= Integrating-factor equation for a planar vector field
{title2=$\psi(A_{1,y}-A_{2,x})=A_2\psi_x-A_1\psi_y$}
If $\psi(A_1,A_2)$ is a <conservative vector field>, equality of the mixed derivatives of its potential gives $\partial_y(\psi A_1)=\partial_x(\psi A_2)$. Expanding proves the displayed necessary equation. For continuously differentiable data on the whole plane, the vanishing planar curl is also sufficient for a potential. A useful <integrating factor> is nonzero on the region under consideration.
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