[PDF] 5/7/2004 CALCULUS BEYOND CALCULUS Math21a, O Knill

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5/7/2004CALCULUSBEYONDCALCULUSMath21a,O.KnillINTRODUCTION.Topicsbeyondmulti-variablecalculusareusuallylabeledwithspecialnameslike"linear

algebra","ordinarydierentialequations","numericalanalysis","partialdierentialequations","functional

analysis"or"complexanalysis".Whereonewoulddrawthelinebetweencalculusandnon-calculustopicsis notclearbutifcalculusisaboutlearningthebasicsoflimits,dierentiation,integrationandsummation,then

multi-variablecalculusisthe"blackbelt"ofcalculus.Arethereotherwaystoplaythissport?HOWWOULDALIENSCOMPUTE?Onanotherplanet,calculusmightbetaughtinacompletelydierent

way.Thelackoffreshideasinthecurrenttextbookoerings,whereallbooksareessentiallyclones(*)ofeach otherandwhereinnovationisfakedbyejectingneweditionseveryyear(whichofcoursehasthemainpurpose topreventtheretailofsecondhandbooks),onecouldthinkthatintherestoftheuniverse,multi-variable calculuswouldhavetobetaughtinthesameway,andwherechapter12isalwaysthechapteraboutmultiple integrals.Actually,aswewanttoshowhere,eventhehumanspecieshascomeupwithawealthofdierent waystodealwithcalculus.Itisverylikelythatcalculustextbookswouldlookverydierentinotherpartsof ourgalaxy.Thisweekbrokethenewsthatonehasdiscovereda5'tharmofthemilkywaygalaxy.Itis77'000 lightyearslongandshouldincreasethechancethatthereareothertextbooksinourhomegalaxy.Inthistext, wewanttogiveanideathatthecalculustopicsinthiscoursecouldbeextendedorbuiltcompletelydierently. Actually,evennumberscanbedeneddierently.JohnConwayintroducedoncenumbersaspairsfLjRg whereLandRaresetsofnumbersdenedpreviously.Forexamplef;j;g=0andf0j;g=1,f;j0g=1,f0j1g. Theadvantageofthisconstructionisthatitallowstosee"numbers"aspartof"games".DonaldE.Knuth, (thegiantofacomputerscientist,whoalsodesigned"TeX",atypesettingsysteminwhichthistextiswritten), wroteabookcalled"surrealnumbers"inwhichtwostudentsndthemselvesonanisland.Theyndastone withtheaxiomsforanewnumbersystemiswrittenanddevelopfromthatanentirelynewnumbersystem whichcontainsthereallineandmore.Thebookisauniquecase,wheremathematicaldiscoveryisdescribed asanovel.Thisissototallydierentfromwhatweknowtraditionallyaboutnumbersthatonecouldexpect

Conwaytobeanalienhimselfiftherewerenotmanyotherproofsofhisingeniouscreativity.(*)OneofthefewexceptionsismaybethebookofMarsdenandTromba,whichisoriginal,preciseandwellwritten.

Itisunfortunatelyslightlytoomathematicalformostcalculusconsumersandsuersfromthesamediseasethatother textbookshave:ascandalousprize.Adenitecounterexampleisthebook"howtoacecalculus"whichisfunny, originalandcontainstheessentialstu.Anditcomesasapaperback.TogetherwithaSchaumoutlinevolume(also inpaperback),itwouldsuceasarudimentarytextbookcombination(andwouldcosttogetherhalfandweightone

fourth)ofthestandarddoorstoppers.NONSTANDARDCALCULUS.AtthetimeofLeonardEuler,peoplethoughtaboutcalculusinamoreintuitive,

butlessformalway.Forexample (1+x=n) n =e x withinnitelylargenwouldbeperfectlyne.Amodern approachwhichcatchesthisspiritis"nonstandardanalysis",wherethenotionof"innitesimal"isgivenaprecise meaning.Thesimplestapproachistoextendthelanguageandintroduceinnitesimalasobjectswhichare smallerthanallstandardobjects.Wesayxyifjxyjisinnitesimal.Allnumbersaretraditionallydened likeorp

