Curl of a cross product ($\nabla \times (U\times v)=U\nabla \cdot v-v\nabla \cdot U+(v\cdot \nabla )U-(U\cdot \nabla )v$)

The displayed vector calculus identity follows by contracting the two alternating tensors in the component formula for curl. If both fields have zero divergence, only the two directional derivatives remain. It rewrites the transport terms in the adjoint linearized Navier-Stokes evolution.