Exact differential

In thermodynamics, exact differential is a type of differential such that integration depends only on the end points and satisfies the condition on integrability (Euler reciprocity relation). [1] Differentials that do not fit this criteria are called inexact diffferentials.

Terminology
Synonyms of exact differential include “full differential” or “complete differential”, such as used in the 1865 and 1875 English editions of Clausius' The Mechanical Theory of Heat, but not “total differential”, which has a different meaning in mathematics.


History
The concept of the exact differential, supposedly, originated in the work of Leonhard Euler, in particular his Euler reciprocity relation.

The subject of the exact and inexact differential has a peculiar usage in thermodynamics as was first introduced in the 1858 article “
On the Treatment of Differential Equations which are not Directly Integrable” by German physicist Rudolf Clausius; an article that formed the basis of the chapter Mathematical Introduction to the first (1865) and second (1875) editions of Clausius' 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]

References
1. 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.
2. (a) Clausius, Rudolf. (1858). On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
(b) Clausius, Rudolf. (1875). 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. Oxford University Press.
4. ChE Screencasts (University of Colorado Boulder) – LearnChemE.com.

External links
Exact differential – Wikipedia.
Exact Differential – from Wolfram MathWorld

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