Nabla in verschillende assenstelsels

Uit Wikipedia, de vrije encyclopedie

In de onderstaande tabel staat een overzicht van de vorm die de de operator nabla aanneemt in de drie assenstelsels:

**Tabel met de operator $\nabla$ in cilinder- en bolcoördinaten**
Operatie	Cartesiaanse coördinaten (x,y,z)	Cilindercoördinaten (ρ,φ,z)	Bolcoördinaten (r,θ,φ)
Relatie	zie:	Cilindercoördinaten	Bolcoördinaten
eenheids- vectoren	$\mathbf{\hat x}, \mathbf{\hat y}, \mathbf{\hat z}$	$\boldsymbol{\hat \rho}, \boldsymbol{\hat \phi}, \boldsymbol{\hat z}$	$\boldsymbol{\hat r}, \boldsymbol{\hat \theta}, \boldsymbol{\hat \phi}$
scalair veld $f\,$	$f(x,y,z)\,$	$f_c(\rho,\phi,z)=f(\rho\cos(\phi),\rho\sin(\phi),z)\,$	$f_b(r,\phi,\theta)=f(r\sin(\theta)\cos(\phi),r\sin(\theta)\sin(\phi),r\cos(\theta))\,$
vectorveld $A\,$	$A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}$	$A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}$	$A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}$
$\nabla f$	${\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} + {\partial f \over \partial z}\mathbf{\hat z}$	${\partial f_c \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f_c \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f_c \over \partial z}\boldsymbol{\hat z}$	${\partial f_b \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f_b \over \partial \theta}\boldsymbol{\hat\theta} + {1 \over r\sin\phi}{\partial f_b \over \partial \phi}\boldsymbol{\hat\phi}$
$\nabla \cdot A$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	$\frac 1{\rho}{\partial \rho A_\rho \over \partial \rho} + \frac 1{\rho}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial r^2 A_r \over \partial r} + {1 \over r\sin\theta}{\partial A_\theta\sin\theta \over \partial \theta} + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}$
$\nabla \times A$	$\begin{matrix} ({\partial A_z \over \partial y} - {\partial A_y \over \partial z}) \mathbf{\hat x} & + \\ ({\partial A_x \over \partial z} - {\partial A_z \over \partial x}) \mathbf{\hat y} & + \\ ({\partial A_y \over \partial x} - {\partial A_x \over \partial y}) \mathbf{\hat z} & \ \end{matrix}$	$\begin{matrix} ({1 \over \rho}{\partial A_z \over \partial \phi} - {\partial A_\phi \over \partial z}) \boldsymbol{\hat \rho} & + \\ ({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}) \boldsymbol{\hat \phi} & + \\ {1 \over \rho}({\partial \rho A_\phi \over \partial \rho} - {\partial A_\rho \over \partial \phi}) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} {1 \over r\sin\theta}({\partial A_\phi\sin\theta \over \partial \theta} - {\partial A_\theta \over \partial \phi}) \boldsymbol{\hat r} & + \\ ({1 \over r\sin\theta}{\partial A_r \over \partial \phi} - {1 \over r}{\partial r A_\phi \over \partial r}) \boldsymbol{\hat \theta} & + \\ {1 \over r}({\partial r A_\theta \over \partial r} - {\partial A_r \over \partial \theta}) \boldsymbol{\hat \phi} & \ \end{matrix}$
$\Delta f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}(\rho {\partial f_c \over \partial \rho}) + {1 \over \rho^2}{\partial^2 f_c \over \partial \phi^2} + {\partial^2 f_c \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}(r^2 {\partial f_b \over \partial r}) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f_b \over \partial \theta}) + {1 \over r^2\sin^2\theta}{\partial^2 f_b \over \partial \phi^2}$
$\Delta A = \nabla^2 A$	$\mathbf{\hat x}\Delta A_x + \mathbf{\hat y}\Delta A_y + \mathbf{\hat z}\Delta A_z$	$\begin{matrix} \boldsymbol{\hat\rho}(\Delta A_\rho - {A_\rho \over \rho^2} - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\phi}(\Delta A_\phi - {A_\phi \over \rho^2} + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) & + \\ \boldsymbol{\hat z} \Delta A_z & \ \end{matrix}$	$\begin{matrix} \boldsymbol{\hat r} & (\Delta A_r - {2 A_r \over r^2} - {2 A_\theta\cos\theta \over r^2\sin\theta} \\ \ & - {2 \over r^2}{\partial A_\theta \over \partial \theta} - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\theta} & (\Delta A_\theta - {A_\theta \over r^2\sin^2\theta} \\ \ & + {2 \over r^2}{\partial A_r \over \partial \theta} - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\phi} & (\Delta A_\phi - {A_\phi \over r^2\sin^2\theta} \\ \ & + {2 \over r^2\sin^2\theta}{\partial A_r \over \partial \phi} + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) & \ \end{matrix}$
Niet evidente rekenregels: $\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f$ (Laplaciaan) $\operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = \mathbf{0}$ $\operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ $\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$

