##### Spherical coordinates
c.m.uf.sph

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

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42