Definition
An inexact differential is commonly defined as a differential form dx where there is no corresponding function x such that: . More precisely, an inexact differential is a differential form that cannot be expressed as the differential of a function. In the language of calculus, for a given vector field F, is an inexact differential if there is no function f such that
The fundamental theorem of calculus for line integrals requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative. This is because there can be no unique representation of an antiderivative for inexact differentials since their variation is inconsistent along different paths. This stipulation of path independence is a necessary addendum to the fundamental theorem of calculus because in one-dimensional calculus there is only one path in between two points defined by a function.
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