Inhoud

1 Afleiding
- 1.1 Cilindercoördinaten
- 1.2 Bolcoördinaten

[bewerk] Afleiding

[bewerk] Cilindercoördinaten

[bewerk] Gradiënt van een scalaire functie

$\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\,$

De component van grad f in de richting van $ρ$ is:

$(\nabla f)_\rho = \frac{\partial f}{\partial x}\cos(\phi)+ \frac{\partial f}{\partial y}\sin(\phi)= \frac{\partial f_c}{\partial \rho}\,$

De component van grad f in de richting van $φ$ is:

$(\nabla f)_\phi = - \frac{\partial f}{\partial x}\sin(\phi)+ \frac{\partial f}{\partial y}\cos(\phi)=\frac 1\rho\frac{\partial f_c}{\partial \phi} \,$

[bewerk] Divergentie van een vectorveld

$\nabla\cdot A= \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\,$

We drukken de divergentie uit in de componenten van de polaire voorstelling:

$A_x =A_\rho\cos(\phi)-A_\phi \sin(\phi)\,$

$A_y =A_\rho\sin(\phi)+A_\phi \cos(\phi)\,$

dus

$\frac{\partial A_x}{\partial x}= \frac{\partial}{\partial x} \left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)=$

$\frac{\partial}{\partial \rho}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)\frac{\partial \rho}{\partial x}+ \frac{\partial}{\partial \phi}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)\frac{\partial \phi}{\partial x} = \,$

$\frac{\partial}{\partial \rho}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)\cos(\phi)+ \frac{\partial}{\partial \phi}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)(-\frac 1\rho\sin(\phi))= \,$

$\frac{\partial A_\rho}{\partial \rho}\cos^2(\phi)+ \frac 1\rho A_\rho\sin^2(\phi)+ \frac 1\rho \frac{\partial A_\phi}{\partial \phi}\sin^2(\phi)+ \frac 1\rho A_\phi \sin(\phi)\cos(\phi) \,$

$\frac{\partial A_y}{\partial y}= \frac{\partial}{\partial y} \left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right)=$

$\frac{\partial}{\partial \rho}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right) \frac{\partial \rho}{\partial y}+ \frac{\partial}{\partial \phi}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right) \frac{\partial \phi}{\partial y} = \,$

$\frac{\partial}{\partial \rho}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right)\sin(\phi)+ \frac{\partial}{\partial \phi}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right)(\frac 1\rho\cos(\phi)) = \,$

$\frac{\partial A_\rho}{\partial \rho}\sin^2(\phi)+ \frac 1\rho A_\rho\cos^2(\phi)+ \frac 1\rho \frac{\partial A_\phi}{\partial \phi}\cos^2(\phi)- \frac 1\rho A_\phi \cos(\phi)\sin(\phi) \,$

Samen leidt dat tot:

$\nabla\cdot A= \frac{\partial A_\rho}{\partial \rho}+ \frac{A_\rho}{\rho}+ \frac 1{\rho}\frac{\partial A_\phi}{\partial \phi}+ \frac{\partial A_z}{\partial z}\,$

[bewerk] Rotatie van een vectorveld

$(\nabla\times A)_x = \frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z} = \frac{\partial A_z}{\partial \rho}\frac{\partial \rho}{\partial y} +\frac{\partial A_z}{\partial \phi}\frac{\partial \phi}{\partial y} -\frac{\partial A_y}{\partial z} = \,$

$\frac{\partial A_z}{\partial \rho}\sin(\phi) +\frac{\partial A_z}{\partial \phi}\frac 1\rho\cos(\phi) -\frac{\partial A_\rho}{\partial z}\sin(\phi) -\frac{\partial A_\phi}{\partial z}\cos(\phi) \,$

$(\nabla\times A)_y = \frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x} = \frac{\partial A_x}{\partial z} -\frac{\partial A_z}{\partial \rho}\frac{\partial \rho}{\partial x} -\frac{\partial A_z}{\partial \phi}\frac{\partial \phi}{\partial x} = \,$

