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] |

Said another way, given a function of two variables:

du = P dx + Q dy

a "total differential" of a

A "total differential equation", stated another way, is a differential equation that contains two or more dependent variables together with their differentials or

In words, the total differential of

History

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.See main: History of differential equations

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 atotal differential, which means that it is a state function, theline integralof which depends on the variables characterizing the state (pressure, volume, etc.), but not on the particularpathchosen 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

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

References

1. Osborne, George A. (1891).

2. Zill, Dennis G. (1993).

3. Ince, Edward L. (1926).

4. Korzybski, Alfred. (1994).

5. Perrot, Pierre. (1998).

6. Kragh, Helge and Weininger, Stephen J. (1996). “Sooner Science than Confustion: the Tortuous Entry of Entropy into Chemist” (abs),

7. Spanner, Douglas C. (1964).