= Isotropic fourth moment of a ball
{title2=$\int_{|x|\le R}x_ix_jx_kx_l\,d^dx=\frac{V_dR^{d+4}}{(d+2)(d+4)}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$}
Here $V_d$ is the volume of the unit ball in $d$-dimensional <Euclidean space>. Rotational invariance makes the integral an <isotropic tensor>, while complete symmetry in its four indices equates the three coefficients. Contracting $i=j$ and $k=l$ gives $d(d+2)$ times their common value. The same contraction of the integral is $\int_{|x|\le R}|x|^4\,d^dx=dV_dR^{d+4}/(d+4)$, proving the formula. For $d=3,R=1$ the coefficient is $4\pi/105$. This is the fourth-order counterpart of the <isotropic second moment of a ball>.
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