$\frac{\partial A_\rho}{\partial z}\cos(\phi)-\frac{\partial A_\phi}{\partial z}\sin(\phi) -\frac{\partial A_z}{\partial \rho}\cos(\phi) +\frac{\partial A_z}{\partial \phi}\frac 1\rho\sin(\phi) \,$

$(\nabla\times A)_z = \frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y} = \frac{\partial A_y}{\partial \rho}\frac{\partial \rho}{\partial x} +\frac{\partial A_y}{\partial \phi}\frac{\partial \phi}{\partial x} -\frac{\partial A_x}{\partial \rho}\frac{\partial \rho}{\partial y} -\frac{\partial A_x}{\partial \phi}\frac{\partial \phi}{\partial y} = \,$

$\frac{\partial (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \rho}\cos(\phi) -\frac{\partial (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi}\frac 1\rho\sin(\phi) \,$

$-\frac{\partial (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \rho}\sin(\phi) -\frac{\partial (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \phi}\frac 1\rho\cos(\phi) = \,$

$\frac{\partial A_\rho}{\partial \rho}\sin(\phi)\cos(\phi)+\frac{\partial A_\phi}{\partial \rho}\cos^2(\phi) -\frac{\partial A_\rho\sin(\phi)}{\partial \phi}\frac 1\rho\sin(\phi) -\frac{\partial A_\phi \cos(\phi)}{\partial \phi}\frac 1\rho\sin(\phi) \,$

$-\frac{\partial A_\rho}{\partial \rho}\cos(\phi)\sin(\phi) +\frac{\partial A_\phi}{\partial \rho}\sin^2(\phi) -\frac{\partial A_\rho \cos(\phi)}{\partial \phi}\frac 1\rho\cos(\phi) +\frac{\partial A_\phi \sin(\phi))}{\partial \phi}\frac 1\rho\cos(\phi) = \,$

$\frac{\partial A_\phi}{\partial \rho} -\frac{\partial A_\rho}{\partial \phi}\frac 1\rho +A_\phi \frac 1\rho \,$

Transformeren naar ρ en φ:

$(\nabla\times A)_\rho = (\nabla\times A)_x\cos(\phi)+(\nabla\times A)_y\sin(\phi) = \frac 1\rho\frac{\partial A_z}{\partial \phi} -\frac{\partial A_\phi}{\partial z} \,$

$(\nabla\times A)_\phi = -(\nabla\times A)_x\sin(\phi)+(\nabla\times A)_y\cos(\phi) = \frac{\partial A_\rho}{\partial z} -\frac{\partial A_z}{\partial \rho} \,$

[bewerk] Laplaciaan van een scalaire functie

Uit het bovenstaande volgt voor de Laplaciaan van een scalaire functie f:

$\Delta f = \nabla\cdot\nabla f = \frac{\partial (\nabla f)_\rho}{\partial \rho}+ \frac{(\nabla f)_\rho}{\rho}+ \frac 1{\rho}\frac{\partial (\nabla f)_\phi}{\partial \phi}+ \frac{\partial (\nabla f)_z}{\partial z}=\,$

$\frac{\partial \frac{\partial f_c}{\partial \rho}}{\partial \rho}+ \frac{\frac{\partial f_c}{\partial \rho}}{\rho}+ \frac 1{\rho}\frac{\partial \frac 1\rho\frac{\partial f_c}{\partial \phi}}{\partial \phi}+ \frac{\partial \frac{\partial f_c}{\partial z}}{\partial z}= \frac{\partial^2 f_c}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial f_c}{\partial \rho}+ \frac 1{\rho^2}\frac{\partial^2 f_c}{\partial \phi^2}+ \frac{\partial^2 f_c}{\partial z^2}\,$

[bewerk] Laplaciaan van een vectorveld

Uit het bovenstaande volgt voor de Laplaciaan van een vectorveld A:

$(\Delta A)_\rho = (\Delta A)_x\cos(\phi)+(\Delta A)_y\sin(\phi)=\Delta A_x\cos(\phi)+\Delta A_y\sin(\phi)= \,$

$\left(\frac{\partial^2 A_x}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_x}{\partial \rho}+ \frac 1{\rho^2}\frac{\partial^2 A_x}{\partial \phi^2}+ \frac{\partial^2 A_x}{\partial z^2}\right)\cos(\phi) + \,$

