Ebook: Advanced Calculus

This book is a basic text in advanced calculus, providing a clear and well motivated,
yet precise and rigorous, treatment of the essential tools of
mathematical analysis at a level immediately following that of a first course
in calculus. It is designed to satisfy many needs; it fills gaps that almost
always, and properly, occur in elementary calculus courses; it contains all
of the material in the standard classical advanced calculus course; and it
provides a solid foundation in the "deltas and epsilons" of a modern rigorous
advanced calculus. It is well suited for courses of considerable diversity,
ranging from "foundations of calculus" to "critical reasoning in mathematical
analysis." There is even ample material for a course having a standard
advanced course as prerequisite.
Throughout the book attention is paid to the average or less-than-average
student as well as to the superior student. This is done at every stage of
progress by making maximally available whatever concepts and discussion
are both relevant and understandable. To illustrate: limit and continuity
theorems whose proofs are difficult are discussed and worked with before
they are proved, implicit functions are treated before their existence is
established, and standard power series techniques are developed before the
topic of uniform convergence is studied. Whenever feasible, if both an
elementary and a sophisticated proof of a theorem are possible, the elementary
proof is given in the text, with the sophisticated proof possibly called for in
an exercise, with hints. Generally speaking, the more subtle and advanced
portions of the book are marked with stars ( *), prerequisite for which is
preceding starred material. This contributes to an unusual flexibility of the
book as a text.
The author believes that most students can best appreciate the more difficult
and advanced aspects of any field of study if they have thoroughly mastered
the relatively easy and introductory parts first. In keeping with this philosophy,
the book is arranged so that progress moves from the simple to the
complex and from the particular to the general. Emphasis is on the concrete,
with abstract concepts introduced only as they are relevant, although the
general spirit is modern. The Riemann integral, for example, is studied first
with emphasis on relatively direct consequences of basic definitions, and
then with more difficult results obtained with the aid of step functions.
Later some of these ideas are extended to multiple integrals and to the
Riemann-Stieltjes integral. Improper integrals are treated at two levels of
sophistication; in Chapter 4 the principal ideas are dominance and the
"big 0" and "little o" concepts, while in Chapter 14 uniform convergence
becomes central, with applications to such topics as evaluations and the
gamma and beta functions.
Vectors are presented in such a way that a teacher using this book may
almost completely avoid the vector parts of advanced calculus if he wishes
to emphasize the "real variables" content. This is done by restricting the
use of vectors in the main part of the book to the scalar, or dot, product,
with applications to such topics as solid analytic geometry, partial differentiation,
and Fourier series. The vector, or cross, product and the differential
and integral calculus of vectors are fully developed and exploited in the last
three chapters on vector analysis, line and surface integrals, and differential
geometry. The now-standard Gibbs notation is used. Vectors are designated
by means of arrows, rather than bold-face type, to conform with
handwriting custom.
Special attention should be called to the abundant sets of problems-there
are over 2440 exercises! These include routine drills for practice, intermediate
exercises that extend the material of the text while retaining its
character, and advanced exercises that go beyond the standard textual subject
matter. Whenever guidance seems desirable, generous hints are included.
In this manner the student is led to such items of interest as limits superior
and inferior, for both sequences and real-valued functions in general, the
construction of a continuous nondifferentiable function, the elementary
theory of analytic functions of a complex variable, and exterior differential
forms. Analytic treatment of the logarithmic, exponential, and trigonometric
functions is presented in the exercises, where sufficient hints are given to
make these topics available to all. Answers to all problems are given in the
back of the book. Illustrative examples abound throughout.
Standard Aristotelian logic is assumed; for example, frequent use is made
of the indirect method of proof. An implication of the form p implies q is
taken to mean that it is impossible for p to be true and q to be false simultaneously;
in other words, that the conjunction of the two statements p
and not q leads to a contradiction. Any statement of equality means simply
that the two objects that are on opposite sides of the equal sign are the same
thing. Thus such statements as "equals may be added to equals," and "two
things equal to the same thing are equal to each other," are true by definition.
