Differentiability 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 ... 

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... 

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... 

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 ... 

When we apply Euler s reciprocity relation to obtain the total differential of entropy by using Equation (176) for a constant interfacial flat boundary area, A, between them, we have... 

As discussed earlier, mixed second derivatives of well-behaved functions are independent of the order of differentiation. This leads to Euler s reciprocity relation... 

Euler s reciprocity relation states that the two mixed second partial derivatives of a differentiable function must be equal to each other ... 

Euler s Reciprocity Relation. A second derivative is the derivative of a first derivative. If / is a differentiable function of two independent variables, x and y, there are four second derivatives ... 

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