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. PREFACE CONTENTS Chapter I THE REAL NUMBER SYSTEM 101. Introduction 102. Axioms of a field 103. Exercises 104. Axioms of an ordered field 105. Exercises 106. Positive integers and mathematical induction 107. Exercises 108. Integers and rational numbers 109. Exercises 110. Geometrical representation and absolute value 111. Exercises 112. Axiom of completeness 113. Consequences of completeness 114. Exercises Chapter 2 FUNCTIONS, SEQUENCES, LIMITS, CONTINUITY 201. Functions and sequences 202. Limit of a sequence 203. Exercises 204. Limit theorems for sequences 205. Exercises 206. Limits of functions 207. Limit theorems for functions 208. Exercises 209. Continuity 210. Types of discontinuity 211. Continuity theorems 212. Exercises 213. More theorems on continuous functions 214. Existence of √2 and other roots 215. Monotonic functions and their inverses 216. Exercises *217. A fundamental theorem on bounded sequences *218. Proofs of some theorems on continuous functions *219. The Cauchy criterion for convergence of a sequence *220. Exercises *221. Sequential criteria for continuity and existence of limits *222. The Cauchy criterion for functions *223. Exercises *224. Uniform continuity *225. Exercises Chapter 3 DIFFERENTIATION 301. Introduction 302. The derivative 303. One-sided derivatives 304. Exercises 305. Rolle's theorem and the Law of the Mean 306. Consequences of the Law of the Mean 307. The Extended Law of the Mean 308. Exercises 309. Maxima and minima 310. Exercises 311. Differentials 312. Approximations by differentials 313. Exercises 314. L'Hospital's Rule. Introduction 315. The indeterminate form 0/0 316. The indeterminate form ∞/∞ 317. Other indeterminate forms 318. Exercises 319. Curve tracing 320. Exercises *321. Without loss of generality *322. Exercises Chapter 4 INTEGRATION 401. The definite integral 402. Exercises *403. More integration theorems *404. Exercises 405. The Fundamental Theorem of Integral Calculus 406. Integration by substitution 407. Exercises 408. Sectional continuity and smoothness 409. Exercises 410. Reduction formulas 411. Exercises 412. Improper integrals, introduction 413. Improper integrals, finite interval 414. Improper integrals, infinite interval 415. Comparison tests. Dominance 416. Exercises *417. The Riemann-Stieltjes integral *418. Exercises Chapter 5 SOME ELEMENTARY FUNCTIONS *501. The exponential and logarithmic functions *502. Exercises *503. The trigonometric functions *504. Exercises 505. Some integration formulas 506. Exercises 507. Hyperbolic functions 508. Inverse hyperbolic functions 509. Exercises *510. Classification of numbers and functions *511. The elementary functions *512. Exercises Chapter 6 FUNCTIONS OF SEVERAL VARIABLES 601. Introduction 602. Neighborhoods in the Euclidean plane 603. Point sets in the Euclidean plane 604. Sets in higher-dimensional Euclidean spaces 605. Exercises 606. Functions and limits 607. Iterated limits 608. Continuity 609. Limit and continuity theorems 610. More theorems on continuous functions 611. Exercises 612. More general functions. Mappings *613. Sequences of points *614. Point sets and sequences *615. Compactness and continuity *616. Proofs of two theorems *617. Uniform continuity 618. Exercises Chapter 7 SOLID ANALYTIC GEOMETRY AND VECTORS 701. Introduction 702. Vectors and scalars 703. Addition and subtraction of vectors. Magnitude 704. Linear combinations of vectors 705. Exercises 706. Direction angles and cosines 707. The scalar or inner or dot product 708. Vectors orthogonal to two vectors 709. Exercises 710. Planes 711. Lines 712. Exercises 713. Surfaces. Sections, traces, intercepts 714. Spheres 715. Cylinders 716. Surfaces of revolution 717. Exercises 718. The standard quadric surfaces 719. Exercises Chapter 8 ARCS AND CURVES 801. Duhamel's principle for integrals *802. A proof with continuity hypotheses 803. Arcs and curves 804. Arc length 805. Integral form for arc length *806. Remark concerning the trigonometric