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Section A.1 Formulas for Div, Grad, Curl
Subsection A.1.1 Rectangular coordinates
\begin{align*}
d\rr \amp= dx\,\xhat + dy\,\yhat + dz\,\zhat\\
\FF \amp= F_x\,\xhat + F_y\,\yhat + F_z\,\zhat
\end{align*}
\begin{align*}
\grad f \amp=
\Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat
+ \Partial{f}{z}\,\zhat\\
\grad\cdot\FF \amp=
\Partial{F_x}{x} + \Partial{F_y}{y} + \Partial{F_z}{z}\\
\grad\times\FF \amp=
\left(\Partial{F_z}{y}-\Partial{F_y}{z}\right)\xhat
+ \left(\Partial{F_x}{z}-\Partial{F_z}{x}\right)\yhat
+ \left(\Partial{F_y}{x}-\Partial{F_x}{y}\right)\zhat\\
\nabla^2 f \amp=
\frac{\partial^2 f}{dx^2}+\frac{\partial^2 f}{dy^2}
+\frac{\partial^2 f}{dz^2}
\end{align*}
Subsection A.1.2 Cylindrical coordinates
\begin{align*}
d\rr \amp= ds\,\shat + s\,d\phi\,\phat + dz\,\zhat\\
\FF \amp= F_s\,\shat + F_\phi\,\phat + F_z\,\zhat
\end{align*}
\begin{align*}
\grad f \amp=
\Partial{f}{s}\,\shat
+ \frac{1}{s}\Partial{f}{\phi}\,\phat
+ \Partial{f}{z}\,\zhat\\
\grad\cdot\FF \amp=
\frac{1}{s}\Partial{}{s}\left({s}F_{s}\right)
+ \frac{1}{s}\Partial{F_\phi}{\phi}
+ \Partial{F_z}{z}\\
\grad\times\FF \amp=
\left(
\frac{1}{s}\Partial{F_z}{\phi} - \Partial{F_\phi}{z}
\right) \shat
+ \left( \Partial{F_s}{z}-\Partial{F_z}{s}\right) \phat
+ \frac{1}{s} \left(
\Partial{}{s}\left({s}F_{\phi}\right) - \Partial{F_s}{\phi}
\right) \zhat\\
\nabla^2 f \amp=
\frac{1}{s}\Partial{}{s}\left(s\Partial{f}{s}\right)
+\frac{1}{s^2}\frac{\partial^2 f}{d\phi^2}
+\frac{\partial^2 f}{dz^2}
\end{align*}
Subsection A.1.3 Spherical coordinates
\begin{align*}
d\rr \amp=
dr\,\rhat + r\,d\theta\,\that + r\,\sin\theta\,d\phi\,\phat\\
\FF \amp= F_r\,\rhat + F_\theta\,\that + F_\phi\,\phat
\end{align*}
\begin{align*}
\grad f \amp=
\Partial{f}{r}\,\rhat
+ \frac{1}{r}\Partial{f}{\theta}\,\that
+ \frac{1}{r\sin\theta}\Partial{f}{\phi}\,\phat\\
\grad\cdot\FF \amp=
\frac{1}{r^2}\Partial{}{r}\left({r^2}F_{r}\right)
+ \frac{1}{r\sin\theta}\Partial{}{\theta}
\left({\sin\theta}F_{\theta}\right)
+ \frac{1}{r\sin\theta}\Partial{F_\phi}{\phi}\\
\grad\times\FF \amp=
\frac{1}{r\sin\theta} \left(
\Partial{}{\theta}\left({\sin\theta}F_{\phi}\right)
- \Partial{F_\theta}{\phi}
\right) \rhat
+ \frac{1}{r} \left(
\frac{1}{\sin\theta} \Partial{F_r}{\phi}
- \Partial{}{r}\left({r}F_{\phi}\right)
\right) \that\\
\amp\qquad
+ \frac{1}{r} \left(
\Partial{}{r}\left({r}F_{\theta}\right) - \Partial{F_r}{\theta}
\right) \phat\\
\nabla^2 f \amp=
\frac{1}{r^2}\frac{\partial}{\partial r}
\left(r^2\Partial{f}{r}\right)
+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}
\left(\sin\theta\Partial{f}{\theta}\right)
+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial\phi^2}
\end{align*}