Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 3 / 9A / d / Solution

Part (c) givesThe half-space $x_1>0$ is a simply connected domain, so the Poincare lemma makes this curl-free field a conservative vector field: it is the gradient of a single-valued scalar potential.

Using $\tan u+\cot u=2/\sin (2u)$,The spherical-coordinate formula from part (b) shows that a potential must satisfy $f_{\varphi }=2$ and may be independent of $r,\theta$. On $x_1>0$ the azimuthal angle has the single-valued branchHence one solution, up to an additive constant, is