= Solution
Write the conservative body force per unit mass as $-\nabla\Phi$. For a fluid of constant density $\rho$, the <Euler equations for an inviscid fluid> and the <incompressible flow> condition are
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla\left(\frac p\rho+\Phi\right),\qquad \nabla\cdot\mathbf u=0.
$$
Using part (a), these become
$$
\partial_t\mathbf u-\mathbf u\times\boldsymbol\omega=-\nabla H,\qquad H=\tfrac12|\mathbf u|^2+\frac p\rho+\Phi.
$$
For <irrotational flow>, locally choose a <velocity potential> with $\mathbf u=\nabla\phi$. Then $\nabla(\partial_t\phi+H)=0$, so \b[the unsteady Bernoulli integral] is
$$
\boxed{\partial_t\phi+\tfrac12|\mathbf u|^2+\frac p\rho+\Phi=C(t).}
$$
The constant is spatially uniform on a connected region where the <velocity potential> is defined; its time dependence can be absorbed into the additive time-dependent gauge of $\phi$. A global single-valued <velocity potential> requires an appropriate topological condition, such as a <simply connected> domain; without it, the integral is locally valid. In a connected steady irrotational region, $\nabla H=0$ directly gives one spatial Bernoulli constant even without choosing a global potential.
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