, an inexact differential
or imperfect differential
is any quantity, particularly heat
Q and work
W, that are not state functions
, in that their values depend on how the process
is carried out. The symbol , or δ (in the modern sense), which originated from the work of German mathematician Carl Gottfried Neumann
in his 1875 Vorlesungen über die mechanische Theorie der Wärme
, indicates that Q and W are path dependent. In terms of infinitesimal quantities, the first law of thermodynamics
is thus expressed as:
where δQ and δW are "inexact", i.e. path-dependent, and dU is "exact", i.e. path-independent.
In general, an inexact differential, as contrasted with an exact differential
, of a function f
; as is true of point functions. In fact, F(b) and 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.
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.
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
is a mistake, since heat
is not a state function having initial and final values. It would, however, be correct to use lower case
in the inexact differential
expression for heat. The offending
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
in place of
. Continuing with the same instance of
, for example, removing the
, the equation
is true for constant pressure.