Rectangular Coordinates

\begin{eqnarray*} d\rr &=& dx\,\xhat + dy\,\yhat + dz\,\zhat \\ \FF &=& F_x\,\xhat + F_y\,\yhat + F_z\,\zhat \end{eqnarray*}

\begin{eqnarray*} \grad f &=& \Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat + \Partial{f}{z}\,\zhat \\ \grad\cdot\FF &=& \Partial{F_x}{x} + \Partial{F_y}{y} + \Partial{F_z}{z} \\ \grad\times\FF &=& \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 \end{eqnarray*}

Cylindrical Coordinates

\begin{eqnarray*} d\rr &=& dr\,\rhat + r\,d\phi\,\phat + dz\,\zhat \\ \FF &=& F_r\,\rhat + F_\phi\,\phat + F_z\,\zhat \end{eqnarray*}

\begin{eqnarray*} \grad f &=& \Partial{f}{r}\,\rhat + \frac{1}{r}\Partial{f}{\phi}\,\phat + \Partial{f}{z}\,\zhat \\ \grad\cdot\FF &=& \frac{1}{r}\Partial{}{r}\left({r}F_{r}\right) + \frac{1}{r}\Partial{F_\phi}{\phi} + \Partial{F_z}{z} \\ \grad\times\FF &=& \left( \frac{1}{r}\Partial{F_z}{\phi} - \Partial{F_\phi}{z} \right) \rhat + \left(\Partial{F_r}{z}-\Partial{F_z}{r}\right) \phat + \frac{1}{r} \left( \Partial{}{r}\left({r}F_{\phi}\right) - \Partial{F_r}{\phi} \right) \zhat \end{eqnarray*}

Spherical Coordinates

\begin{eqnarray*} d\rr &=& dr\,\rhat + r\,d\theta\,\that + r\,\sin\theta\,d\phi\,\phat \\ \FF &=& F_r\,\rhat + F_\theta\,\that + F_\phi\,\phat \end{eqnarray*}

\begin{eqnarray*} \grad f &=& \Partial{f}{r}\,\rhat + \frac{1}{r}\Partial{f}{\theta}\,\that + \frac{1}{r\sin\theta}\Partial{f}{\phi}\,\phat \\ \grad\cdot\FF &=& \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 &=& \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 \\ && \quad + \frac{1}{r} \left( \Partial{}{r}\left({r}F_{\theta}\right) - \Partial{F_r}{\theta} \right) \phat \end{eqnarray*}