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

The integration domain is invariant under every rotation matrix, and its volume element is unchanged. The substitution $\mathbf{y}=R\mathbf{x}$ therefore provesThus $I$ 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 $\alpha$ and $\beta$, and exchanging $j,l$ equates $\alpha$ and $\gamma$. Therefore

The contraction in the PDF is $I_{iijj}$. By Einstein notation, its integrand is $({\sum }_i{x}_i^2)({\sum }_j{x}_j^2)=r^4$. In spherical coordinates,

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.