📂Mathematical Physics

Gradient, Divergence and Curl in Curvilinear Coordinates

Formula

The del operator in the spherical coordinates is as follows.

$$\nabla = \dfrac{\partial}{\partial r} \widehat{\mathbf{r}} + \dfrac{1}{r}\dfrac{\partial}{\partial \theta} \widehat{\boldsymbol{\theta}} + \dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial \phi} \widehat{\boldsymbol{\phi}}$$

Description

The del operator is not a vector, but for convenience, it is represented as above.

• Gradient:

  $$\nabla f= \frac{\partial f}{\partial r} \mathbf{\hat r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \boldsymbol{\hat \theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\boldsymbol{\hat \phi}$$

• Divergence:

  $$\begin{align*} \nabla \cdot \mathbf{F} &= \frac{1}{r^{2}\sin\theta}\left( \frac{\partial (r^{2}\sin\theta F_{r})}{\partial r}+\frac{\partial (r\sin\theta F_{\theta})}{\partial \theta}+\frac{\partial (rF_{\phi})}{\partial \phi} \right) \\ &= \frac{1}{r^{2}}\frac{\partial (r^{2} F_{r})}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (\sin\theta F_{\theta})}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial F_{\phi}}{\partial \phi} \end{align*}$$

• Curl:

  $$\nabla \times \mathbf{F} = \frac{1}{r\sin\theta}\left(\dfrac{\partial (F_{\phi} \sin\theta)}{\partial \theta} - \dfrac{\partial F_{\theta}}{\partial \phi} \right)\mathbf{\hat r} + \frac{1}{r}\left(\frac{1}{\sin\theta}\dfrac{\partial F_{r}}{\partial \phi} - \dfrac{\partial (r F_{\phi})}{\partial r} \right)\boldsymbol{\hat \theta} + \frac{1}{r}\left(\dfrac{\partial (r F_{\theta})}{\partial r} - \dfrac{\partial F_{r}}{\partial \theta} \right)\boldsymbol{\hat \phi}$$

• Laplacian:

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