The vector fields $\mathbf{A}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey the evolution equations

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

where $\mathbf{u}$ is a given vector field and $\psi$ is a given scalar field. Use suffix notation to show that the scalar field $h=\mathbf{A} \cdot \mathbf{B}$ obeys an evolution equation of the form

$$\frac{\partial h}{\partial t}=\mathbf{B} \cdot \nabla f-\mathbf{u} \cdot \nabla h$$

where the scalar field $f$ should be identified.

Suppose that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$ and $\boldsymbol{\nabla} \cdot \mathbf{u}=0$. Show that, if $\mathbf{u} \cdot \mathbf{n}=\mathbf{B} \cdot \mathbf{n}=0$ on the surface $S$ of a fixed volume $V$ with outward normal $\mathbf{n}$, then

$$\frac{\mathrm{d} H}{\mathrm{~d} t}=0, \quad \text { where } H=\int_{V} h \mathrm{~d} V .$$

Suppose that $\mathbf{A}=a r^{2} \sin \theta \mathbf{e}_{\theta}+r\left(a^{2}-r^{2}\right) \sin \theta \mathbf{e}_{\phi}$ with respect to spherical polar coordinates, and that $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that

$$h=a r^{2}\left(a^{2}+r^{2}\right) \sin ^{2} \theta$$

and calculate the value of $H$ when $V$ is the sphere $r \leqslant a$.

$$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta}\left|\begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right|\right.$$