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]
In mathematical thermodynamics, Euler reciprocity relation or "reciprocity relation" is the following relational criterion; namely:

If this holds:

for the following two dimensional function:

$dF = Mdx + Ndy \,$

then F is an exact differential (i.e. state function).

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 x and y are any two of the independent variables, M is ∂f /∂x, N is ∂f/∂y, and each partial derivative has all independent variables held constant except the variable shown in the denominator. [1] In plain speak, the reciprocity relation, which is difficult to explain fully in words, is a type of symmetry differentiation criterion that determines whether or not a function is immediately Integrable.

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 mathematician Pierre Fermat (1601-1665), discussed in correspondence letters to other mathematicians or written as notes in the margins of his books.

In 1729, Swiss mathematician Leonhard Euler began to read Fermat’s work seriously following correspondence with German mathematician Christian Goldbach (1690-1764) on some of Fermat’s observations; which thus acted to launch the long drawn-out study of “reciprocity laws”, upward through the works of Adrien-Marie Legendre, Carl Gauss (1777-1855), Johann Dirichlet (1805-1859), Carl Jacobi (1804-1851), and Sergei Eisenstein (1898-1948). [4]

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 Johann Bernoulli (1667–1748), and into the following year (1740), Euler began using the integrating factor as an aid to derive differential equations that were integrable in finite form. [3]

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 “path-independentstate function (one that meets the reciprocity relation) as compared to a “path-dependent” state function (one that does not meet the reciprocity relation).

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). Thermodynamics and Chemistry (Euler reciprocity relation, pg. 413). Prentice Hall.
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). Reciprocity Laws: From Euler to Eisenstein. Springer.