Second spherical moment integral (${\int }_{S^2}{n}_i{n}_{j}d\Omega =(4\pi /3){\delta }_{ij}$)

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 ${\int }_{S^2}|\mathbf{n}{|}^{2}d\Omega =4\pi$. 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.