State (without proof) the divergence theorem for a vector field $\mathbf{F}$ with continuous first-order partial derivatives throughout a volume $V$ enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface $S$.

By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field

$$\mathbf{F}=\left(x^{3}+2 x y^{2}, y^{3}+2 y z^{2}, z^{3}+2 z x^{2}\right)$$

defined within a sphere of radius $R$ centred at the origin.

Suppose that functions $\phi, \psi$ are continuous and that their first and second partial derivatives are all also continuous in a region $V$ bounded by a smooth surface $S$.

Show that

$$\begin{aligned} \int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S} \\ \int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S}-\int_{S} \psi \boldsymbol{\nabla} \phi \cdot \mathbf{d} \mathbf{S} \end{aligned}$$

Hence show that if $\rho(\mathbf{x})$ is a continuous function on $V$ and $g(\mathbf{x})$ a continuous function on $S$ and $\phi_{1}$ and $\phi_{2}$ are two continuous functions such that

$$\begin{aligned} \nabla^{2} \phi_{1}(\mathbf{x}) &=\nabla^{2} \phi_{2}(\mathbf{x})=\rho(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { in } V, \text { and } \\ \phi_{1}(\mathbf{x}) &=\phi_{2}(\mathbf{x})=g(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { on } S \end{aligned}$$

then $\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x})$ for all $\mathbf{x}$ in $V$.