Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 9

For $r>0$, use the gradient of a radial function with $g(r)=1/r$:Equivalently each component is $-x_j/r^3$ by differentiating $r^2=x^2+y^2+z^2$. The origin is excluded from the function's domain.

The product rule for divergence, $\nabla \cdot \mathbf{r}=3$, and the result in (a) give, for $r>0$,This is a pointwise identity away from the origin, where the radial unit vector is undefined.