Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 3 / b / Solution

Apply Green's second identity to $\Phi (\mathbf{r})$ and $G(\mathbf{r},{\mathbf{r}}^{\prime })$, differentiating in $\mathbf{r}$:The volume terms become $\Phi ({\mathbf{r}}^{\prime })-{\int }_{V}G\rho dV$. On the boundary $G=0$ and $\Phi =f$, soHere $\partial /\partial n$ is the outward normal derivative in the integration variable. One may justify the identity at the singularity by excising a small sphere around ${\mathbf{r}}^{\prime }$; its unit flux supplies the evaluation $\Phi ({\mathbf{r}}^{\prime })$.