 Inexact differential

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.

An **inexact differential** refers to a differential quantity that cannot be expressed as the total differential of a state function (or exact function). In thermodynamics, for example, the distinction between exact and inexact differentials is crucial for understanding the nature of different physical quantities.