Vorticity cross-product identity ($(\nabla \times U)\times v=(v\cdot \nabla )U-(\nabla U{)}^{T}v$)

For differentiable vector fields, the displayed identity relates vorticity to the antisymmetric part of the velocity gradient. Together with the curl of a cross product, it converts the adjoint linearized Navier-Stokes evolution between component and rotational forms.

The local vector-calculus identityseparates the convective acceleration into a kinetic-energy gradient and a vorticity cross-product contribution. Substitution into the steady Euler equations for an inviscid fluid yields the vorticity form of the Bernoulli equation.