(i) Starting with the divergence theorem, derive Green's first theorem

$$\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S$$

(ii) The function $\phi(\mathbf{x})$ satisfies Laplace's equation $\nabla^{2} \phi=0$ in the volume $V$ with given boundary conditions $\phi(\mathbf{x})=g(\mathbf{x})$ for all $\mathbf{x} \in \partial V$. Show that $\phi(\mathbf{x})$ is the only such function. Deduce that if $\phi(\mathbf{x})$ is constant on $\partial V$ then it is constant in the whole volume $V$.

(iii) Suppose that $\phi(\mathbf{x})$ satisfies Laplace's equation in the volume $V$. Let $V_{r}$ be the sphere of radius $r$ centred at the origin and contained in $V$. The function $f(r)$ is defined by

$$f(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S$$

By considering the derivative $d f / d r$, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that $f(r)$ is constant and that $f(r)=\phi(\mathbf{0})$.

(iv) Let $M$ denote the maximum of $\phi$ on $\partial V_{r}$ and $m$ the minimum of $\phi$ on $\partial V_{r}$. By using the result from (iii), or otherwise, show that $m \leqslant \phi(\mathbf{0}) \leqslant M$.