Mathematical Analysis. Functions of Several Real Variables and Applications PDF
Preview Mathematical Analysis. Functions of Several Real Variables and Applications
UNITEXT 137 Nicola Fusco · Paolo Marcellini Carlo Sbordone Mathematical Analysis Functions of Several Real Variables and Applications UNITEXT + La Matematica per il 3 2 Volume 137 Editor-in-Chief AlfioQuarteroni,PolitecnicodiMilano,Milan,Italy;ÉcolePolytechniqueFédérale deLausanne(EPFL),Lausanne,Switzerland SeriesEditors LuigiAmbrosio,ScuolaNormaleSuperiore,Pisa,Italy PaoloBiscari,PolitecnicodiMilano,Milan,Italy CiroCiliberto,UniversitàdiRoma“TorVergata”,Rome,Italy CamilloDeLellis,InstituteforAdvancedStudy,Princeton,NewJersey,USA MassimilianoGubinelli,HausdorffCenterforMathematics,Rheinische Friedrich-Wilhelms-Universität,Bonn,Germany VictorPanaretos,InstituteofMathematics,ÉcolePolytechniqueFédéralede Lausanne(EPFL),Lausanne,Switzerland LorenzoRosasco,DIBRIS,UniversitàdegliStudidiGenova,Genova,Italy CenterforBrainsMindandMachines,MassachusettsInstituteofTechnology, Cambridge,Massachusetts,USA;IstitutoItalianodiTecnologia,Genova,Italy TheUNITEXT-LaMatematicaperil3+2seriesisdesignedforundergraduate andgraduateacademiccourses,andalsoincludesadvancedtextbooksataresearch level. Originally released in Italian, the series now publishes textbooks in English addressedtostudentsinmathematicsworldwide. Some of the most successful books in the series have evolved through several editions,adaptingtotheevolutionofteachingcurricula. Submissions must include at least 3 sample chapters, a table of contents, and a preface outlining the aims and scope of the book, how the book fits in with the currentliterature,andwhichcoursesthebookissuitablefor. For any further information, please contact the Editor at Springer: [email protected] THESERIESISINDEXEDINSCOPUS Nicola Fusco • Paolo Marcellini • Carlo Sbordone Mathematical Analysis Functions of Several Real Variables and Applications NicolaFusco PaoloMarcellini DipartimentodiMatematicaeApplicazioni DipartimentodiMatematicaeInformatica “RenatoCaccioppoli” “U.Dini” UniversitàdiNapoliFedericoII UniversitàdiFirenze Napoli,Italy Firenze,Italy CarloSbordone DipartimentodiMatematicaeApplicazioni “RenatoCaccioppoli” Università di Napoli Federico II Napoli, Italy Translatedby SimonG.Chiossi DepartamentodeMatemáticaAplicada UniversidadeFederalFluminense Niterói,RiodeJaneiro,Brazil ISSN2038-5714 ISSN2532-3318 (electronic) UNITEXT ISSN2038-5722 ISSN2038-5757 (electronic) LaMatematicaperil3+2 ISBN978-3-031-04150-1 ISBN978-3-031-04151-8 (eBook) https://doi.org/10.1007/978-3-031-04151-8 Translation fromtheItalian language edition: “Lezioni diAnalisi Matematica Due”byNicola Fusco etal.,©Zanichelli2020.PublishedbyZanichellieditoreS.p.A.AllRightsReserved. ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Coverillustration: Abstractdigitalminimalinstallation,whitespherewithcubicalcutsectorcorner.3d renderingillustration.©EugeneSergeev/GettyImages/iStockphoto ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface There is a long and established tradition of printed manuals on Mathematical Analysis I. For lecturersin the faculties of sciences, engineeringand architecture, it is therefore easy to select the most suitable ones, according to their own mathematical taste and depending on the objectives of the classes to be offered. There is, on the contrary, general agreement as regards the lack of a similar availabilityoftextbookson MathematicalAnalysisII. Whilst preparing the present book, we thought that we should allow for the possibility of choosing between two levels of presentation. For this reason, the contents are first discussed in an elementary way, and at a successive stage, they areexaminedfromseveral,morepenetrating,angles. The agile organisation of the subject matter helps instructors to determine effortlesslywhichpartstopresentduringlecturesandwheretostop. Inacourseaddressedtoengineeringorcomputersciencestudents,forexample, it may be appropriate to discuss multiple integrals from a more concrete point of view, starting from normal domains in the plane and in space. For mathematics andphysicsstudents,ontheotherhand,amorerigorousapproachtoRiemann’sor Lebesgue’stheorymightbemorefitting. Anotherpossibility is to leave the chapteron the Lebesgueintegralfor a third- yearclass. Similarly,we areconvincedthatthetheoryofregularsurfacesinspace shouldbeenoughforthe majorityofundergraduatedegrees,while some lecturers mightthinkitusefultoteach,inanelementaryway,k-dimensionalmanifoldsand thegeneralisationsofStokes’stheoremandofthedivergencetheoremtoRn. In the same line of thought, the various chapters’ appendices provide further opportunitiestogodeeperintocertaintopics:amongthem,theAscoli-Arzelàtheo- rem,theregularityofconvexfunctionsinRn,theGammafunctionorLp spaces,all topicsthatareparamountinmodernmathematicalanalysis.Otherinstancesinclude the Weierstrass theorem on polynomialapproximation of continuousfunctions or Peano’sexistencetheorem(typicallyonly existence,withoutuniqueness)fornon- linearODEsandsystemsundergeneralassumptions. v vi Preface The authorsbelievethatanytextbookcancontributetothe successofa lecture course only to a point, and the choices made by lecturers are decisive in this respect. Nonetheless, they still hope that this book will be welcomed by students andcolleagues. Napoli,Italy NicolaFusco Firenze,Italy PaoloMarcellini Napoli,Italy CarloSbordone Contents 1 SequencesandSeriesofFunctions ....................................... 