2are"standard".Thenotionwhichtellsthateveryboundedsequencehasanaccumulationpointis

expressedbythefactthatthereexitsanitesetAsuchthatallx2Dareinnitesimallyclosetoanelementin A.ThefactthatacontinuousfunctiononacompactsettakesitsmaximumisseenbytakingM=max a2A f(a). Whatimpressedmeasanundergraduatestudentlearningnonstandardcalculus(inaspecialcoursecompletely devotedtothatsubject)wastheeleganceofthelanguageaswellasthecompactnessinwhichtheentirecalculus storycouldbepacked.Forexample,toexpressthatafunctionfiscontinuous,onewouldsayxythen f(x)f(y).ThisismoreintuitivethantheWeierstrassdenition8>09>0jxyj<)jf(x)f(y)j< whichisunderstoodtodayprimarilybyintimidation.Toillustratethis,EdNelson,thefounderofoneof thenonstandardanalysis avors,asksthemeaningof

8>09>0jxyj<)jf(x)f(y)j<

inorderto demonstratehowunintuitivethisdenitionreallyis.Thederivativeofafunctionisdenedasthestandard partoff 0 (x)=(f(x+dx)f(x))=dx,wheredxisinnitesimal.Dierentiabilitymeansthatthisexpression isniteandindependentoftheinnitesimaldxchosen.IntegrationR x a f(y)dyisdenedasthestandardpart ofP x j

2[a;x]

f(x i )dxwherex k =kdxanddxisaninnitesimal.Thefundamentaltheoremofcalculusisthe trivialityF(x+dx)F(x))=f(x)dx. Thereasonsthatpreventednonstandardcalculustogomainstreamwerelackofmarketing,badluck,being belowacertaincriticalmassandtheinitialbelievethatstudentswouldhavetoknowsomeofthefoundations ofmathematicstojustifythegame.Itisalsoquiteasharpknifeanditseasytocutthengertooanddoing mistakes.Thename"nonstandardcalculus"certainlywasnotfortunatetoo.Peoplecallitnow"innitesimal calculus".Introducingthesubjectusingnameslike"hyper-reals"andusing"ultra-lters"certainlydidnothelp topromotetheideas(weusuallyalsodon'tteachcalculusbyintroducingDedekindcutsorcompleteness)but therearebookslikebyAlainRobertwhichshowthatitispossibletoteachnonstandardcalculusinanatural way. DISCRETESPACECALCULUS.Manyideasincalculusmakesenseinadiscretesetup,wherespaceisagraph, curvesarecurvesinthegraphandsurfacesarecollectionsof"plaquettes",polygonsformedbyedgesofthe graph.Onecanlookatfunctionsonthisgraph.Scalarfunctionsarefunctionsdenedontheverticesofthe graphs.Vectoreldsarefunctionsdenedontheedges,othervectoreldsaredenedasfunctionsdenedon plaquettes.Thegradientisafunctiondenedonanedgeasthedierencebetweenthevaluesoffattheend points. Consideranetworkmodeledbyaplanargraphwhichformstriangles.Ascalarfunctionassignsavaluef n toeachnoden.Anareafunctionassignsvaluesf T toeachtriangleT.AvectoreldassignvaluesF nm to eachedgeconnectingnodenwithnodem.ThegradientofascalarfunctionisthevectoreldF nm =f n f m . ThecurlofavectoreldFisattachestoeachtriangle(k;m;n)thevaluecurl(F) kmn =F km +F mn +F nk . Itisameasureforthecirculationoftheeldaroundatriangle.Acurve inourdiscreteworldisasetof pointsr j ;j=1;:::;nsuchthatnodesr j andr j+1 areadjacent.ForavectoreldFandacurve ,theline integralisP n j=1 F r(j)r j+1 .AregionRintheplaneisacollectionoftrianglesT.Thedoubleintegralof anareafunctionf T isP T2R f T .Theboundaryofaregionisthesetofedgeswhichareonlysharedbyone triangle.Theorientationof isasusual.Greenstheoremisnowalmosttrivial.Summingupthecurlovera regionisthelineintegralalongtheboundary. Onecanpushthediscretisationfurtherbyassumingthatthefunctionstakevaluesinaniteset.Theintegral theoremsstillworkinthatcasetoo. QUANTUMMULTIVARIABLECALCULUS.Quantumcalculusis"calculuswithouttakinglimits".Thereare indicationsthatspaceandtimelookdierentatthemicroscopicsmall,thePlanckscaleoftheorderh.Oneof theideastodealwiththissituationistointroducequantumcalculuswhichcomesindierenttypes.Wediscuss q-calculuswherethederivativeisdenedas D q f(x)=d q f(x)=d q (x) with d q f(x)=f(qx)f(x) .Youcan seethatD q x n =[n]x n1 ,where[n]= q n 1 q1 .Asq!1whichcorrespondstothedeformationofquantum mechanicswithh!0toclassicalmechanics,wehave[n]!n.Therearequantumversionsfordierentiation ruleslikeD q (fg)(x)=D q f(x)g(x)+f(qx)D q g(x)butquantumcalculusismorefriendlytostudentsbecause thereisnosimplechainrule. Once,wecandierentiate,wecantakeanti-derivatives.ItisdenotedbyRf(x)d q (x).Asweknowthederivative ofx n ,wehaveRx n dq(x)=P n a n x n+1 =[n+1]+CwithaconstantC.Theanti-derivativeofageneralfunction isaseries

