Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae of general use in working with various coordinate systems.

[edit] Note

  • This page uses standard physics notation. For spherical coordinates, θ is the angle between the z axis and the radius vector connecting the origin to the point in question. φ is the angle between the projection of the radius vector onto the x-y plane and the x axis. Some (American mathematics) sources reverse this definition.
  • The function atan2(y, x) is used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π].


Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates
Operation Cartesian coordinates (x,y,z) Cylindrical coordinates (s,φ,z) Spherical coordinates (r,θ,φ) Parabolic cylindrical coordinates (ο,τ,z)
Definition
of
coordinates		 \begin{matrix} s & = & \sqrt{x^2+y^2} \\ \phi & = & \arctan(y/x) \\ z & = & z \end{matrix} \begin{matrix} x & = & s\cos\phi \\ y & = & s\sin\phi \\ z & = & z \end{matrix} \begin{matrix} x & = & r\sin\theta\cos\phi \\ y & = & r\sin\theta\sin\phi \\ z & = & r\cos\theta \end{matrix} \begin{matrix} x & = & \sigma \tau\\ y & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z & = & z \end{matrix}
\begin{matrix} r & = & \sqrt{x^2+y^2+z^2} \\ \theta & = & \arctan{\left(\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}\right)}\\ \phi & = & \arctan(y/x) \\ \end{matrix} \begin{matrix} r & = & \sqrt{s^2 + z^2} \\ \theta & = & \arctan{(s/z)}\\ \phi & = & \phi \end{matrix} \begin{matrix} s & = & r\sin(\theta) \\ \phi & = & \phi\\ z & = & r\cos(\theta) \end{matrix} \begin{matrix} s\cos\phi & = & \sigma \tau\\ s\sin\phi & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z & = & z \end{matrix}
Definition
of
unit
vectors		 \begin{matrix} \boldsymbol{\hat s} & = & \frac{x}{s}\mathbf{\hat x}+\frac{y}{s}\mathbf{\hat y} \\ \boldsymbol{\hat\phi} & = & -\frac{y}{s}\mathbf{\hat x}+\frac{x}{s}\mathbf{\hat y} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix} \begin{matrix} \mathbf{\hat x} & = & \cos\phi\boldsymbol{\hat s}-\sin\phi\boldsymbol{\hat\phi} \\ \mathbf{\hat y} & = & \sin\phi\boldsymbol{\hat s}+\cos\phi\boldsymbol{\hat\phi} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix} \begin{matrix} \mathbf{\hat x} & = & \sin\theta\cos\phi\boldsymbol{\hat r}+\cos\theta\cos\phi\boldsymbol{\hat\theta}-\sin\phi\boldsymbol{\hat\phi} \\ \mathbf{\hat y} & = & \sin\theta\sin\phi\boldsymbol{\hat r}+\cos\theta\sin\phi\boldsymbol{\hat\theta}+\cos\phi\boldsymbol{\hat\phi} \\ \mathbf{\hat z} & = & \cos\theta \boldsymbol{\hat r}-\sin\theta \boldsymbol{\hat\theta} \\ \end{matrix} \begin{matrix} \boldsymbol{\hat \sigma} & = & \frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}-\frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\ \boldsymbol{\hat\tau} & = & \frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}+\frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\ \mathbf{\hat z} & = & \mathbf{\hat z} \end{matrix}
\begin{matrix} \mathbf{\hat r} & = & \frac{x\mathbf{\hat x}+y\mathbf{\hat y}+z\mathbf{\hat z}}{r} \\ \boldsymbol{\hat\theta} & = & \frac{xz\mathbf{\hat x}+yz\mathbf{\hat y}-s^2\mathbf{\hat z}}{r s} \\ \boldsymbol{\hat\phi} & = & \frac{-y\mathbf{\hat x}+x\mathbf{\hat y}}{s} \end{matrix} \begin{matrix} \mathbf{\hat r} & = & \frac{s}{r}\boldsymbol{\hat s}+\frac{ z}{r}\mathbf{\hat z} \\ \boldsymbol{\hat\theta} & = & \frac{z }{r}\boldsymbol{\hat s}-\frac{s}{r}\mathbf{\hat z} \\ \boldsymbol{\hat\phi} & = & \boldsymbol{\hat\phi} \end{matrix} \begin{matrix} \boldsymbol{\hat s} & = & \sin\theta\mathbf{\hat r}+\cos\theta\boldsymbol{\hat\theta} \\ \boldsymbol{\hat\phi} & = & \boldsymbol{\hat\phi} \\ \mathbf{\hat z} & = & \cos\theta\mathbf{\hat r}-\sin\theta\boldsymbol{\hat\theta} \\ \end{matrix} \begin{matrix} \end{matrix}
A vector field \mathbf{A} A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} A_s\boldsymbol{\hat s} + 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} A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\phi\boldsymbol{\hat z}
Gradient \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 \over \partial s}\boldsymbol{\hat s} + {1 \over s}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z} {\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} \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat \tau} + {\partial f \over \partial z}\boldsymbol{\hat z}
Divergence \nabla \cdot \mathbf{A} {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z} {1 \over s}{\partial \left( s A_s \right) \over \partial s} + {1 \over s}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z} {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_\phi \over \partial \phi} \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\sigma \over \partial \sigma} + \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\tau \over \partial \tau} + {\partial A_z \over \partial z}


