1. (30%) For a general second order tensor T, show that using index notation,
(a) $\epsilon$-$\delta$ identity $\epsilon_{mnr}\epsilon_{mqs} = \delta_{nq}\delta_{rs} - \delta_{ns}\delta_{rq}$. (Note that $m$ is summed.)
(b) $\text{det}(T) = \frac{1}{6}\epsilon_{pqr}\epsilon_{ijk}T_{ip}T_{jq}T_{kr}$ for a general second order tensor T.
(Hint: $\epsilon_{mnr}\text{det}(T) = \begin{vmatrix} T_{m1} & T_{m2} & T_{m3} \\ T_{n1} & T_{n2} & T_{n3} \\ T_{r1} & T_{r2} & T_{r3} \end{vmatrix}$, $\text{det}(T) = \epsilon_{ijk}T_{i1}T_{j2}T_{k3}$)