THE GEOMETRY OF VECTOR CALCULUS

\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*}

\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*}

\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}{d\phi^2} \end{align*}