In physics, an inexact differential, as contrasted with an exact differential, of a function f is denoted: The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β...

; 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

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. In mathematics, one solves certain ordinary differential equations by using an integrating factor. ...

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.

Example

As an example, the use the inexact differential in thermodynamics is a way to mathematically quantify functions that are not state function and thus path dependent. In thermodynamic calculations, the use of the symbol ΔQ is a mistake, since heat is not a state function having initial and final values. It would, however, be correct to use lower case δQ in the inexact differential expression for heat. The offending Δ belongs further down in the Thermodynamics section in the equation :, which should be : (Baierlein, p. 10, equation 1.11, though he denotes internal energy by E in place of U.^[1] Continuing with the same instance of ΔQ, for example, removing the Δ, the equation ‹ The template below has been proposed for deletion. ... In thermodynamics, a state function (or state quantity) is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... Path-dependence exists when the outcome of a process depends on its past history, on the entire sequence of decisions made by agents and resulting outcomes, and not just on contemporary conditions. ... In physics, heat, symbolized by Q, is defined as energy in transit. ... In physics, heat, symbolized by Q, is defined as energy in transit. ...

is true for constant pressure.

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

In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... In mathematics, one solves certain ordinary differential equations by using an integrating factor. ...

References

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