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

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- Solution i

Solution

Both the Levi-Civita symbol and

T

are invariant under proper rotations. Their tensor contraction

A_{l} = ϵ_{ijk} T_{ijk l}

is consequently an isotropic vector. The introductory proof shows that it must vanish, so $ϵ_{ijk} T_{ijk l} = 0$ .

For the displayed combination of Kronecker deltas, the three terms are

α ϵ_{ijk} δ_{ij} δ_{k l}, β ϵ_{ijk} δ_{ik} δ_{j l}, γ ϵ_{ijk} δ_{i l} δ_{jk} .

In each term the tensor contraction repeats two indices of the Levi-Civita symbol; each term is zero. This verifies the identity for arbitrary

α, β, γ

.

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