Past exam of the mathematical methods II course of the University of Oxford / 2019 / 10 / a / Solution

Use $s$ for cylindrical radius, to distinguish it from spherical radius $r$. In cylindrical coordinates $(s,\varphi ,z)$, the azimuthal angle $\varphi$ is measured from positive $x$ in the horizontal plane:Here $0\le \varphi <2\pi$ and $z\in \mathbb{R}$.

In spherical polar coordinates $(r,\theta ,\varphi )$, take $\theta$ as the polar angle from positive $z$ and use the same azimuthal angle $\varphi$:The ranges are $r\ge 0$, $0\le \theta \le \pi$ and $0\le \varphi <2\pi$. The radial direction has horizontal projection of length $r\sin \theta$, which explains the extra factor in $x,y$. The azimuth is undefined on the polar axis, and both angles are nonunique at the origin; these are coordinate degeneracies rather than geometric singularities.