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

$\backslash partial\; f.$ $\backslash int\_\{a\}^\{b\}\; \backslash left\; (\backslash frac\{df\}\{dx\}\; \backslash right)\; \backslash ne\; f(b)\; -\; f(a)$; 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 $\backslash \; If\; \backslash \; df\; \backslash ;\; =\; P(x,y)\; dx\; \backslash ;\; +\; Q(x,y)\; dy,\backslash \; then\backslash \; \backslash frac\{\backslash partial\; P\}\{\backslash partial\; y\}\; \backslash \; \backslash ne\; \backslash \; \backslash frac\{\backslash partial\; Q\}\{\backslash partial\; x\}.$

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.

thermodynamics | mathematics