Curl \nabla \times \mathbf{A} \begin{matrix} \displaystyle\left({\partial A_z \over \partial y} - {\partial A_y \over \partial z}\right) \mathbf{\hat x} & + \\ \displaystyle\left({\partial A_x \over \partial z} - {\partial A_z \over \partial x}\right) \mathbf{\hat y} & + \\ \displaystyle\left({\partial A_y \over \partial x} - {\partial A_x \over \partial y}\right) \mathbf{\hat z} & \ \end{matrix} \begin{matrix} \displaystyle\left({1 \over s}{\partial A_z \over \partial \phi} - {\partial A_\phi \over \partial z}\right) \boldsymbol{\hat s} & + \\ \displaystyle\left({\partial A_s \over \partial z} - {\partial A_z \over \partial s}\right) \boldsymbol{\hat \phi} & + \\ \displaystyle{1 \over s}\left({\partial \left( s A_\phi \right) \over \partial s} - {\partial A_s \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix} \begin{matrix} \displaystyle{1 \over r\sin\theta}\left({\partial \over \partial \theta} \left( A_\phi\sin\theta \right) - {\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat r} & + \\ \displaystyle{1 \over r}\left({1 \over \sin\theta}{\partial A_r \over \partial \phi} - {\partial \over \partial r} \left( r A_\phi \right) \right) \boldsymbol{\hat \theta} & + \\ \displaystyle{1 \over r}\left({\partial \over \partial r} \left( r A_\theta \right) - {\partial A_r \over \partial \theta}\right) \boldsymbol{\hat \phi} & \ \end{matrix} \begin{matrix} \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \tau} - {\partial A_\tau \over \partial z}\right) \boldsymbol{\hat \sigma} & - \\ \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \sigma}- {\partial A_\sigma \over \partial z}\right) \boldsymbol{\hat \tau} & + \\ \displaystyle\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}\left({\partial \left( s A_\phi \right) \over \partial s} - {\partial A_s \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix}
Laplace operator \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 s}{\partial \over \partial s}\left(s {\partial f \over \partial s}\right) + {1 \over s^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2} {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 \phi^2} \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} f}{\partial \sigma^{2}} + \frac{\partial^{2} f}{\partial \tau^{2}} \right) + \frac{\partial^{2} f}{\partial z^{2}}
Vector Laplacian \Delta \mathbf{A} = \nabla^2 \mathbf{A} \Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z} \begin{matrix} \displaystyle\left(\Delta A_s - {A_s \over s^2} - {2 \over s^2}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat s} & + \\ \displaystyle\left(\Delta A_\phi - {A_\phi \over s^2} + {2 \over s^2}{\partial A_s \over \partial \phi}\right) \boldsymbol{\hat\phi} & + \\ \displaystyle\left(\Delta A_z \right) \boldsymbol{\hat z} & \ \end{matrix} \begin{matrix} \left(\Delta A_r - {2 A_r \over r^2} - {2 \over r^2\sin\theta}{\partial \left(A_\theta \sin\theta\right) \over \partial\theta} - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat r} & + \\ \left(\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}\right) \boldsymbol{\hat\theta} & + \\ \left(\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}\right) \boldsymbol{\hat\phi} & \end{matrix}
Differential displacement d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z} d\mathbf{l} = ds\boldsymbol{\hat s} + s d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z} d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi} d\mathbf{l} = \sqrt{\sigma^{2} + \tau^{2}} d\sigma\boldsymbol{\hat \sigma} + \sqrt{\sigma^{2} + \tau^{2}} d\tau\boldsymbol{\hat \tau} + dz\boldsymbol{\hat z}
Differential normal area \begin{matrix}d\mathbf{S} = &dy\,dz\,\mathbf{\hat x} + \\ &dx\,dz\,\mathbf{\hat y} + \\ &dx\,dy\,\mathbf{\hat z}\end{matrix} \begin{matrix} d\mathbf{S} = & s\, d\phi\, dz\,\boldsymbol{\hat s} + \\ & ds \,dz\,\boldsymbol{\hat \phi} + \\ & s \,ds d\phi \,\mathbf{\hat z} \end{matrix} \begin{matrix} d\mathbf{S} = & r^2 \sin\theta \,d\theta \,d\phi \,\mathbf{\hat r} + \\ & r\sin\theta \,dr\,d\phi \,\boldsymbol{\hat \theta} + \\ & r\,dr\,d\theta\,\boldsymbol{\hat \phi} \end{matrix} \begin{matrix} d\mathbf{S} = & \sqrt{\sigma^{2} + \tau^{2}}, d\tau\, dz\,\boldsymbol{\hat \sigma} + \\ & \sqrt{\sigma^{2} + \tau^{2}} d\sigma\,dz\,\boldsymbol{\hat \tau} + \\ & \sigma^{2} + \tau^{2} d\sigma, d\tau \,\mathbf{\hat z} \end{matrix}
Differential volume d\tau = dx\,dy\,dz \, d\tau = s\, ds\, d\phi\, dz\, d\tau = r^2\sin\theta \,dr\,d\theta\, d\phi\, d\tau = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,
Non-trivial calculation rules:
  1. \operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f (Laplacian)
  2. \operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = 0
  3. \operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0
  4. \operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} (using Lagrange's formula for the cross product)
  5. \Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f

[edit] See also