= Inexact differential
{title2=$\eta\ne d\Phi$}
A <differential one-form> on a specified domain is inexact if it is not the differential of a single-valued <scalar potential> there. For $P\,dx+Q\,dy$, unequal cross derivatives $P_y\ne Q_x$ prove inexactness locally. Equality is sufficient on a <simply connected> plane domain under the usual smoothness hypotheses, but holes can obstruct a global potential. The form $42y\,dx+x\,dy$ has cross derivatives $42$ and $1$ and counterclockwise unit-square integral $-41$, so it is inexact. A <conservative vector field> has an <exact differential> as its work form.
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