Let $\mathbf{A}(t, \mathbf{x})$ and $\mathbf{B}(t, \mathbf{x})$ be time-dependent, continuously differentiable vector fields on $\mathbb{R}^{3}$ satisfying

$$\frac{\partial \mathbf{A}}{\partial t}=\nabla \times \mathbf{B} \quad \text { and } \quad \frac{\partial \mathbf{B}}{\partial t}=-\nabla \times \mathbf{A}$$

Show that for any bounded region $V$,

$$\frac{d}{d t}\left[\frac{1}{2} \int_{V}\left(\mathbf{A}^{2}+\mathbf{B}^{2}\right) d V\right]=-\int_{S}(\mathbf{A} \times \mathbf{B}) \cdot d \mathbf{S}$$

where $S$ is the boundary of $V$.