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 / iii / Solution

Contract the isotropic expression from (i). Since ${\delta }_{ii}=3$ and ${\delta }_{ij}{\delta }_{ij}=3$,The value in (ii) gives $A=4\pi /105$, so the isotropic fourth moment of a ball isFor example ${\int }_B{x}_1^{4}dV=4\pi /35$ and ${\int }_B{x}_1^2{x}_2^{2}dV=4\pi /105$, illustrating the factor of three between these two fourth moments.