 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-3/4b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

A second-rank tensor is antisymmetric when

T_{ij} = - T_{ji} .

Under an orthogonal Cartesian coordinate change with matrix

R

T_{ij}^{'} = R_{ik} R_{j ℓ} T_{k ℓ} .

Therefore

T_{ji}^{'} = R_{jk} R_{i ℓ} T_{k ℓ} = - R_{jk} R_{i ℓ} T_{ℓ k} = - T_{ij}^{'},

after relabelling the dummy indices. Hence the antisymmetric second-rank tensor property is independent of Cartesian coordinates.

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