Ebook: Functions of Several Variables

This book is an exposition of selected topics from the calculus of functions
of several variables. It is intended for undergraduate mathematics stu-
dents in the third or fourth year analysis program, who have had several
semesters of the calculus and at least an introduction to linear algebra.
Specifically, the prerequisites include knowledge of the real numbers
and functions of one variable plus some introductory experience with
multivariate calculus of the type that is usually included in the first two
years of college mathematics. The linear algebra needed, which is approxi-
mately the content of Linear Algebra by Ross A. Beaumont, includes the
concept of a finite dimensional vector space, some experience with the
idea of a basis for a vector space, and some elementary concepts and
properties associated with linear transformations, such as those of rank
and determinants. Aside from the fact that the fundamental existence and
uniqueness theorem for ordinary differential equations is used without
proof, the results used are proved in the body of the text.
The topics treated in this book were selected with two primary objec-
tives: (1) these topics cover the notions usually referred to as "vector
analysis," and (2) they cover concepts that can be easily generalized to
differentiable manifolds in a relatively coordinate-free manner.
The book divides naturally into three sections. The first two chapters
are rather standard, treating respectively the point set topology of Rn
and differentiation on Rn. In the second chapter the inverse function
theorem and the theorem on change of variables in multiple integrals are
proved and several important implications are discussed in detail. The
latter include the concept of local coordinates and the rank of a differentia-
ble map from Rm to Rn. Basically preparatory, these two chapters con-
stitute the theoretical foundations of the material developed in the re-
mainder of the book.
Chapters Three, Four, and Five constitute the next unit. They are
basically manipulative. In Chapter Three the notion of a (tangent)
vector at p E Rn and the dual notion of covectors at p are developed. With
this introduction, Chapter Four is devoted to exposition of the multilinear
algebra necessary to construct and verify the properties of exterior
multiplication. This chapter actually includes a little more than is needed,
however, since the exterior product is constructed by antisymmetrization
of multilinear forms rather than by the somewhat more elementary
method of giving a multiplication table with respect to a specific basis and
showing that the resulting properties of the product imply uniqueness.
Chapter Five treats differential forms on Rn, k-chains, Stokes theorem,
and some related integral expressions involving the metric, such as
Green's identities and Poisson's integral formula for harmonic functions.
Chapter Six treats the concept of a flow with velocity field X and the
related derivations on vector fields and differential forms. It includes
Frobenius' theorem on completely integrable systems of first-order partial
differential equations and Poincare's lemma that a closed differential form
is locally exact.
Chapter Seven shows how the notation and ideas developed earlier
can be used in the theory of functions of a complex variable. After a dis-
cussion of terminology and of the concept of an analytic coordinate system
in the first two sections, the remainder of the chapter is devoted to develop-
ing some of the standard material centering around Cauchy's integral
formula and power series expansions. The nature of these last two chapters
is again somewhat more theoretical than manipulative.