A summary of the "Euler reciprocity relation", a synonym for what Rudolf Clausius calls the "condition for an exact differential", which would seem to indicate that this logic was originated by Swiss mathematician Leonhard Euler. [5]
In
mathematical thermodynamics, the condition for an exact differential is that given some function u of two or more variables, such as:

du = P dx + Q dy

the expression on the right side of the equation Pdx + Qdy is an exact differential only when P and Q satisfy the following condition or criterion:

$\left( \frac{\partial P}{\partial y} \right)_{x} = \left( \frac{\partial Q}{\partial x} \right)_{y}$

meaning that the partial derivative of the function P with respect to y at constant x equals the partial derivative of the function P with respect to x at constant y.

Examples
The expression ydx + xdy is an exact differential, because it is the differential of xy; whereas, 2ydx + xdy is not an exact differential, because it is not the differential of any function of x and y.

Proof
To prove or derive this condition, we first start with the function of two variables:

du = P dx + Q dy

and note that the definition of 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:

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

hence, by comparison of these two expressions, we find that:

 $P = \left( \frac{\partial u}{\partial x} \right)_{y}$ and $Q = \left( \frac{\partial u}{\partial y} \right)_{x}$

Differentiating the first of these with respect to y, and the second with respect to x, we have:

 $\left( \frac{\partial P}{\partial y} \right)_{x} = \left( \frac{\partial^2 u}{\partial y \partial x} \right)$ and $\left( \frac{\partial Q}{\partial x} \right)_{y} = \left( \frac{\partial^2 u}{\partial x \partial y} \right)$

Whence, by the calculus principle that the order of differentiation is immaterial, we have:

$\left( \frac{\partial^2 u}{\partial x \partial y} \right) = \left( \frac{\partial^2 u}{\partial y \partial x} \right)$

whereby it is proved that:

$\left( \frac{\partial P}{\partial y} \right)_{x} = \left( \frac{\partial Q}{\partial x} \right)_{y}$

and that this is called the "condition for an exact differential".

Maxwell's equations
See main: Maxwell's relations
The application of the condition for an exact differential for functions of two variables that are already known to exist or be actual functions, such as internal energy U, enthalpy H, Helmholtz energy A, and Gibbs energy G, gives the quick derivation of what are called "Maxwell's equations". [2] In short, applying the results of the condition for an exact differential to the four exact differentials dU, dH, dA, and dG, gives the following relations as tabulated below to the right:

 Potential function Relation dU = T dS – P dV hence $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V \qquad$ dH = T dS + V dP hence $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P \qquad$ dA = –P dV – S dT hence $\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T$ dG = dH – S dT hence $\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T$

These relations are sometimes also called Maxwell relations. [3] It is not necessary, according to American physicist Mark Zemansky, to memorize these relations since they are so easily derived. The Maxwell equations do not refer to a process but merely express relations that hold at any equilibrium state of a chemical system. [2] The great interest of Maxwell's equations according French thermodynamicst Pierre Perrot, is that they lead to the partial derivatives of entropy as a function of physical quantities directly available by experiment. [4]

References
1. Osborne, George A. (1891). An Elementary Treatise on Differential and Integral Calculus (sections 62-63: Principle that the Order of Differentiation is Immaterial, pgs. 81-82; section 65: Total Differential of a Function of Several Independent Variables, pgs. 83-84; section 66: Condition for an Exact Differential, pgs. 85-86). Leach, Shewell, and Sanborn.
2. Zemansky, Mark W. (1943). Heat and Thermodynamics - an Intermediate Textbook for Students of Physics, Chemistry, and Engineering (exact differential, pg. 27, condition for an exact differential, pg. 220). McGraw-Hill Book Co., Inc.
3. Maxwell relations - Wikipedia.
4. Perrot, Pierre. (1998). A to Z of Thermodynamics (pg. 195-97). New York: Oxford University Press.
5. The Terrible Beauty of Thermodynamics (2010) – ScribD.com.