Integrating-factor equation for a planar vector field ($\psi (A_{1,y}-A_{2,x})=A_2{\psi }_x-A_1{\psi }_y$)

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.