Nabla in cylindrical and spherical coordinates

From Academic Kids

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

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)
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>
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:

  1. <math>\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f<math> (Laplacian)
  2. <math>\operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = 0<math>
  3. <math>\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0<math>
  4. <math>\operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}<math>
  5. <math>\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f<math>
  6. 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>
  • Note: This page uses standard physics notation, some (American mathematics) sources define θ as the angle with the <math>xy <math>-plane.

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