= Kelvin's circulation theorem
{c}
{title2=$d\Gamma/dt=0$}
{wiki}
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.
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