 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/i/solution

R

R_{ia} R_{jb} R_{m c} ϵ_{ab c} = (det R) ϵ_{ijm} .

Use this twice, and use

R_{m c} R_{m f} = δ_{c f}

for the contracted index. It follows that

R_{ia} R_{jb} R_{k c} R_{l d} ϵ_{ab e} ϵ_{c d e} = (det R)^{2} ϵ_{ijm} ϵ_{k l m} = T_{ijk l} .

Therefore

T_{ijk l}^{'} = R_{ia} R_{jb} R_{k c} R_{l d} T_{ab c d} = T_{ijk l}

: it is a fourth-rank isotropic tensor. Although a single alternating symbol is a pseudotensor under reflections, the two determinant factors multiply to one, so this product is an ordinary tensor even for improper orthogonal transformations.

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