Description:

This book is a basic text in advanced calculus, providing a clear and well motivated,yet precise and rigorous, treatment of the essential tools ofmathematical analysis at a level immediately following that of a first coursein calculus. It is designed to satisfy many needs; it fills gaps that almostalways, and properly, occur in elementary calculus courses; it contains allof the material in the standard classical advanced calculus course; and itprovides a solid foundation in the "deltas and epsilons" of a modern rigorousadvanced calculus. It is well suited for courses of considerable diversity,ranging from "foundations of calculus" to "critical reasoning in mathematicalanalysis." There is even ample material for a course having a standardadvanced course as prerequisite.Throughout the book attention is paid to the average or less-than-averagestudent as well as to the superior student. This is done at every stage ofprogress by making maximally available whatever concepts and discussionare both relevant and understandable. To illustrate: limit and continuitytheorems whose proofs are difficult are discussed and worked with beforethey are proved, implicit functions are treated before their existence isestablished, and standard power series techniques are developed before thetopic of uniform convergence is studied. Whenever feasible, if both anelementary and a sophisticated proof of a theorem are possible, the elementaryproof is given in the text, with the sophisticated proof possibly called for inan exercise, with hints. Generally speaking, the more subtle and advancedportions of the book are marked with stars ( *), prerequisite for which ispreceding starred material. This contributes to an unusual flexibility of thebook as a text.The author believes that most students can best appreciate the more difficultand advanced aspects of any field of study if they have thoroughly masteredthe relatively easy and introductory parts first. In keeping with this philosophy,the book is arranged so that progress moves from the simple to thecomplex and from the particular to the general. Emphasis is on the concrete,with abstract concepts introduced only as they are relevant, although thegeneral spirit is modern. The Riemann integral, for example, is studied firstwith emphasis on relatively direct consequences of basic definitions, andthen with more difficult results obtained with the aid of step functions.Later some of these ideas are extended to multiple integrals and to theRiemann-Stieltjes integral. Improper integrals are treated at two levels ofsophistication; in Chapter 4 the principal ideas are dominance and the"big 0" and "little o" concepts, while in Chapter 14 uniform convergencebecomes central, with applications to such topics as evaluations and thegamma and beta functions.Vectors are presented in such a way that a teacher using this book mayalmost completely avoid the vector parts of advanced calculus if he wishesto emphasize the "real variables" content. This is done by restricting theuse 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 differentialand integral calculus of vectors are fully developed and exploited in the lastthree chapters on vector analysis, line and surface integrals, and differentialgeometry. The now-standard Gibbs notation is used. Vectors are designatedby means of arrows, rather than bold-face type, to conform withhandwriting custom.Special attention should be called to the abundant sets of problems-thereare over 2440 exercises! These include routine drills for practice, intermediateexercises that extend the material of the text while retaining itscharacter, and advanced exercises that go beyond the standard textual subjectmatter. Whenever guidance seems desirable, generous hints are included.In this manner the student is led to such items of interest as limits superiorand inferior, for both sequences and real-valued functions in general, theconstruction of a continuous nondifferentiable function, the elementarytheory of analytic functions of a complex variable, and exterior differentialforms. Analytic treatment of the logarithmic, exponential, and trigonometricfunctions is presented in the exercises, where sufficient hints are given tomake these topics available to all. Answers to all problems are given in theback of the book. Illustrative examples abound throughout.Standard Aristotelian logic is assumed; for example, frequent use is madeof the indirect method of proof. An implication of the form p implies q istaken 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 pand not q leads to a contradiction. Any statement of equality means simplythat the two objects that are on opposite sides of the equal sign are the samething. Thus such statements as "equals may be added to equals," and "twothings 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 usedfor equations, both conditional and identical, and the triple bar - is reservedfor 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 usesare "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 grantedthe 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 functionalnotation, and the tradition (y == f(x)) established by Dirichlet has beenfollowed. When there can be no reasonable misinterpretation the notationf(x) is used both to indicate the value of the function f corresponding to aparticular value x of the independent variable and also to represent thefunction f itself (and similarly for f(x, y), f(x, y, z), and the like). Thispermissiveness has two merits. In the first place it indicates in a simple waythe number of independent variables and the letters representing them. Inthe second place it avoids such elaborate constructions as "the function fdefined 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 "theline x + y == 2 · · · ," instead of "the line that is the graph of the equationx + y == 2 · · ·," and "this is John Smith," instead of "this is a man whosename is John Smith."In a few places parentheses are used to indicate alternatives. The principalinstances of such uses are heralded by announcements or footnotes in thetext. 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 thata < b," whereas a sentence like "A function having a positive (negative)derivative over an interval is strictly increasing (decreasing) there" is acompression of two statements into one, the parentheses indicating analternative formulation.Although this text is almost completely self-contained, it is impossiblewithin the compass of a book of this size to pursue every topic to the extentthat might be desired by every reader. Numerous references to other booksare inserted to aid the intellectually ambitious and curious. Since many ofthese references are to the author's Real Variables (abbreviated here to RV),of this same Appleton-Century Mathematics Series, and since the presentAdvanced Calculus (AC for short) and RV have a very substantial body ofcommon material, the reader or potential user of either book is entitled toat least a short explanation of the differences in their objectives. In brief,A C is designed principally for fairly standard advanced calculus courses, ofeither the "vector analysis" or the "rigorous" type, while RV is designedprincipally for courses in introductory real variables at either the advancedcalculus or the post-advanced calculus level. Topics that are in both AC andRV include all those of the basic "rigorous advanced calculus." Topics thatviii PREFACEare in AC but not in RV include solid analytic geometry, vector analysis,complex variables, extensive treatment of line and surface integrals, anddifferential geometry. Topics that are in RVbut not in ACinclude a thoroughtreatment of certain properties of the real numbers, dominated convergenceand measure zero as related to the Riemann integral, bounded variation asrelated to the Riemann-Stieltjes integral and to arc length, space-filling arcs,independence of parametrization for simple arc length, the Moore-Osgooduniform convergence theorem, metric and topological spaces, a rigorousproof of the transformation theorem for multiple integrals, certain theoremson improper integrals, the Gibbs phenomenon, closed and complete orthonormalsystems of functions, and the Gram-Schmidt process.One note of caution is in order. Because of the rich abundance of materialavailable, complete coverage in one year is difficult. Most of the unstarredsections can be completed in a year's sequence, but many teachers will wishto sacrifice some of the later unstarred portions in order to include some ofthe earlier starred items. Anybody using the book as a text should be advisedto give some advance thought to the main emphasis he wishes to give hiscourse and to the selection of material suitable to that emphasis.