \nabla )\mathbf {v} =\nabla {\frac {\mathbf {v} ^{2}}{2}}-\mathbf {v} \times (\nabla \times \mathbf {v} )$ $\mathbf {A} \times (\nabla \times \mathbf {C} )=\nabla _{\mathbf {C} }(\mathbf {A} \cdot \mathbf {C} )-(\mathbf {A} \cdot \nabla )\mathbf {C} =(\nabla \mathbf {C} )\cdot \mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {C}$ $\nabla \times (\mathbf {A} \times \mathbf {B} )=\mathbf {A} \,(\nabla \cdot \mathbf {B} )-\mathbf {B} \,(\nabla \cdot \mathbf {A} )+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B}$ $\nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\nabla \times \mathbf {A} )-\mathbf {A} \cdot (\nabla \times \mathbf {B} )$