Preface ix
a certain philosophy concerning abstract verses concrete concepts. We briefly in-
troduce abstract metric spaces, inner product spaces, and normed linear spaces,
but only as an aside. We emphasize that Euclidean space is the object of study in
this text, but we do point out now and then when a theorem concerning Euclidean
space does or does not hold in a general metric space or inner product space or
normed vector space. That is, the course is grounded in the concrete world of Rd,
but the student is made aware that there are more exotic worlds in which these
concepts are important.
Chapter 8 is devoted to the study of continuous functions between Euclidean
spaces. We study the basic properties of continuous functions as they relate to open
and closed sets and compact and connected sets. The third section is devoted to
sequences and series of functions and the concept of uniform convergence. The last
two sections comprise a review of the topic of linear functions between Euclidean
spaces and the corresponding matrices. This includes the study of rank, dimension
of image and kernel, and invertible matrices. We also introduce representations of
linear or affine subspaces in parametric form as well as solution sets of systems of
equations.
The most important topic in the second half of the course is probably the
study, in Chapter 9, of the total differential of a function from
Rp
to
Rq.
This is
introduced in the context of affine approximation of a function near a point in its
domain. The Chain Rule for the total differential is proved in what we believe is
a novel and intuitively satisfying way. This is followed by applications of the total
differential and the Chain Rule, including the multivariable Taylor formula and the
inverse and implicit function theorems.
Chapter 10 is devoted to integration over Jordan regions in Rd. The develop-
ment, using upper and lower sums, is very similar to the development of the single
variable integral in Chapter 5. Where the proofs are virtually identical to those
in Chapter 5, they are omitted. The really new and different material here is that
on Fubini’s Theorem and the change of variables formula. We give rigorous and
detailed proofs of both results along with a number of applications.
The chapter on vector calculus, Chapter 11, uses the modern formalism of
differential forms. In this formalism, the major theorems of the subject Green’s
Theorem, Stokes’s Theorem, and Gauss’s Theorem all have the same form. We
do point out the classical forms of each of these theorems, however. Each of the
main theorems is proved first on a rectangle or cube and then extended to more
complicated domains through the use of transformation laws for differential forms
and the change of variables formula for multiple integrals. Most of the chapter
focuses on integration over sets in R,
R2,
or
R3
which can be parameterized by
smooth maps from an interval, a square or a cube, or sets which can be partitioned
into sets of this form. However, in an optional section at the end, we introduce
integrals over p-chains and p-cycles and state the general form of Stokes’s Theorem.
There are topics which could have been included in this text but were not. Some
of our colleagues suggested that we include an introductory chapter or section on
formal logic. We considered this but decided against it. Our feeling is that logic
at this simple level is just language used with precision. Students have been using
language for most of their lives, perhaps not always with precision, but that doesn’t
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