$\left(\frac{\partial^2 A_y}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_y}{\partial \rho}+ \frac 1{\rho^2}\frac{\partial^2 A_y}{\partial \phi^2}+ \frac{\partial^2 A_y}{\partial z^2}\right)\sin(\phi)=\,$

$\frac{\partial^2 A_\rho}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\rho}{\partial \rho}+ \frac 1{\rho^2} \left( \frac{\partial^2 A_x}{\partial \phi^2}\cos(\phi)+ \frac{\partial^2 A_y}{\partial \phi^2}\sin(\phi) \right)+ \frac{\partial^2 A_\rho}{\partial z^2}= \,$

$\frac{\partial^2 A_\rho}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\rho}{\partial \rho}+ \frac{\partial^2 A_\rho}{\partial z^2} + \,$

$\frac 1{\rho^2} \left( \frac{\partial^2 (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \phi^2}\cos(\phi)+ \frac{\partial^2 (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi^2}\sin(\phi) \right) =\,$

$\frac{\partial^2 A_\rho}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\rho}{\partial \rho}+ \frac{\partial^2 A_\rho}{\partial z^2} + \frac 1{\rho^2}\frac{\partial^2 A_\rho}{\partial \phi^2} - \frac 1{\rho^2}A_\rho - \frac 2{\rho^2}\frac{\partial A_\phi}{\partial \phi} =\,$

$\Delta A_\rho - \frac 1{\rho^2}A_\rho - \frac 2{\rho^2}\frac{\partial A_\phi}{\partial \phi} \,$

En analoog:

$(\Delta A)_\phi = -(\Delta A)_x\sin(\phi)+(\Delta A)_y\cos(\phi)=-\Delta A_x\sin(\phi)+\Delta A_y\cos(\phi)= \,$

$-\left(\frac{\partial^2 A_x}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_x}{\partial \rho}+ \frac 1{\rho^2}\frac{\partial^2 A_x}{\partial \phi^2}+ \frac{\partial^2 A_x}{\partial z^2}\right)\sin(\phi) + \,$

$\frac{\partial^2 A_\phi}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\phi}{\partial \rho}- \frac 1{\rho^2} \left( \frac{\partial^2 A_x}{\partial \phi^2}\sin(\phi)- \frac{\partial^2 A_y}{\partial \phi^2}\cos(\phi) \right)+ \frac{\partial^2 A_\phi}{\partial z^2}= \,$

$\frac{\partial^2 A_\phi}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\phi}{\partial \rho}+ \frac{\partial^2 A_\phi}{\partial z^2}- \,$

$\frac 1{\rho^2} \left( \frac{\partial^2 (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \phi^2}\sin(\phi)- \frac{\partial^2 (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi^2}\cos(\phi) \right) =\,$

$\frac{\partial^2 A_\phi}{\partial \rho^2}+ \frac{1}{\rho}\frac{\partial A_\phi}{\partial \rho}+ \frac{\partial^2 A_\phi}{\partial z^2}+ \frac 1{\rho^2}\frac{\partial^2 A_\phi}{\partial \phi^2}- \frac 1{\rho^2}A_\phi^2+ \frac 2{\rho^2}\frac{\partial A_\rho}{\partial \phi} =\,$

$\Delta A_\phi - \frac 1{\rho^2}A_\phi + \frac 2{\rho^2}\frac{\partial A_\rho}{\partial \phi} \,$

[bewerk] Bolcoördinaten

[bewerk] Gradiënt van een scalaire functie

$\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\,$

De component van grad f in de richting van r is:

$(\nabla f)_r = \frac{\partial f}{\partial x}\cos(\phi)\sin(\theta)+ \frac{\partial f}{\partial y}\sin(\phi)\sin(\theta)+ \frac{\partial f}{\partial z}\cos(\theta) = \frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial r}+ \frac{\partial f}{\partial z}\frac{\partial z}{\partial r} = \frac{\partial f_b}{\partial r} \,$

De component van grad f in de richting van $φ$ is:

$(\nabla f)_\phi = - \frac{\partial f}{\partial x}\sin(\phi)+ \frac{\partial f}{\partial y}\cos(\phi) = \frac 1{r\sin(\theta)}\left( \frac{\partial f}{\partial x}\frac{\partial x}{\partial \phi}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial \phi} \right) = \frac 1{r\sin(\theta)}\frac{\partial f_b}{\partial \phi} \,$

De component van grad f in de richting van $θ$ is:

