
English biophysicist Douglas Spanner’s diagrams 3.7 and 3.8, in his An Introduction to Thermodynamics: Experimental Botany (1964), from his chapter section on “Total Differential of Z”, wherein he explains the graphical nature of the total derivative of a function z = f(x, y). [7]

English biophysicist Douglas Spanner’s diagrams 3.7 and 3.8, in his An Introduction to Thermodynamics: Experimental Botany (1964), from his chapter section on “Total Differential of Z”, wherein he explains the graphical nature of the total derivative of a function z = f(x, y). [7]

In mathematics, a total differential, not to be confused with a complete differential (or exact differential), see below, is the sum of partial differentials arising from the separate variation of the variables. [1] In other words, given a function z = f(x, y), having continuous first partial derivatives in a region of the xy-plane, its "total differential" is: [2]

$dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$

Said another way, given a function of two variables:

du = P dx + Q dy

a "total differential" of a function of several independent variables is the sum of its partial differentials arising from the separate variation of the variables, whereby the derivative of u can be written also as:

$du = \left( \frac{\partial u}{\partial x} \right)_{y} dx + \left( \frac{\partial u}{\partial y} \right)_{x} dy$

A "total differential equation", stated another way, is a differential equation that contains two or more dependent variables together with their differentials or differential coefficients with respect to a single independent variable which may, or may not, enter explicitly into the equation. [3] To state yet another way, using another set of notation, given a function f(x, y), of x and y variables, the "total differential" of f(x, y) is:

$df = d_x f + d_y f = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$

In words, the total differential of f(x, y) is found by finding the partial derivatives with respect to x and y, multiplying them respectively by dx and dy, and adding. [4]

History

See main: History of differential equations

Differential equations developed from calculus. Differential equations differ from ordinary equations of mathematics in that in addition to variables and constants they also contain derivatives of one or more of the variables involved.

English physicist Isaac Newton solved his first differential equation in 1676 by the use of infinite series, eleven years after his discovery of calculus in 1665. German mathematician Gottfried Leibniz solved his first differential equation in 1693, the year in which Newton first published his results. Hence, 1693 marks the inception for the differential equations as a distinct field in mathematics. [4]

Complete differential | Exact differential
In thermodynamics, a total differential is not to be confused with a complete differential or exact differential; or, sometimes, a "total exact differential", which is considered as a neoplasm. [5] This is confusion is most predominant in the case of entropy, the most-complex of all exact differentials. To exemplify a case of mis-labeling, in the 1996 article “Sooner Silence than Confusion: the Tortuous entry of Entropy into Chemistry”, American physical organic chemist Stephen Weininger and Danish chemical thermodynamics historian Helge Kragh state the following: [6]

“In an important paper of 1854 Clausius showed that for reversible processes the quantity δq/T is a total differential, which means that it is a state function, the line integral of which depends on the variables characterizing the state (pressure, volume, etc.), but not on the particular path chosen for the integration.”

A number of errors exist in this statement, which is ironic in that it is in the opening pages of an article aiming to clear the cloud of confusion in regards to entropy for chemistry students and teachers.

Firstly, in regards to these errors, Clausius did not use the term “total differential”, which as as explained is a mislabeling, but rather he used the term “completed differential” (Thomas Hirst, 1865 translation; Walter Browne, 1879 translation), a term to which he affixed a very specific meaning, namely that it is a function that satisfies the Euler reciprocity relation or as Clausius called it the “condition of immediate integrability”.

Secondly, Clausius did not use the small delta symbol “δ” for heat, but rather used the notation “dQ”, which he says he adopts from the notation of Leonhard Euler. Clausius does, however, note that German mathematician Carl Jacobi used the “∂” notation or "curly d" symbol, as it has come to be called, in place of the d in the numerator and denominator of the fraction which represents a partial differential coefficient..

Thirdly, although in Clausius’ 1854 fourth memoir “On a Modified form of the Second Fundamental Theorem in the Mechanical Theory of Heat” he does introduce the dQ/T function, it is his 1858 publication “On the Treatment of Differential Equations which are Not Directly Integrable” wherein he, in effect, proved that entropy is a path-independent state function for a reversible process, not necessarily the former memoir.

References
1. Osborne, George A. (1891). An Elementary Treatise on Differential and Integral Calculus (section 65: Total Differential of a Function of Several Independent Variables, pgs. 83-84). Leach, Shewell, and Sanborn.
2. Zill, Dennis G. (1993). A First Course in Differential Equations (total differential, pg. 54). PWS-KENT Publishing.
3. Ince, Edward L. (1926). Ordinary Differential Equations (pg. 3). Dover.
4. Korzybski, Alfred. (1994). Science and Sanity: an Introduction to non-Aristotelian Systems and General Semantics (Section: Differential equations, pg. 595-). Institute of General Semantics.
5. Perrot, Pierre. (1998). A to Z of Thermodynamics (pg. 105). New York: Oxford University Press.
6. Kragh, Helge and Weininger, Stephen J. (1996). “Sooner Science than Confustion: the Tortuous Entry of Entropy into Chemist” (abs), Historical Studies in the Physical and Biological Sciences, 27(1): 91-130.
7. Spanner, Douglas C. (1964). An Introduction to Thermodynamics: Experimental Botany (§: Total Differential of z, pgs. 29-31). Academic Press.