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Nabla in verschillende assenstelsels: verschil tussen versies

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Regel 1:
Dit is een lijst van enkele formules uit vectoranalyse voor het werken met veelvoorkomende kromlijnige coördinatenstelsels: [[Cartesiaanse coördinaten|Cartesische coördinaten]], [[cilindercoördinaten]], [[bolcoördinaten]].
In de onderstaande '''tabel''' staat een overzicht van de vorm die de operator [[nabla]] aanneemt in de '''drie assenstelsels''':
 
* [[Cartesiaanse coördinaten]]
== Conversies tussen stelsels ==
* [[Cilindercoördinaten]]
* [[Bolcoördinaten]]
 
== Tabel ==
{| class="wikitable"
|+ Conversies tussen Cartesische, cilinder- en bolcoördinaten<ref name="griffiths">{{Cite book|title=Introduction to Electrodynamics|last=Griffiths|first=David J.|publisher=Pearson|year=2012|isbn=978-0-321-85656-2}}</ref>
<!-- Header -->
! colspan="2" rowspan="2" |
|+ '''Tabel met de operator <math>\nabla</math> in cilinder- en bolcoördinaten'''
! colspan="3" | From
|-
! Cartesian
! Cylindrical
! Spherical
|-
! rowspan="3" |To
! Cartesian
| <math>\begin{align}
x &= x \\
y &= y \\
z &= z
\end{align}</math>
| <math>\begin{align}
x &= \rho \cos\varphi \\
y &= \rho \sin\varphi \\
z &= z
\end{align}</math>
| <math>\begin{align}
x &= r \sin\theta \cos\varphi \\
y &= r \sin\theta \sin\varphi \\
z &= r \cos\theta
\end{align}</math>
|-
! Cylindrical
| <math>\begin{align}
\rho &= \sqrt{x^2 + y^2} \\
\varphi &= \arctan\left(\frac{y}{x}\right) \\
z &= z
\end{align}</math>
| <math>\begin{align}
\rho &= \rho \\
\varphi &= \varphi \\
z &= z
\end{align}</math>
| <math>\begin{align}
\rho &= r \sin\theta \\
\varphi &= \varphi \\
z &= r\cos\theta
\end{align}</math>
|-
! Spherical
| <math>\begin{align}
r &= \sqrt{x^2 + y^2 + z^2} \\
\theta &= \arctan\left(\frac{\sqrt{x^2 + y^2}}{z}\right) \\
\varphi &= \arctan\left(\frac{y}{x}\right)
\end{align}</math>
| <math>\begin{align}
r &= \sqrt{\rho^2 + z^2} \\
\theta &= \arctan{\left(\frac{\rho}{z}\right)} \\
\varphi &= \varphi
\end{align}</math>
| <math>\begin{align}
r &= r \\
\varphi &= \varphi \\
\theta &= \theta \\
\end{align}</math>
|}
== Conversies tussen eenheidsvectoren ==
 
