= Radial tensor moments in three dimensions
{title2=$\int F(r)x_ix_jx_kx_l\,d^3x=\frac{4\pi}{15}\int_0^\infty r^6F(r)\,dr\,(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$}
For a radial weight $F$ with the required absolutely integrable moments, <rotational symmetry> makes the second and fourth moments <isotropic tensors>. The second moment is $A\delta_{ij}$; its <trace> gives $3A=4\pi\int_0^\infty r^4F(r)\,dr$. Complete symmetry of the fourth moment forces equal coefficients of the three <Kronecker delta> pairings. Its double <tensor contraction> gives $15B=4\pi\int_0^\infty r^6F(r)\,dr$, proving the displayed expression. Odd moments vanish under $x\mapsto-x$. Choosing $F$ to be the <indicator function> of a <spherical shell> gives all the corresponding shell moments without separate angular integrations.
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