Nabla in cylindrical and spherical coordinates
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This is a list of some vector calculus formulae of general use in working with standard coordinate systems.
| Operation | Cartesian coordinates (x,y,z) | Cylindrical coordinates (ρ,φ,z) | Spherical coordinates (r,θ,φ) |
|---|---|---|---|
| Definition of coordinates | <math>\left[\begin{matrix}
x & = & \rho\cos\phi \\
y & = & \rho\sin\phi \\
z & = & z \end{matrix}\right.<math>
| <math>\left[\begin{matrix}
x & = & r\sin\theta\cos\phi \\
y & = & r\sin\theta\sin\phi \\
z & = & r\cos\theta \end{matrix}\right.<math>
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<math>\left[\begin{matrix}
\rho & = & \sqrt{x^2 + y^2} \\
\phi & = & \operatorname{atan2}(y, x) \\
z & = & z \end{matrix}\right.<math>
| <math>\left[\begin{matrix}
r & = & \sqrt{x^2 + y^2 + z^2} \\
\theta & = & \arccos(z / r) \\
\phi & = & \operatorname{atan2}(y, x) \end{matrix}\right.<math>
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| <math>\mathbf{A}<math> | <math>A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}<math> | <math>A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}<math> | <math>A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}<math> |
| <math>\nabla f<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>
| <math>{\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}<math>
| <math>{\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}<math>
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| <math>\nabla \cdot \mathbf{A}<math> | <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 \rho A_\rho \over \partial \rho}
+ {1 \over \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>
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| <math>\nabla \times \mathbf{A}<math> | <math>\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}<math>
| <math>\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}<math>
| <math>\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}<math>
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| <math>\Delta f = \nabla^2 f<math> | <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}(\rho {\partial f \over \partial \rho})
+ {1 \over \rho^2}{\partial^2 f \over \partial \phi^2}
+ {\partial^2 f \over \partial z^2}<math>
| <math>{1 \over r^2}{\partial \over \partial r}(r^2 {\partial f \over \partial r})
+ {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f \over \partial \theta})
+ {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \phi^2}<math>
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| <math>\Delta \mathbf{A} = \nabla^2 \mathbf{A}<math> | <math>\mathbf{\hat x}\Delta A_x + \mathbf{\hat y}\Delta A_y + \mathbf{\hat z}\Delta A_z<math> | <math>\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}<math>
| <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^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}<math>
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| Differential Displacement | <math>d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}<math> | <math>d\mathbf{l} = d\rho\boldsymbol{\hat \rho} + \rho d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z}<math> | <math>d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi}<math> |
| Differential Normal Area | <math>\begin{matrix}d\mathbf{S} = &dydz\mathbf{\hat x} \\
&dxdz\mathbf{\hat y} \\ &dxdy\mathbf{\hat z}\end{matrix}<math> | <math>\begin{matrix}
d\mathbf{S} = & \rho d\phi dz\boldsymbol{\hat \rho} \\ & d\rho dz\boldsymbol{\hat \phi} \\ & \rho d\rho d\phi \mathbf{z} \end{matrix}<math> | <math>\begin{matrix}
d\mathbf{S} = & r^2 \sin\theta d\theta d\phi \mathbf{\hat r} \\ & r\sin\theta drd\phi \boldsymbol{\hat \theta} \\ & rdrd\theta\boldsymbol{\hat \phi} \end{matrix}<math> |
| Differential Volume | <math>dv = dxdydz\,\!<math> | <math>dv = \rho d\rho d\phi dz\,\!<math> | <math>dv = r^2\sin\theta drd\theta d\phi\,\!<math> |
Non-trivial calculation rules:
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- Note: This page uses standard physics notation, some (American mathematics) sources define θ as the angle with the <math>xy <math>-plane.