 Solution

Incompressibility and $\mathbf{u}=\nabla \varphi$ give $\boxed{{\nabla }^2\varphi =0}$. The permitted vector identity, with zero vorticity, gives $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)$. Substituting into Euler's equation yieldsThe bracket is spatially constant in a connected fluid region, soAdding a function of time to the potential changes only $F(t)$ and leaves the velocity unchanged.