Inexact differential
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In physics, an inexact differential, as contrasted with an exact differential, of a function f is denoted:
<math>\partial f.</math> <math>\int_{a}^{b} df \ne F(b) - F(a)</math>; as is true of point functions. In fact, <math>F(b), F(a)</math>, 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 <math>\ \mbox{If}\ df = P(x,y) dx \; + Q(x,y) dy,\ \mbox{then}\ \frac{\partial P}{\partial y} \ \ne \ \frac{\partial Q}{\partial x}.</math>
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.
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[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 <math>\Delta Q</math> is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case <math>\delta Q</math> in the inexact differential expression for heat. The offending <math>\Delta</math> belongs further down in the Thermodynamics section in the equation :<math>q = U - w \ </math>, which should be :<math>q = \Delta U - w \ </math> (Baierlein, p. 10, equation 1.11, though he denotes internal energy by <math>E</math> in place of <math>U</math>.<ref>Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.</ref> Continuing with the same instance of <math>\Delta Q</math>, for example, removing the <math>\Delta</math>, the equation
- <math>Q = \int_{T_0}^{T_f}C_p\,dT \,\!</math>
is true for constant pressure.
[edit] See also
- Closed and exact differential forms for a higher-level treatment
- Differential
- Exact differential
- Integrating factor for solving non-exact differential equations by making them exact
[edit] References
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[edit] External links
- Inexact Differential – from Wolfram MathWorld
- Exact and Inexact Differentials – University of Arizona
- Exact and Inexact Differentials – University of Texas
- Exact Differential – from Wolfram MathWorld
