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.
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A differential one-form on a specified domain is inexact if it is not the differential of a single-valued scalar potential there. For , unequal cross derivatives 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 has cross derivatives and and counterclockwise unit-square integral , so it is inexact. A conservative vector field has an exact differential as its work form.