|
<!-- Header --><!-- Definition of A --><!-- grad f --><!-- div A --><!-- curl A --><!-- Laplacian f --><!-- vector Laplacian A --><!-- Material derivative (A dot del)B --><!-- Tensor divergence del dot T --><!-- Differentials displacement --><!-- Differentials normal area --><!-- Differentials volume --><!-- Cartesian -->
Dit is een lijst van enkele formules uit vectoranalyse voor het werken met veelvoorkomende kromlijnige coördinatenstelsels: [[Cartesiaanse coördinaten|Cartesische coördinaten]], [[cilindercoördinaten]], [[bolcoördinaten]].
<!-- Cylindrical \frac{\partial B_}{\partial } -->
<!-- Sp -->
<!-- Cartesian -->
<!-- cylindrical -->
<!-- spherical -->ovenstaande tabel.
=== Niet-triviale rekenregelsabla) ===
{{Appendix|2=
{{References}}}}
== Conversies tussen stelsels ==
{| 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>
! colspan="2" rowspan="2" |
! 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"
|+ Conversies tussen eenheidsvectoren in Cartesische, cilindrische en sferische coördinaten in termen van ''bestemmingscoördinaten''<ref name="griffiths"/>
|-
!
! Cartesische
! Cilindrische
! Sferische
|-
! Cartesische
| {{n/a}}
| <math>\begin{align}
\hat{\mathbf x} &= \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf y} &= \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}</math>
| <math>\begin{align}
\hat{\mathbf x} &= \sin\theta \cos\varphi \hat{\mathbf r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf y} &= \sin\theta \sin\varphi \hat{\mathbf r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
\end{align}</math>
|-
! Cilindrische
| <math>\begin{align}
\hat{\boldsymbol \rho} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}</math>
| {{n/a}}
| <math>\begin{align}
\hat{\boldsymbol \rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol \theta} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta}
\end{align}</math>
|-
! Sferische
| <math>\begin{align}
\hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2}} \\
\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}} \\
\hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}}
\end{align}</math>
| <math>\begin{align}
\hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol \rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \theta} &= \frac{z \hat{\boldsymbol \rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}</math>
| {{n/a}}
|}
{| class="wikitable"
|+ Conversies tussen eenheidsvectoren in Cartesische, cilindrische en sferische coördinaten in termen van ''oorsprongscoördinaten''
|-
!
! Cartesische
! Cilindrische
! Sferische
|-
! Cartesische
| {{n/a}}
| <math>\begin{align}
\hat{\mathbf x} &= \frac{x \hat{\boldsymbol \rho} - y \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf y} &= \frac{y \hat{\boldsymbol \rho} + x \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}</math>
| <math>\begin{align}
\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}} \\
\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}} \\
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \sqrt{x^2 + y^2} \hat{\boldsymbol \theta}}{\sqrt{x^2 + y^2 + z^2}}
\end{align}</math>
|-
! Cilindrische
| <math>\begin{align}
\hat{\boldsymbol \rho} &= \cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y} \\
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \\
\hat{\mathbf z} &= \hat{\mathbf z}
\end{align}</math>
| {{n/a}}
| <math>\begin{align}
\hat{\boldsymbol \rho} &= \frac{\rho \hat{\mathbf r} + z \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\
\hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \rho \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}}
\end{align}</math>
|-
! Sferische
| <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} \\
\hat{\boldsymbol \theta} &= \cos\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) - \sin\theta \hat{\mathbf z} \\
\hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y}
\end{align}</math>
| <math>\begin{align}
\hat{\mathbf r} &= \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{\mathbf z} \\
\hat{\boldsymbol \theta} &= \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{\mathbf z} \\
\hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi}
\end{align}</math>
| {{n/a}}
|}
== Nabla formule ==<!-- Cartesian -->
<!-- Cylindrical \frac{\partial B_}{\partial } -->
<!-- Sp -->
<!-- Cartesian -->
<!-- cylindrical -->
<!-- spherical -->{| class="wikitable" style="background: white"
|+ Tabel met nabla-operator in Cartesische, cilindrische en sferische coördinaten
<!-- Header -->
|-
! style="background: white" | Operatie
! style="background: white" | Cartesische coördinaten {{math|(''x'', ''y'', ''z'')|x}}
! style="background: white" | Cilindercoördinaten {{math|(''ρ'', ''φ'', ''z'')|x}}
! style="background: white" | Bolcoördinaten {{math|(''r'', ''θ'', ''φ'')|x}}, waar ''φ'' de azimutale en {{math|θ}} de polaire hoek is{{ref|Alpha|α}}
<!-- Definition of A -->
|- align="center"
! style="background: white" | [[Vectorveld]] <span style="font-weight: normal">{{math|'''A'''|x}}</span>
| <math>A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}</math>
| <math>A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z}</math>
| <math>A_r \hat{\mathbf r} + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}</math>
