Graphics Reference
In-Depth Information
CHAPTER 4
Advanced Calculus Topics
4.1
Introduction
The object of this chapter is to introduce basic topological concepts as they apply to
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n
and to cover some important topics in advanced calculus. The reader is assumed
to have had the basic three-semester sequence of calculus and it is not our intent to
redo that material here. Our emphasis will be on multivariable functions and their
properties, the assumption being that the reader has a reasonable understanding of
functions of a single variable. Proofs are given in those cases where it was thought
to be helpful in understanding some new ideas or if they involved some geometric
insights.
Section 4.2 introduces the topological concepts. We limit the discussion to those
that are specifically needed for advanced calculus and leave the more general study
of topology to Chapter 5. Section 4.3 describes the derivative of vector-valued func-
tions of several variables and related results that generalize well-known properties of
real-valued functions of a single variable. The inverse function theorem and the
implicit function theorem are discussed in Section 4.4. These are such important the-
orems and get used so often that we give a fairly detailed outline of their proofs. Many
results in differential topology and algebraic geometry would be impossible without
them. Next, Section 4.5 develops the basic results regarding critical points of func-
tions and this leads to Morse theory, which is described in Section 4.6. The problem
of finding zeros of functions is addressed in Section 4.7. Section 4.8 reviews basic
facts about integrating functions of several variables. Finally, in Sections 4.9 and 4.9.1,
we start a brief overview of the topic of differential forms and their integrals that will
be continued in Section 8.12.
4.2
The Topology of Euclidean Space
Topology is the study of the most basic properties of point sets such as, what is meant
by a neighborhood of a point, what open and closed sets are, and what makes a func-
