Vector Fields In Cylindrical And Spherical Coordinates

Vector fields in cylindrical coordinates

• ρ is the length of the vector projected onto the X-Y-plane,
• φ is the angle of the projected vector with the positive X-axis (0 <= φ < 2π),
• z is the regular z-coordinate.

$\backslash left[\backslash begin\{matrix\}$

     \rho & = & \sqrt{x^2 + y^2} \\     \phi & = & \operatorname{atan2}(y, x) \\     z & = & z \end{matrix}\right. 

$\backslash left[\backslash begin\{matrix\}$

     x & = & \rho\cos\phi \\     y & = & \rho\sin\phi \\     z & = & z \end{matrix}\right.  

$\backslash mathbf\; A\; =\; A\_x\backslash mathbf\{\backslash hat\; x\}\; +\; A\_y\backslash mathbf\{\backslash hat\; y\}\; +\; A\_z\backslash mathbf\{\backslash hat\; z\}$

                  = A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z} 

$\backslash begin\{bmatrix\}\backslash boldsymbol\backslash hat\backslash rho\; \backslash \backslash \; \backslash boldsymbol\backslash hat\backslash phi\; \backslash \backslash \; \backslash boldsymbol\backslash hat\; z\backslash end\{bmatrix\}$

   = \begin{bmatrix} \cos\phi & \sin\phi & 0 \\                    -\sin\phi & \cos\phi & 0 \\                    0 & 0 & 1 \end{bmatrix}     \begin{bmatrix} \mathbf\hat x \\ \mathbf\hat y \\ \mathbf\hat z \end{bmatrix} 

• Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

Time derivative of a vector field in cylindrical coordinates

$\backslash mathbf\backslash dot\; A\; =\; \backslash dot\; A\_x\; \backslash mathbf\backslash hat\; x\; +\; \backslash dot\; A\_y\; \backslash mathbf\backslash hat\; y\; +\; \backslash dot\; A\_z\; \backslash mathbf\backslash hat\; z$

$\backslash mathbf\backslash dot\; A\; =\; \backslash dot\; A\_\backslash rho\; \backslash boldsymbol\backslash hat\backslash rho\; +\; A\_\backslash rho\; \backslash boldsymbol\backslash dot\backslash hat\backslash rho$

   + \dot A_\phi \boldsymbol\hat\phi + A_\phi \boldsymbol\dot\hat\phi   + \dot A_z \boldsymbol\hat z + A_z \boldsymbol\dot\hat z 

$\backslash left[\backslash begin\{matrix\}$

   \boldsymbol\dot\hat\rho & = & \dot\phi \boldsymbol\hat\phi \\   \boldsymbol\dot\hat\phi & = & - \dot\phi \boldsymbol\hat\rho \\   \boldsymbol\dot\hat z   & = & 0 \end{matrix}\right. 

$\backslash mathbf\backslash dot\; A\; =\; \backslash boldsymbol\backslash hat\backslash rho\; (\backslash dot\; A\_\backslash rho\; -\; A\_\backslash phi\; \backslash dot\backslash phi)$

   + \boldsymbol\hat\phi (\dot A_\phi + A_\rho \dot\phi)   + \boldsymbol\hat z \dot A_z 

Gradient, divergence, curl, and laplacian in cylindrical coordinates

Vector fields in spherical coordinates

• r is the length of the vector,
• θ is the angle with the positive Z-axis (0 <= θ <= π),
• φ is the angle with the X-Z-plane (0 <= φ < 2π).

$\backslash left[\backslash begin\{matrix\}$

     r & = & \sqrt{x^2 + y^2 + z^2} \\     \theta & = & \arccos(z / r) \\     \phi & = & \operatorname{atan2}(y, x) \end{matrix}\right. 

$\backslash left[\backslash begin\{matrix\}$

     x & = & r\sin\theta\cos\phi \\     y & = & r\sin\theta\sin\phi \\     z & = & r\cos\theta \end{matrix}\right.  

$\backslash mathbf\; A\; =\; A\_x\backslash mathbf\{\backslash hat\; x\}\; +\; A\_y\backslash mathbf\{\backslash hat\; y\}\; +\; A\_z\backslash mathbf\{\backslash hat\; z\}$

                  = A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi} 

$\backslash begin\{bmatrix\}\backslash boldsymbol\backslash hat\; r\; \backslash \backslash \; \backslash boldsymbol\backslash hat\backslash theta\; \backslash \backslash \; \backslash boldsymbol\backslash hat\backslash phi\; \backslash end\{bmatrix\}$

   = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\                     \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\                     -\sin\phi          & \cos\phi           & 0 \end{bmatrix}     \begin{bmatrix} \mathbf\hat x \\ \mathbf\hat y \\ \mathbf\hat z \end{bmatrix} 

• Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

Time derivative of a vector field in spherical coordinates

$\backslash mathbf\backslash dot\; A\; =\; \backslash dot\; A\_x\; \backslash mathbf\backslash hat\; x\; +\; \backslash dot\; A\_y\; \backslash mathbf\backslash hat\; y\; +\; \backslash dot\; A\_z\; \backslash mathbf\backslash hat\; z$

$\backslash mathbf\backslash dot\; A\; =\; \backslash dot\; A\_r\; \backslash boldsymbol\backslash hat\; r\; +\; A\_r\; \backslash boldsymbol\backslash dot\backslash hat\; r$

   + \dot A_\theta \boldsymbol\hat\theta + A_\theta \boldsymbol\dot\hat\theta   + \dot A_\phi \boldsymbol\hat\phi + A_\phi \boldsymbol\dot\hat\phi 

$\backslash begin\{bmatrix\}\backslash boldsymbol\backslash dot\backslash hat\; r\; \backslash \backslash \; \backslash boldsymbol\backslash dot\backslash hat\backslash theta\; \backslash \backslash \; \backslash boldsymbol\backslash dot\backslash hat\backslash phi\; \backslash end\{bmatrix\}$

   = \begin{bmatrix} 0           & \dot\theta & \dot\phi \sin\theta \\                     -\dot\theta & 0          & \dot\phi \cos\theta \\                     -\dot\phi \sin\theta & -\dot\phi \cos\theta & 0 \end{bmatrix}     \begin{bmatrix} \boldsymbol\hat r \\ \boldsymbol\hat\theta \\ \boldsymbol\hat\phi \end{bmatrix} 

$\backslash mathbf\backslash dot\; A\; =\; \backslash boldsymbol\backslash hat\; r\; (\backslash dot\; A\_r\; -\; A\_\backslash theta\; \backslash dot\backslash theta\; -\; A\_\backslash phi\; \backslash dot\backslash phi\; \backslash sin\backslash theta)$

   + \boldsymbol\hat\theta (\dot A_\theta + A_r \dot\theta - A_\phi \dot\phi \cos\theta)   + \boldsymbol\hat\phi (\dot A_\phi + A_r \dot\phi \sin\theta + A_\phi \dot\phi \cos\theta) 

Gradient, divergence, curl, and laplacian in spherical coordinates

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