Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 4 / 16H (Methods)

An isotropic tensor has unchanged components under every proper orthogonal change of basis: $T_{i_1\dots i_n}=R_{i_1{j}_1}\cdots R_{i_n{j}_n}{T}_{j_1\dots j_n}$ for every $R\in SO(3)$. Orthogonality gives $R_{ia}{R}_{jb}{\delta }_{ab}={\delta }_{ij}$, so the Kronecker delta is isotropic. The determinant identity gives $R_{ia}{R}_{jb}{R}_{kc}{ϵ}_{abc}=(\det R){ϵ}_{ijk}={ϵ}_{ijk}$, so the Levi-Civita symbol is isotropic too. If reflections are included, this last object is a pseudotensor: it changes sign. The question's isotropy convention therefore refers to proper rotations.

For the sphere moment $T_{i_1\dots i_n}={\int }_{S^2}{\hat{x}}_{i_1}\cdots {\hat{x}}_{i_n}dS$, change variables to $R\hat{x}$. Rotations preserve the sphere and its area measure; inserting the component transformation proves that $T$ is an isotropic tensor. For the second moment, half-turns about coordinate axes kill the off-diagonal components, while rotations interchanging axes make the diagonal entries equal. Thus it is a multiple $a{\delta }_{ij}$. Contracting the indices gives $3a={\int }_{S^2}|\hat{x}{|}^{2}dS=4\pi$. Odd moments vanish by $\hat{x}\mapsto -\hat{x}$, including the third moment. The supplied general fourth-rank form and symmetry under every permutation of the four indices make its three coefficients equal. A double contraction then gives $4\pi =b(9+3+3)$. ThusTo explain the weighted volume integrals, put $v=(1,i,0)$; its bilinear square is $v\cdot v=1+i^2=0$, without complex conjugation. Separating radial and angular factors givesFor $n=2$ the angular factor is $(4\pi /3)v\cdot v=0$. For $n=3$ it is zero by oddness. For $n=4$ it is $(4\pi /15)3(v\cdot v{)}^2=0$. Therefore all three requested integrals vanish, assuming the radial integrals exist. More generally the null-vector cancellation of radial moments follows from rotation about the third axis, which multiplies the integral by $e^{in\varphi }$ without otherwise changing it.