The circulation  around a closed material contour  is defined by:
where u is the velocity vector, and ds is an element along the closed contour.
The governing equation for an inviscid fluid with a conservative body force is
where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.
The condition of barotropicity implies that the density is a function only of the pressure, i.e. .
Taking the convective derivative of circulation gives
For the first term, we substitute from the governing equation, and then apply Stokes' theorem, thus:
The final equality arises since  owing to barotropicity. We have also made use of the fact that the curl of any gradient is necessarily 0, or  for any function .
For the second term, we note that evolution of the material line element is given by
Hence
The last equality is obtained by applying gradient theorem.
Since both terms are zero, we obtain the result