Theory and Problems of ADVANCED CALCULUS Second Edition ROBERT WREDE, Ph.D. MURRAY R. SPIEGEL, Ph.D. Former Professor and Chairman of Mathematics Rensselaer Polytechnic Institute Hartford Graduate Center Schaum’s Outline Series McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright ©2002, 1963 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-139834-1 The material in this eBook also appears in the print version of this title: 0-07-137567-8 All trademarks are trademarks of their respective owners. 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DOI: 10.1036/0071398341 Akeyingredientinlearningmathematicsisproblemsolving.Thisisthestrength,andnodoubt the reason for the longevity of Professor Spiegel’s advanced calculus. His collection of solved and unsolved problems remains a part of this second edition. Advanced calculus is not a single theory. However, the various sub-theories, including vector analysis, infinite series, and special functions, have in common a dependency on the fundamentalnotionsofthecalculus.Animportantobjectiveofthissecondeditionhasbeento modernize terminology andconcepts, so that the interrelationships becomeclearer. For exam- ple, in keeping with present usage fuctions of a real variable are automatically single valued; differentials are defined as linear functions, and the universal character of vector notation and theory are given greater emphasis. Further explanations have been included and, on occasion, the appropriate terminology to support them. The order of chapters is modestly rearranged to provide what may be a more logical structure. A brief introduction is provided for most chapters. Occasionally, a historical note is included; however, for the most part the purpose of the introductions is to orient the reader to the content of the chapters. IthankthestaffofMcGraw-Hill.Formereditor,GlennMott,suggestedthatItakeonthe project. Peter McCurdy guided me in the process. Barbara Gilson, Jennifer Chong, and Elizabeth Shannon made valuable contributions to the finished product. Joanne Slike and Maureen Walker accomplished the very difficult task of combining the old with the new and, in the process, corrected my errors. The reviewer, Glenn Ledder, was especially helpful in the choice of material and with comments on various topics. ROBERT C. WREDE iii Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. This page intentionally left blank. For more information about this title, click here. CHAPTER 1 NUMBERS 1 Sets.Realnumbers.Decimalrepresentationofrealnumbers.Geometric representationofrealnumbers.Operationswithrealnumbers.Inequal- ities.Absolutevalueofrealnumbers.Exponentsandroots.Logarithms. Axiomatic foundations of the real number system. Point sets, intervals. Countability. Neighborhoods. Limit points. Bounds. Bolzano- Weierstrass theorem. Algebraic and transcendental numbers. The com- plex number system. Polar form of complex numbers. Mathematical induction. CHAPTER 2 SEQUENCES 23 Definition of a sequence. Limit of a sequence. Theorems on limits of sequences.Infinity.Bounded,monotonicsequences.Leastupperbound and greatest lower bound of a sequence. Limit superior, limit inferior. Nested intervals. Cauchy’s convergence criterion. Infinite series. CHAPTER 3 FUNCTIONS, LIMITS, AND CONTINUITY 39 Functions. Graph of a function. Bounded functions. Montonic func- tions. Inverse functions. Principal values. Maxima and minima. Types of functions. Transcendental functions. Limits of functions. Right- and left-handlimits.Theoremsonlimits.Infinity.Speciallimits.Continuity. Right-andleft-handcontinuity.Continuityinaninterval.Theoremson continuity. Piecewise continuity. Uniform continuity. CHAPTER 4 DERIVATIVES 65 Theconceptanddefinitionofaderivative.Right-andleft-handderiva- tives. Differentiability in an interval. Piecewise differentiability. Differ- entials. The differentiation of composite functions. Implicit differentiation.Rulesfordifferentiation.Derivativesofelementaryfunc- tions. Higher order derivatives. Mean value theorems. L’Hospital’s rules. Applications. v Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. vi CONTENTS CHAPTER 5 INTEGRALS 90 Introductionofthedefiniteintegral.Measurezero.Propertiesofdefinite integrals. Mean value theorems for integrals. Connecting integral and differentialcalculus.Thefundamentaltheoremofthecalculus.General- ization of the limits of integration. Change of variable of integration. Integrals of elementary functions. Special methods of integration. Improperintegrals.Numericalmethodsforevaluatingdefiniteintegrals. Applications. Arc length. Area. Volumes of revolution. CHAPTER 6 PARTIAL DERIVATIVES 116 Functions of two or more variables. Three-dimensional rectangular coordinate systems. Neighborhoods. Regions. Limits. Iterated limits. Continuity. Uniform continuity. Partial derivatives. Higher order par- tial derivatives. Differentials. Theorems on differentials. Differentiation of composite functions. Euler’s theorem on homogeneous functions. Implicit functions. Jacobians. Partial derivatives using Jacobians. The- orems on Jacobians. Transformation. Curvilinear coordinates. Mean value theorems. CHAPTER 7 VECTORS 150 Vectors. Geometric properties. Algebraic properties of vectors. Linear independence and linear dependence of a set of vectors. Unit vectors. Rectangular(orthogonalunit)vectors.Componentsofavector.Dotor scalar product. Cross or vector product. Triple products. Axiomatic approach to vector analysis. Vector functions. Limits, continuity, and derivatives of vector functions. Geometric interpretation of a vector derivative. Gradient, divergence, and curl. Formulas involving r. Vec- tor interpretation of Jacobians, Orthogonal curvilinear