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)

Answer

One of the most difficult things about this exercise was finding a copy of Jackson (1999). It is a 'classic' text, so, unlike most English mathematical text books in the Berlin library, the only copies available were translation into German. Luckily it wasn't too hard to find the required results. They were on the very last page. The heading is 'Description of vector operations in various coordinate systems'. Simple really!

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