Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 6 / a / Solution

Split the Cartesian second-rank tensor into symmetric and antisymmetric parts:Define $u_k=\frac{1}{2}{ϵ}_{kij}{T}_{ij}$. The symmetric part contributes zero to this contraction. The epsilon-delta identity then givesThereforeThe symmetric part and the three independent antisymmetric components determine these quantities uniquely. Under proper rotations $u$ is a vector; under reflections it is an axial vector.