$(\nabla f)_\theta = \frac{\partial f}{\partial x}\cos(\phi)\cos(\theta)+ \frac{\partial f}{\partial y}\sin(\phi)\cos(\theta)- \frac{\partial f}{\partial z}\sin(\theta) = \frac 1r\left( \frac{\partial f}{\partial x}\frac{\partial x}{\partial \theta}+ \frac{\partial f}{\partial y}\frac{\partial y}{\partial \theta}+ \frac{\partial f}{\partial z}\frac{\partial z}{\partial \theta} \right) = \frac 1r\frac{\partial f_b}{\partial \theta} \,$

[bewerk] Divergentie van een vectorveld

$\nabla\cdot A= \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\,$

We drukken de divergentie uit in de voorstelling in bolcoördinaten:

$A_x =A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta)\,$

$A_y =A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)\,$

$A_z =A_r\cos(\theta)-A_\theta\sin(\theta)\,$

Nu is:

$\frac{\partial A_x}{\partial x}= \frac{\partial A_x}{\partial r}\frac{\partial r}{\partial x}+ \frac{\partial A_x}{\partial \phi}\frac{\partial \phi}{\partial x}+ \frac{\partial A_x}{\partial \theta}\frac{\partial \theta}{\partial x}$

en analoog voor y en z, zodat:

$\nabla\cdot A=\,$

$\frac{\partial }{\partial r} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) \cos(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) (-\frac 1{r\sin(\theta)})\sin(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) \frac 1{r}\cos(\phi)\cos(\theta) + \,$

$\frac{\partial }{\partial r} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \sin(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \frac 1{r\sin(\theta)}\cos(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \frac 1{r}\sin(\phi)\cos(\theta) + \,$

$\frac{\partial }{\partial r} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \cos(\theta)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) (-\frac 1r)\sin(\theta) \,$

We verzamelen apart:

de termen waarin $A r$ voorkomt:

$A_r \left( \frac 1{r}\sin^2(\phi)+ \frac 1{r}\cos^2(\phi)+ \frac 1{r}\cos^2(\phi)\cos^2(\theta)+ \frac 1{r}\sin^2(\phi)\cos^2(\theta)+ \frac 1{r}\sin^2(\theta) \right) =\frac 2{r}A_r \,$

$\frac{\partial A_r}{\partial r} \left( \cos^2(\phi)\sin^2(\theta) + \sin^2(\phi)\sin^2(\theta) + \cos^2(\theta) \right) = \frac{\partial A_r}{\partial r} \,$

$\frac{\partial A_r}{\partial \phi} \left( -\sin(\phi)\cos(\phi)\sin(\theta) + \cos(\phi)\sin(\phi)\sin(\theta) \right) =0 \,$

$\frac{\partial A_r}{\partial \theta} \left( \cos(\phi)\sin(\theta)(-\frac 1{r})\cos(\phi)\cos(\theta) + \sin(\phi)\sin(\theta)\frac 1{r}\sin(\phi)\cos(\theta) + \cos(\theta)\frac 1{r}\sin(\theta) \right) =0 \,$

de termen waarin $A φ$ voorkomt:

$A_\phi \left( \frac 1{r\sin(\theta)}\cos(\phi)\sin(\phi) -\frac 1{r\sin(\theta)}\sin(\phi)\cos(\phi) \right) =0 \,$

$\frac{\partial A_\phi}{\partial r} \left( -\sin(\phi)\cos(\phi)\sin(\theta) + \cos(\phi)\sin(\phi)\sin(\theta) \right) =0 \,$

$\frac{\partial A_\phi}{\partial \phi} \left( \sin(\phi)\frac 1{r\sin(\theta)}\sin(\phi) + \cos(\phi)\frac 1{r\sin(\theta)}\cos(\phi) \right) = \frac 1{r\sin(\theta)}\frac{\partial A_\phi}{\partial \phi} \,$

$\frac{\partial A_\phi}{\partial \theta} \left( -\frac 1{r}\sin(\phi)\cos(\phi)\cos(\theta) + \frac 1{r}\cos(\phi)\sin(\phi)\cos(\theta) \right) =0 \,$

en de termen waarin $A θ$ voorkomt:

$A_\theta \left( \frac 1{r\sin(\theta)}\sin^2(\phi)\cos(\theta)+ \frac 1{r\sin(\theta)}\cos^2(\phi)\cos(\theta)- \frac 1{r}\cos^2(\phi)\sin(\theta)\cos(\theta)- \frac 1{r}\sin^2(\phi)\sin(\theta)\cos(\theta)+ \frac 1{r}\sin(\theta)\cos(\theta) \right) \,$

