Calculus in 3D Geometry Vectors and Multivariate Calculus 1st Edition Zbigniew Nitecki
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Calculusin3D
Geometry,Vectors, andMultivariateCalculus
ZbigniewNitecki
CommitteeonBooks
JenniferJ.Quinn,Chair
MAATextbooksEditorialBoard
StanleyE.Seltzer,Editor
WilliamRobertGreen,Co-Editor
BelaBajnok SuzanneLynneLarsonJeffreyL.Stuart MatthiasBeck JohnLorch RonD.Taylor,Jr. HeatherAnnDyeMichaelJ.McAseyElizabethThoren CharlesR.HamptonVirginiaNoonburgRuthVanderpool
2010 MathematicsSubjectClassification.Primary26-01.
Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/text-40
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Names:Nitecki,Zbigniew,author.
Title:Calculusin3D:Geometry,vectors,andmultivariatecalculus/ZbigniewNitecki.
Description:Providence,RhodeIsland:MAAPress,animprintoftheAmericanMathematicalSociety, [2018] | Series:AMS/MAAtextbooks;volume40 | Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018020561 | ISBN9781470443603(alk.paper)
Subjects:LCSH:Calculus–Textbooks. | AMS:Realfunctions–Instructionalexposition(textbooks, tutorialpapers,etc.).msc
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5.5OrientedSurfacesandFluxIntegrals
A.1DifferentiabilityintheImplicitFunctionTheorem
A.3ThePrincipalAxisTheorem
A.4DiscontinuitiesandIntegration
A.5LinearTransformations,Matrices,andDeterminants
Preface
Thepresentvolumeisasequeltomyearlierbook, CalculusDeconstructed:ASecond CourseinFirst-YearCalculus,publishedbytheMathematicalAssociationofAmerica in2009.IhaveusedversionsofthispairofbooksforseveralyearsintheHonorsCalculuscourseatTufts,atwo-semester“bootcamp”intendedformathematicallyinclined freshmenwhohavebeenexposedtocalculusinhighschool.Thefirstsemesterofthis course,usingtheearlierbook,coverssingle-variablecalculus,whilethesecondsemester,usingthepresenttext,coversmultivariatecalculus.However,thepresentbookis designedtobeabletostandaloneasatextinmultivariatecalculus.
Thetreatmentherecontinuesthebasicstanceofitspredecessor,combininghandsondrillintechniquesofcalculationwithrigorousmathematicalarguments.Nonetheless,therearesomedifferencesinemphasis.Ononehand,thepresenttextassumes ahigherlevelofmathematicalsophisticationonthepartofthereader:thereisnoexplicitguidanceintherhetoricalpracticesofmathematicians,andthetheorem-proof formatisfollowedalittlemorebrusquelythanbefore.Ontheotherhand,thematerial beingdevelopedhereisunfamiliarterritoryfortheintendedaudiencetoafargreater degreethanintheprevioustext,somoreeffortisexpendedonmotivatingvariousapproachesandprocedures,andasubstantialnumberoftechnicalargumentshavebeen separatedfromthecentraltext,asexercisesorappendices.
Wherepossible,Ihavefollowedmyownpredilectionforgeometricargumentsover formalones,althoughthetwoperspectivesarenaturallyintertwined.Attimes,this mayfeellikeananalysistext,butIhavestudiouslyavoidedthetemptationtogivethe general, ��-dimensionalversionsofargumentsandresultsthatwouldseemnaturalto amaturemathematician:thebookis,afterall,aimedatthemathematicalnovice,and Ihavetakenseriouslythelimitationimpliedbythe“3D”inmytitle.Thishastheadvantage,however,thatmanyideascanbemotivatedbynaturalgeometricarguments. Ihopethatthisapproachlaysagoodintuitivefoundationforfurthergeneralization thatthereaderwillseeinlatercourses.
Perhapsthefundamentalsubtextofmytreatmentisthewaythatthetheorydevelopedearlierforfunctionsofonevariableinteractswithgeometrytohandlehigherdimensionsituations.Theprogressionhere,afteraninitialchapterdevelopingthe toolsofvectoralgebraintheplaneandinspace(includingdotproductsandcross products),istofirstviewvector-valuedfunctionsofasinglerealvariableintermsof parametrizedcurves—here,muchofthetheorytranslatesverysimplyinacoordinatewiseway—thentoconsiderreal-valuedfunctionsofseveralvariablesbothasfunctions withavectorinputandintermsofsurfacesinspace(andlevelcurvesintheplane), andfinallytovectorfieldsasvector-valuedfunctionsofvectorvariables.
