(a) Suppose that a tensor $T_{i j}$ can be decomposed as

$$T_{i j}=S_{i j}+\epsilon_{i j k} V_{k}$$

where $S_{i j}$ is symmetric. Obtain expressions for $S_{i j}$ and $V_{k}$ in terms of $T_{i j}$, and check that $(*)$ is satisfied.

(b) State the most general form of an isotropic tensor of rank $k$ for $k=0,1,2,3$, and verify that your answers are isotropic.

(c) The general form of an isotropic tensor of rank 4 is

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

Suppose that $A_{i j}$ and $B_{i j}$ satisfy the linear relationship $A_{i j}=T_{i j k l} B_{k l}$, where $T_{i j k l}$ is isotropic. Express $B_{i j}$ in terms of $A_{i j}$, assuming that $\beta^{2} \neq \gamma^{2}$ and $3 \alpha+\beta+\gamma \neq 0$. If instead $\beta=-\gamma \neq 0$ and $\alpha \neq 0$, find all $B_{i j}$ such that $A_{i j}=0$.

(d) Suppose that $C_{i j}$ and $D_{i j}$ satisfy the quadratic relationship $C_{i j}=T_{i j k l m n} D_{k l} D_{m n}$, where $T_{i j k l m n}$ is an isotropic tensor of rank 6 . If $C_{i j}$ is symmetric and $D_{i j}$ is antisymmetric, find the most general non-zero form of $T_{i j k l m n} D_{k l} D_{m n}$ and prove that there are only two independent terms. [Hint: You do not need to use the general form of an isotropic tensor of rank 6.]