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.