Steady planar vorticity as a local function of stream function ($\Delta \psi =F(\psi )\text{locally where }\nabla \psi \ne 0$)

In a steady inviscid planar incompressible flow, vorticity is constant along streamlines. A regular stream function supplies a local transverse coordinate, so locally $\Delta \psi$ depends only on $\psi$. A global single-valued function needs the same value on every component of each level set. This can fail: $\psi =x^3-x$ gives the steady Euler shear $(u,v)=(0,1-3x^2)$ with constant pressure, but $\psi (0)=\psi (1)=0$ while $\Delta \psi$ is respectively $0$ and $6$.