The Euler Reciprocity Relation

In addition, four Maxwell equations result from application of the Euler reciprocity relation (Equation (101) was derived for internal energy, U) and the exact differentials rule... 

The Euler reciprocity relation is an identity relating the mixed second partial derivatives. If z = z(x, y) is a differentiable function, then the two different mixed second partial derivatives must equal each other ... 

An important set of identities obtained from the Euler reciprocity relation and thermodynamic equations is the set of Maxwell relations. These relations allow you to replace a partial derivative that is difficult or impossible to measure with one that can be measured. One of the Maxwell relations is ... 

In order to tell whether some differential form is an exact differential or not, we can use the Euler reciprocity relation. If there exists a function u = u x,y) such that... 

Differential forms with three or more terms can also either be exact or inexact. The Euler reciprocity relation provides a test for such differentials. For example, if... 

Find the partial derivatives and write the expression for dU using T, V, and n as independent variables. Show that the partial derivatives obey the Euler reciprocity relations in Eqs. (7.33-7.35). [5]... 

Some functions depend on more than a single variable. To find extrema of such functions, it is necessary to find where all the partial derivatives are zero. To find extrema of multivariate functions that are subject to constraints, the Lagrange multiplier method is useful. Integrating multivariate functions is different from integrating single-variable functions multivariate functions require the concept of a pathway. State functions do not depend on the pathway of integration. The Euler reciprocal relation relation can be used to distinguish state functions from path-dependent functions. In the next three chapters, we will combine the First and Second Laws with multivariate calculus to derive the principles of thermodynamics. 

Some such differential forms are exact, which means that they are differentials of functions, as explained in Chapter 8. Other differentials are inexact, which means that they are not differentials of any function. If the differential is exact, the equation is called an exact differential equation. The test for exactness is based on the Euler reciprocity relation, as inEq. (8.29) If... 

From the Euler reciprocity relation shown in Eq. (B-13) of Appendix B, we can write... 

Using the Euler reciprocity relation, we obtain a second Maxwell relation ... 

We refer to the second partial derivatives in Eqs. (B-12a) and (B-12b) as mixed second partial derivatives. The Euler reciprocity relation is a theorem of mathematics If / is differentiable, then the two mixed second partial derivatives in Eq. (B-12a) and (B-12b) are the same function ... 

To test the differential du for exactness, we can see if the appropriate derivatives of L, M, and N are mixed second derivatives of the same function and obey the Euler reciprocity relation ... 

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