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Mathematical Analysis. A Very Short Introduction Richard Earl

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RISK BaruchFischhoffandJohnKadvany

RITUAL BarryStephenson

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THEROMANEMPIRE

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THEROMANREPUBLIC

DavidM.Gwynn

ROMANTICISM MichaelFerber

ROUSSEAU RobertWokler

RUSSELL A.C.Grayling

THERUSSIANECONOMY

RichardConnolly

RUSSIANHISTORY GeoffreyHosking

RUSSIANLITERATURE CatrionaKelly

THERUSSIANREVOLUTION

S.A.Smith

SAINTS SimonYarrow

SAMURAI MichaelWert

SAVANNAS PeterA.Furley

SCEPTICISM DuncanPritchard

SCHIZOPHRENIA ChrisFrithand EveJohnstone

SCHOPENHAUER

ChristopherJanaway

SCIENCEANDRELIGION

ThomasDixonandAdamR.Shapiro

SCIENCEFICTION DavidSeed

THESCIENTIFICREVOLUTION

LawrenceM.Principe

SCOTLAND RabHouston

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SEXUALSELECTION MarleneZukand

LeighW.Simmons

SEXUALITY VéroniqueMottier

WILLIAMSHAKESPEARE

StanleyWells

SHAKESPEARE’SCOMEDIES

BartvanEs

SHAKESPEARE’SSONNETSAND POEMS JonathanF.S.Post

SHAKESPEARE’STRAGEDIES

StanleyWells

GEORGEBERNARDSHAW

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MARYSHELLEY CharlotteGordon

THESHORTSTORY AndrewKahn

SIKHISM EleanorNesbitt

SILENTFILM DonnaKornhaber

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SLANG JonathonGreen

SLEEP StevenW.Lockleyand RussellG.Foster

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SOCIALANDCULTURAL ANTHROPOLOGY

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JonathanScourfield

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SOCIOLOGY SteveBruce

SOCRATES C.C.W.Taylor

SOFTMATTER TomMcLeish

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SOUTHEASTASIA JamesR.Rush

THESOVIETUNION StephenLovell

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SPANISHLITERATURE JoLabanyi

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SPINOZA RogerScruton

SPIRITUALITY PhilipSheldrake

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STARS AndrewKing

STATISTICS DavidJ.Hand

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STRUCTURALENGINEERING

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SUPERSTITION StuartVyse

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SYNAESTHESIA JuliaSimner

SYNTHETICBIOLOGY JamieA.Davies

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TAXATION StephenSmith

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TIDES DavidGeorgeBowersand EmyrMartynRoberts

TIME JenannIsmael

TOCQUEVILLE HarveyC.Mansfield

LEOTOLSTOY LizaKnapp

TOPOLOGY RichardEarl

TRAGEDY AdrianPoole

TRANSLATION MatthewReynolds

THETREATYOFVERSAILLES

MichaelS.Neiberg

TRIGONOMETRY

GlenVanBrummelen

THETROJANWAR EricH.Cline

TRUST KatherineHawley

THETUDORS JohnGuy

TWENTIETH-CENTURYBRITAIN

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THEUNITEDNATIONS

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UNIVERSITIESANDCOLLEGES

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THEU.S.CIVILWAR LouisP.Masur

THEU.S.CONGRESS DonaldA.Ritchie

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THEU.S.SUPREMECOURT

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UTILITARIANISM

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UTOPIANISM LymanTowerSargent

VATICANII ShaunBlanchardand StephenBullivant

VETERINARYSCIENCE JamesYeates THEVIKINGS JulianD.Richards

VIOLENCE PhilipDwyer

THEVIRGINMARY

MaryJoanWinnLeith

THEVIRTUES CraigA.Boydand KevinTimpe

VIRUSES DorothyH.Crawford

VOLCANOES MichaelJ.Branneyand JanZalasiewicz

VOLTAIRE NicholasCronk

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WEATHER StormDunlop

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WORK StephenFineman

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THEWORLDTRADE

ORGANIZATION AmritaNarlikar

WORLDWARII GerhardL.Weinberg

WRITINGANDSCRIPT

AndrewRobinson

ZIONISM MichaelStanislawski

ÉMILEZOLA BrianNelson

Mathematical Analysis

AVeryShortIntroduction

RichardEarl

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

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Forcherishedfriends: Gareth,Natalie, Owen,andMatthew

Acknowledgements xix

Listofillustrations xxi

1 Taming infinity 1

2

3

4

5

6

All change...thecalculusofFermat,Newton, and Leibniz 17

Tothe limit:analysis in the 18th and 19th centuries 39

Should I believe mycomputer? 65

Dimensionsaplenty 83

I’llnamethattune in ... 112

7 Puttingthe i in analysis 130

8 Butthere ’ s more... 154

Appendix 167

Historical timeline 187

Referencesand furtherreading 191

Index 193

Acknowledgements

Thanksgoto JonahBlain, Daniel Claydon, Martin Galpin,Natalie

Lane, PatrickMcDermott, James Munro,and EndreSu ¨ lifor their

commentson,and helpwith,draftchapters.