$=\frac 1{r\sin(\theta)}A_\theta\cos(\theta) \,$

$\frac{\partial A_\theta}{\partial r} \left( \cos^2(\phi)\cos(\theta)\sin(\theta) + \sin^2(\phi)\cos(\theta)\sin(\theta) - \sin(\theta)\cos(\theta) \right) =0 \,$

$\frac{\partial A_\theta}{\partial \phi} \left( -\frac 1{r\sin(\theta)}\cos(\phi)\sin(\phi)\cos(\theta) + \frac 1{r\sin(\theta)}\sin(\phi)\cos(\phi)\cos(\theta) \right) = 0 \,$

$\frac{\partial A_\theta}{\partial \theta} \left( \frac 1r\cos^2(\phi)\cos^2(\theta) + \frac 1r\sin^2(\phi)\cos^2(\theta) + \frac 1r\sin^2(\theta) \right) = \frac{\partial A_\theta}{\partial \theta} \,$

Samen geeft dat:

$\nabla\cdot A= \frac 2{r}A_r + \frac{\partial A_r}{\partial r} + \frac 1{r\sin(\theta)}\frac{\partial A_\phi}{\partial \phi} + \frac 1{r\sin(\theta)}A_\theta\cos(\theta) + \frac{\partial A_\theta}{\partial \theta} = \,$

$\frac 1{r^2}\frac{\partial r^2A_r}{\partial r} + \frac 1{r\sin(\theta)}\frac{\partial A_\phi}{\partial \phi} + \frac 1{r\sin(\theta)}\frac{\partial \sin(\theta)A_\theta}{\partial \theta} \,$

[bewerk] Rotatie van een vectorveld

Voor de divergentie bepaalden we $\frac{\partial A_x}{\partial x}, \frac{\partial A_y}{\partial y}, \frac{\partial A_z}{\partial z} \,$ . Nu moeten de andere afgeleiden bepaald worden.

Voor de x-component:

$\frac{\partial A_z}{\partial y} = \frac{\partial A_z}{\partial r }\frac{\partial r }{\partial y} +\frac{\partial A_z}{\partial \phi }\frac{\partial \phi }{\partial y} +\frac{\partial A_z}{\partial \theta}\frac{\partial \theta}{\partial y} = \,$

$\frac{\partial }{\partial r} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \sin(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \frac 1{r\sin(\theta)}\cos(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \frac 1{r}\sin(\phi)\cos(\theta) \,$

$\frac{\partial A_y}{\partial z} = \frac{\partial A_y}{\partial r }\frac{\partial r }{\partial z} +\frac{\partial A_y}{\partial \phi }\frac{\partial \phi }{\partial z} +\frac{\partial A_y}{\partial \theta}\frac{\partial \theta}{\partial z} = \,$

$\frac{\partial }{\partial r} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \cos(\theta)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) (-\frac 1r)\sin(\theta) \,$

Daaruit volgt:

$(\nabla\times A)_x = \frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z} = \,$

$\frac{\partial A_r}{\partial \phi} \frac 1{r\sin(\theta)}\cos(\phi)\cos(\theta) +\frac{\partial A_r}{\partial \theta} \frac 1{r}\sin(\phi) \,$

$-\frac{\partial A_\phi}{\partial r} \cos(\phi)\cos(\theta) +\frac{\partial A_\phi}{\partial \theta} \frac 1r\cos(\phi)\sin(\theta) \,$

$-\frac{\partial A_\theta}{\partial r} \sin(\phi) -\frac{\partial A_\theta}{\partial \phi} \frac 1{r}\cos(\phi) -A_\theta \frac 1{r}\sin(\phi) \,$

Voor de y-component:

$\frac{\partial A_x}{\partial z} = \frac{\partial A_x}{\partial r }\frac{\partial r }{\partial z} +\frac{\partial A_x}{\partial \phi }\frac{\partial \phi }{\partial z} +\frac{\partial A_x}{\partial \theta}\frac{\partial \theta}{\partial z} = \,$

$\frac{\partial }{\partial r} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) \cos(\theta)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) (-\frac 1r)\sin(\theta) \,$

$\frac{\partial A_z}{\partial x} = \frac{\partial A_z}{\partial r }\frac{\partial r }{\partial x} +\frac{\partial A_z}{\partial \phi }\frac{\partial \phi }{\partial x} +\frac{\partial A_z}{\partial \theta}\frac{\partial \theta}{\partial x} = \,$