Rf(x)d

q (x)=(1q)xP 1 j=0 q j f(q j x)

Forfunctionsf(x)M=x

 with0<1,theintegral isdened.Withananti-derivative,therearedeniteintegrals.Thefundamentaltheoremofq-calculusR b a f(x)d q (x)=F(b)F(a)whereFistheanti-derivativeoffstillholds.Manyresultsgeneralizetoq-calculus liketheTaylortheorem.AbookofKacandCheungisabeautifulreadaboutthat. Multivariablecalculusanddierentialequationscanbedevelopedtoo.Ahandicapisthelackofachainrule. Forexample,todenealineintegral,wewouldhavetodeneRFd q rinsuchawaythatRrfdr q =f(r(b)) f(r(a)).Inquantumcalculus,thenaivedenitionofthelengthofacurvedependsontheparameterizationof thecurve.SurpriseswithquantumdierentialequationsQODEd q f=fwhichisf(qx)f(x)=(q1)f(x) simpliestof(qx)=qf(x).Ithassolutionsf(x)=ax,whereaisaconstantaswellasfunctionsobtained bytakinganarbitraryfunctiong(t)ontheinterval[1;q)satisfyingf(q)=qf(1)andextendingittotheother intervals[q k ;q k+1 ]usingtherulef(q k x)=q k f(x).Thesesolutionsgrowlinearly.Wehaveinnitelymany solutions. INFINITEDIMENSIONALCALCULUS.Calculusininnitedimensionsiscalledfunctionalanalysis.Func- tionsoninnitedimensionalspaceswhicharealsocalledfunctionalsforwhichthegradientcanbedened.The lateristheanalogueofD u f.OnedoesnotalwayshaveD u f=rf
u.Anexampleofaninnitedimensional spaceisthesetXofallcontinuousfunctionsontheunitinterval.Anexampleofatwodimensionalsurfacein thatspacewouldber(u;v)=(cos(u)sin(v)x 2 +sin(u)sin(v)cos(x)+cos(v)=(1+x 2 ).Thissurfaceisactually atwodimensionalsphere.OnthisspaceXonecandeneadotproductf
g=Rf(x)g(x)dx. Thetheorywhichdealswiththeproblemofextremizingfunctionalsininnitedimensionsiscalledcalculusof variations.Thereareproblemsinthiseld,whichactuallycanbeansweredwithintherealmofmultivariable calculus.Forexample:aclassicalproblemistondamongallclosedregionswithboundaryoflength1theone whichhasmaximalarea.Thesolutionisthecircle.Oneoftheproofswhichwasfoundmorethan100yearsago usesGreenstheorem.Onecanalsolookattheproblemtondthepolygonwithnedgesandlength1which hasmaximalarea.ThisisaLagrangeextremizationproblemwithregularpolygonsassolutions.Inthelimit whenthenumberofpointsgotoinnity,oneobtainstheisoperimetricinequality.Otherproblemsinthe calculusofvariationsarethesearchfortheshortestpathbetweentwopointsinahillyregion.Thisshortest pathiscalledageodesic.Alsohere,onecanndapproximatesolutionsbyconsideringpolygonsandsolving anextremizationproblembutdirectmethodsinthattheoryarebetter. GENERALIZEDCALCULUS.Inphysics,onewantstodealwithobjectswhicharemoregeneralthanfunctions.

