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:
\begin{flalign}
\nabla f &= \hat{\bf x}\frac{\partial f}{\partial x} + \hat{\bf y}\frac{\partial f}{\partial y} + \hat{\bf z}\frac{\partial f}{\partial z} & \tag{B.6}\end{flalign}
Gradient in cylindrical coordinates:
\begin{flalign}
\nabla f &= \hat{\bf \rho}\frac{\partial f}{\partial \rho}
+\hat{\bf \phi}\frac{1}{\rho}\frac{\partial f}{\partial \phi}
+ \hat{\bf z}\frac{\partial f}{\partial z} & \tag{B.7}\end{flalign}
Gradient in spherical coordinates:
\begin{flalign}
\nabla f &= \hat{\bf r}\frac{\partial f}{\partial r}
+\hat{\bf \theta}\frac{1}{r}\frac{\partial f}{\partial \theta}
+\hat{\bf \phi}\frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi} & \tag{B.8}\end{flalign}
Divergence
Divergence in Cartesian coordinates:
\begin{flalign}
\nabla \cdot {\bf A} &= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} & \tag{B.9}\end{flalign}
Divergence in cylindrical coordinates:
\begin{flalign}
\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} & \tag{B.10}\end{flalign}
Divergence in spherical coordinates:
\begin{flalign}
\nabla \cdot {\bf A} &= ~~\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 A_r\right) & \\
&~~ +\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left(A_{\theta}\sin\theta\right)& \\
&~~ +\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial \phi} & \tag{B.11}\end{flalign}
Curl
Curl in Cartesian coordinates:
\begin{flalign}
\nabla \times {\bf A} &= ~~\hat{\bf x}\left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) & \\
&~~ +\hat{\bf y}\left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) & \\
&~~ +\hat{\bf z}\left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) & \tag{B.12}\end{flalign}
Curl in cylindrical coordinates:
\begin{flalign}
\nabla \times {\bf A} &= ~~\hat{\bf \rho}\left( \frac{1}{\rho}\frac{\partial A_z}{\partial \phi} - \frac{\partial A_{\phi}}{\partial z} \right) & \\
&~~ +\hat{\bf \phi}\left( \frac{\partial A_{\rho}}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) & \\
&~~ +\hat{\bf z}\frac{1}{\rho}\left[ \frac{\partial}{\partial \rho}\left(\rho A_{\phi}\right) - \frac{\partial A_{\rho}}{\partial \phi} \right] & \tag{B.13}\end{flalign}
Curl in spherical coordinates:
\begin{flalign}
\nabla \times {\bf A} &= ~~\hat{\bf 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{\bf \theta}\frac{1}{r}\left[ \frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial}{\partial r}\left(rA_{\phi}\right) \right] & \\
&~~ +\hat{\bf \phi}\frac{1}{r}\left[ \frac{\partial}{\partial r}\left(r A_{\theta}\right) - \frac{\partial A_r}{\partial \theta} \right] & \tag{B.14}\end{flalign}
Laplacian
Laplacian in Cartesian coordinates:
\begin{flalign}
\nabla^2 f &= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} & \tag{B.15}\end{flalign}
Laplacian in cylindrical coordinates:
\begin{flalign}
\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} & \tag{B.16}\end{flalign}
Laplacian in spherical coordinates:
\begin{flalign}
\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} & \tag{B.17}\end{flalign}