functions 807. Exercises 808. Cylindrical and spherical coordinates 809. Arc length in rectangular, cylindrical, and spherical coordinates 810. Exercises 811. Curvature and radius of curvature in two dimensions 812. Circle of curvature *813. Evolutes and involutes 814. Exercises Chapter 9 PARTIAL DIFFERENTIATION 901. Partial derivatives 902. Partial derivatives of higher order *903. Equality of mixed partial derivatives 904. Exercises 905. The fundamental increment formula 906. Differentials 907. Change of variables. The chain rule *908. Homogeneous functions. Euler's theorem 909. Exercises *910. Directional derivatives. Tangents and normals *911. Exercises 912. The Law of the Mean 913. Approximations by differentials 914. Maxima and minima 915. Exercises 916. Differentiation of an implicit function 917. Some notational pitfalls 918. Exercises 919. Envelope of a family of plane curves 920. Exercises 921. Several functions defined implicitly. Jacobians 922. Coordinate transformations. Inverse transformations 923. Functional dependence 924. Exercises 925. Extrema with one constraint. Two variables 926. Extrema with one constraint. More than two variables 927. Extrema with more than one constraint 928. Lagrange multipliers 929. Exercises *930. Differentiation under the integral sign. Leibnitz's rule *931. Exercises *932. The Implicit Function Theorem *933. Existence theorem for inverse transformations *934. Sufficiency conditions for functional dependence *935. Exercises Chapter 10 MULTIPLE INTEGRALS 1001. Introduction 1002. Double integrals 1003. Area 1004. Second formulation of the double integral *1005. Inner and outer area. Criterion for area *1006. Theorems on double integrals *1007. Proof of the second formulation 1008. Iterated integrals, two variables *1009. Proof of the Fundamental Theorem 1010. Exercises 1011. Triple integrals. Volume 1012. Exercises 1013. Double integrals in polar coordinates 1014. Volumes with double integrals in polar coordinates 1015. Exercises 1016. Mass of a plane region of variable density 1017. Moments and centroid of a plane region 1018. Exercises 1019. Triple integrals, cylindrical coordinates 1020. Triple integrals, spherical coordinates 1021. Mass, moments, and centroid of a space region 1022. Exercises 1023. Mass, moments, and centroid of an arc 1024. Attraction 1025. Exercises 1026. Jacobians and transformations of multiple integrals 1027. General discussion 1028. Exercises Chapter 11 INFINITE SERIES OF CONSTANTS 1101. Basic definitions 1102. Three elementary theorems 1103. A necessary condition for convergence 1104. The geometric series 1105. Positive series 1106. The integral test 1107. Exercises 1108. Comparison tests. Dominance 1109. The ratio test 1110. The root test 1111. Exercises *1112. More refined tests *1113. Exercises 1114. Series of arbitrary terms 1115. Alternating series 1116. Absolute and conditional convergence 1117. Exercises 1118. Groupings and rearrangements 1119. Addition, subtraction, and multiplication of series *1120. Some aids to computation 1121. Exercises Chapter 12 POWER SERIES 1201. Interval of convergence 1202. Exercises 1203. Taylor series 1204. Taylor's formula with a remainder 1205. Expansions of functions 1206. Exercises 1207. Some Maclaurin series 1208. Elementary operations with power series 1209. Substitution of power series 1210. Integration and differentiation of power series 1211. Exercises 1212. Indeterminate expressions 1213. Computations 1214. Exercises 1215. Taylor series, several variables 1216. Exercises *Chapter 13 UNIFORM CONVERGENCE AND LIMITS *1301. Uniform convergence of sequences *1302. Uniform convergence of series *1303. Dominance and the Weierstrass M-test *1304. Exercises *1305. Uniform convergence and continuity *1306. Uniform convergence and integration *1307. Uniform convergence and differentiation *1308. Exercises *1309. Power series. Abel's theorem *1310. Proof of Abel's theorem *1311. Exercises *1312. Functions defined by power series. Exercises *1313. Uniform