(a) Prove that

$$\begin{aligned} &\nabla \times(\psi \mathbf{A})=\psi \mathbf{\nabla} \times \mathbf{A}+\boldsymbol{\nabla} \psi \times \mathbf{A} \\ &\nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot \boldsymbol{\nabla} \times \mathbf{A}-\mathbf{A} \cdot \boldsymbol{\nabla} \times \mathbf{B} \end{aligned}$$

where $\mathbf{A}$ and $\mathbf{B}$ are differentiable vector fields and $\psi$ is a differentiable scalar field.

(b) Find the solution of $\nabla^{2} u=16 r^{2}$ on the two-dimensional domain $\mathcal{D}$ when

(i) $\mathcal{D}$ is the unit disc $0 \leqslant r \leqslant 1$, and $u=1$ on $r=1$;

(ii) $\mathcal{D}$ is the annulus $1 \leqslant r \leqslant 2$, and $u=1$ on both $r=1$ and $r=2$.

[Hint: the Laplacian in plane polar coordinates is:

$$\left.\nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} . \quad\right]$$