Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 10C / iii / Solution

The tensor contraction $C_{mkl}={ϵ}_{ijm}{T}_{ijkl}$ is an isotropic third-rank tensor under proper rotations. It therefore equals $\nu {ϵ}_{mkl}$. Cyclically permuting three indices is even, so ${ϵ}_{mkl}={ϵ}_{klm}$ and ${ϵ}_{ijm}{T}_{ijkl}=\nu {ϵ}_{klm}$.

For the given Kronecker delta expression, the $\alpha$ term vanishes by antisymmetry, while the $\beta$ and $\gamma$ terms give respectively $\beta {ϵ}_{klm}$ and $\gamma {ϵ}_{lkm}=-\gamma {ϵ}_{klm}$. HenceThese computations also illustrate isotropic contractions of a fourth-rank tensor.