$\frac{\partial }{\partial r} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \cos(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) (-\frac 1{r\sin(\theta)})\sin(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\theta)-A_\theta\sin(\theta) \right) \frac 1{r}\cos(\phi)\cos(\theta) \,$

Daaruit volgt:

$(\nabla\times A)_y = \frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x} = \,$

$\frac{\partial A_r}{\partial \phi} \frac 1{r\sin(\theta)}\sin(\phi)\cos(\theta) -\frac{\partial A_r}{\partial \theta} \frac 1r\cos(\phi) \,$

$- \frac{\partial A_\phi}{\partial r} \sin(\phi)\cos(\theta) + \frac{\partial A_\phi}{\partial \theta} \frac 1r\sin(\phi)\sin(\theta) \,$

$+ A_\theta \frac 1r\cos(\phi) + \frac{\partial A_\theta}{\partial r} \cos(\phi) - \frac{\partial A_\theta}{\partial \phi} \frac 1{r}\sin(\phi) \,$

Voor de z-component:

$\frac{\partial A_x}{\partial y} = \frac{\partial A_x}{\partial r }\frac{\partial r }{\partial y} +\frac{\partial A_x}{\partial \phi }\frac{\partial \phi }{\partial y} +\frac{\partial A_x}{\partial \theta}\frac{\partial \theta}{\partial y} = \,$

$\frac{\partial }{\partial r} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) \sin(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta)) \right) \frac 1{r\sin(\theta)}\cos(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta) \right) \frac 1{r}\sin(\phi)\cos(\theta) \,$

$\frac{\partial A_y}{\partial x}= \frac{\partial A_y}{\partial r }\frac{\partial r }{\partial x} +\frac{\partial A_y}{\partial \phi }\frac{\partial \phi }{\partial x} +\frac{\partial A_y}{\partial \theta}\frac{\partial \theta}{\partial x} = \,$

$\frac{\partial }{\partial r} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \cos(\phi)\sin(\theta)+ \,$

$\frac{\partial }{\partial \phi} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) (-\frac 1{r\sin(\theta)})\sin(\phi)+ \,$

$\frac{\partial }{\partial \theta} \left( A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta) \right) \frac 1{r}\cos(\phi)\cos(\theta) \,$

Daaruit volgt:

$(\nabla\times A)_z = \frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y} = \,$

$\frac{\partial A_r}{\partial \phi} \frac 1{r} - A_\phi \frac 1{r\sin(\theta)} - \frac{\partial A_\phi}{\partial r} \sin(\theta) - \frac{\partial A_\phi}{\partial \theta} \frac 1{r}\cos(\theta) + \frac{\partial A_\theta}{\partial \phi} \frac 1{r\sin(\theta)}\cos(\theta) \,$

Transformeren naar r, φ en θ:

$(\nabla\times A)_r = (\nabla\times A)_x\cos(\phi)\sin(\theta) +(\nabla\times A)_y\sin(\phi)\sin(\theta) +(\nabla\times A)_z\cos(\theta) = \,$

$A_\phi \frac 1{r\sin(\theta)}\cos(\theta) + \frac 1r \frac{\partial A_\phi}{\partial \theta} - \frac{\partial A_\theta}{\partial \phi} \frac 1{r\sin(\theta)} \,$

$(\nabla\times A)_\phi = -(\nabla\times A)_x\sin(\phi) +(\nabla\times A)_y\cos(\phi) = \,$

$\frac{\partial A_\theta}{\partial r} - \frac 1r \frac{\partial A_r}{\partial \theta} + \frac 1r A_\theta \,$

$(\nabla\times A)_\theta = (\nabla\times A)_x\cos(\phi)\cos(\theta) +(\nabla\times A)_y\sin(\phi)\cos(\theta) -(\nabla\times A)_z\sin(\theta) = \,$

$\frac 1r A_\phi - \frac 1{r\sin(\theta)} \frac{\partial A_r}{\partial \phi} + \frac{\partial A_\phi}{\partial r} \,$

[bewerk] Laplaciaan van een scalaire functie

Uit het bovenstaande volgt voor de Laplaciaan van een scalaire functie f:

$\Delta f = \nabla\cdot\nabla f = \frac 1{r^2}\frac{\partial r^2\frac{\partial f_b}{\partial r}}{\partial r} + \frac 1{r\sin(\theta)}\frac{\partial \frac 1{r\sin(\theta)}\frac{\partial f_b}{\partial \phi}}{\partial \phi} + \frac 1{r\sin(\theta)}\frac{\partial \sin(\theta)\frac 1r\frac{\partial f_b}{\partial \theta}}{\partial \theta} = \,$

