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Concept
Inexact differential
Summary
An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential. I.e., its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths such that , and
In this case, we denote the integrals as and respectively to make explicit the path dependence of the change of the quantity we are considering as .
More generally, an inexact differential is a differential form which is not an exact differential, i.e., for all functions ,
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.
Instead of the differential symbol d, the symbol δ is used, a convention which originated in the 19th century work of German mathematician Carl Gottfried Neumann, indicating that Q (heat) and W (work) are path-dependent, while U (internal energy) is not.
Official source
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