Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 16 / c / Solution

From the preceding gradient, ${\partial }_x^2(r^{-1})=-r^{-3}+3x^2{r}^{-5}$. Summing the three Cartesian second partial derivatives givesThus the reciprocal radius is harmonic away from the origin. Its negative gradient isThe curl vanishes because $E$ is a gradient field and mixed second derivatives commute. The exclusion of the origin is essential: the potential and field are singular there.