= Solution
For constant viscosity, differentiating $\sigma=-pI+\mu(\nabla u+\nabla u^T)$ gives $\nabla\cdot\sigma=-\nabla p+\mu\nabla^2u$, since $\nabla\cdot u=0$. This is zero by the body-force-free <Stokes equations>.
Apply the <divergence theorem> to the annular fluid. Its outer normal is $+n_R$, but its inner normal is minus the sphere's outward normal. Therefore $0=\int_{S_R}\sigma n_RdS-\int_{S_a}\sigma n_adS$. The second integral is the hydrodynamic force on the inner sphere, $-F$, because that sphere exerts $F$ on the fluid. Thus again \b[$G=-F$], without differentiating the far field. Local momentum conservation carries the same total force through every enclosing surface.
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