Idiosyncracies
Thereareanumberofways,someapparent,someperhapsmoresubtle,inwhichthis treatmentdiffersfromthestandardones:
ConicSections: Ihaveincludedin§2.1atreatmentofconicsections,startingwitha versionofApollonius’sformulationintermsofsectionsofadoublecone(andexplainingtheoriginofthenames parabola, hyperbola,and ellipse),thendiscussing
Preface thefocus-directrixformulationfollowingPappus,andfinallysketchinghowthis leadstothebasicequationsforsuchcurves.Ihavetakenaquasi-historicalapproachhere,tryingtogiveanideaoftheclassicalGreekapproachtocurveswhich contrastssomuchwithourcontemporarycalculus-basedapproach.ThisisanexampleofaplacewhereIthinksomehistoricalcontextenrichesourunderstanding ofthesubject.Thiscanbetreatedasoptionalinclass,butIpersonallyinsiston spendingatleastoneclassonit.
Parametrization: Ihavestressedtheparametricrepresentationofcurvesandsurfacesfarmore,andbeginningsomewhatearlier,thanmanymultivariatetexts. Thisapproachisessentialforapplyingcalculustogeometricobjects,anditisalso abeautifulandsatisfyinginterplaybetweenthegeometricandanalyticpointsof view.WhileChapter 2 beginswithatreatmentoftheconicsectionsfromaclassicalpointofview,thisisfollowedbyacatalogueofparametrizationsofthese curvesand,in§ 2.4,byamorecarefulstudyofregularcurvesintheplaneand theirrelationtographsoffunctions.Thisleadsnaturallytotheformulationof pathintegralsin§ 2.5.Similarly,quadricsurfacesareintroducedin§ 3.4 aslevel setsofquadraticpolynomialsinthreevariables,andthe(three-dimensional)ImplicitFunctionTheoremisintroducedtoshowthatanysuchsurfaceislocallythe graphofafunctionoftwovariables.Thenotionofparametrizationofasurface isthenintroducedandexploitedin§ 3.6 toobtainthetangentplanesofsurfaces. Whenwegettosurfaceintegralsin§4.4,thisgivesanaturalwaytodefineandcalculatesurfaceareaandsurfaceintegralsoffunctions.Thisapproachcomestofull fruitioninChapter5intheformulationoftheintegraltheoremsofvectorcalculus.
LinearAlgebra: Linearalgebraisnotstrictlynecessaryforproceduralmasteryof multivariatecalculus,butsomeunderstandingoflinearity,linearindependence, andthematrixrepresentationoflinearmappingscanilluminatethe“hows”and “whys”ofmanyprocedures.Most(butnotall)ofthestudentsinmyclasshave alreadyencounteredvectorsandmatricesintheirhighschoolcourses,butfewof themunderstandthesemoreabstractconcepts.Inthecontextoftheplaneand 3-spaceitispossibletointerpretmanyofthesealgebraicnotionsgeometrically, andIhavetakenfulladvantageofthispossibilityinmynarrative.Ihaveintroducedtheseideaspiecemeal,andincloseconjunctionwiththeirapplicationin multivariatecalculus.
Forexample,in§ 3.2,thederivative,asalinearreal-valuedfunction,canbe representedasahomogeneouspolynomialofdegreeoneinthecoordinatesofthe input(asinthefirstTaylorpolynomial),asthedotproductofthe(vector)input withafixedvector(thegradient),orasmultiplyingthecoordinatecolumnofthe inputbyarow(a1×�� matrix,thematrixofpartials).Thenin§4.3and§4.5,substitutioninadoubleortripleintegralisinterpretedasacoordinatetransformation whoselinearizationisrepresentedbytheJacobianmatrix,andwhosedeterminant reflectstheeffectofthistransformationonareaorvolume.InChapter5,differentialformsareconstructedas(alternating)multilinearfunctionals(buildingonthe differentialofareal-valuedfunction)andinvestigationoftheireffectonpairsor triplesofvectors—especiallyinviewofindependenceconsiderations—ultimately leadstothestandardrepresentationoftheseformsviawedgeproducts.
