Chapter 13 Introduction

This chapter extends the principles of differential calculus for functions of one variable to functions of several variables. We begin in Section 13.1 with the geometry of real-valued functions and study the graphs of these functions as an aid in visualizing them. Section 13.2 gives some basic definitions relating to limits and continuity. This subject is treated briefly, because it requires time and mathematical maturity to develop fully and is therefore best left to a more advanced course. Fortunately, a complete understanding of all the subtleties of the limit concept is not necessary for our purposes; the student who has difficulty with this section should bear this in mind. However, we hasten to add that the notion of a limit is central to the definition of the derivative, but not to the computation of most derivatives in specific problems, as we already know from one-variable calculus. Section 13.3 and Section 13.4 deal with the definition of the derivative, and establish some basic rules of calculus: namely, how to differentiate a sum, product, quotient, or composition. In Section 13.5, we study directional derivatives and tangent planes, relating these ideas to those in Section 13.1.

In generalizing calculus from one dimension to several, it is often convenient to use the language of matrix algebra. What we shall need has been summarized in Section 11.7.