Pre-Quantum Electrodynamics
Spherical coordinatesc.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*}

Created: 2024-02-27 Tue 10:31