coordinates. Gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates. Special curvilinear coordinates. CHAPTER 8 APPLICATIONS OF PARTIAL DERIVATIVES 183 Applicationstogeometry.Directionalderivatives.Differentiationunder the integral sign. Integration under the integral sign. Maxima and minima. Method of Lagrange multipliers for maxima and minima. Applications to errors. CHAPTER 9 MULTIPLE INTEGRALS 207 Double integrals. Iterated integrals. Triple integrals. Transformations of multiple integrals. The differential element of area in polar coordinates, differential elements of area in cylindrical and spherical coordinates. CONTENTS vii CHAPTER 10 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 229 Line integrals. Evaluation of line integrals for plane curves. Properties oflineintegralsexpressedforplanecurves.Simpleclosedcurves,simply and multiply connected regions. Green’s theorem in the plane. Condi- tionsforalineintegraltobeindependentofthepath.Surfaceintegrals. The divergence theorem. Stoke’s theorem. CHAPTER 11 INFINITE SERIES 265 Definitionsofinfiniteseriesandtheirconvergenceanddivergence.Fun- damental facts concerning infinite series. Special series. Tests for con- vergence and divergence of series of constants. Theorems on absolutely convergent series. Infinite sequences and series of functions, uniform convergence. Special tests for uniform convergence of series. Theorems onuniformlyconvergentseries.Powerseries.Theoremsonpowerseries. Operations with power series. Expansion of functions in power series. Taylor’stheorem.Someimportantpowerseries.Specialtopics.Taylor’s theorem (for two variables). CHAPTER 12 IMPROPER INTEGRALS 306 Definition of an improper integral. Improper integrals of the first kind (unbounded intervals). Convergence or divergence of improper integrals of the first kind. Special improper integers of the first kind. Convergence tests for improper integrals of the first kind. Improper integrals of the second kind. Cauchy principal value. Special improper integrals of the second kind. Convergence tests for improper integrals of the second kind. Improper integrals of the third kind. Improper integrals containing a parameter, uniform convergence. Special tests for uniform convergence of integrals. Theorems on uniformly conver- gent integrals. Evaluation of definite integrals. Laplace transforms. Linearity. Convergence. Application. Improper multiple integrals. CHAPTER 13 FOURIER SERIES 336 Periodicfunctions.Fourierseries.Orthogonalityconditionsforthesine and cosine functions. Dirichlet conditions. Odd and even functions. HalfrangeFouriersineorcosineseries.Parseval’sidentity.Differentia- tion and integration of Fourier series. Complex notation for Fourier series. Boundary-value problems. Orthogonal functions. viii CONTENTS CHAPTER 14 FOURIER INTEGRALS 363 The Fourier integral. Equivalent forms of Fourier’s integral theorem. Fourier transforms. CHAPTER 15 GAMMA AND BETA FUNCTIONS 375 Thegammafunction.Tableofvaluesandgraphofthegammafunction. The beta function. Dirichlet integrals. CHAPTER 16 FUNCTIONS OF A COMPLEX VARIABLE 392 Functions. Limits and continuity. Derivatives. Cauchy-Riemann equa- tions.Integrals.Cauchy’stheorem.Cauchy’sintegralformulas.Taylor’s series. Singular points. Poles. Laurent’s series. Branches and branch points. Residues. Residue theorem. Evaluation of definite integrals. INDEX 425 Numbers Mathematicshasitsownlanguagewithnumbersasthealphabet. Thelanguageisgivenstructure withtheaidofconnectivesymbols,rulesofoperation,andarigorousmodeofthought(logic). These concepts,whichpreviouslywereexploredinelementarymathematicscoursessuchasgeometry,algebra, and calculus, are reviewed in the following paragraphs. SETS Fundamental in mathematics is the concept of a set, class, or collection of objects having specified characteristics. For example, we speak of the set of all university professors, the set of all letters A;B;C;D;...;Z of the English alphabet, and so on. The individual objects of the set are called members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of A;B;C;D;...;Z. The set consisting of no elements is called the empty set or null set. REAL NUMBERS The following types of numbers are already familiar to the student: 1. Natural numbers 1;2;3;4;...; also called positive integers, are used in counting members of a set. Thesymbolsvariedwiththetimes,e.g.,theRomansusedI,II,III,IV, . . . Thesumaþb and product a(cid:1)b or ab of any two natural numbers a and b is also a natural number. This is often expressed by saying that the set of natural numbers is closed under the operations of addition and multiplication, or satisfies the closure property with respect to these operations. 2. Negativeintegersandzerodenotedby(cid:2)1;(cid:2)2;(cid:2)3;...and0,respectively,arosetopermitsolu- tionsofequationssuchasxþb¼a,whereaandbareanynaturalnumbers. Thisleadstothe operation of subtraction, or inverse of addition, and we write x¼a(cid:2)b. The set of positive and negative integers and zero is called the set of integers. 3. Rationalnumbersorfractionssuchas2,(cid:2)5, . . .arosetopermitsolutionsofequationssuchas 3 4 bx¼aforallintegersaandb,whereb6¼0. Thisleadstotheoperationofdivision,orinverseof multiplication, and we write x¼a=b or a(cid:3)b where a is the numerator and b the denominator. Thesetofintegersisasubsetoftherationalnumbers,sinceintegerscorrespondtorational numbers where b¼1. pffiffiffi 4. Irrational numbers such as 2 and (cid:1) are numbers which are not rational, i.e., they cannot be expressed as a=b (called the quotient of a and b), where a and b are integers and b6¼0. The set of rational and irrational numbers is called the set of real numbers. 1 Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.