Euler criterion

It is easy to recognize that the Euler criterion will be satisfied if the integral or graphical criteria (1.16) are satisfied. Suppose that z(x, y) indeed exists (e.g., displayed as a graph), so that (1.10) is assured. Comparison of (1.10) with the assumed form of the differential then shows that... 

The Euler criterion is therefore equivalent to the familiar mixed partials of a function are equal rule of calculus. This cross-differentiation rule is also the condition for the function z(x, y) to have well-defined (single-valued) first derivatives at each point, and thus to be graphable. 

Exercises Use the Euler criterion (1.17) to determine whether each of the following differentials dz is exact or inexact ... 

For arbitrary n, the more general statement of the Euler criterion can be formulated in terms of a general n-term differential form... 

The first law establishes that (8.71) is an exact differential, satisfying the Euler criterion [cf. (1.21)]... 

This result, sometimes referred to as the Euler criterion, or the reciprocity relationship, will now be used to derive some thermodynamic expressions. 

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