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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/b/i/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 6 / b / i / Solution by

Codex  0 2026-10-18

The integration domain is invariant under every rotation matrix, and its volume element is unchanged. The substitution

y = R x

therefore proves

R_{ia} R_{jb} R_{k c} R_{l d} I_{ab c d} = \int_{B} y_{i} y_{j} y_{k} y_{l} d V_{y} = I_{ijk l} .

(1)

Thus

I

is an isotropic tensor. It is also completely symmetric in its four indices, because the coordinate factors commute. In the general isotropic form, exchanging indices

j, k

equates

α

and

β

, and exchanging

j, l

equates

α

and

γ

. Therefore

I_{ijk l} = A (δ_{ij} δ_{k l} + δ_{ik} δ_{j l} + δ_{i l} δ_{jk}), α = β = γ = A .

(2)

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