Listofillustrations

1Dividing upasquare in two

differ entways 4

2 Apentagonin acircle in a

hexagon 5

3 ThreeproofsofThales ’

theor em 20

4 An alyticgeometry 22

5 Fermat’ swork on

tangen ts 24

(a) PortraitofPierrede Fermat.

WikimediaCommons/Public Domain

6 Fermat’ swork on extremaand

areas 26

7 The modulus function 28

8Signedarea 29

9FundamentalTheorem of

Calculus 31

10 Kepler’s firsttwo laws 36

11Exponen tialsan d

logarithms 40

12Trigonometry 49

13 Euler’swork 53

PortraitofLeonhard

Euler. WikimediaCommons/Public

Domain

14 S-likegrowthforVerhulst’ s model 57

15 Biologicalmodels 58

16 Howsolutionsdevelop 60

17Continuity 61

18 Riemannintegration 63

19Two fixedpoints 66

20Cobwebbing 67

21Further cobwebbing 68

22Plotof data 70

23 Interpolatingwith

polynomials 71

24Linear approximation 72

25 Euler’s method—first

iteration 74

26 Euler’s method successive

iterations 74

27 Trapeziumrule 76

28Simpson’s rule 77

29 Newton’s method 79

30 Modellingapendulum 80

31Vector sandvector

fields 85

32 Curvesbetween points 90

33 Thebrachistochrone problem 91

34 Acatenoid 92

35 Volume integrals 94

36 Latitudeand longitude 96

37Soli dangle in acone 98

38 Flux 100

39Work 101

40 Divergenceof aplanar

flow 104

41 Anincompressiblewhirlpool

flow 106

42 Otherflows 107

43 InterpretingStokes ’ theorem

physically 108

44 Gradient flow is normal toa surface 110

45 Harpsichordandpiano strings 113

46 Wavespropagating 114

47 Derivingthewave

equation 115

48 First harmonicatdifferent

times 117

49 Secondandthird harmonics 118

50 Twowaves initially 119

51 Gibbs’ phenomenon 121

52Different homophonic

drummembranes 123

53Applyingthespectral

theorem 124

54Detailfrom a lithograph of

Augustin-Louis, Baron

Cauchy 133

Wellcome Library, London.

Copyrightedwork available under

CreativeCommonsAttribution2.0

Generic licenceCC BY 2.0(https://

creativecommons.org/licenses/by/2.0/ deed.en),via WikimediaCommons

55 Thecomplex plane 134

56 Transformationsof the

complex plane 136

57 Differentwaysthata

complexnumber can becomesmall 137

58 Discsof convergence 143

59 Apath and itsendpoints 146

60 Parameterizingthecircle 148

61Theeffectof sine 151

62 Applicationsto fluid flow 152

63 Flowaroundan aerofoil 153

64Dirichlet’ s function 155

65 Lebesgue integration 157

Photograph ofHenriLebesgue.

WikimediaCommons/Public

Domain

66Centrallimittheorem 160

67 Logarithm on thecircle 184

Chapter1 Taminginfinity

Throughoutthehistoryofmathematics,infinityandthe infinitesimal—theinfinitelylargeandtheinfinitelysmall—have raiseddifficultiesandparadoxes.Formathematicians,thestudyof infinityarisesnaturally,butindifferentwaysfromhowanartistor theologianmightcontemplateit.Anartistmightwelcomethe challengeofrepresentingsomeparadoxofinfinityintheirwork; forthedevout,aninfinitedeitymaybecentraltotheirfaith.But themathematicianseekstodefine,andultimatelylimit,infinity,as mathematicscannotfullyprogresswithoutrigorousandcareful definitions.Notsurprisingly,thishasbeenatortuousandrocky journeythathastakencenturies,ifnotmillennia,toresolve.Ifa studyofthehistoryofmathematicsteachesanything,itisan appreciationoftheendeavourinvolvingmanypeople,overmany generations,tofindrightwaysofthinkingaboutmathematics.And thisisespeciallytrueinthefieldofanalysis.

