= Solution
Use the <Euler equations for an inviscid fluid> with no magnetic or gravitational force. Let $w$ denote <specific enthalpy>. For a <homentropic flow>, $dw=dp/\rho$, so the pressure acceleration is $-\nabla w$. The velocity identity
$$
(\mathbf u\cdot\nabla)\mathbf u=\nabla\frac{u^2}{2}-\mathbf u\times\boldsymbol\omega,\qquad\boldsymbol\omega=\nabla\times\mathbf u,
$$
therefore gives
$$
\partial_t\mathbf u=\mathbf u\times\boldsymbol\omega-\nabla\mathcal B,\qquad\mathcal B=w+\frac{u^2}{2}.
$$
Taking the <curl>, using that the curl of a <gradient> is zero and that differentiation commutes for smooth fields, proves <barotropic vorticity transport>:
$$
\boxed{\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega).}
$$
More generally the same proof works for any <barotropic fluid>, replacing $w$ by a pressure potential $\int^\rho p'(s)\,ds/s$. Here “isentropic” must supply this barotropic closure, for example through one common <specific entropy> throughout the fluid. Merely imposing <entropy advection equation> $Ds/Dt=0$ allows spatial <specific entropy> gradients; in that case the <vorticity equation> contains the additional <baroclinic vorticity generation> term $\rho^{-2}\nabla\rho\times\nabla p$.
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