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

Let $M_{ijkl}={\int }_V{x}_i{x}_j{x}_k{x}_l{d}^{3}x$. 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:A double tensor contraction givesHere the three contracted Kronecker delta products contribute $9$, $3$ and $3$. Contracting instead with the four unit vectors yieldsThis agrees with subtracting two instances of the isotropic fourth moment of a ball.