Table with the del or nabla in cylindrical and spherical coordinates
| Operation
| Cartesian coordinates (x,y,z)
| Cylindrical coordinates (ρ,φ,z)
| Spherical coordinates (r,θ,φ)
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|---|
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>
|
|---|
<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>
|
| <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>
|
|---|
| <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>
|
|---|
| <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>
|
|---|
| <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>
|
|---|
| <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>
|
Non-trivial calculation rules:
- <math>\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f<math> (Laplacian)
- <math>\operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = 0<math>
- <math>\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0<math>
- <math>\operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A})
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}<math>
- <math>\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f<math>
- Lagrange's formula for the cross product:
<math>\mathbf{A} \times (\mathbf{B} \times \mathbf{C})
= \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B})<math>
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