Integrating factor for a differential one-form (${\partial }_x(\mu f)={\partial }_y(\mu g)$)

For the differential one-form $fdy+gdx$, a nonzero integrating factor $\mu$ makes $\mu fdy+\mu gdx$ an exact differential form locally. If it equals $dF$, then $F_y=\mu f$, $F_x=\mu g$, and equality of mixed partial derivatives gives the displayed compatibility condition. Conversely this condition is local sufficiency for smooth coefficients on a small rectangle; global exactness additionally depends on the domain. Zeros of the multiplying factor can change the solution set, so equivalence is asserted only where the factor is nonzero.