Pre-Quantum Electrodynamics

\begin{align*} \mbox{Gradient:} &{\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r} + \frac{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi} + \frac{\partial T}{\partial z} ~\hat{\bf z} \nonumber\\ \mbox{Divergence:} &{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r) + \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}. \nonumber\\ \mbox{Curl:} &{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r} + \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi} + \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z} \nonumber\\ \mbox{Laplacian:} &{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} \end{align*}