 Solution

An irrotational flow satisfies $\nabla \times \mathbf{u}=0$. On a simply connected region, or locally on any small regular region, this gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$. The incompressibility condition is $\nabla \cdot \mathbf{u}=0$, soThus the potential is a harmonic function. The locality qualification matters: a curl-free circulation around an excluded vortex core need not admit a globally single-valued potential. No such circulation is needed for the sphere flow below.