Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 13 / c / Solution

Write the scalar field as $u=R(r)Y(\theta ,\varphi )$, where $R=r+r^{-2}$ and $Y=\sin \theta \cos \varphi$. Its radial contribution to the Laplacian in spherical coordinates isFor the angular contribution, $Y_{\theta }=\cos \theta \cos \varphi$ and $Y_{\varphi \varphi }=-\sin \theta \cos \varphi$, soCombining the radial and angular terms provesThe calculation uses $0<\theta <\pi$ to avoid coordinate singularities. In Cartesian coordinates, $u=x(1+r^{-3})$, which extends smoothly across the polar axis for $r>0$. It therefore satisfies Laplace's equation there as well. The origin is a genuine singularity and is excluded.