Inexact differential

In mathematics, an inexact differential, as contrasted with an exact differential, of a function f is denoted:

$\partial f.$ $\int _{a}^{b}\left({\frac {df}{dx}}\right)\neq f(b)-f(a)$; as is true of point functions.

An inexact differential is one whose integral is path dependent. This may be expressed mathematically for a function of two variables as $\ If\ df\;=P(x,y)dx\;+Q(x,y)dy,\ then\ {\frac {\partial P}{\partial y}}\ \neq \ {\frac {\partial Q}{\partial x}}.$

A differential dQ that is not exact is said to be integrable when there is a function 1/τ such that the new differential dQ/τ is exact. The function 1/τ is called the integrating factor, τ being the integrating denominator.

Differentials which are not exact are often denoted with a δ rather than a d. For example, in thermodynamics, δQ and δW denote infinitesimal amounts of heat energy and work, respectively.