Asecondexampleisthedefinitionof2×2and3×3determinants.Thereseemto betwoprevalentapproachesintheliteraturetointroducingdeterminants:oneis formal,dogmaticandbrief,simplygivingarecipeforcalculationandproceeding fromtherewithlittlemotivationforit;theotherisevenmoreformalbutelaborate, usuallyinvolvingthetheoryofpermutations.IbelieveIhavecomeupwithanapproachtointroducing 2×2 and 3×3 determinants(alongwithcross-products) whichisbothmotivatedandrigorous,in§1.6.Startingwiththeproblemofcalculatingtheareaofaplanartrianglefromthecoordinatesofitsvertices,wededucea formulawhichisnaturallywrittenastheabsolutevalueofa2×2determinant;investigationofthedeterminantitselfleadstothenotionofsigned(i.e.,oriented) area(whichhasitsowncharm,andprophesiestheintroductionof2-formsin Chapter 5).Goingtotheanalogousprobleminspace,weintroducethenotionof anorientedarea,representedbyavector(whichweultimatelytakeasthedefinitionofthecross-product,anapproachtakenforexamplebyDavidBressoud).We notethatorientedareasprojectnicely,andfromtheprojectionsofanorientedarea vectorontothecoordinateplanescomeupwiththeformulaforacross-productas theexpansionbyminorsalongthefirstrowofa 3×3 determinant.Inthepresent treatment,variousalgebraicpropertiesofdeterminantsaredevelopedasneeded, andtherelationtolinearindependenceisarguedgeometrically.
VectorFieldsvs.DifferentialForms: Anumberofrelativelyrecenttreatmentsof vectorcalculushavebeenbasedexclusivelyonthetheoryofdifferentialforms, ratherthanthetraditionalformulationusingvectorfields.Ihavetriedthisapproachinthepast,butinmyexperienceitconfusesthestudentsatthislevel,so thattheyendupdealingwiththetheoryonablindlyformalbasis.Bycontrast,I finditeasiertomotivatetheoperatorsandresultsofvectorcalculusbytreatinga vectorfieldasthevelocityofamovingfluid,andsohaveusedthisasmyprimary approach.However,theformalismofdifferentialformsisveryslickasacalculationaldevice,andsoIhavealsointroduceditinterwovenwiththevectorfield approach.Themainstrengthofthedifferentialformsapproach,ofcourse,isthat itgeneralizestodimensionshigherthan3;whileIhintatthis,itisoneplacewhere myself-imposedlimitationto“3D”isevident.
Appendices: Mygoalinthisbook,asinitspredecessor,istomakeavailabletomy studentsanessentiallycompletedevelopmentofthesubjectfromfirstprinciples, inparticularpresenting(oratleastsketching)proofsofallresults.Ofcourse,it isphysically(andcognitively)impossibletoeffectivelypresenttoomanytechnicalargumentsaswellasnewideasintheavailableclasstime.Ihavetherefore (adoptingapracticeusedbyamongothersJerryMarsdeninhisvarioustextbooks) relegatedtoexercisesandappendices1 anumberoftechnicalproofswhichcanbest beapproachedonlyaftertheresultsbeingprovenarefullyunderstood.Thishas theadvantageofstreamliningthecentralnarrative,and—toberealistic—bringing itclosertowhatthestudentwillexperienceintheclassroom.Itismyexpectation that(dependingonthepreferenceoftheteacher)mostoftheseappendiceswillnot bedirectlytreatedinclass,buttheyarethereforreferenceandmaybereturned tolaterbythecuriousstudent.Thisformatcomportswiththeactualpracticeof mathematicianswhenconfrontinganewresult:weallbeginwithaquickskim
1Specifically,Appendices A.1-A.2, A.4, A.6-A.7, A.9-A.10,and A.12.
Preface focusedonunderstandingthestatementoftheresult,followedbyseveral(often, verymany)re-readingsfocusedonunderstandingtheargumentsinitsfavor. Theotherappendicespresentextramaterialwhichfillsoutthecentralnarrative:
• AppendixA.3presentsthePrincipalAxisTheorem,thateverysymmetricmatrixhasanorthonormalbasisofeigenvectors.Togetherwiththe(optional) lastpartof§ 3.9,thiscompletesthetreatmentofquadraticformsinthree variablesandsojustifiestheSecondDerivativeTestforfunctionsofthreevariables.Thetreatmentofquadraticformsintermsofmatrixalgebra,whichis notnecessaryforthebasictreatmentofquadraticformsintheplane(where completionofthesquaresuffices),doesallowfortheproof(inExercise 4)of thefactthatthelocusofaquadraticequationintwovariableshasasitslocus aconicsection,apoint,aline,twointersectinglinesortheemptyset. IamparticularlyfondoftheproofofthePrincipalAxisTheoremitself,which isawonderfulexampleofsynergybetweenlinearalgebraandcalculus(Lagrangemultipliers).
• AppendixA.5presentsthebasicfactsaboutthematrixrepresentation,invertibility,andoperatornormofalineartransformation,andageometricargumentthatthedeterminantofaproductofmatricesistheproductoftheir determinants.
• Appendix A.8 presentstheexampleofH.SchwartzandG.Peanoshowing howthe“natural”extensiontosurfaceareaofthedefinitionofarclengthvia piecewiselinearapproximationsfails.
