If $\mathbf{E}$ and $\mathbf{B}$ are vectors in $\mathbb{R}^{3}$, show that

$$T_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)$$

is a second rank tensor.

Now assume that $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey Maxwell's equations, which in suitable units read

$$\begin{aligned} &\nabla \cdot \mathbf{E}=\rho \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$

where $\rho$ is the charge density and $\mathbf{J}$ the current density. Show that

$$\frac{\partial}{\partial t}(\mathbf{E} \times \mathbf{B})=\mathbf{M}-\rho \mathbf{E}-\mathbf{J} \times \mathbf{B} \quad \text { where } \quad M_{i}=\frac{\partial T_{i j}}{\partial x_{j}}$$