Inexact differential ($\eta \ne d\Phi$)

A differential one-form on a specified domain is inexact if it is not the differential of a single-valued scalar potential there. For $Pdx+Qdy$, 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 $42ydx+xdy$ 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.