Skew-symmetry of incompressible transport

For a periodic or boundary-tangent divergence-free vector field $u$, integration by parts givesConsequently $\int (u\cdot \nabla f)f=0$, the cancellation underlying many fluid energy estimates.

For a velocity field that need not be divergence free, the corrected transportsatisfies $⟨\tilde{B}(u,v),w⟩=-⟨\tilde{B}(u,w),v⟩$ and therefore preserves the quadratic energy cancellation.