Introduction to Functions of Several Real Variables

This chapter is concerned with a number of topics in multivariable calculus. The basic notions of limits and continuity for a function of several variables are recalled. Then, the notions of partial derivatives, differentiability and differentiability of composite functions are recalled. Section 2.4 introduces the classical notion of directional derivative, which is important for further developments in the so-called “nonsmooth analysis”. Section 2.5 is concerned with the notion of homogeneous functions, particularly important in production theory, such as the economic properties of the so-called “Cobb-Douglas function”. Section 2.6 is concerned with implicit functions, important not only in constrained optimization problems with equality constraints, but also in several economic questions, such as, for instance, the “comparative statics problems” considered by P. A. Samuelson. The final Sect. 2.7 is concerned with the classical Taylor’s formula for functions of several variables and with the mean value theorem.