Mathematicalanalysis,tosomeextent,seekstorigorouslydefine infiniteprocessesthatariseinmathematics,sothatlogical argumentscanbemadeandtheoremsproven.Iwrite‘tosome extent’becauseanalysisismuchmorethanmakingpreviously informalmathematicsprecise;wewillseethatanalysishasideas andconceptsallofitsown,manyofwhichwerebeingstudied centuriesbeforetherewasanotionofanalysisasasubjectinits ownright.Weshallalsoseethattheapplicationsofanalysisare

numerousacrossmathematicsandscience;indeed,manyofthe rootsofanalysisarearesultofhumankind’seffortstomodelthe worldaroundus.

Actualandpotentialinfinities

The countingnumbers begin1,then2,then3,andsoon,andare commonlylistedas

1; 2; 3; ...

Heretheellipsis‘...’signifiesthatthelistgoesonforever. Wecancontinuecountingwithoutend,neverrunningout ofnumbers.Eachcountingnumber,howeverlarge,is itselffinite,butwerecognizethatthereareinfinitelymany countingnumbers.

Theabovecapturestwodifferenttypesofinfinity.A potential infinity isaninfiniteprocessthatgoesonwithoutend.Muchof mathematicalanalysisfocusesonsuchinfiniteprocesses.Whenwe saythereareinfinitelymanycountingnumbers,wearereferringto an actualinfinity asifinfinityitselfwereanumber.

Earlyon,weimplicitlymeetpotentialinfinities.Welearnthat numberscanberepresentedasdecimalexpansionssuchas:

Butwhat exactly doesallthismean?Whatdetailsarehiddenby thoseellipses?Thisisdefinitelyamathematician’squestion.No experimentalscientistorengineerhaseverneededtoknowthe accuracyofavaluetomorethan15decimalplaces.Thevalueof π givenaboveissufficientlyaccurateforspacemissionsexploringthe

solarsystem.So,whilstmoderncomputershavecalculated π to trillionsofplaces,therearefewbenefitsofsuchknowledgeinthe physicalworld.

Oneargumentforverifyingthefirstdecimalexpansionmightgoas follows.Ifwewrite x ¼ 0 333333 andthenmultiplybothsidesby10,weget 10x ¼ 3 3333333 (asmultiplyingby10movesthedecimalpointoneplacetothe right).Subtractingthefirstequationfromthesecond,wefind 9x ¼ 3andhence x ¼ 3 9 ¼ 1 3 asclaimed.

Sofarasitgoes,theaboveiscorrect,butnotthe‘wholetruth’, untilwegiveaprecisedefinitionofwhatadecimalexpansion represents;withoutthat,howcanwebecertainthatthealgebraic manipulationsabovearevalid?

Thefollowinginfinitesum,oftennamedaftertheItalian mathematicianGuidoGrandi,givesfurtherevidenceoftheneed forrigour.Ifweset

thenwemightarguethat

andsowe’veshownthat0 ¼ 1.Withoutproperdefinitions, seeminglyreasonablealgebraicmanipulationsofaninfinitesum canleadtononsensicalconclusions.

Thefirstcalculationmaynotseemlikeaninfinitesum,butthe notation x ¼ 0:3333 ... isjustshorthandfortheinfinitesum

Weclearlyneedanexplicitdefinitionofwhataninfinitesum means,sowecanshowthat x ¼ 1 3 butexcludecontradictorysums likeGrandi’s.Let’snowlookatadifferentwayofapproachingthe problem.

Howis π calculated?

Inadditiontothepreviousalgebraicmethods,geometricmethods canalsobeusedtodetermineinfinitesums.Asquareofside1, andsoarea1,canbedividedupintwodifferentways(Figure1).

ForexampleFigure1(a)showsthat

(a)(b)

1.Dividingupasquareintwodifferentways(1(a),1(b)).

andFigure1(b)showsthat

InFigure1(a),thesquareisdividedupintorectanglesandsquares, thefirstregionhavingarea 1 2,andeachsubsequentregionbeing halfthesizeofthepreviousregion.Thesquaresandrectangles ultimatelycoverthewholesquare,andsotheinfinitesumequals1.

InFigure1(b),wecoverthesquarewiththreecollectionsofsquares havingareas 1 4 ; 1 16 ; 1 64 ; ... ; andsothesumofeachcollection’sareas isonethirdthatofthesquare,namely 1 3.

Fornow,we’llfocusonapproximating π,whichnaturallylends itselftogeometricmethods.Recallthat π istheratioofacircle’s circumferencetoitsdiameter,andalsoequalstheareaofacircle withradius1.

Hereapentagonhasbeeninscribedinacircleandahexagon circumscribesthecircle(Figure2).Asthepentagonisinsidethe circle,thepentagonhasasmallerareathanthecircle;asthe 2.Apentagoninacircleinahexagon.

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