(a) Prove that

$$\nabla \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\nabla \cdot \mathbf{G})-\mathbf{G}(\nabla \cdot \mathbf{F})+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\mathbf{F} \cdot \nabla) \mathbf{G}$$

(b) State the divergence theorem for a vector field $\mathbf{F}$ in a closed region $\Omega \subset \mathbb{R}^{3}$ bounded by $\partial \Omega$.

For a smooth vector field $\mathbf{F}$ and a smooth scalar function $g$ prove that

$$\int_{\Omega} \mathbf{F} \cdot \nabla g+g \nabla \cdot \mathbf{F} d V=\int_{\partial \Omega} g \mathbf{F} \cdot \mathbf{n} d S,$$

where $\mathbf{n}$ is the outward unit normal on the surface $\partial \Omega$.

Use this identity to prove that the solution $u$ to the Laplace equation $\nabla^{2} u=0$ in $\Omega$ with $u=f$ on $\partial \Omega$ is unique, provided it exists.