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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/2019/paper-2/6/b/iii/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 / 2019 / Paper 2 / 6 / b / iii / Solution by

Codex  0 2026-10-18

Let

M_{ijk l} = \int_{V} x_{i} x_{j} x_{k} x_{l} d^{3} x

. It is an isotropic tensor and is invariant under every permutation of its four indices. Thus the three coefficients in the general rank-four isotropic form must coincide:

M_{ijk l} = C (δ_{ij} δ_{k l} + δ_{ik} δ_{j l} + δ_{i l} δ_{jk}) .

(1)

A double tensor contraction gives

15 C = M_{iijj} = \int_{V} r^{4} d^{3} x = 4 π \int_{R_{1}}^{R_{2}} r^{6} d r = \frac{4 π}{7} (R_{2}^{7} - R_{1}^{7}) .

(2)

Here the three contracted Kronecker delta products contribute

9

3

and

3

. Contracting instead with the four unit vectors yields

\int_{V} a = 1 \prod 4 (x \cdot n_{a}) d^{3} x = \frac{4 π}{105} (R_{2}^{7} - R_{1}^{7}) [(n_{1} \cdot n_{2}) (n_{3} \cdot n_{4}) + (n_{1} \cdot n_{3}) (n_{2} \cdot n_{4}) + (n_{1} \cdot n_{4}) (n_{2} \cdot n_{3})] .

(3)

This agrees with subtracting two instances of the isotropic fourth moment of a ball.

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