Past exam of the fluids and waves course of the University of Oxford / 2015 / 1 / a / Solution

Write the planar velocity field as $\mathbf{u}=(u,v,0)$, with $\mathbf{k}=(0,0,1)$. Expanding the curl givesConsequently, for a piecewise smooth path $\gamma$ from the origin to $\mathbf{x}=(x,y)$,The arbitrary additive constant $\psi (0)$ may be set to zero. The integrand is exactly $\nabla \psi \cdot d\mathbf{r}$, so the fundamental theorem for line integrals proves path independence immediately: the difference between two paths is the line integral of a gradient around a closed curve, which is zero.

One can also see the local integrability condition from Green's theorem:because the given representation enforces incompressibility. This second proof applies whenever the region between the paths lies in the regular fluid domain. The question already assumes a globally defined streamfunction; without that assumption, incompressibility alone would guarantee a global streamfunction on a simply connected domain, but a hole carrying nonzero volume flux could obstruct it.