ABSTRACT
This chapter reviews the tools from multivariate calculus that we need to describe processes in which multiple degrees of freedom change together. An analyst need these methods to solve two main problems: to find the extrema of multivariate functions and to integrate them. In mathematics, a critical point is where a first derivative equals zero. It could be a maximum, a minimum, or a saddle point. In the chapter critical point is used only in its mathematical sense. Constraints hold material systems at constant temperature or pressure, or at constant concentrations of chemical species. State functions do not depend on the pathway of integration. The Euler reciprocal relation can be used to distinguish state functions from path-dependent functions. 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.
