A snapshot summary of the Euler reciprocity relation or what Rudolf Clausius synonymously referred to as the "condition for an exact differential", both varieties of which seem to have originated by Swiss mathematician Leonhard Euler. [2] |

If this holds:

for the following two dimensional function:

then

This, however, is for two dimensions (as can be extended to three dimensions), that applies to any a function of any number of independent variables, where

Synonyms

Synonyms include: condition for an exact differential, condition for a complete differential, or condition of immediate integrability (Rudolf Clausius, 1858-1875), Euler’s criterion (James Partington, 1911), integrability condition (Philip Mirowski, 1989), or Euler’s reciprocity relation.

History

The first to begin studying reciprocity questions, supposedly, was French mathematicianSee also: History of differential equations

In 1729, Swiss mathematician Leonhard Euler began to read Fermat’s work seriously following correspondence with German mathematician

Euler's exact publication remains to be tracked down among his collected works of 60 to 80 quarto volumes (900 books).

It is known, however, that in 1739, in correspondence with Swiss mathematician

Thermodynamics | Entropy

In 1854, within the framework of launching the science of thermodynamics, German physicist Rudolf Clausius, in his fourth memoir "On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat", began to employ, in an uncited manner, Euler’s reciprocity relation (although he never called it by this name, not cited Euler explicitly has having introduced this proof) to develop the physical concept of a “

Clausius expanded on the underlying mathematics of this in his 1858 "Mathematical Introduction: on the Treatment of Differential Equations Which Are Not Directly Integrable", wherein he cites both Euler and Jacobi.

The foremost example of Clausius’ use of Euler’s reciprocity relation is in the derivation of the mathematical function of entropy (S), on the logic that the integrating factor of the inverse of the absolute temperature of the body (1/T) makes the inexact differential of a quantity of heat (dQ) an exact differential (dQ/T). This is the one aspect of thermodynamics that makes the subject notoriously difficult and its roots deep.

Maxwell’s relations

Euler’s reciprocity relation is used to derive the Maxwell relations.

References

1. DeVoe, Howard. (2001).

2. The Terrible Beauty of Thermodynamics (2010) – ScribD.com.

3. Sasser, John E. (1992). “History of Ordinary Differential Equations: the First Hundred Years”, Proceedings of the Midwest Mathematics History Society.

4. Lemmermeyer, Franz. (2000).

Further reading

● Partington, James R. (1911).

● Mortimer, Robert G. (1999). (Euler reciprocity relation, pg. 136). Academic Press.

● Blinder, S.M. (1966). “Mathematical Methods in Elementary Thermodynamics”,

● Blinder, Seymour Michael. (1969).

External links

● Neira, Maurico E. (2009). “Thermodynamics: Formulas and Constants”, Dec 4, Mesguerra.net.