= Solution
An <irrotational flow> has $\nabla\times\mathbf u=0$. Hence a <velocity potential> $\phi$ exists locally, and globally on a <simply connected domain>, with $\mathbf u=\nabla\phi$. The exterior of a sphere in three dimensions has this property. Constant <mass density> reduces the <continuity equation> to $\nabla\cdot\mathbf u=0$, so
$$
\boxed{\nabla^2\phi=0}.
$$
Thus the <velocity potential> is a <harmonic function>. For the <Euler equations>, use the <vorticity cross-product identity>
$$
(\mathbf u\cdot\nabla)\mathbf u
=\nabla\!\left(\frac12|\mathbf u|^2\right)
-\mathbf u\times(\nabla\times\mathbf u).
$$
The last term vanishes. Since $\partial_t\mathbf u=\nabla\phi_t$, the <Euler momentum equation> becomes
$$
\nabla\!\left(\phi_t+\frac12|\nabla\phi|^2+\frac p\rho\right)=0.
$$
On each connected fluid region, a function with zero <gradient> is spatially constant. Therefore the <Unsteady Bernoulli equation> is
$$
\boxed{\frac p\rho+\phi_t+\frac12|\nabla\phi|^2=F(t)}.
$$
The freedom to add a function of time to the <velocity potential> changes $F(t)$ but leaves the <velocity field> unchanged. On a multiply connected region, zero <vorticity> alone guarantees only local potentials; a global single-valued <velocity potential> additionally requires zero <circulation> around every closed path.
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