Inexact differential

In thermodynamics, inexact differential is defined such that the integration of the differential "depends" on the path between end points (see: path dependent). [1] Differentials that do not fit this criteria are called exact diffferentials. Synonyms of inexact differential include “imperfect differential”, or “incomplete differential”, among others.

The subject of the exact and inexact differentials seems to be unique to the science of thermodynamics; introduced in the 1858 article “On the Treatment of Differential Equations which are not Directly Integrable” by German physicist Rudolf Clausius, later used an introductory section to the first (1865) and second (1875) editions of his thermodynamics textbook The Mechanical Theory of Heat. [2]

The symbol đ (d-crossbar)
, or δ (in the modern sense), originated from the work of German mathematician Carl Neumann, specifically in his 1875 Lectures on the Mechanical Theory of Heat, indicating, as Clausius did, that δQ and δW are path dependent (inexact differentials), whereas internal energy dU is not (exact differential). [3]

German physicist Georg Helm seems to have been the first to point this origin out, specifically in his 1898 History of Energetics, wherein he comments Neumann was the one that suggested that the special differential sign  \bar{d}Q \, (d hat Q) should be used as a way of always keeping in mind the differentials, such as heat, that do not fit the condition for an exact differential. [4]

Note: the original German text needs to be checked to see if it is  \bar{d} \,(d-hat) or đ (d-crossbar), although the Helm references seems to indicated the former was used.

1. (a) Potter, Merle C. and Scott, Elaine P. (2004). Thermal Sciences - an Introduction to Thermodynamics, Fluid Mechanics, and Heat Transfer, (pg. 67). U.S.: Brooks/Cole.
(b) Inexact Differential – from Wolfram MathWorld.
2. (a) Clausius, Rudolf. (1958). On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
(b) Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (Section: Mathematical Introduction, pgs. 1-20). London: Macmillan & Co.
3. (a) Neumann, Carl. (1875). Lectures on the Mechanical Theory of Heat (Vorlesungen über die mechanische Theorie der Wärme), Germany.
(b) Laider, Keith, J. (1993). The World of Physical Chemistry (pg. 98). Oxford University Press.
4. (a) Helm, Georg F. (1898). Die Energetik: Nach Ihrer Geschichtlichen Entwickelung (Energetics: Historical Development). Leipzig.
(b) Helm, Georg F. (2000). The Historical Development of Energetics (Carl Neumann: special differential sign, pg. 130). Kluwer Academic Press.

External links
Inexact differential – Wikipedia.

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