(a) Show, using Cartesian coordinates, that $\psi=1 / r$ satisfies Laplace's equation, $\nabla^{2} \psi=0$, on $\mathbb{R}^{3} \backslash\{0\} .$

(b) $\phi$ and $\psi$ are smooth functions defined in a 3-dimensional domain $V$ bounded by a smooth surface $S$. Show that

$$\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\int_{S}(\phi \nabla \psi-\psi \nabla \phi) \cdot d \mathbf{S}$$

(c) Let $\psi=1 /\left|\mathbf{r}-\mathbf{r}_{0}\right|$, and let $V_{\varepsilon}$ be a domain bounded by a smooth outer surface $S$ and an inner surface $S_{\varepsilon}$, where $S_{\varepsilon}$ is a sphere of radius $\varepsilon$, centre $\mathbf{r}_{0}$. The function $\phi$ satisfies

$$\nabla^{2} \phi=-\rho(\mathbf{r}) .$$

Use parts (a) and (b) to show, taking the limit $\varepsilon \rightarrow 0$, that $\phi$ at $\mathbf{r}_{0}$ is given by

$$4 \pi \phi\left(\mathbf{r}_{0}\right)=\int_{V} \frac{\rho(\mathbf{r})}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} d V+\int_{S}\left(\frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} \frac{\partial \phi}{\partial n}-\phi(\mathbf{r}) \frac{\partial}{\partial n} \frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|}\right) d S,$$

where $V$ is the domain bounded by $S$.