(i) Starting with Poisson's equation in $\mathbb{R}^{3}$,

$$\nabla^{2} \phi(\mathbf{x})=f(\mathbf{x})$$

derive Gauss' flux theorem

$$\int_{V} f(\mathbf{x}) d V=\int_{\partial V} \mathbf{F}(\mathbf{x}) \cdot \mathbf{d} \mathbf{S}$$

for $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ and for any volume $V \subseteq \mathbb{R}^{3}$.

(ii) Let

$$I=\int_{S} \frac{\mathbf{x} \cdot \mathbf{d} \mathbf{S}}{|\mathbf{x}|^{3}} .$$

Show that $I=4 \pi$ if $S$ is the sphere $|\mathbf{x}|=R$, and that $I=0$ if $S$ bounds a volume that does not contain the origin.

(iii) Show that the electric field defined by

$$\mathbf{E}(\mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}-\mathbf{a}}{|\mathbf{x}-\mathbf{a}|^{3}}, \quad \mathbf{x} \neq \mathbf{a}$$

satisfies

$$\int_{\partial V} \mathbf{E} \cdot \mathbf{d} \mathbf{S}= \begin{cases}0 & \text { if } \mathbf{a} \notin V \\ \frac{q}{\epsilon_{0}} & \text { if } \mathbf{a} \in V\end{cases}$$

where $\partial V$ is a surface bounding a closed volume $V$ and $\mathbf{a} \notin \partial V$, and where the electric charge $q$ and permittivity of free space $\epsilon_{0}$ are constants. This is Gauss' law for a point electric charge.

(iv) Assume that $f(\mathbf{x})$ is spherically symmetric around the origin, i.e., it is a function only of $|\mathbf{x}|$. Assume that $\mathbf{F}(\mathbf{x})$ is also spherically symmetric. Show that $\mathbf{F}(\mathbf{x})$ depends only on the values of $f$ inside the sphere with radius $|\mathbf{x}|$ but not on the values of $f$ outside this sphere.