\begin{eqnarray*} d\rr &=& dx\,\ii + dy\,\jj + dz\,\kk \\ \FF &=& F_x\,\ii + F_y\,\jj + F_z\,\kk \end{eqnarray*}

\begin{eqnarray*} \grad f &=& \Pf{x}\,\ii + \Pf{y}\,\jj + \Pf{z}\,\kk \\ \grad\cdot\FF &=& \PF{x}{x} + \PF{y}{y} + \PF{z}{z} \\ \grad\times\FF &=& \CF{z}{y}\ii + \CF{x}{z}\jj + \CF{y}{x}\kk \end{eqnarray*}

\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 &=& \Pf{r}\,\rhat + \frac{1}{r}\Pf{\phi}\,\phat + \Pf{z}\,\zhat \\ \grad\cdot\FF &=& \frac{1}{r}\QF{r}{r}{r} + \frac{1}{r}\PF{\phi}{\phi} + \PF{z}{z} \\ \grad\times\FF &=& \left( \frac{1}{r}\PF{z}{\phi} - \PF{\phi}{z} \right) \rhat + \CF{r}{z} \phat + \frac{1}{r} \left( \QF{r}{\phi}{r} - \PF{r}{\phi} \right) \zhat \end{eqnarray*}

\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 &=& \Pf{r}\,\rhat + \frac{1}{r}\Pf{\theta}\,\that + \frac{1}{r\sin\theta}\Pf{\phi}\,\phat \\ \grad\cdot\FF &=& \frac{1}{r^2}\QF{r^2}{r}{r} + \frac{1}{r\sin\theta}\QF{\sin\theta}{\theta}{\theta} + \frac{1}{r\sin\theta}\PF{\phi}{\phi} \\ \grad\times\FF &=& \frac{1}{r\sin\theta} \left( \QF{\sin\theta}{\phi}{\theta} - \PF{\theta}{\phi} \right) \rhat + \frac{1}{r} \left( \frac{1}{\sin\theta} \PF{r}{\phi} - \QF{r}{\phi}{r} \right) \that \\ && \quad + \frac{1}{r} \left( \QF{r}{\theta}{r} - \PF{r}{\theta} \right) \phat \end{eqnarray*}