Translation-rotation reciprocal coupling ($\mathbf{F}=-A\mathbf{U}-B\mathbf{\Omega },\mathbf{T}=-B^T\mathbf{U}-C\mathbf{\Omega }$)

For a fixed rigid-body shape and origin in an unbounded stationary Stokes flow, the Lorentz reciprocal theorem makes the off-diagonal blocks of the six-dimensional hydrodynamic resistance matrix transposes. Here $\mathbf{F},\mathbf{T}$ are the fluid's force and torque on the body. If every pure translation has zero torque about the chosen origin, then $B^T=0$, hence $B=0$: every pure rotation about that origin has zero resultant force. This conditional statement does not require such an origin to exist for every body.