Forexample,thevectoreldF(x;y)=(y;x)=(x

2 +y 2 )hasitscurlconcentratedontheorigin(0;0).Thisis anexampleofanobjectwhichisadistribution.AnexampleofsuchaSchwartzdistributionisa"function" fwhichisinniteat0,zeroeverywhereelse,butwhichhasthepropertythatRfddx=1.Itiscalled theDiracdeltafunction.Mathematically,onedenesdistributionsasalinearmaponaspaceofsmooth "testfunctions"whichdecayfastatinnity.Onewrites(f;)forthis.Forcontinuousfunctionsonehas (f;)=R 1 1 f(x)(x)dx,fortheDiracdistributiononehas(f;)=(0).Onewoulddenethederivative ofadistributionas(f 0 ;)=f( 0 ).ForexamplefortheHeavysidefunctionH(x)=0x<0

1x>0onehas

(H;)=R 1 1 (x)dxandbecause(H; 0 )=R 1 1  0 (x)dx= 0 (0)onehas(H 0 ;)=(0)theDiracdelta function.TheDiracdeltafunctionisstillwhatonecallsameasure,anobjectwhichappearsalsoinprobability theory.However,ifwedierentiatetheDiracdeltafunction,weobtain(D 0 ;)=(D; 0 )= 0 (0).Thisis anobjectwhichcannomorebeseenasaprobabilitydistributionandisanewtruly"generalized"function. COMPLEXCALCULUSCalculusinthecomplexiscalledcomplexanalysis.Manythingswhichareabit mysteriousintherealbecomemoretransparentwhenconsideredinthecomplex.Forexample,complexanalysis helpstosolvesomeintegrals,itallowstosolvesomeproblemsintherealplanebetter.Herearesomeplaces wherecomplexanalysiscouldhavehelpedus:tondharmonicfunctions,onecantakeanicefunctioninthe complexlikez 4 =(x+iy) 4 =x 4 x 2 y 2 +y 4 +i(x 3 y+xy 3 )andlookatitsrealandcomplexpart.These areharmonicfunctions.Dierentiationinthecomplexisdenedasintherealbutsincethecomplexplaneis twodimensional,oneasksmore:f(z+dz)f(z)=dzhastoexistandbeequalforeverydz!0.Onewrites @ z =(@ x i@ y )=2andcomplexdierentiationsatisesalltheknownpropertiesfromdierentiationonthe realline.Forexampled=dzz n =nz n1 .Multivariablecalculusverymuchhelpsalsotointegratefunctions inthecomplex.Again,becausethecomplexplaneistwodimensional,onecanintegratealongpathsandthe complexintegralR C f(z)dzisactuallyalineintegral.Ifz(t)=x(t)+iy(t)isthepathandf=u+ivthen R C f(z)dz=R b a (ux 0 vy 0 )dt+iR b a (uy 0 +vx 0 )dt .Greenstheoremforexampleistheeasiestwaytoderive theCauchyintegraltheoremwhichsaysthatR C f(z)dz=0ifCistheboundaryofaregioninwhichfis dierentiable.Integrationinthecomplexisusefulforexampletocomputedeniteintegrals. Onecanask,whethercalculuscanalsobedoneinothernumbersystemsbesidestherealsandthecomplex numbers.Theanswerisyes,onecandocalculususingquaternions,octonionsoroverniteeldsbuteachof themhasitsowndiculty.QuaternionmultiplicationalreadydoesnomorecommuteAB6=BAandoctonions multiplicationisevennomoreassociative(AB)C6=A(BC).Inordertodocalculusinniteelds,dierentiation willhavetobereplacedbydierences.AFinnishmathematicianKustaanheimopromotedaround1950anite geometricalapproachwiththeaimtodorealphysicsusingaverylargeprimenumber.Theseideaswerelater mainlypickedupbyphilosophers.Itisactuallyamatteroffactthatthenaturalnumbersarethemost complexandthecomplexnumbersarethemostnatural. CALCULUSINHIGHERDIMENSIONS.Whenextendingcalculustohigherdimensions,the conceptofvectorelds,functionsanddierentiationsgrad,curl,divarereorganizedbyintroduc- ingdierentialforms.Foraninteger0kn,denek-formsasobjectsoftheform =P I a I dx I =P i 1 Inthreedimensions,wherewewritedx=dx 1 ;dy=dx 2 anddz=dx 3 : k=form=d=0ff x dx+f y dy+f z dz

