In mathematics, a differential refers to a variable that relates to or constitutes a difference or small change. [1]

See also: History of differential equations, Symbols
There are three types of differential symbols used in thermodynamics: ordinary differential, partial differential, and exact differential. The notation for these three types of differentials is generally consistent, although notation sometimes varies. These types of differentials are summarized below:


d d \,English letter "d"Ordinary differential

The symbols dx, dy, and dx/dy were all introduced by German mathematician Gottfried Leibniz in his 1675 notebook. [5]
 \partial \,"curly d"
rounded d
curved d
Jacobi's delta
Partial differential
Used to denote a partial derivative, such as:
\frac{\partial z}{\partial x}
which is read as "the partial derivative of z with respect to x" or "the partial of z with respect to x".
In 1770, French mathematician Marquis Condorcet, in his “Memoir on Partial Differential Equations”, introduced the so-called “curly d” symbol, as follows: [6]

“Throughout this paper, both dz and z will either denote two partial differences of z, where one of them is with respect to x, and the other, with respect to y, or dz and z will be employed as symbols of total difference, and of partial difference, respectively.”

To note, it is difficult to see, in this translation, if this corresponds to the modern concept of total differential and partial differential?

The modern standard curly d (∂) notation in equation form was first used in 1786 by French mathematician Adrien-Marie Legendre who stated the following: [7]

“Pour éviter toute ambiguité, je répresentarie par ∂u/∂x le coefficient de x dans la différence de u, and par du/dx la différence complète de u divisée par dx.”
"To avoid any ambiguity, I represent ∂u/∂x as the coefficient of x in the difference of u, and by du/dx the complete difference of u divided by dx."

Again, the translation of "différence complète" to complete differential, needs to be fact-checked. In any event, Legendre is said to have abandoned this symbol usage. The curly d symbol (∂) was later re-introduced in 1841 by German mathematician Carl Jacobi, who stated in Latin: [3]

“Sed quia uncorum accumulatio et legenti et scribenti molestior fieri solet, praetuli characteristica d differentialia vulgaria, differentialia autem partialia characteristica ∂ denotare.”

which translates as: we will denote the partial differential by the character " ∂ ", or something to this effect.

In 1882, German physicist Hermann Helmholtz was using both the partial differential and the standard differential d in his "On the Thermodynamics of Chemical Processes". [9]

Inexact differential
Synonyms: imperfect differential, incomplete differential.In 1858, German physicist Rudolf Clausius published his his 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable” and followup chapter section "Mathematical Introduction" (1865, 1875), wherein he laid out the thermodynamic version of the so-called "exact differential" as compared to the "inexact differential".

 \bar{d} \,d hat(1875)Synonyms: total differential, completed differentialThe d hat notation ( \bar{d} \,) was introduced in 1875 by German mathematician Carl Neumann to represent an inexact differential, i.e. one that is path dependent, which is the case with differential units of heat  \bar{d}Q \,and work \bar{d}W \,. [4]
d crossbarInexact differential Often used, in 20th century thermodynamics textbooks, to represent an exact differential; inconsistent use.The first person to use or introduce this this notation needs to be tracked down.
δ \delta \,Greek lower case letter deltaExact differential Sometimes used to represent an exact differential; inconsistent use. Used by Johann Bernoulli in 1706 to denote the difference of functions; used in 1821 by Augustin-Louis Cauchy to denote “difference”. [5]

In 1924, James Partington was using δ to denote that “δQ and δA are not perfect differentials, since each depends on the path of change.” Partington footnotes this statement, directing readers to his 1920 Higher Mathematics for Chemical Students, in which he says that he gives a proper treatment of the properties of perfect differentials, as presented by Rudolf Clausius (1875). [8]
Δ \Delta \,Greek capital letter DeltaDifference

Introduced in 1923 by Gilbert Lewis to represent a change of state of a system in going from an initial state to final state, such as:

 \Delta G = G_f - G_i \,

or rather the "difference" in the values between the final state and the initial state.

The term "differential" was first used in 1647. [1]

Cartesian systems
The differential dF of a Cartesian coordinates defined function F(x,y,z) is given by: [2]

 dF = \Bigg( \frac{\partial F}{\partial x} \Bigg)_{y,z}  + \Bigg( \frac{\partial F}{\partial y} \Bigg)_{x,z}    + \Bigg( \frac{\partial F}{\partial z} \Bigg)_{x,y}    \,

1. Differential – Merriam-Webster.com.
2. Perrot, Pierre. (1998). A to Z of Thermodynamics. Oxford University Press.
3. Jacobi, Carl G.J. (1841). “On the Determinant Functions” (De determinantibus Functionalibus)”, in: Crelle’s Journal, Band 22, pp. 319-352, (pp. 393-438 of vol. 1 of the Collected Works).
4. (a) Neumann, Carl. (1875). Vorlesungen über die mechanische Theorie der Wärme (Lectures on the Mechanical Theory of Heat). Germany.
(c) Anon. (1877). “Science (Review: Lectures on the Mechanical Theory of Heat by Carl Neumann, 1875)”, The Westminster Review (pg. 250-), Jan-Apr.
(b) Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press.
5. (a) Leibniz, Gottfried. (1675). “manuscript of November 11”; (Cajori vol. 2, page 204).
(b) Cajori, Florian. A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929.
5. Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.
6. Condorcet, Marquis de. (1770). “Memoir on Partial Differential Equations” (“Memoire sur les Equations aux différence partielles”); in: in Histoire de L'Academie Royale des Science (quote, pg. 152), pgs. 151-178, Annee M. DCCLXXIII, 1773.
7. Legendre, Adrien-Marie. (1786). “Memoir on how to distinguish the Maxima and Minima in Calculus of Variations” (Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations), in: Histoire de l'Academie Royale des Sciences, Annee M. DCCLXXXVI (quote, pg. 8), pp. 7-37, Paris, M. DCCXXXVIII (1788).
8. Partington, James. (1920). Higher Mathematics for Chemical Students (§§ 55, 115). London.
9. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes”, in: Physical Memoirs Selected and Translated from Foreign Sources, 1: 43-97. Physical Society of London, Taylor and Francis, 1888.

External links
– Wikipedia.

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