10.5: Mathematical Formulas - Vector Operators
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This section contains a summary of vector operators expressed in each of the three major coordinate systems:
- Cartesian (\(x\),\(y\),\(z\))
- cylindrical (\(\rho\),\(\phi\),\(z\))
- spherical (\(r\),\(\theta\),\(\phi\))
Associated basis vectors are identified using a caret (\(\hat{~}\)) over the symbol. The vector operand \({\bf A}\) is expressed in terms of components in the basis directions as follows:
- Cartesian: \({\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z\)
- cylindrical: \({\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z\)
- spherical: \({\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}\)
Gradient
Gradient in Cartesian coordinates:
\[\nabla f = \hat { \mathbf { x } } \frac { \partial f } { \partial x } + \hat { \mathbf { y } } \frac { \partial f } { \partial y } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber \]
Gradient in cylindrical coordinates:
\[\nabla f = \hat { \rho } \frac { \partial f } { \partial \rho } + \hat { \phi } \frac { 1 } { \rho } \frac { \partial f } { \partial \phi } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber \]
Gradient in spherical coordinates:
\[\nabla f = \hat { \mathbf { r } } \frac { \partial f } { \partial r } + \hat { \theta } \frac { 1 } { r } \frac { \partial f } { \partial \theta } + \hat { \phi } \frac { 1 } { r \sin \theta } \frac { \partial f } { \partial \phi } \nonumber \]
Divergence
Divergence in Cartesian coordinates:
\[\nabla \cdot \mathbf { A } = \frac { \partial A _ { x } } { \partial x } + \frac { \partial A _ { y } } { \partial y } + \frac { \partial A _ { z } } { \partial z } \nonumber \]
Divergence in cylindrical coordinates:
\[\begin{aligned} \nabla \cdot {\bf A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_{\rho}\right) +\frac{1}{\rho}\frac{\partial A_{\phi}}{\partial \phi} + \frac{\partial A_z}{\partial z} & \end{aligned} \nonumber \] Divergence in spherical coordinates: \[\begin{aligned} \nabla \cdot {\bf A} &= ~~\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 A_r\right) & \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left(A_{\theta}\sin\theta\right)& \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial \phi} & \end{aligned} \nonumber \]
Curl
Curl in Cartesian coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } = & \hat { \mathbf { x } } \left( \frac { \partial A _ { z } } { \partial y } - \frac { \partial A _ { y } } { \partial z } \right) \\ & + \hat { \mathbf { y } } \left( \frac { \partial A _ { x } } { \partial z } - \frac { \partial A _ { z } } { \partial x } \right) \\ & + \hat { \mathbf { z } } \left( \frac { \partial A _ { y } } { \partial x } - \frac { \partial A _ { x } } { \partial y } \right) \end{aligned} \nonumber \]
Curl in cylindrical coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } = & \hat { \rho } \left( \frac { 1 } { \rho } \frac { \partial A _ { z } } { \partial \phi } - \frac { \partial A _ { \phi } } { \partial z } \right) \\ & + \hat { \phi } \left( \frac { \partial A _ { \rho } } { \partial z } - \frac { \partial A _ { z } } { \partial \rho } \right) \\ & + \hat { \mathbf { z } } \frac { 1 } { \rho } \left[ \frac { \partial } { \partial \rho } \left( \rho A _ { \phi } \right) - \frac { \partial A _ { \rho } } { \partial \phi } \right] \end{aligned} \nonumber \]
Curl in spherical coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } & = \hat { \mathbf { r } } \frac { 1 } { r \sin \theta } \left[ \frac { \partial } { \partial \theta } \left( A _ { \phi } \sin \theta \right) - \frac { \partial A _ { \theta } } { \partial \phi } \right] \\ & + \hat { \theta } \frac { 1 } { r } \left[ \frac { 1 } { \sin \theta } \frac { \partial A _ { r } } { \partial \phi } - \frac { \partial } { \partial r } \left( r A _ { \phi } \right) \right] \\ & + \hat { \phi } \frac { 1 } { r } \left[ \frac { \partial } { \partial r } \left( r A _ { \theta } \right) - \frac { \partial A _ { r } } { \partial \theta } \right] \end{aligned} \nonumber \]
Laplacian
Laplacian in Cartesian coordinates:
\[\nabla ^ { 2 } f = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial z ^ { 2 } } \nonumber \]
Laplacian in cylindrical coordinates:
\[\nabla ^ { 2 } f = \frac { 1 } { \rho } \frac { \partial } { \partial \rho } \left( \rho \frac { \partial f } { \partial \rho } \right) + \frac { 1 } { \rho ^ { 2 } } \frac { \partial ^ { 2 } f } { \partial \phi ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial z ^ { 2 } } \nonumber \]
Laplacian in spherical coordinates:
\[\begin{aligned} \nabla ^ { 2 } f = & \frac { 1 } { r ^ { 2 } } \frac { \partial } { \partial r } \left( r ^ { 2 } \frac { \partial f } { \partial r } \right) \\ & + \frac { 1 } { r ^ { 2 } \sin \theta } \frac { \partial } { \partial \theta } \left( \frac { \partial f } { \partial \theta } \sin \theta \right) \\ & + \frac { 1 } { r ^ { 2 } \sin ^ { 2 } \theta } \frac { \partial ^ { 2 } f } { \partial \phi ^ { 2 } } \end{aligned} \nonumber \]