1Mdx+Ndy+Pdz(N

x M y )dx^dy+(P x M z )dx^dz+(P y N z )dy^dz 2Adx 1 ^dx 2 +Bdx 1 ^dx 3 +Cdx 2 ^dx 3 (A z B y +C x )dx^dy^dz 3gdx 1 ^dx 2 ^dx 3 0. (ThecomputationofthecurlwasN x dx^dy+P x dx^dz+M y dy^dx+P y dy^dz+M z dz^dx+N z dz^dy= (N x M y )dx^dy+(P x M z )dx^dz+(P y N z )dy^dz.)Afteranidenticationof0with3formsand

1with2forms(calledHodge*operation)onecanseetheexteriorderivativesasgradient,curlanddivergence:

Itisusefulalsotoclarifythenotionofasurfacebydeningmanifolds.Manydenitionswhichwehave seenforcurvesorsurfacescanbeextendedtomanifolds.Thedotproductcanbedenedmoregenerallyon manifoldsandleadstoatheorycalledRiemanniangeometryusedinthetheoryofgeneralrelativity.When learningrelativity,onedealswith4-dimensionalmanifoldswhichincorporatebothspaceandtime.There,the languageofdierentialformsisalreadymandatory.Theexteriorderivativeofa1formisa2formwhichhas

6component.Anexampleofsucha2-formistheelectromagneticeldF=dAcombining3electricand3

magneticcomponents.TheMaxwellequationsaredF=0;d 

F=I,whered

 =disdenedusingthe

Hodge*operation.Theconsequenced

 dA=Ibecomesthewaveequationd  dA=0intheabsenceofIa vectorincorporatingbothelectricchargeandcurrenti. FRACTALCALCULUS.Wehavedealtinthiscourseswith0-dimensionalobjects(points),1-dimensional objects(curves),2-dimensionalobjects(surfaces)aswellas3-dimensionalobjects(solids).Sincemorethan

100years,mathematiciansalsostudiedfractals,objectswithnon-integerdimensionarecalledfractals.An

