Inexact differential

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

$\partial f.$ $\int_{a}^{b} df \ne F(b) - F(a)$ ; as is true of point functions. In fact, $F (b), F (a)$ , in general, are not defined.

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

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.

1 Example
2 See also
3 References
4 External links

[edit] Example

As an example, the use of the inexact differential in thermodynamics is a way to mathematically quantify functions that are not state functions and thus path dependent. In thermodynamic calculations, the use of the symbol $Δ Q$ is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case $δ Q$ in the inexact differential expression for heat. The offending $Δ$ belongs further down in the Thermodynamics section in the equation : $q = U - w \$ , which should be : $q = \Delta U - w \$ (Baierlein, p. 10, equation 1.11, though he denotes internal energy by $E$ in place of $U$ .^[1] Continuing with the same instance of $Δ Q$ , for example, removing the $Δ$ , the equation