Extrema of Multivariate Functions

The Extrema of Multivariate Functions Occur Where the Partial Derivatives Are Zero 

To use extremum principles, which can identify states of equilibrium, we find points at which derivatives are zero. In mathematics, a critical point is w here a first derivative equals zero. It could be a maximum, a minimum, or a saddle point. In statistical thermodynamics, critical point has a different meaning, but in this chapter critical point is used only in its mathematical sense. 

The critical point x occurs at the value / = fix ) that satisfies Equation (5.8). If the second derivative df jdx- is positive e er where, the critical point is a minimum. If the second derivative is negative everyw-here, the critical point is a maximum. 

Now let s find an extremum of a multivariate function fix,y). A critical point is where small changes in both x and y together cause the total differential df to equal zero  

Since x and y are independent variables, the small non-zero quantities dx and dy are imrelated to each other. This means that we cannot suppose that the dx and dy terms in Equation (5.9) balance perfectly to cause df to equal zero. The independence of the v u-iables x and y requires that both terms (df/dx)dx and (dzldy)dy equal zero simultaneously to satisfy Equation (5.9)  

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Here we review the tools from multivariate calculus that we need to describe processes in which multiple degrees of freedom change together. We need these methods to solve two main problems to find the extrema of multivariate functions, and to integrate them. Here we introduce the mathematics. 

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. 

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