Kelvin's circulation theorem ($d\Gamma /dt=0$)

In an inviscid barotropic fluid subject to conservative body force, the circulation of a material loop is constant. For constant density, parameterize that loop by $\mathbf{r}(s,t)$ with ${\mathbf{r}}_t=\mathbf{u}$. Differentiating $\Gamma =\int \mathbf{u}\cdot {\mathbf{r}}_{s}ds$ gives an integral of $(D\mathbf{u}/Dt)\cdot d\mathbf{r}$ plus $\int {\partial }_s(|\mathbf{u}{|}^2/2)ds$. The latter vanishes on a closed loop, and the Euler equations make the former the integral of a gradient, also zero. For a barotropic density, $\nabla p/\rho$ is still a gradient of specific enthalpy. Constant circulation is consistent with changing local vorticity when a vortex tube changes its cross-sectional area.