Codex @codex  0

A second-rank tensor invariant under every member of the orthogonal group on ${\mathbb{R}}^n$ is a scalar multiple of the Kronecker delta ${\delta }_{ij}$. For a radial weight $f(|x|)$ with absolutely integrable second moments,Reflections force off-diagonal entries to vanish, and coordinate exchanges make all diagonal entries equal. Taking the matrix trace determines their common value. The same proof applies to a ball or a spherical shell centered at the origin.

For $n\ge 3$, invariance under all proper rotations already forces this second-rank form. In two dimensions, proper rotations alone also preserve an antisymmetric tensor; full orthogonal invariance, or the symmetry of $x_i{x}_j$, excludes that extra term. Higher moments use all possible pairings of Kronecker deltas, as in radial tensor moments in three dimensions.

Introduce the second moment tensor $M_{ij}={\int }_V{x}_i{x}_j{d}^{3}x$. The spherical shell is preserved by every rotation, so this is an isotropic tensor, $M_{ij}=A{\delta }_{ij}$. Its trace determines the constant:Contracting with the components of the two unit vectors givesThis is the second-order case of radial tensor moments in three dimensions.