Inexact differential

An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another function and is therefore path independent. Consequently, an inexact differential cannot be expressed in terms of its antiderivative for the purpose of integral calculations; i.e. its value cannot be inferred just by looking at the initial and final states of a given system. It is primarily used in calculations involving heat and work because they are path functions, not state functions.

An inexact differential is commonly defined as a differential form dx where there is no corresponding function x such that: ${\displaystyle x=\int {\text{d}}x\,}$. 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, ${\displaystyle \delta F=F\,{\text{d}}r}$ is an inexact differential if there is no function f such that

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