## Friday, 20 September 2019

### Exercise 3.02 Spherical gradient divergence curl as covariant derivatives

 Top of last page in German version of Jackson

### Question

You are familiar with the operations of gradient ($\nabla\phi$), divergence ($\nabla\bullet\mathbf{V}$) and curl ($\nabla\times\mathbf{V}$) in ordinary vector analysis in three-dimensional Euclidean space. Using covariant derivatives, derive formulae for these operations in spherical polar coordinates $\left\{r,\theta,\phi\right\}$ defined by
\begin{align}
x&=r\sin{\theta}\cos{\phi}&\phantom {10000}(1)\nonumber\\
y&=r\sin{\theta}\sin{\phi}&\phantom {10000}(2)\nonumber\\
z&=r\cos{\theta}&\phantom {10000}(3)\nonumber
\end{align}Compare your results to those in Jackson (1999) or an equivalent text. Are they identical? Should they be? (JD Jackson, Classical Electrodynamics, Wiley 1999)