Explain what is meant by an isotropic tensor.

Show that the fourth-rank tensor

$$A_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$$

is isotropic for arbitrary scalars $\alpha, \beta$ and $\gamma$.

Assuming that the most general isotropic tensor of rank 4 has the form $(*)$, or otherwise, evaluate

$$B_{i j k l}=\int_{r<a} x_{i} x_{j} \frac{\partial^{2}}{\partial x_{k} \partial x_{l}}\left(\frac{1}{r}\right) d V$$

where $\mathbf{x}$ is the position vector and $r=|\mathbf{x}|$.