$\frac 1{r^2}\frac{\partial r^2\frac{\partial f_b}{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2 f_b}{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial f_b}{\partial \theta}}{\partial \theta} \,$

[bewerk] Laplaciaan van een vectorveld

Uit het bovenstaande volgt in bolcoördinaten voor de Laplaciaan van een vectorveld A:

$(\Delta A)_r = (\Delta A)_x\cos(\phi)\sin(\theta) +(\Delta A)_y\sin(\phi)\sin(\theta) +(\Delta A)_z\cos(\theta) = \,$

$\Delta A_x\cos(\phi)\sin(\theta) +\Delta A_y\sin(\phi)\sin(\theta) +\Delta A_z\cos(\theta) = \,$

$\left( \frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta} \right) (A_x) \cos(\phi)\sin(\theta) + \,$

$\frac 1{r^2}\frac{\partial r^2\frac{\partial A_r}{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_r}{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial A_r}{\partial \theta}}{\partial \theta} - \frac {2}{r^2}A_r - \frac 2{r^2\sin(\theta)}\frac{\partial A_\phi}{\partial \phi} - \frac {2\cos(\theta)}{r^2\sin(\theta)}A_\theta - \frac 2{r^2}\frac{\partial A_\theta}{\partial \theta} = \,$

$\Delta A_r - \frac{2}{r^2}A_r -\frac{2\cos(\theta)}{r^2\sin(\theta)}A_\theta -\frac{2}{r^2}\frac{\partial A_\theta}{\partial \theta} -\frac{2}{r^2\sin(\theta)}\frac{\partial A_\phi}{\partial \phi} \,$

En analoog voor de φ-component:

$(\Delta A)_\phi = -(\Delta A)_x\sin(\phi) +(\Delta A)_y\cos(\phi)= -\Delta A_x\sin(\phi)+\Delta A_y\cos(\phi)= \,$

$\left( \frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2}{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta} \right) (A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)) \cos(\phi) = \,$

$\frac 1{r^2}\frac{\partial r^2\frac{\partial A_\phi}{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_\phi}{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial A_\phi}{\partial \theta}}{\partial \theta} - \frac{A_\phi}{(r\sin(\theta))^2} + \frac{2}{(r\sin(\theta))^2}\frac{\partial A_r}{\partial \phi} + \frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta }{\partial \phi} = \,$

$\Delta A_\phi - \frac{A_\phi}{(r\sin(\theta))^2} + \frac{2}{(r\sin(\theta))^2}\frac{\partial A_r}{\partial \phi} + \frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta}{\partial \phi} \,$

En analoog voor de θ-component:

$(\Delta A)_\theta = (\Delta A)_x\cos(\phi)\cos(\theta) +(\Delta A)_y\sin(\phi)\cos(\theta) -(\Delta A)_z\sin(\theta) = \,$

$\Delta A_x\cos(\phi)\cos(\theta) +\Delta A_y\sin(\phi)\cos(\theta) -\Delta A_z\sin(\theta) = \,$

$\frac 1{r^2}\frac{\partial r^2\frac{\partial A_\theta }{\partial r}}{\partial r} + \frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_\theta }{\partial \phi^2} + \frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial A_\theta }{\partial \theta}}{\partial \theta} - \frac{A_\theta}{(r\sin(\theta))^2} + \frac{2}{r^2}\frac{\partial A_r}{\partial \theta} - \frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi} = \,$

$\Delta A_\theta - \frac{A_\theta}{(r\sin(\theta))^2} + \frac{2}{r^2}\frac{\partial A_r}{\partial \theta} - \frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi}$

Categorie: Analyse

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[bewerk] Afleiding

[bewerk] Cilindercoördinaten

[bewerk] Gradiënt van een scalaire functie

[bewerk] Divergentie van een vectorveld

[bewerk] Rotatie van een vectorveld

[bewerk] Laplaciaan van een scalaire functie

[bewerk] Laplaciaan van een vectorveld

[bewerk] Bolcoördinaten

[bewerk] Gradiënt van een scalaire functie

[bewerk] Divergentie van een vectorveld

[bewerk] Rotatie van een vectorveld

[bewerk] Laplaciaan van een scalaire functie

[bewerk] Laplaciaan van een vectorveld

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