• Appendix A.11 clarifiestheneedfororientabilityassumptionsbypresenting theMöbiusband.
Format
Ingeneral,Ihavecontinuedtheformatofmypreviousbookinthisone. Asbefore, exercises comeinfourflavors:
PracticeProblems: serveasdrillincalculation.
TheoryProblems: involvemoreideas,eitherfillingingapsintheargumentinthe textorextendingargumentstoothercases.Someoftheseareabitmoresophisticated,givingdetailsofresultsthatarenotsufficientlycentraltotheexpositionto deserveexplicitproofinthetext.
ChallengeProblems: requiremoreinsightorpersistencethanthestandardtheory problems.Inmyclass,theyareentirelyoptional,extra-creditassignments.
HistoricalNotes: exploreargumentsfromoriginalsources.Therearemuchfewerof thesethaninthepreviousvolume,inlargepartbecausethehistoryofmultivariate calculusisnotnearlyaswelldocumentedandstudiedasisthehistoryofsinglevariablecalculus.Nonetheless,Istronglyfeelthatweshouldstrivemorethanwe havetopresentmathematicsinatleastacertainamountofhistoricalcontext:I believethatitisveryhelpfultostudentstorealizethatmathematicsisanactivity byrealpeopleinrealtime,andthatourunderstandingofmanymathematical phenomenahasevolvedovertime.
Acknowledgments
Aswiththepreviousbook,IwanttothankJasonRichards,whoasmygraderinthis courseoverseveralyearscontributedmanycorrectionsandusefulcommentsabout thetext.Afterhegraduated,severalotherstudentgraders—ErinvanErp,Thomas Snarsky,WenyuXiong,andKiraSchuman—madefurtherhelpfulcomments.Ialso affectionatelythankmystudentsoverthepastfewyears,particularlyMattRyan,who notedalargenumberoftyposandminorerrorsinthe“beta”versionofthisbook.I havebenefitedgreatlyfrommuchhelpwithTEXpackagesespeciallyfromthee-forum onpstricksandpst-3DsolidsrunbyHerbertVoss,aswellasthe“TeXonMacOSX” elist.MycolleagueLoringTuhelpedmebetterunderstandtheroleoforientationinthe integrationofdifferentialforms.Onthehistoryside,SandroCapparinihelpedintroducemetotheearlyhistoryofvectors,andLenoreFeigenbaumandespeciallyMichael N.FriedhelpedmewithsomevexingquestionsconcerningApollonius’classification oftheconicsections.ScottMaclachlanhelpedmethinkthroughseveralsomewhat esotericbutusefulresultsinvectorcalculus.Asalways,whatispresentedhereismy owninterpretationoftheircomments,andisentirelymypersonalresponsibility.
CoordinatesandVectors
1.1LocatingPointsinSpace
RectangularCoordinatesinthePlane. Thegeometryofthenumberline ℝ isquitestraightforward:thelocationofarealnumber �� relativetoothernumbersis determined—andspecified—bytheinequalitiesbetweenitandothernumbers ��′:if ��<��′,then�� istotheleft of��′,andif��>��′,then�� istotheright of��′.Furthermore, the distance between �� and ��′ isjustthedifference △��=��′−�� (resp. ��−��′)inthe first(resp.second)case,asituationsummarizedasthe absolutevalue
Whenitcomestopointsintheplane,moresubtleconsiderationsareneeded.The mostfamiliarsystemforlocatingpointsintheplaneisa rectangular or Cartesian coordinatesystem.Wepickadistinguishedpointcalledtheorigin,denoted��,and drawtwomutuallyperpendicularlinesthroughtheorigin,eachregardedasacopyof therealline,withtheorigincorrespondingtozero.Thefirstline,the ��-axis,isby convention horizontal withthe“increasing”directiongoingleft-to-right;thesecond, or ��-axis,is vertical,with“up”increasing.
Givenapoint��intheplane,wedrawarectanglewith��and��asoppositevertices, andthetwoedgesemanatingfrom �� lyingalongouraxes.Thetwoedgesemanating from �� areparalleltotheaxes;eachofthemintersectsthe“other”axisatthepoint correspondingtoanumber �� (resp. ��)onthe ��-axis(resp. ��-axis).1 Wesaythatthe (rectangularorCartesian) coordinates of �� arethetwonumbers (��,��).
Weadoptthenotation ℝ2 forthecollectionofallpairsofrealnumbers,andthis withthecollectionofallpointsintheplane,referringto“thepoint ��(��,��)”whenwe mean“thepoint �� intheplanewhose(rectangular)coordinatesare (��,��)”.
