= Second spherical moment integral
{title2=$\int_{S^2}n_in_j\,d\Omega=(4\pi/3)\delta_{ij}$}
For the outward unit normal on a three-dimensional <sphere>, the off-diagonal integrals vanish by <reflection symmetry>. The diagonal integrals are equal by <rotational symmetry>, and their sum is $\int_{S^2}|\mathbf n|^2d\Omega=4\pi$. This proves the displayed <matrix> identity, where $\delta_{ij}$ is the <Kronecker delta>. Contracting it with a constant <vector> gives $\int_{S^2}(\mathbf U\cdot\mathbf n)\mathbf n\,d\Omega=(4\pi/3)\mathbf U$, useful for integrating far-field <stress> forces.
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