exampleistheKochsnow akewhichisobtainedasalimitbyrepeatedstellationofanequilateraltriangleof initialarc-lengthA=3.Afteronestellation,thelengthhasincreasedbyafactor4=3.Afternsteps,thelength ofthecurveisA(4=3) n .Thedimensionislog(4=3)>1.Therearethingswhichdonomoreworkhere.For example,thelengthofthiscurveisinnite.Alsothecurvehasnodenedvelocityatallplaces.Onecanask whetheronecouldapplyGreenstheoremstillinthiscase.Insomesense,thisispossible.Aftereverynitestep ofthisconstructiononecancomputethelineintegralalongthecurveandGreenstheoremtellsthatthisisa doubleintegralofcurl(F)overtheregionenclosedbythecurve.Sinceforlargerandlargen,lessandlessregion isaddedtothecurve,theintegralRRcurl(F)dxdyisdenedinthelimit.So,onecandenethelineintegral alongthecurve.Closelyrelatedtofractaltheoryisgeometricmeasuretheory,whichisageneralization ofdierentialgeometrytosurfaceswhicharenomoresmooth.Merginginideasfromgeneralizedfunctions anddierentialforms,onedenescurrents,functionalsonsmoothdierentialforms.Thetheoryisusefulfor studyingminimalsurfaces.Fractalsappearnaturallyindierentialequationsasattractors.Themostinfamous fractalisprobablytheMandelbrotset,whichisdenedasthesetofcomplexnumberscforwhichtheiteration ofthemapf(z)=z 2 +cstartingwith0leadstoaboundedsequence0!c!c 2 +c!(c 2 +c) 2 +c:::.The boundaryoftheMandelbrotsetisactuallynotafractal.Itissowigglythatitsdimensionisactually2.One canmodifytheKochsnow aketogetdimension2too.Ifoneaddsnewtriangleseachtimesothatthelength ofthenewcurveisdoubledeachtime,thenthedimensionoftheKochcurveis2too. THEFOUNDATIONSOFCALCULUS.Woulditbepossiblethatalienssomewhereelsewouldbuildup mathematicsradicallydierent,bystartingwithadierentaxiomsystem?Itislikely.Thereasonisthat alreadyweknowthatthereisnotasinglewaytobuildupmathematicaltruth.Wehavesomechoice:it cameasashockaroundthemiddleofthelastcenturythatforanystrongenoughmathematicaltheory,one canndstatementswhicharenotprovablewithinthatsystemandwhichonecaneitheracceptasanew axiomoracceptthenegationasanewaxiom.Evensomeoftherespectedaxiomsarealreadyindependent ofmoreelementaryones.OneofthemistheaxiomofchoicewhichsaysthatforanycollectionCof nonemptyset,onecanchoosefromeachsetanelementandformanewset.Aconsequenceofthisaxiom whichisclosetocalculusisa"compactnessproperty"forfunctionsontheinterval.Ifwetakethedistance d(f;g)=max(f(x)g(x))betweentwofunctionsandtakeasequenceofdierentiablefunctionssatisfying jf n (x)jMandjf 0n (x)jMthenthereexistsasubsequencef n k whichconvergestoacontinuousfunction. ArathercounterintuitiveconsequenceoftheAxiomofchoiceisthatonecandecomposetheunitballinto

5piecesE

i ,movethosepiecesusingrotationsandtranslationsandreassemblethemtoformtwocopiesof theunitball.WecannotintegrateRRR E i dVoverthisregions:thesumoftheirvolumeswouldbeeither

4=3or8=3dependingonwhetherweintegratebeforeorafterthearrangement.Thisissuchaparadoxical

constructionthatonecallsitaparadox:theBanach-Taskiparadox. Anotheraxiomforwhichisnotclear,whetheritmattersincalculusisthecontinuumhypothesiswhichtellsthat thereisnocardinalitybetweenthe"innity"ofthenaturalnumbers@ 0 andthe"innity"oftherealnumbers 1 .

Ithadbeenrealizedinthe19'thcenturybyCantor

thatthesetwoinnitiesaredierent.Cantorsar- gumentistoassumethatonecouldenumerateall numbersbetween0and1:thenlookatthediagonal number,inwhicheachdigitisaltered.Forexample: x=0:35285:::: .Thisrealnumberxdisagreeswith allthenumbersinthelistandwasthereforenotac- countedforinthecounting.10: 2

4231423412341234134:::

20:3 4

4223498273413904173:::

30:69
1

4341074147346738874:::

40:999

7

4283382464200104131:::

50:3620

4

4747389238934211147:::

Thepossibilitiestoquestionoralterthemathematicalbuildupdoesnotstoponthesettheoreticallevel. Peopleeventriedtochangelogic.Amongthingswhichhavebeenproposedarefuzzylogic,wheretruthcan takeavaluebetween0and1orquantumlogicoralogicinwhichonehasthreepossibilities,true,nottrueor "undecided". Onenotonlyhasthepossibilitytoextendmathematicsdierently.Aratherrichplaygroundcanbecovered byrestrictingtools.Onecanforexampleask,whatpartofcalculuscanstillbedoneinwithouttheaxiom ofchoice.Onecanalsoaskthatoneonlyisallowedtotalkaboutobjectswhichcanbeconstructedexplicitly. Therewerepeople,strictnitists,whowentevenfurtherandsuggestedtodisqualifytoolargenumberslike 10 10 10 .Calculusteachershaveeventogofurtherandproduceproblemssothattheansweris1or0oror

1=3.Problemsinnalexams,wheretheansweris1234=12isunthinkable.Thephilosophicaldirectionteachers

areforcedtofollowiscalledultrastrictnitism...

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