Theideaofusingapairofnumbersinthiswaytolocateapointintheplanewas pioneeredintheearlyseventeenthcenurybyPierredeFermat(1601-1665)andRené Descartes(1596-1650).Bymeansofsuchascheme,aplanecurvecanbeidentified withthe locus ofpointswhosecoordinatessatisfysomeequation;thestudyofcurves byanalysisofthecorrespondingequations,called analyticgeometry,wasinitiated intheresearchofthesetwomen.2
Oneparticularadvantageofarectangularcoordinatesystem(inwhichtheaxes areperpendiculartoeachother)overanobliqueone(axesnotmutuallyperpendicular)isthecalculationofdistances.If �� and �� arepointswithrespectiverectangular coordinates(��1,��1)and(��2,��2),thenwecanintroducethepoint��whichsharesitslast
1Traditionally, �� (resp. ��)iscalledthe abcissa (resp. ordinate)of ��.
2Actually,itisabitofananachronismtorefertorectangularcoordinatesas“Cartesian”,sincebothFermatandDescartesoftenusedobliquecoordinates,inwhichtheaxesmakeanangleotherthanarightone. WeshallexploresomeofthedifferencesbetweenrectangularandobliquecoordinatesinExercise 13.Furthermore,Descartesinparticulardidn’treallyconsiderthemeaningofnegativevaluesforeithercoordinate.
Chapter1.CoordinatesandVectors coordinatewith �� anditsfirstwith ��—thatis, �� hascoordinates (��2,��1).The“legs” ���� and ���� oftherighttriangle △������ areparalleltothecoordinateaxes,whilethe hypotenuse ���� exhibitsthedistancefrom �� to ��;Pythagoras’Theoremthengivesthe distanceformula
Inanobliquesystem,theformulabecomesmorecomplicated(Exercise 13).
RectangularCoordinatesinSpace. Therectangularcoordinateschemeextendsnaturallytolocatingpointsinspace.Weagaindistinguishonepoint �� asthe origin,andconstructarectangularcoordinatesystemonthehorizontalplanethrough it(the����-plane),anddrawathird��-axisverticallythrough��.Apoint�� islocatedby thecoordinates �� and �� ofthepoint ������ inthe ����-planethatliesontheverticalline through ��,togetherwiththenumber �� correspondingtotheintersectionofthe ��-axis withthehorizontalplanethrough��.The“increasing”directionalongthe��-axisisdefinedbythe right-handrule:ifourrighthandisplacedattheoriginwiththe ��-axis comingoutofthepalmandthefingerscurlingtowardthepositive ��-axis,thenour rightthumbpointsinthe“positive ��”direction.Notethestandingconventionthat, whenwedrawpicturesofspace,weregardthe��-axisaspointingtowardus(orslightly toourleft)outofthepage,the ��-axisaspointingtotherightalongthepage,andthe ��-axisaspointingupalongthepage(Figure 1.1).
PicturesofSpace
Thisleadstotheidentificationofthesetℝ3 oftriples(��,��,��)ofrealnumberswith thepointsofspace,whichwesometimesrefertoas three-dimensionalspace (or 3-space).
Asintheplane,thedistancebetweentwopoints ��(��1,��1,��1) and ��(��2,��2,��2) in ℝ3 canbecalculatedbyapplyingPythagoras’Theoremtotherighttriangle������,where ��(��2,��2,��1) sharesitslastcoordinatewith �� anditsothercoordinateswith ��.Details arelefttoyou(Exercise 11);theresultingformulais
Inwhatfollows,wewilldenotethedistancebetween �� and �� by dist(��,��)
Figure1.1.
PolarandCylindricalCoordinates. Rectangularcoordinatesarethemostfamiliarsystemforlocatingpoints,butinproblemsinvolvingrotations,itissometimes convenienttouseasystembasedonthedirectionanddistancetoapointfromthe origin.
Intheplane,thisleadsto polarcoordinates.Givenapoint �� intheplane,think ofthelineℓthrough��and��asacopyoftherealline,obtainedbyrotatingthe��-axis�� radianscounterclockwise;then �� correspondstotherealnumber �� on ℓ.Therelation ofthe polar coordinates (��,��) of �� to rectangular coordinates (��,��) isillustratedin
Figure 1.2,fromwhichweseethat
Figure1.2. PolarCoordinates
ThederivationofEquation(1.3)fromFigure 1.2 requiresapinchofsalt:wehave drawn �� asanacuteangleand ��, ��,and �� aspositive.Butinourinterpretationof ℓ asarotatedcopyofthe ��-axis(and �� asthenetcounterclockwiserotation)allpossible configurationsareaccountedfor,andtheformularemainstrue.
