Euler reciprocal relationship

The Maxwell equations derive from the Euler reciprocity relationship ... 

Figure 5.12 For proving the Euler reciprocal relationship, two paths from. 4 to C through B, or through D.

In Chapter 9 w e will use the Euler relationship to establish the Maxwell relations between the thermodynamic quantities. Here we derive the Euler relationship. Figure 5.12 shows four points at the vertices of a rectangle in the xy plane. Using a Taylor series expansion, Equation (4.22), compute the change in a function Af through tw o different routes. First integrate from point A to point B to point C. Then integrate from point A to point D to point C. Compare the results to find the Euler reciprocal relationship. For Af = f(x + Ax,y + Ay) -f(x,y), the hrst terms of the Taylor series are... 

Equations (5.44) and (5.45) both give /(C), so you can set the right-hand sides equal and cancel identical terms. The remaining terms give the Euler reciprocal relationship (Equation (5.39)) ... 

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

We now derive the Maxwell relations. These are relationships between partial derivatives that follow from Euler s reciprocal relation. Equation (5.39) (page 75). 

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