= Flux of a linear field through a sphere
{title2=$\int_{S_a}(M\mathbf x+\mathbf b)\cdot\mathbf n\,dS=(4\pi a^3/3)\operatorname{tr}M$}
On an origin-centered radius-$a$ sphere, $\mathbf n=\mathbf x/a$. Spherical symmetry gives $\int_{S_a}x_i\,dS=0$ and $\int_{S_a}x_ix_j\,dS=(4\pi a^4/3)\delta_{ij}$: off-diagonal moments vanish by reflection, diagonal moments are equal, and their sum is $a^2$ times the surface area. Contracting these identities with $M_{ij}/a$ proves the displayed <flux> directly, including a zero contribution from a constant field $\mathbf b$. It agrees with the <divergence theorem>, since $\nabla\cdot(M\mathbf x+\mathbf b)=\operatorname{tr}M$.
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