Advanced Calculus of a Single Variable

391 Pages·2016·2.389 MB·English

by  Tunc Geveci

Tunc Geveci
Advanced
Calculus
of a Single
Variable
Advanced Calculus of a Single Variable

Tunc Geveci
Advanced Calculus
of a Single Variable
123
TuncGeveci
DepartmentofMathematicsandStatistics
SanDiegoStateUniversity
SanDiego,CA,USA
ISBN978-3-319-27806-3 ISBN978-3-319-27807-0 (eBook)
DOI10.1007/978-3-319-27807-0
LibraryofCongressControlNumber:2015959713
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www.
springer.com)
DedicatedtoSimla
Mugeand Zehra

Preface
Thisbookisbasedonaone-semestersinglevariableadvancedcalculuscoursethat
I have been teaching at San Diego State University for many years. Mathematics
departmentsinmanyschoolsoffersuchacourse.Theaimisarigorousdiscussion
oftheconceptsandtheoremsthataredealtwithinformallyinthefirsttwosemesters
of a beginning calculus course. As such, students are expected to gain a deeper
understandingof the fundamentalconceptsof calculus, such as limits, continuity,
thederivativeandtheRiemannintegral.Successinthiscourseisexpectedtoprepare
themformoreadvancedcoursesinrealandcomplexanalysis.
The first semester of advanced calculus can be followed by a rigorous course
in multivariable calculus and an introductory real analysis course that treats the
Lebesgueintegralandmetricspaces,withspecialemphasisonBanachandHilbert
spaces.Ibelievethateachcourserequiresaseparatetext.
Chapter1beginswithaquickreviewofthepropertiesofthesetofrealnumbers
asanorderedfield.Theconceptofthelimitofasequenceandtherelevantrulesare
discussedrigorously.Thecompletenessofthefieldofrealnumbersisintroducedas
theexistenceofthelimitofaCauchysequence.Ibelievethatthisisbetterthanthe
introductionofthenotionofcompletenessviatheexistenceoftheleastupperbound
of a subset of real numbers that is bounded above. After all, students have been
dealing with Cauchy sequences in the form of decimal approximationsall along.
Anaddedadvantageis thefactthatthenotionofcompletenessastheexistenceof
thelimitofaCauchysequenceappearstimeandagainwithinthegeneralframework
ofametricspacethatmaynothaveanorderrelation,asinthecasesofthefieldof
complexnumbers,Banach or Hilbert spaces. The least upper boundprinciple and
thespecialnatureoftheconvergenceordivergenceofamonotonesequencearealso
treatedinChapter1.Thenotionofaninfinitelimitisdiscussedcarefullysincethe
convenientsymbol1canbemisunderstoodandmistreatedbythestudent.
Chapter2discussesthecontinuityandlimitsoffunctions.Ihavechosentolimit
thediscussiontofunctionsdefinedonintervals.Ibelievethatthepointsettopology
ofmoregeneralsetsbelongstoamoreadvancedrealanalysiscourse.Manystudents
encounter serious difficulties in the transition from informal to rigorous calculus
anyway.
vii
viii Preface
I emphasizethe "-ı definitions. Such emphasisis essentialfor the appreciation
of the difference between mere pointwise continuity and uniform continuity on a
set. One of the highlights of the chapter is the Intermediate Value Theorem that
hasbearingonthedefinitionsofbasicinversefunctionsthatfigureprominentlyin
beginningcalculus.
Chapter 3 takes up the derivative. The emphasis is on the nature of the error
in local linear approximations to functions. That renders a rigorous proof of the
celebrated chain rule quite straightforward. The student is also prepared for the
generalization of the concept of the derivative to functions of more than a single
real variable, even to functions between normed vector spaces. I have included a
detaileddiscussionofconvexityasaniceapplicationoftheMeanValueTheorem.
RigorousproofsofvariousversionsofL’Hôpital’srulearenotneglected.
Chapter 4 is on the Riemann integral. Integrability criteria in terms of upper
andlowersumsandtheoscillationofafunctionarediscussed.Thetwoapproaches
complementeachother.Iestablishthelinkwiththeusualintroductionoftheintegral
via arbitrary Riemann sums in a beginning calculus course, unlike some popular
