Vector Operators in Spherical Coordinates

Formulae for vector operators in the spherical coordinate system $(r,\varphi ,\theta )$.

Gradient

$$\nabla T=\frac{\partial T}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial T}{\partial \theta }\hat{\theta }+\frac{1}{r\sin \theta }\frac{\partial T}{\partial \varphi }\hat{\varphi }$$

Divergence

$$\nabla \cdot v=\frac{1}{r^2}\frac{\partial }{\partial r}(r^{2}v\_r)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta v\_\theta )+\frac{1}{r\sin \theta }\frac{\partial v\_\varphi }{\partial \varphi }$$

Curl

\nabla \times v = \frac{1}{r\sin \theta} \left\[ \frac{\partial}{\partial \theta} (\sin \theta v\_{\phi})-\frac{\partial v\_{\theta}}{\partial \phi} \right]\hat{r} + \frac{1}{r} \left\[ \frac{1}{\sin \theta} \frac{\partial v\_{r}}{\partial \phi} - \frac{\partial}{\partial r}(rv\_{\phi})\right]\hat{\theta}+\frac{1}{r}\left\[ \frac{\partial}{\partial r}(rv\_{\theta})-\frac{\partial v\_{r}}{\partial \theta} \right]\hat{\phi}

Laplacian

$${\nabla }^{2}T=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{{\partial }^{2}T}{\partial {\varphi }^2}\right)$$