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For constant density $\rho$, the Euler equations for an inviscid fluid with a conservative force per unit mass areSet $\mathbf{\omega }=\nabla \times \mathbf{u}$. The vector identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times \mathbf{\omega }$ and vanishing curl of a gradient givewhere both divergence terms vanish. Thus the vorticity equation isThe partial time derivative measures Eulerian change; $(\mathbf{u}\cdot \nabla )\mathbf{\omega }$ is advection; the right-hand side describes stretching and tilting of vortex lines by velocity gradients. A conservative body force supplies no curl source, and constant density removes any baroclinic pressure source.

For planar velocity independent of $z$, vorticity is perpendicular to the plane, $\mathbf{\omega }=(0,0,\omega )$, and $(\mathbf{\omega }\cdot \nabla )\mathbf{u}=\omega {\partial }_z\mathbf{u}=0$. Hence each material particle conserves its vorticity, $D\omega /Dt=0$. Choose the stream function convention $u={\psi }_y$, $v=-{\psi }_x$. ThenIn a steady flow, ${\psi }_y{\omega }_x-{\psi }_x{\omega }_y=0$, so vorticity is constant along streamlines. Where $\nabla \psi \ne 0$, use $\psi$ as a local transverse coordinate: this equality makes $\omega$ independent of the coordinate along a streamline. ConsequentlyThis is steady planar vorticity as a local function of stream function.

A global single-valued $F$ additionally requires compatible vorticity values on all components of each stream-function level set. That condition is not supplied by steady Euler flow alone. For example, $\psi =x^3-x$ gives $(u,v)=(0,1-3x^2)$, a smooth steady incompressible shear with zero advective acceleration and constant pressure. Yet $\psi (0,y)=\psi (1,y)=0$ while $\Delta \psi (0,y)=0$ and $\Delta \psi (1,y)=6$. Thus the unrestricted global deduction is false; the local result, or a suitable connected-level-set hypothesis, is the valid conclusion.