<!-- grad f -->
|- align="center"
! style="background: white" | [[Gradiënt (wiskunde)|Gradiënt]] <span style="font-weight: normal">{{math|∇''f''|x}}</span><ref name="griffiths" />
| <math>{\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y}
+ {\partial f \over \partial z}\hat{\mathbf z}</math>
| <math>{\partial f \over \partial \rho}\hat{\boldsymbol \rho}
+ {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}
+ {\partial f \over \partial z}\hat{\mathbf z}</math>
| <math>{\partial f \over \partial r}\hat{\mathbf r}
+ {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}
+ {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}</math>
<!-- div A -->
|- align="center"
! style="background: white" | [[Divergentie (vectorveld)|Divergentie]] <span style="font-weight: normal">{{math|∇ ⋅ '''A'''|x}}</span><ref name="griffiths" />
| <math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
| <math>{1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho}
+ {1 \over \rho}{\partial A_\varphi \over \partial \varphi}
+ {\partial A_z \over \partial z}</math>
| <math>{1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r}
+ {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right)
+ {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}</math>
<!-- curl A -->
|- align="center"
! style="background: white" | [[Rotatie (vectorveld)|Rotatie]] <span style="font-weight: normal">{{math|∇ × '''A'''|x}}</span><ref name="griffiths" />
| <math>\begin{align}
\left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\
+ \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\
+ \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z}
\end{align}</math>
| <math>\begin{align}
\left(
\frac{1}{\rho} \frac{\partial A_z}{\partial \varphi}
- \frac{\partial A_\varphi}{\partial z}
\right) &\hat{\boldsymbol \rho} \\
+ \left(
\frac{\partial A_\rho}{\partial z}
- \frac{\partial A_z}{\partial \rho}
\right) &\hat{\boldsymbol \varphi} \\
{}+ \frac{1}{\rho} \left(
\frac{\partial \left(\rho A_\varphi\right)}{\partial \rho}
- \frac{\partial A_\rho}{\partial \varphi}
\right) &\hat{\mathbf z}
\end{align}</math>
| <math>\begin{align}
\frac{1}{r\sin\theta} \left(
\frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right)
- \frac{\partial A_\theta}{\partial \varphi}
\right) &\hat{\mathbf r} \\
{}+ \frac{1}{r} \left(
\frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi}
- \frac{\partial}{\partial r} \left( r A_\varphi \right)
\right) &\hat{\boldsymbol \theta} \\
{}+ \frac{1}{r} \left(
\frac{\partial}{\partial r} \left( r A_{\theta} \right)
- \frac{\partial A_r}{\partial \theta}
\right) &\hat{\boldsymbol \varphi}
\end{align}</math>
<!-- Laplacian f -->
|- align="center"
! style="background: white" | [[Laplace-operator]] <span style="font-weight: normal">{{math|∇<sup>2</sup>''f'' ≡ ∆''f''|x}}</span><ref name="griffiths" />
| <math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
| <math>{1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right)
+ {1 \over \rho^2}{\partial^2 f \over \partial \varphi^2}
+ {\partial^2 f \over \partial z^2}</math>
| <math>{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 \varphi^2}
</math>
<!-- vector Laplacian A -->
|- align="center"
! style="background: white" | Vector Laplaciaan <span style="font-weight: normal">{{math|∇<sup>2</sup>'''A''' ≡ ∆'''A'''}}</span>
| <math>\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z} </math>
| <math>\begin{align}
\mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\
+ \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\
{}+ \nabla^2 A_z &\hat{\mathbf z}
\end{align}</math>
| align="center" | <math>\begin{align}
\left(\nabla^2 A_r - \frac{2 A_r}{r^2}
- \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta}
- \frac{2}{r^2\sin\theta}{\frac{\partial A_\varphi}{\partial \varphi}}\right) &\hat{\mathbf r} \\
+ \left(\nabla^2 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}\right) &\hat{\boldsymbol \theta} \\
+ \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta}
+ \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi}
+ \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi}
\end{align}</math>
<!-- Material derivative (A dot del)B --><!-- Tensor divergence del dot T --><!-- Differentials displacement --><!-- Differentials normal area --><!-- Differentials volume -->|}
{{Note|Alfa|α}}Deze pagina gebruikt <math>\theta</math> voor de polaire hoek en <math>\varphi</math> voor de azimutale hoek; dit is de gebruikelijke notatie voor natuurkunde. De bron voor deze formules gebruikt <math>\theta</math> voor de azimutale hoek en <math>\varphi</math> voor de polaire hoek; dat is de gebruikelijke wiskundige notatie. Om de wiskundige variant te verkrijgen, wissel <math>\theta</math> en <math>\varphi</math> in de bovenstaande tabel.
=== Niet-triviale rekenregels ===
# <math>\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f</math>
# <math>\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0</math>
# <math>\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
# <math>\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math> (Lagrange's formule voor nabla)
# <math>\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f</math>
{{Appendix|2=
{{References}}}}
[[Categorie:Wiskundige analyse]]
| 