Whileagivengeometricpoint��hasonlyonepairofrectangularcoordinates(��,��), ithasmanypairsofpolar coordinates.Thusif(��,��)isonepairofpolarcoordinatesfor ��thensoare(��,��+2����)and(−��,��+(2��+1)��)foranyinteger��(positiveornegative). Also, ��=0 preciselywhen �� istheorigin,sothentheline ℓ isindeterminate: ��=0 togetherwith any valueof �� satisfiesEquation(1.3),andgivestheorigin.
Forexample,tofindthepolarcoordinatesofthepoint �� withrectangularcoordinates (−2√3,2),wefirstnotethat ��2 =(−2√3)2 +(2)2 =16.Usingthepositive solutionofthis, ��=4,wehave
Thefirstequationsaysthat �� is,uptoaddingmultiplesof 2��,oneof ��=5��/6 or ��=7��/6;thefactthat sin�� ispositivepicksoutthefirstofthesevalues.Soonesetof polarcoordinatesfor �� is (��,��)=(4,5�� 6 +2����), where �� isanyinteger.Replacing �� withitsnegativeandadding �� totheangle,weget thesecondset,whichismostnaturallywrittenas (−4,−�� 6 +2����).
Chapter1.CoordinatesandVectors
Forproblemsin space involvingrotations(orrotationalsymmetry)aboutasingle axis,aconvenientcoordinatesystemlocatesapoint �� relativetotheoriginasfollows (Figure1.3):if��isnotonthe��-axis,thenthisaxistogetherwiththeline����determine
Figure1.3. CylindricalCoordinates
a(vertical)plane,whichcanberegardedasthe ����-planerotatedsothatthe ��-axis moves��radianscounterclockwise(inthehorizontalplane);wetakeasourcoordinates theangle �� togetherwiththecoordinatesof �� in this plane,whichequalthedistance �� ofthepointfromthe ��-axisandits(signed)distance �� fromthe ����-plane.Wecan thinkofthisasahybrid:combinethe polar coordinates (��,��) oftheprojection ������ withthevertical rectangular coordinate �� of �� toobtainthe cylindricalcoordinates (��,��,��) of ��.Eventhoughinprinciple �� couldbetakenasnegative,inthissystem itiscustomarytoconfineourselvesto ��≥0.Therelationbetweenthecylindrical coordinates (��,��,��) andtherectangularcoordinates (��,��,��) ofapoint �� isessentially givenbyEquation(1.3): ��=��cos��,��=��sin��,��=��. (1.4)
Wehaveincludedthelastrelationtostressthefactthatthiscoordinateisthesamein bothsystems.Theinverserelationsaregivenby
and,forcylindricalcoordinates,thetrivialrelation ��=��.
Thename“cylindricalcoordinates”comesfromthegeometricfactthatthelocus oftheequation ��=�� (whichinpolarcoordinatesgivesacircleofradius �� aboutthe origin)givesaverticalcylinderwhoseaxisofsymmetryisthe ��-axis,withradius ��.
Cylindricalcoordinatescarrytheambiguitiesofpolarcoordinates:apointonthe ��-axishas ��=0 and �� arbitrary,whileapointoffthe ��-axishas �� determinedupto adding even multiplesof �� (since �� istakentobepositive).
SphericalCoordinates. Anothercoordinatesysteminspace,whichisparticularly usefulinproblemsinvolvingrotationsaroundvariousaxesthroughtheorigin(forexample,astronomicalobservations,wheretheoriginisatthecenteroftheearth)isthe systemof sphericalcoordinates.Here,apoint �� islocatedrelativetotheorigin ��
1.1.LocatingPointsinSpace 5
bymeasuringthedistance ��=|����| of �� fromtheorigintogetherwithtwoangles:the angle�� betweenthe����-planeandtheplanecontainingthe��-axisandtheline����,and theangle �� betweenthe(positive) ��-axisandtheline ���� (Figure 1.4).Ofcourse,the
spherical coordinate �� of �� isidenticaltothe cylindrical coordinate ��,andweusethe samelettertoindicatethisidentity.3 While �� issometimesallowedtotakeonallreal values,itiscustomaryinsphericalcoordinatestorestrict �� to 0≤��≤��.Therelation betweenthecylindricalcoordinates (��,��,��) andthesphericalcoordinates (��,��,��) of apoint �� isillustratedinFigure 1.5 (whichisdrawnintheverticalplanedetermined by ��):
Spherical vs. CylindricalCoordinates ��=��sin��,��=��,��=��cos��. (1.6)
Toinverttheserelations,wenotethat,since��≥0and0≤��≤�� byconvention,�� and �� completelydetermine �� and ��: ��=√��2+��2,��=��,��=arccos�� �� . (1.7)
Theambiguitiesinsphericalcoordinatesarethesameasthoseforcylindricalcoordinates:theoriginhas ��=0 andboth �� and �� arbitrary;anyotherpointonthe ��-axis
3Bewarnedthatinsomeoftheengineeringandphysicsliteraturethenamesofthetwosphericalangles arereversed,leadingtopotentialconfusionwhenconvertingbetweensphericalandcylindricalcoordinates.
