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Undergraduate Texts in Mathematics Miklós Laczkovich Vera T. Sós Real Analysis Series, Functions of Several Variables, and Applications Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California Berkeley, CA, USA Advisory Board: Colin Adams, Williams College David A. Cox Amherst College Pamela Gorkin, Bucknell University Roger E. Howe, Yale University Michael E. Orrison, Harvey Mudd College Lisette G. de Pillis, Harvey Mudd College Jill Pipher, Brown University Fadil Santosa, University of Minnesota Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamong different aspectsofthesubject.Theyfeature examples that illustrate key concepts as well as exercises that strengthen understanding. More information about this series at http://www.springer.com/series/666 ó ó Mikl s Laczkovich Vera T. S s (cid:129) Real Analysis Series, Functions of Several Variables, and Applications 123 MiklósLaczkovich Vera T. Sós Department ofAnalysis Alfréd Rényi Institute of Mathematics EötvösLorándUniversity—ELTE Hungarian Academy of Sciences Budapest Budapest Hungary Hungary Translated byGergely Bálint ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts inMathematics ISBN978-1-4939-7367-5 ISBN978-1-4939-7369-9 (eBook) https://doi.org/10.1007/978-1-4939-7369-9 LibraryofCongressControlNumber:2017949163 MathematicsSubjectClassification(2010): MSC2601,MSC26BXX,MSC2801,MSC28AXX ©SpringerScience+BusinessMediaLLC2017 TranslationfromthesecondHungarianlanguageedition:ValósAnalízisIIbyMiklósLaczkovichand VeraT.Sós,©LaczkovichMiklós,T.SósVera,Typotex,2013 1stedition:AnalízisIIbyLaczkovichMiklósandT.SósVera,©LaczkovichMiklós,T.SósVera,Nemzeti TankönykiadóZrt.,Budapest,2007 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditors giveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsor omissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaLLC Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A. Preface Analysisformsanessentialbasisofbothmathematicsandstatistics,aswellasmost ofthenaturalsciences.Moreover,andtoaneverincreasingextent,mathematicshas been used to underpin our understanding of the social sciences. It was Galileo’s insightthat“Nature’sgreatbookiswritteninthelanguageofmathematics.”Andit is the theory of analysis (specifically, differentiation and integration) that was created for the express purpose of describing the universe in the language of mathematics. Working out the precise mathematical theory took almost 300 years, withalargeportionofthistimedevotedtocreatingdefinitionsthatencapsulatethe essence of limit and continuity. This task was neither easy nor self-evident. In postsecondary education, analysis is a foundational requirement whenever mathematicsisanintegralcomponentofadegreeprogram.Masteringtheconcepts of analysis can be a difficult process. This is one of the reasons why introductory analysis courses and textbooks introduce the material at many different levels and employvariousmethodsofpresentingthemainideas.Thisbookisnotmeanttobe a first course in analysis, for we assume that the reader already knows the funda- mental definitions and basic results of one-variable analysis, as is discussed, for example, in [7]. In most of the cases we present the necessary definitions and theorems of one-variable analysis, and refer to the volume [7], where a detailed discussion of the relevant material can be found. In this volume we discuss the differentiation and integration of functions of several variables, infinite numerical series, and sequences and series offunctions. We place strong emphasis on presenting applications and interpretations of the results, both in mathematics itself, like the notion and computation of arc length, area, and volume, and in physics, like the flow of fluids. In several cases, the applications or interpretations serve as motivation for formulating relevant mathe- maticaldefinitionsandinsights.InChapter8wepresentapplicationsofanalysisin apparently distant fields of mathematics. It is important to see that although the classical theory of analysis is now more than100yearsold,theresultsdiscussedherestillinspireactiveresearchinabroad spectrumofscientificareas.Duetothenatureofthebookwecannotdelveintosuch v vi Preface matters with any depth; we shall mention only a small handful of unsolved problems. Many of the definitions, statements, and arguments of single-variable analysis can be generalized to functions of several variables in a straightforward manner, andwe occasionallyomittheproofofatheoremthatcanbeobtainedbyrepeating the analogous one-variable proof. In general, however, the study of functions of several variables is considerably richer than simple generalizations of one-variable theorems. In the realm offunctions of several variables, new phenomena and new problemsarise,andtheinvestigationsoftenlead tootherbranchesofmathematics, suchasdifferentialgeometry,topology,andmeasuretheory.Ourintentistopresent therelevantdefinitions,theorems,andtheirproofsinfulldetail.However,insome cases the seemingly intuitively obvious facts about higher-dimensional geometry and functions of several variables prove remarkably difficult to prove in full gen- erality. When this occurs (for example, in Chapter 5, during the discussion of the so-calledintegraltheorems)withresultsthataretooimportantforeitherthetheory or its applications, we present the facts, but not the full proofs. Our explicit intent isto presentthe material gradually, and todevelop precision basedonintuitionwiththehelpofwell-designedexamples.Masteringthismaterial demands full student involvement, and to this end we have included about 600 exercises.Someof these areroutine, butseveral ofthem are problems that call for anincreasinglydeepunderstandingofthemethodsandresultsdiscussedinthetext. The most difficult exercises require going beyond the text to develop new ideas; these are marked by ð(cid:2)Þ. Hints and/or complete solutions are provided for many exercises, and these are indicated by (H) and (S), respectively. Budapest, Hungary Miklós Laczkovich February 2017 Vera T. Sós Contents 1 Rp !R functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Real Functions of Several Variables and Their Graphs . . . . . . . . . 