Formelsammlung Nabla-Operator

$\sqrt[n]{x}$

Dieser Artikel ist eine Formelsammlung zum Thema Nabla-Operator. Es werden mathematische Symbole verwendet, die im Artikel Mathematische Symbole erläutert werden.

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; atan2 ist der Arkustangens mit zwei Argumenten.

Tabelle mit Nabla-Operator in Zylinder und Kugelkoordinaten

Operation	Kartesische Koordinaten (x,y,z)	Zylinderkoordinaten (ρ,φ,z)	Kugelkoordinaten (r,θ,φ)
Definition der Koordinaten		$\left[\begin{matrix} x & = & \rho\cos\phi \\ y & = & \rho\sin\phi \\ z & = & z \end{matrix}\right.$	$\left[\begin{matrix} x & = & r\sin\theta\cos\phi \\ y & = & r\sin\theta\sin\phi \\ z & = & r\cos\theta \end{matrix}\right.$
		$\left[\begin{matrix} \rho & = & \sqrt{x^2 + y^2} \\ \phi & = & \operatorname{atan2}(y, x) \\ z & = & z \end{matrix}\right.$	$\left[\begin{matrix} r & = & \sqrt{x^2 + y^2 + z^2} \\ \theta & = & \arccos(z / r) \\ \phi & = & \operatorname{atan2}(y, x) \end{matrix}\right.$
$\mathbf{A}$	$A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}$	$A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}$	$A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}$
$\nabla f$	${\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} + {\partial f \over \partial z}\mathbf{\hat z}$	${\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z}$	${\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$
$\nabla \cdot \mathbf{A}$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho}{\partial ( \rho A_\rho ) \over \partial \rho} + {1 \over \rho}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial ( r^2 A_r ) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} ( A_\theta\sin\theta ) + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}$
$\nabla \times \mathbf{A}$	$\begin{matrix} \left({\partial A_z \over \partial y} - {\partial A_y \over \partial z}\right) \mathbf{\hat x} & + \\ \left({\partial A_x \over \partial z} - {\partial A_z \over \partial x}\right) \mathbf{\hat y} & + \\ \left({\partial A_y \over \partial x} - {\partial A_x \over \partial y}\right) \mathbf{\hat z} & \ \end{matrix}$	$\begin{matrix} \left({1 \over \rho}{\partial A_z \over \partial \phi} - {\partial A_\phi \over \partial z}\right) \boldsymbol{\hat \rho} & + \\ \left({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}\right) \boldsymbol{\hat \phi} & + \\ {1 \over \rho}\left({\partial ( \rho A_\phi ) \over \partial \rho} - {\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} {1 \over r\sin\theta}\left({\partial \over \partial \theta} ( A_\phi\sin\theta ) - {\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat r} & + \\ {1 \over r}\left({1 \over \sin\theta}{\partial A_r \over \partial \phi} - {\partial \over \partial r} ( r A_\phi ) \right) \boldsymbol{\hat \theta} & + \\ {1 \over r}\left({\partial \over \partial r} ( r A_\theta ) - {\partial A_r \over \partial \theta}\right) \boldsymbol{\hat \phi} & \ \end{matrix}$
$\Delta f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}\left(\sin\theta {\partial f \over \partial \theta}\right) + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \phi^2}$
$\Delta \mathbf{A} = \nabla^2 \mathbf{A}$	$\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z}$	$\begin{matrix} (\Delta A_\rho - {A_\rho \over \rho^2} - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\rho} & + \\ (\Delta A_\phi - {A_\phi \over \rho^2} + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) \boldsymbol{\hat\phi} & + \\ (\Delta A_z ) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} (\Delta A_r - {2 A_r \over r^2} - {2 A_\theta\cos\theta \over r^2\sin\theta} - {2 \over r^2}{\partial A_\theta \over \partial \theta} - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat r} & + \\ (\Delta A_\theta - {A_\theta \over r^2\sin^2\theta} + {2 \over r^2}{\partial A_r \over \partial \theta} - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}) \boldsymbol{\hat\theta} & + \\ (\Delta A_\phi - {A_\phi \over r^2\sin^2\theta} + {2 \over r^2\sin^2\theta}{\partial A_r \over \partial \phi} + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) \boldsymbol{\hat\phi} & \end{matrix}$
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) $\operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = 0$ $\operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \Delta \mathbf{A}$ $\Delta ( f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$ $\nabla\cdot(f \mathbf A)=f \nabla\cdot\mathbf A+\mathbf A\cdot\nabla f$ $\nabla\times f \mathbf A= f \nabla\times \mathbf A-\mathbf A\times \nabla f$ $\nabla ( \mathbf{A} \cdot \mathbf{B} ) = ( \mathbf{A} \cdot \nabla ) \mathbf{B} + ( \mathbf{B} \cdot \nabla ) \mathbf{A} + \mathbf{A} \times ( \nabla \times \mathbf{B} ) + \mathbf{B} \times ( \nabla \times \mathbf{A} ),$ woraus mit $\mathbf{A} = \mathbf{B} = \mathbf{v}$ unmittelbar die für die Strömungslehre wichtige Weber-Transformation folgt: $( \mathbf{v} \cdot \nabla ) \mathbf{v} = \nabla \frac{\mathbf{v}^2}{2} - \mathbf{v} \times ( \nabla \times \mathbf{v} )$ $\mathbf{A}\times(\nabla\times\mathbf{C}) = \nabla_{\mathbf{C}}(\mathbf{A}\cdot\mathbf{C})-(\mathbf{A}\cdot\nabla)\mathbf{C}=(\nabla\mathbf{C})\cdot\mathbf{A}-(\mathbf{A}\cdot\nabla )\mathbf{C}$ $\nabla\times (\mathbf A\times\mathbf B)= \mathbf A\,(\nabla\cdot\mathbf B) - \mathbf B\,(\nabla\cdot\mathbf A) + (\mathbf B\cdot\nabla)\mathbf A - (\mathbf A\cdot\nabla)\mathbf B$ $\nabla\cdot(\mathbf A\times\mathbf B)=\mathbf B\cdot(\nabla\times\mathbf A)-\mathbf A\cdot(\nabla\times\mathbf B)$