A few words regarding notation should be given. The equal sign == is used
for equations, both conditional and identical, and the triple bar - is reserved
for definitions. For simplicity, if the meaning is clear from the context,
neither symbol is restricted to the indicative mood as in "(a + b )2 ==
a2 + 2ab + b2," or "where f(x) - x2 + 5." Examples of subjunctive uses
are "let x == n," and "let e - 1," which would be read "let x be equal ton,"
and "let e be defined to be 1," respectively. A similar freedom is granted
the inequality symbols. For instance, the symbol > in the following constructions
"if e > 0, then · · · ," "let e > 0," and "let e > 0 be given,"
could be translated "is greater than," "be greater than," and "greater than,"
respectively. A relaxed attitude is also adopted regarding functional
notation, and the tradition (y == f(x)) established by Dirichlet has been
followed. When there can be no reasonable misinterpretation the notation
f(x) is used both to indicate the value of the function f corresponding to a
particular value x of the independent variable and also to represent the
function f itself (and similarly for f(x, y), f(x, y, z), and the like). This
permissiveness has two merits. In the first place it indicates in a simple way
the number of independent variables and the letters representing them. In
the second place it avoids such elaborate constructions as "the function f
defined by the equation f(x) == sin 2x is periodic," by permitting simply,
"sin 2x is periodic." This practice is in the spirit of such statements as "the
line x + y == 2 · · · ," instead of "the line that is the graph of the equation
x + y == 2 · · ·," and "this is John Smith," instead of "this is a man whose
name is John Smith."
In a few places parentheses are used to indicate alternatives. The principal
instances of such uses are heralded by announcements or footnotes in the
text. Here again it is hoped that the context will prevent any ambiguity.
Such a sentence as "The function j"(x) is integrable from a to b (a < b)"
would mean that ''f(x) is integrable from a to b, where it is assumed that
a < b," whereas a sentence like "A function having a positive (negative)
derivative over an interval is strictly increasing (decreasing) there" is a
compression of two statements into one, the parentheses indicating an
alternative formulation.
Although this text is almost completely self-contained, it is impossible
within the compass of a book of this size to pursue every topic to the extent
that might be desired by every reader. Numerous references to other books
are inserted to aid the intellectually ambitious and curious. Since many of
these references are to the author's Real Variables (abbreviated here to RV),
of this same Appleton-Century Mathematics Series, and since the present
Advanced Calculus (AC for short) and RV have a very substantial body of
common material, the reader or potential user of either book is entitled to
at least a short explanation of the differences in their objectives. In brief,
A C is designed principally for fairly standard advanced calculus courses, of
either the "vector analysis" or the "rigorous" type, while RV is designed
principally for courses in introductory real variables at either the advanced
calculus or the post-advanced calculus level. Topics that are in both AC and
RV include all those of the basic "rigorous advanced calculus." Topics that
viii PREFACE
are in AC but not in RV include solid analytic geometry, vector analysis,
complex variables, extensive treatment of line and surface integrals, and
differential geometry. Topics that are in RVbut not in ACinclude a thorough
treatment of certain properties of the real numbers, dominated convergence
and measure zero as related to the Riemann integral, bounded variation as
related to the Riemann-Stieltjes integral and to arc length, space-filling arcs,
independence of parametrization for simple arc length, the Moore-Osgood
uniform convergence theorem, metric and topological spaces, a rigorous
proof of the transformation theorem for multiple integrals, certain theorems
on improper integrals, the Gibbs phenomenon, closed and complete orthonormal
systems of functions, and the Gram-Schmidt process.
One note of caution is in order. Because of the rich abundance of material
available, complete coverage in one year is difficult. Most of the unstarred
sections can be completed in a year's sequence, but many teachers will wish
to sacrifice some of the later unstarred portions in order to include some of
the earlier starred items. Anybody using the book as a text should be advised
to give some advance thought to the main emphasis he wishes to give his
course and to the selection of material suitable to that emphasis.