1 1.1 SequencesofFunctions:PointwiseandUniformConvergence.... 1 1.2 FirstTheoremsonUniformConvergence........................... 4 1.3 TheoremsonInterchangingLimitsandIntegralsorDerivatives... 7 1.4 UniformConvergenceandMonotonicity ........................... 14 1.5 SeriesofFunctions................................................... 17 1.6 PowerSeries.......................................................... 22 1.7 TaylorSeries.......................................................... 28 1.8 FourierSeries......................................................... 36 1.9 TheConvergenceofFourierSeries.................................. 42 AppendixtoChap.1......................................................... 48 1.10 TheAscoli-ArzelàTheorem......................................... 48 1.11 TheWeierstrassApproximationTheorem .......................... 50 1.12 Abel’sTheoremonPowerSeries.................................... 52 2 MetricSpacesandBanachSpaces........................................ 59 2.1 Introduction........................................................... 59 2.2 MetricSpaces......................................................... 59 2.3 SequencesinaMetricSpace:ContinuousFunctions............... 65 2.4 VectorSpaces:LinearMaps ......................................... 69 2.5 TheVectorSpaceRn andItsDual................................... 72 2.6 NormedVectorSpaces............................................... 76 2.7 TheNormedVectorSpaceRn....................................... 78 2.8 CompleteMetricSpaces:BanachSpaces........................... 83 2.9 LipschitzFunctions:TheContractionTheorem.................... 86 2.10 CompactSets:ContinuousFunctionsonCompactSets............ 89 2.11 ConnectedOpenSubsetsofRn...................................... 92 AppendixtoChap.2......................................................... 94 2.12 FurtherCompactnessTheorems:GeneralisedWeierstrass Theorem .............................................................. 94 vii viii Contents 3 FunctionsofSeveralVariables............................................ 101 3.1 Round-UpofTopologyinRn........................................ 101 3.2 LimitsandContinuity................................................ 103 3.3 PartialDerivatives.................................................... 105 3.4 HigherDerivatives.Schwarz’sTheorem............................ 109 3.5 Gradient.Differentiability ........................................... 113 3.6 CompositeFunctions................................................. 118 3.7 DirectionalDerivatives............................................... 122 3.8 FunctionswithVanishingGradientonConnectedSets ............ 127 3.9 HomogeneousFunctions............................................. 129 3.10 FunctionsDefinedbyIntegrals...................................... 131 3.11 TaylorFormulaandHigher-OrderDifferentials.................... 135 3.12 QuadraticForms.Definite,Semi-definiteandIndefinite Matrices............................................................... 140 3.13 LocalMaximaandMinima.......................................... 144 3.14 Vector-ValuedFunctions............................................. 150 AppendixtoChap.3......................................................... 158 3.15 ConvexFunctions .................................................... 158 3.16 ComplementsonQuadraticForms.................................. 173 3.17 TheMaximumPrincipleforHarmonicFunctions.................. 181 4 OrdinaryDifferentialEquations.......................................... 187 4.1 Introduction:TheInitialValueProblem............................. 187 4.2 Cauchy’sLocalExistenceandUniquenessTheorem............... 196 4.3 FirstConsequencesofCauchy’sTheorem.......................... 206 4.4 TheGlobalExistenceandUniquenessTheorem:Extension ofSolutions........................................................... 210 4.5 SolvingFirst-OrderODEsinNormalForm ........................ 216 4.6 SolvingFirst-OrderODEsNotinNormalForm.................... 221 4.7 SolvingHigher-OrderEquations .................................... 224 4.8 QualitativeStudyofSolutions....................................... 226 AppendixtoChap.4......................................................... 232 4.9 Peano’sTheorem..................................................... 232 5 LinearDifferentialEquations............................................. 237 5.1 GeneralProperties.................................................... 237 5.2 GeneralIntegralofLinearODEs.................................... 241 5.3 TheMethodofVariationofParameters............................. 247 5.4 BernoulliEquations.................................................. 250 5.5 HomogeneousEquationswithConstantCoefficients .............. 252 5.6 Equations with Constant Coefficients and Special Right-HandSide...................................................... 257 5.7 LinearEulerEquations............................................... 260 AppendixtoChap.5......................................................... 263 5.8 BoundaryValueProblems ........................................... 263 5.9 LinearSystems....................................................... 268
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