3 1.3 Convergence of Point Sequences. . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Basics of Point Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.8 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.9 Higher-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.10 Applications of Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.11 Appendix: Tangent Lines and Tangent Planes . . . . . . . . . . . . . . . 63 2 Functions from Rp to Rq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1 Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.4 Implicit and Inverse Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 The Jordan Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 Definition and Basic Properties of the Jordan Measure . . . . . . . . . 95 3.2 The measure of a Few Particular Sets . . . . . . . . . . . . . . . . . . . . . 106 3.3 Linear Transformations and the Jordan Measure. . . . . . . . . . . . . . 115 3.4 Appendix: The Measurability of Bounded Convex Sets . . . . . . . . 119 4 Integrals of Multivariable Functions I . . . . . . . . . . . . . . . . . . . . . . . 123 4.1 The Definition of the Multivariable Integral . . . . . . . . . . . . . . . . . 123 4.2 The Multivariable Integral on Jordan Measurable Sets . . . . . . . . . 128 4.3 Calculating Multivariable Integrals . . . . . . . . . . . . . . . . . . . . . . . 135 4.4 First Appendix: Proof of Theorem 4.12 . . . . . . . . . . . . . . . . . . . . 146 4.5 Second Appendix: Integration by Substitution (Proof of Theorem 4.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 vii viii Contents 5 Integrals of Multivariable Functions II. . . . . . . . . . . . . . . . . . . . . . . 155 5.1 The Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Conditions for the Existence of the Primitive Function . . . . . . . . . 163 5.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.4 Surface and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.5 Integral Theorems in Three Dimension . . . . . . . . . . . . . . . . . . . . 187 6 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.1 Basics on Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2 Operations on Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.3 Absolute and Conditionally Convergent Series. . . . . . . . . . . . . . . 202 6.4 Other Convergence Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.5 The Product of Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.6 Summable Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 6.7 Appendix: On the History of Infinite Series . . . . . . . . . . . . . . . . . 227 7 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.1 The Convergence of Sequences of Functions . . . . . . . . . . . . . . . . 229 7.2 The Convergence of Series of Functions . . . . . . . . . . . . . . . . . . . 239 7.3 Taylor Series and Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.4 Abel Summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.5 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.6 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.7 First Appendix: The Cauchy–Hadamard Formula . . . . . . . . . . . . . 292 7.8 Second Appendix: Complex Series . . . . . . . . . . . . . . . . . . . . . . . 295 7.9 Third Appendix: On the History of the Fourier Series. . . . . . . . . . 297 8 Miscellaneous Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.1 Approximation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.2 Approximation of Definite Integrals. . . . . . . . . . . . . . . . . . . . . . . 311 8.3 Parametric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.4 Sets with Lebesgue Measure Zero and the Lebesgue Criterion for Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.5 Two Applications of Lebesgue’s Theorem . . . . . . . . . . . . . . . . . . 343 8.6 Some Applications of Integration in Number Theory . . . . . . . . . . 346 8.7 Brouwer’s Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 352 8.8 The Peano Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 9 Hints, Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Notation.. .... .... .... .... ..... .... .... .... .... .... ..... .... 383 References.... .... .... .... ..... .... .... .... .... .... ..... .... 385 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 387 Functions of Several Variables Functions of several variables are needed in order to describe complex processes. Adetailedmeteorologicalreliefmapindicatingthetemperatureasitchangesduring the day needs four variables: three coordinates of the place (longitude, latitude, altitude) and one coordinate of the time. The mathematical description of complex systems, e.g., the motion of gases or fluids, may need millions of variables. Ifasystemdependsonpparameters,thenwecandescribeaquantitydetermined by the system using a function that assigns the value of the quantity to the sequences of length p that characterize the state of the system. Wesaythatf isafunctionofpvariablesifeveryelementofthedomainoff is asequence oflength p.Forexample,ifwe assign toevery date (year, month,day) thecorrespondingdayoftheweek,thenweobtainafunctionofthreevariables,for which f (2016, July, 18) = Monday. Inthesequelwewillmainlyconsiderfunctionsthatdependonsequencesofreal parameters. ix