Outgoing null oblate spheroidal coordinates

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 37D / b / i / Solution

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With parameter $a\geq0$, oblate spheroidal coordinates on Euclidean three-space are
$$
x=\sqrt{r^2+a^2}\sin\theta\cos\phi,
\quad
y=\sqrt{r^2+a^2}\sin\theta\sin\phi,
\quad
z=r\cos\theta.
$$
Writing $\Sigma=r^2+a^2\cos^2\theta$ and $\Delta=r^2+a^2$, their spatial line element is
$$
dl^2=\frac\Sigma\Delta dr^2+\Sigma d\theta^2
+\Delta\sin^2\theta\,d\phi^2.
$$


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