 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/6/a/ii/solution

Use three components of the proposed isotropic tensor form. Since

T_{1122} = 0

, one gets

α = 0

. Since

T_{1212} = ϵ_{123}^{2} = 1

, one gets

β = 1

. Finally

T_{1221} = ϵ_{123} ϵ_{213} = - 1

gives

γ = - 1

ϵ_{ijm} ϵ_{k l m} = δ_{ik} δ_{j l} - δ_{i l} δ_{jk}, (α, β, γ) = (0, 1, - 1) .

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