Figure1.4. SphericalCoordinates
Figure1.5.
Chapter1.CoordinatesandVectors (��=0 or ��=��)hasarbitrary ��,andforpointsoffthe ��-axis, �� can(inprinciple)be augmentedbyarbitraryevenmultiplesof ��.
Thus,thepoint �� withcylindricalcoordinates (��,��,��)=(4, 5�� 6 ,4) hasspherical coordinates (��,��,��)=(4√2,5�� 6 , ��4).
CombiningEquations(1.4)and(1.6),wecanwritetherelationbetweenthespherical coordinates (��,��,��) ofapoint �� andits rectangular coordinates (��,��,��) as ��=��sin��cos��,��=��sin��sin��,��=��cos��. (1.8)
Theinverserelationsareabitmorecomplicated,butclearly,given ��, ��,and ��,
and �� iscompletelydetermined(if ��≠0)bythelastequationin(1.8),while �� is determinedby(1.5)and(1.4).
Insphericalcoordinates,theequation ��=�� describesthesphereofradius �� centeredattheorigin,while ��=�� describesaconewithvertexattheorigin,making anangle �� (resp. ��−��)withitsaxis,whichisthepositive(resp.negative) ��-axisif 0<��<��/2 (resp. ��/2<��<��).
Exercisesfor§ 1.1
AnswerstoExercises1a,2a,3a,and4aaregiveninAppendix A.13.
Practiceproblems:
(1) Findthedistancebetweeneachpairofpoints(thegivencoordinatesarerectangular):
(a) (1,1),(0,0) (b) (1,−1),(−1,1)
(c) (−1,2),(2,5) (d) (1,1,1),(0,0,0)
(e) (1,2,3),(2,0,−1) (f) (3,5,7),(1,7,5)
(2) Whatconditionsonthe(rectangular)coordinates ��,��,�� signifythat ��(��,��,��) belongsto
(a) the ��-axis? (b)the ��-axis? (c) ��-axis?
(d) the ����-plane?(e)the ����-plane?(f)the ����-plane?
(3) Foreachpointwiththegivenrectangularcoordinates,find (i)itscylindricalcoordinatesand (ii)itssphericalcoordinates:
(a) ��=0, ��=1, ��=−1 (b) ��=1, ��=1, ��=1
(c) ��=1, ��=√3, ��=2 (d) ��=1, ��=√3, ��=−2
(e) ��=−√3, ��=1, ��=1 (f) ��=−√3, ��=−1, ��=1
(4) Giventhesphericalcoordinatesofthepoint,finditsrectangularcoordinates:
(a) (��,��,��)=(2, �� 3, �� 2) (b) (��,��,��)=(1, �� 4, 2�� 3 )
(c) (��,��,��)=(2, 2�� 3 , �� 4) (d) (��,��,��)=(1, 4�� 3 , �� 3)
(5) Whatisthegeometricmeaningofeachtransformation(describedincylindrical coordinates)below?
(a) (��,��,��)→(��,��,−��) (b) (��,��,��)→(��,��+��,��)
(c) (��,��,��)→(−��,��− �� 4,��)
(6) Describethelocusofeachequation(incylindricalcoordinates)below:
(a) ��=1 (b) ��= �� 3 (c) ��=1
(7) Whatisthegeometricmeaningofeachtransformation(describedinsphericalcoordinates)below?
(a) (��,��,��)→(��,��+��,��) (b) (��,��,��)→(��,��,��−��) (c) (��,��,��)→(2��,��+ �� 2,��)
(8) Describethelocusofeachequation(insphericalcoordinates)below:
(a) ��=1 (b) ��= �� 3 (c) ��= �� 3
(9) Expresstheplane ��=�� intermsof(a)cylindricaland(b)sphericalcoordinates.
(10) Whatconditionsonthesphericalcoordinatesofapointsignifythatitlieson:
(a) the ��-axis? (b)the ��-axis? (c) ��-axis?
(d) the ����-plane?(e)the ����-plane?(f)the ����-plane?