advanced calculus texts that neglect to mention that connection as if a new type
of integral is being discussed. I have included a detailed discussion of improper
integrals,includingthecomparisonandDirichlettests.Someofthemostimportant
improperintegralsthatare encounteredin practicerequiresuch tests. I emphasize
Cauchy-typecriteriafortheconvergenceofanimproperintegral.
Chapter5isareviewofaseriesofrealnumbers.Ichosetoprovidedetailssince
thistopicchallengesstudentsinabeginningcalculuscoursewhererigorousproofs
arenotprovided.IemphasizeCauchy-typecriteriafortheconvergenceofseries.
Chapter6discussestheconvergenceofsequencesandseriesofreal-valuedfunc-
tionsonintervals.Thedistinctionbetweenmerepointwiseanduniformconvergence
is emphasized, with ample examples. The nice behavior of sequences and series
of functions with respect to integration and differentiation are not valid unless
certain uniform convergenceconditions are satisfied. The analyticity of functions
definedviapowerseriesfollowssmoothlyoncetheappropriatefoundationinvolving
uniformconvergenceisestablished.Thechapterisconcludedwiththedefinitionof
familiarspecialfunctionsviapowerseries.
Thisbookisanundergraduatetextandnotamonographonaspecialtopic.My
writing has been inspired and influenced by a variety of authors over many years
sincemyinitialencounterwithanalysisasastudent.Ihavebeenfortunatetohave
hadteacherssuchasStefanWarschawski,ErrettBishop,andWilliamF.Lucas.The
followingisashortlistofbooksthatarerelevanttothewayItreatedthetopicsthat
areincludedinthisbook(excellentclassicaltextsinthisclassicalsubject):
1. IntroductiontoCalculusandAnalysis,Vol.1,byRichardCourantandFritzJohn,
Springer,1998
2. TheoryandApplicationofInfiniteSeries,byKonradKnopp,Dover,1990
3. The Elements of Real Analysis, Second Edition, by Robert G. Bartle, Wiley,
1976
SanDiego,CA,USA TuncGeveci
Contents
1 RealNumbers,Sequences,andLimits..................................... 1
1.1 TerminologyandNotation.............................................. 1
1.1.1 SetTheoreticTerminologyandNotation...................... 1
1.1.2 Functions........................................................ 3
1.2 RealNumbers........................................................... 7
1.2.1 RulesofArithmetic............................................. 7
1.2.2 TheOrderAxioms.............................................. 9
1.2.3 TheNumberLine............................................... 11
1.2.4 TheAbsoluteValueandtheTriangleInequality.............. 12
1.2.5 TheArchimedeanPropertyofR............................... 16
1.2.6 Problems ........................................................ 17
1.3 TheLimitofaSequence................................................ 19
1.3.1 TheDefinitionoftheLimitofaSequence .................... 19
1.3.2 TheLimitsofCombinationsofSequences.................... 24
1.3.3 Problems ........................................................ 29
1.4 TheCauchyConvergenceCriterion.................................... 30
1.4.1 BasicFactsaboutCauchySequences.......................... 30
1.4.2 IrrationalNumbersAreUncountable.......................... 37
1.4.3 Problems ........................................................ 41
1.5 TheLeastUpperBoundPrinciple...................................... 42
1.5.1 TheLeastUpperBoundPrinciple ............................. 42
1.5.2 MonotoneSequences........................................... 46
1.5.3 Problems ........................................................ 52
1.6 InfiniteLimits........................................................... 53
1.6.1 TheDefinitionoftheInfiniteLimit............................ 53
1.6.2 PropositionsfortheEvaluationofInfiniteLimits............. 55
1.6.3 Problems ........................................................ 59
2 LimitsandContinuityofFunctions........................................ 61
2.1 Continuity ............................................................... 61
2.1.1 TheDefinitionofContinuity................................... 61
ix