= Two-dimensional potential-vorticity inversion
{title2=$\psi(\mathbf x)=\frac1{2\pi}\int\zeta(\mathbf x')\log|\mathbf x-\mathbf x'|\,d^2x'+\psi_h$}
In <two-dimensional vortex dynamics>, solve the <Poisson equation> for the <streamfunction> with the prescribed <boundary conditions>. On the plane its fundamental solution is $(2\pi)^{-1}\log|\mathbf x|$, so differentiating the displayed inverse gives the <planar Biot-Savart kernel>. The <harmonic function> $\psi_h$ represents boundary and global-flow data. An additive constant changes no <velocity>; circulation around holes and a uniform flow are additional degrees of freedom if their boundary or far-field values are not prescribed. The same inverse acts independently on each level in <nonrotating layerwise-two-dimensional vortex dynamics>.
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