limits of functions *1314. Three theorems on uniform limits *1315. Exercises *Chapter 14 IMPROPER INTEGRALS *1401. Introduction. Review *1402. Alternating integrals. Abel's test *1403. Exercises *1404. Uniform convergence *1405. Dominance and the Weierstrass M-test *1406. The Cauchy criterion and Abel's test for uniform convergence *1407. Three theorems on uniform convergence *1408. Evaluation of improper integrals *1409. Exercises *1410. The gamma function *1411. The beta function *1412. Exercises *1413. Infinite products *1414. Wallis's infinite product for π *1415. Euler's constant *1416. Stirling's formula *1417. Weierstrass's infinite product for 1/Γ(α) *1418. Exercises *1419. Improper multiple integrals *1420. Exercises Chapter 15 COMPLEX VARIABLES 1501. Introduction 1502. Complex numbers 1503. Embedding of the real numbers 1504. The number i 1505. Geometrical representation 1506. Polar form 1507. Conjugates 1508. Roots 1509. Exercises 1510. Limits and continuity 1511. Sequences and series 1512. Exercises 1513. Complex-valued functions of a real variable 1514. Exercises *1515. The Fundamental Theorem of Algebra Chapter 16 FOURIER SERIES 1601. Introduction 1602. Linear function spaces 1603. Periodic functions. The space R_2π 1604. Inner product. Orthogonality. Distance 1605. Least squares. Fourier coefficients 1606. Fourier series 1607. Exercises 1608. A convergence theorem. The space S_2π 1609. Bessel's inequality. Parseval's equation 1610. Cosine series. Sine series 1611. Other intervals 1612. Exercises *1613. Partial sums of Fourier series *1614. Functions with one-sided limits *1615. The Riemann-Lebesgue Theorem *1616. Proof of the convergence theorem *1617. Fejer's summability theorem *1618. Uniform summability *1619. Weierstrass's theorem *1620. Density of trigonometric polynomials *1621. Some consequences of density *1622. Further remarks *1623. Other orthonormal systems *1624. Exercises 1625. Applications of Fourier series. The vibrating string 1626. A heat conduction problem 1627. Exercises Chapter 17 VECTOR ANALYSIS 1701. Introduction 1702. The vector or outer or cross product 1703. The triple scalar product. Orientation in space 1704. The triple vector product 1705. Exercises 1706. Coordinate transformations 1707. Translations 1708. Rotations 1709. Exercises 1710. Scalar and vector fields. Vector functions 1711. Ordinary derivatives of vector functions 1712. The gradient of a scalar field 1713. The divergence and curl of a vector field 1714. Relations among vector operations 1715. Exercises *1716. Independence of the coordinate system *1717. Curvilinear coordinates. Orthogonal coordinates *1718. Vector operations in orthogonal coordinates *1719. Exercises Chapter 18 LINE AND SURFACE INTEGRALS 1801. Introduction 1802. Line integrals in the plane 1803. Independence of path and exact differentials 1804. Exercises 1805. Green's Theorem in the plane 1806. Local exactness 1807. Simply- and multiply-connected regions 1808. Equivalences in simply-connected regions 1809. Exercises *1810. Analytic functions of a complex variable. Exercises 1811. Surface elements 1812. Smooth surfaces 1813. Schwarz's example 1814. Surface area 1815. Exercises 1816. Surface integrals 1817. Orientable smooth surfaces 1818. Surfaces with edges and corners 1819. The divergence theorem 1820. Green's identities 1821. Harmonic functions 1822. Exercises 1823. Orientable sectionally smooth surfaces 1824. Stokes's Theorem 1825. Independence of path. Scalar potential *1826. Vector potential 1827. Exercises *1828. Exterior differential forms. Exercises Chapter 19 DIFFERENTIAL GEOMETRY 1901. Introduction 1902. Curvature. Osculating plane 1903. Applications to kinematics 1904. Torsion. The Frenet formulas 1905. Local behavior 1906. Exercises 1907. Curves on a surface. First fundamental form 1908. Intersections of smooth surfaces 1909. Plane sections. Meusnier's theorem 1910. Normal sections. Mean and total curvature 1911. Second fundamental form 1912. Exercises ANSWERS TO PROBLEMS INDEX