{| class="wikitable"
<!-- kolomopschriften -->
|+ Conversies tussen eenheidsvectoren in Cartesische, cilindrische en sferische coördinaten in termen van ''bestemmingscoördinaten''<ref name="griffiths"/>
|--------
|-
! Operatie
!
! Cartesiaanse coördinaten (x,y,z)
! Cartesische
! Cilindercoördinaten (ρ,φ,z)
! Cilindrische
! Bolcoördinaten (r,θ,φ)
! Sferische
|----- align="center"
|-
! Relatie
! Cartesische
|zie:
| {{n/a}}
|[[Cilindercoördinaten]]
| <math>\begin{align}
|[[Bolcoördinaten]]
\hat{\mathbf x} &= \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\
<!-- eenheidsvectoren-->
\hat{\mathbf y} &= \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\
|----- align="center"
\hat{\mathbf z} &= \hat{\mathbf z}
! eenheids-<br />vectoren
\end{align}</math>
|<math>\mathbf{\hat x}, \mathbf{\hat y}, \mathbf{\hat z}</math>
| <math>\begin{align}
|<math>\boldsymbol{\hat \rho}, \boldsymbol{\hat \phi}, \boldsymbol{\hat z}</math>
\hat{\mathbf x} &= \sin\theta \cos\varphi \hat{\mathbf r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\
|<math>\boldsymbol{\hat r}, \boldsymbol{\hat \theta}, \boldsymbol{\hat \phi}</math>
\hat{\mathbf y} &= \sin\theta \sin\varphi \hat{\mathbf r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\
<!-- scalair veld-->
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
|----- align="center"
\end{align}</math>
! [[scalair veld]]<br /><math>f\,</math>
|-
|<math>f(x,y,z)\,</math>
! Cilindrische
|<math>f_c(\rho,\phi,z)=f(\rho\cos(\phi),\rho\sin(\phi),z)\,</math>
| <math>\begin{align}
|<math>f_b(r,\phi,\theta)=f(r\sin(\theta)\cos(\phi),r\sin(\theta)\sin(\phi),r\cos(\theta))\,</math>
\hat{\boldsymbol \rho} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
<!-- vectorveld A -->
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
|----- align="center"
\hat{\mathbf z} &= \hat{\mathbf z}
! [[vectorveld]]<br /><math>A\,</math>
\end{align}</math>
| <math>A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}</math>
| {{n/a}}
| <math>A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}</math>
| <math>\begin{align}
| <math>A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}</math>
\hat{\boldsymbol \rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol \theta} \\
<!-- grad f -->
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
|----- align="center"
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
! <math>\nabla f</math>
\end{align}</math>
| <math>{\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y}
|-
+ {\partial f \over \partial z}\mathbf{\hat z}</math>
! Sferische
| <math>{\partial f_c \over \partial \rho}\boldsymbol{\hat \rho}
| <math>\begin{align}
+ {1 \over \rho}{\partial f_c \over \partial \phi}\boldsymbol{\hat \phi}
\hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2}} \\
+ {\partial f_c \over \partial z}\boldsymbol{\hat z}</math>
\hat{\boldsymbol \theta} &= \frac{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2} \sqrt{x^2 + y^2}} \\
| <math>{\partial f_b \over \partial r}\boldsymbol{\hat r}
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}}
+ {1 \over r}{\partial f_b \over \partial \theta}\boldsymbol{\hat\theta}
\end{align}</math>
+ {1 \over r\sin\theta}{\partial f_b \over \partial \phi}\boldsymbol{\hat\phi}</math>
| <math>\begin{align}
<!-- div A -->
\hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol \rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
|----- align="center"
\hat{\boldsymbol \theta} &= \frac{z \hat{\boldsymbol \rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
! <math>\nabla \cdot A</math>
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
| <math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
\end{align}</math>
| <math>\frac 1{\rho}{\partial (\rho A_\rho) \over \partial \rho}
| {{n/a}}
+ \frac 1{\rho}{\partial A_\phi \over \partial \phi}
|}
+ {\partial A_z \over \partial z}</math>
| <math>{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}</math>
<!-- rotatie A -->
 
{| class="wikitable"
|----- align="center"
|+ Conversies tussen eenheidsvectoren in Cartesische, cilindrische en sferische coördinaten in termen van ''oorsprongscoördinaten''
! <math>\nabla \times A</math>
|-
| <math>\begin{matrix}
!
({\partial A_z \over \partial y} - {\partial A_y \over \partial z}) \mathbf{\hat x} & + \\
! Cartesische
({\partial A_x \over \partial z} - {\partial A_z \over \partial x}) \mathbf{\hat y} & + \\
! Cilindrische
({\partial A_y \over \partial x} - {\partial A_x \over \partial y}) \mathbf{\hat z} & \ \end{matrix}</math>
! Sferische
| <math>\begin{matrix}
|-
({1 \over \rho}{\partial A_z \over \partial \phi}
! Cartesische
- {\partial A_\phi \over \partial z}) \boldsymbol{\hat \rho} & + \\
| {{n/a}}
({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}) \boldsymbol{\hat \phi} & + \\
| <math>\begin{align}
{1 \over \rho}({\partial \rho A_\phi \over \partial \rho}
\hat{\mathbf x} &= \frac{x \hat{\boldsymbol \rho} - y \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
- {\partial A_\rho \over \partial \phi}) \boldsymbol{\hat z} & \ \end{matrix}</math>
\hat{\mathbf y} &= \frac{y \hat{\boldsymbol \rho} + x \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
| <math>\begin{matrix}
\hat{\mathbf z} &= \hat{\mathbf z}
{1 \over r\sin\theta}({\partial A_\phi\sin\theta \over \partial \theta}
\end{align}</math>
- {\partial A_\theta \over \partial \phi}) \boldsymbol{\hat r} & + \\
| <math>\begin{align}
({1 \over r\sin\theta}{\partial A_r \over \partial \phi}
\hat{\mathbf x} &= \frac{x \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - y \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\
- {1 \over r}{\partial r A_\phi \over \partial r}) \boldsymbol{\hat \theta} & + \\
\hat{\mathbf y} &= \frac{y \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + x \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\
{1 \over r}({\partial r A_\theta \over \partial r}
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \sqrt{x^2 + y^2} \hat{\boldsymbol \theta}}{\sqrt{x^2 + y^2 + z^2}}
- {\partial A_r \over \partial \theta}) \boldsymbol{\hat \phi} & \ \end{matrix}</math>
\end{align}</math>
 
