Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / a

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- Solution a

Solution

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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