= Integrating factor for a differential one-form
{title2=$\partial_x(\mu f)=\partial_y(\mu g)$}
For the <differential one-form> $f\,dy+g\,dx$, a nonzero integrating factor $\mu$ makes $\mu f\,dy+\mu g\,dx$ 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.
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