|-
<!-- Laplaciaan f -->
! Cilindrische
 
|----- <math>\begin{align="center"}
\hat{\boldsymbol \rho} &= \cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y} \\
! <math>\Delta f = \nabla^2 f</math>
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \\
| <math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
\hat{\mathbf z} &= \hat{\mathbf z}
 
\end{align}</math>
| <math>{1 \over \rho}{\partial \over \partial \rho}(\rho {\partial f_c \over \partial \rho})
| {{n/a}}
+ {1 \over \rho^2}{\partial^2 f_c \over \partial \phi^2}
| <math>\begin{align}
+ {\partial^2 f_c \over \partial z^2}</math>
\hat{\boldsymbol \rho} &= \frac{\rho \hat{\mathbf r} + z \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \\
 
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
| <math>{1 \over r^2}{\partial \over \partial r}(r^2 {\partial f_b \over \partial r})
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \rho \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}}
+ {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f_b \over \partial \theta})
\end{align}</math>
+ {1 \over r^2\sin^2\theta}{\partial^2 f_b \over \partial \phi^2}</math>
|-
 
! Sferische
<!-- laplacien A -->
| <math>\begin{align}
 
\hat{\mathbf r} &= \sin\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) + \cos\theta \hat{\mathbf z} \\
|----- align="center"
\hat{\boldsymbol \theta} &= \cos\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) - \sin\theta \hat{\mathbf z} \\
! <math>\Delta A = \nabla^2 A</math>
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y}
| <math>\mathbf{\hat x}\Delta A_x + \mathbf{\hat y}\Delta A_y + \mathbf{\hat z}\Delta A_z</math>
\end{align}</math>
| <math>\begin{matrix}
| <math>\begin{align}
\boldsymbol{\hat\rho}(\Delta A_\rho - {A_\rho \over \rho^2}
\hat{\mathbf r} &= \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{\mathbf z} \\
- {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) & + \\
\hat{\boldsymbol \theta} &= \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{\mathbf z} \\
\boldsymbol{\hat\phi}(\Delta A_\phi - {A_\phi \over \rho^2}
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
+ {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) & + \\
\boldsymbol{\hat z} \Delta A_z & \ \end{matrixalign}</math>
| {{n/a}}
 
| rowspan="2" | <math>\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\theta}{\partial A_r \over \partial \phi}
+ {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) & \ \end{matrix}</math>
 
<!-- les nabla sur les nabla -->
 
|-----
| colspan="3" | '''Niet evidente rekenregels:'''
<ol>
<li><math>\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f</math> ([[Laplace-operator|Laplaciaan]])
<li><math>\operatorname{rot\ grad\ } f = \nabla \times (\nabla f) = \mathbf{0}</math>
<li><math>\operatorname{div\ rot\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
<li><math>\operatorname{rot\ rot\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A})
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math>
<li><math>\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f</math>
</ol>
|}
 
== Afleidingen ==
=== Cilindercoördinaten ===
Overgenomen van "https://nl.wikipedia.org/wiki/Nabla_in_verschillende_assenstelsels"