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Write the first-order surface velocity as ${\mathbf{u}}_1={\mathbf{U}}_1+{\mathbf{\Omega }}_1\times \mathbf{x}+\mathbf{g}$, where the previous derivative givesAs a test field, choose a sphere translating with arbitrary velocity $\hat{\mathbf{V}}$. Its surface velocity is $\hat{\mathbf{V}}$ and its traction is $\hat{\mathbf{\sigma }}\mathbf{n}=-3\mu \hat{\mathbf{V}}/(2a)$. The Lorentz reciprocal theorem for Stokes flow givesTerms at infinity vanish because both velocities decay. Since $\int \mathbf{n}dS=0$, the rotation term has zero surface average. The arbitrary test velocity therefore gives the first-order mobility of a nearly spherical particle:

For the ellipsoidal perturbation, the second spherical moment gives $\int fdS=(4\pi a^3/3)\mathrm{tr}\mathbf{D}=0$. The supplied fourth moment gives, componentwise,Substitution yieldswhere the last notation uses symmetry of the tensor.

One can also use a rotating-sphere test with surface traction $-3\mu \hat{\mathbf{\Omega }}\times \mathbf{n}$. Its reciprocal integral, together with zero first-order torque, givesHere $f(-\mathbf{n})=f(\mathbf{n})$, so the last integrand is odd and its integral vanishes. Thus ${\mathbf{\Omega }}_1=0$. Inversion symmetry forbids a translation-to-rotation coupling for this centred ellipsoidal perturbation.