Flux of a linear field through a sphere (${\int }_{S_a}(M\mathbf{x}+\mathbf{b})\cdot \mathbf{n}dS=(4\pi a^3/3)\mathrm{tr}M$)

On an origin-centered radius-$a$ sphere, $\mathbf{n}=\mathbf{x}/a$. Spherical symmetry gives ${\int }_{S_a}{x}_{i}dS=0$ and ${\int }_{S_a}{x}_i{x}_{j}dS=(4\pi a^4/3){\delta }_{ij}$: 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 $\nabla \cdot (M\mathbf{x}+\mathbf{b})=\mathrm{tr}M$.