If $\mathbf{B}$ is a vector, and

$$T_{i j}=\alpha B_{i} B_{j}+\beta B_{k} B_{k} \delta_{i j}$$

show for arbitrary scalars $\alpha$ and $\beta$ that $T_{i j}$ is a symmetric second-rank tensor.

Find the eigenvalues and eigenvectors of $T_{i j}$.

Suppose now that $\mathbf{B}$ depends upon position $\mathbf{x}$ and that $\nabla \cdot \mathbf{B}=0$. Find constants $\alpha$ and $\beta$ such that

$$\frac{\partial}{\partial x_{j}} T_{i j}=[(\nabla \times \mathbf{B}) \times \mathbf{B}]_{i} .$$

Hence or otherwise show that if $\mathbf{B}$ vanishes everywhere on a surface $S$ that encloses a volume $V$ then

$$\int_{V}(\nabla \times \mathbf{B}) \times \mathbf{B} d V=0$$