Conservation of vorticity flux through a material surface ($d{\int }_{S(t)}\mathbf{\omega }\cdot d\mathbf{S}/dt=0$)

For smooth inviscid barotropic fluid with potential body forces, barotropic vorticity transport and $\nabla \cdot \mathbf{\omega }=0$ give $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}-\mathbf{\omega }\nabla \cdot \mathbf{u}$. Combine this with material surface element evolution to obtain $D(\mathbf{\omega }\cdot d\mathbf{S})/Dt=0$. Integrating over a material surface proves the flux law. Its boundary must move with the fluid; flux through a fixed open surface is generally not conserved. By Stokes theorem the same statement is Kelvin's circulation theorem around the material boundary.