= Steady planar vorticity as a local function of stream function
{title2=$\Delta\psi=F(\psi)\quad\text{locally where }\nabla\psi\ne0$}
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 $6$.
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