Theoryproblems:
(11) Provethedistanceformulafor ℝ3 (Equation(1.2))
asfollows(seeFigure 1.6).Given ��(��1,��1,��1) and ��(��
,��
),let �� bethepoint whichsharesitslastcoordinatewith �� anditsfirsttwocoordinateswith ��.Use thedistanceformulain ℝ2 (Equation(1.1))toshowthat dist(��,��)=√(��
andthenconsiderthetriangle △������.Showthattheangleat �� isarightangle, andhencebyPythagoras’Theoremagain,
Challengeproblems:
(12) UsePythagoras’Theoremandtheangle-summationformulastoprovetheLawof Cosines:If ������ isanytrianglewithsides ��=|����|,��=|����|,��=|����| andtheangleat �� is ∠������=��,then ��2 =��2+��2−2����cos��. (1.10)
Figure1.6. Distancein3-Space
Chapter1.CoordinatesandVectors
Hereisonewaytoproceed(seeFigure 1.7)Dropaperpendicularfrom �� to ����,
Figure1.7. LawofCosines
meeting ���� at ��.Thisdividestheangleat �� intotwoangles,satisfying ��+��=�� anddivides ���� intotwointervals,withrespectivelengths |����|=�� and |����|=��, so ��+��=��.Finally,set |����|=��
Nowshowthefollowing: ��=��sin��,��=��sin��,��=��cos��=��cos�� andusethis,togetherwithPythagoras’Theorem,toconcludethat ��2+��2 =��2+��2+2��2 and ��2 =��2+��2+2���� andhence ��2 =��2+��2−2����cos(��+��).
SeeExercise 15 fortheversionofthiswhichappearsinEuclid. (13) ObliqueCoordinates: Consideranobliquecoordinatesystemonℝ2,inwhich theverticalaxisisreplacedbyanaxismakinganangleof �� radianswiththehorizontalone;denotethecorrespondingcoordinatesby (��,��) (seeFigure 1.8).
Figure1.8. ObliqueCoordinates
(a) Showthattheobliquecoordinates(��,��)andrectangularcoordinates(��,��)of apointarerelatedby ��=��+��cos��,��=��sin��.
(b) Showthatthedistanceofapoint �� withobliquecoordinates (��,��) fromthe originisgivenby dist(��,��)=√��2+��2+2����cos��.
(c) Showthatthedistancebetweenpoints �� (withobliquecoordinates (��1,��1)) and �� (withobliquecoordinates (��2,��2))isgivenby dist(��,��)=√△��2+△��2+2△��△��cos��, where△��≔��2−��1 and△��≔��2−��1.(Hint: Therearetwowaystodothis. Oneistosubstitutetheexpressionsfortherectangularcoordinatesinterms
oftheobliquecoordinatesintothestandarddistanceformula,theotheristo usethelawofcosines.Trythemboth.)
Historynote:
(14) Givenarighttrianglewith“legs”ofrespectivelengths �� and �� andhypotenuseof length �� (Figure 1.9) Pythagoras’Theorem saysthat
Figure1.9. Right-angletriangle
Inthisproblem,weoutlinetwoquitedifferentproofsofthisfact.
FirstProof: ConsiderthepairoffiguresinFigure 1.10.
Figure1.10. Pythagoras’TheorembyDissection
(a) Showthatthewhitequadrilateralontheleftisasquare(thatis,showthatthe anglesatthecornersarerightangles).
(b) ExplainhowthetwofiguresprovePythagoras’theorem.
AvariantofFigure 1.10 wasusedbythetwelfth-centuryIndianwriterBhāskara (b.1114)toprovePythagoras’Theorem.Hisproofconsistedofafigurerelatedto Figure 1.10 (withouttheshading)togetherwiththesingleword“Behold!”.
AccordingtoEves[14,p.158]andMaor[36,p.63],reasoningbasedonFigure1.10appearsinoneoftheoldestChinesemathematicalmanuscripts,theCaho PeiSuangChin,thoughttodatefromtheHandynastyinthethirdcenturyBC.
ThePythagoreanTheoremappearsasProposition47,BookIofEuclid’sElements withadifferentproof(seebelow).Inhistranslationofthe Elements,Heathhasan extensivecommentaryonthistheoremanditsvariousproofs[28,vol.I,pp.350368].Inparticular,he(aswellasEves)notesthattheproofabovehasbeensuggestedaspossiblythekindofproofthatPythagorashimselfmighthaveproduced. Evesconcurswiththisjudgement,butHeathdoesnot.
SecondProof: TheproofaboverepresentsonetraditioninproofsofthePythagoreanTheorem,whichMaor[36]calls“dissectionproofs.”Asecondapproach
Chapter1.CoordinatesandVectors isviathetheoryofproportions.Hereisanexample:again,suppose △������ hasa rightangleat��;labelthesideswithlower-caseversionsofthelabelsoftheopposite vertices(Figure 1.11)anddrawaperpendicular ���� fromtherightangletothe hypotenuse.Thiscutsthehypotenuseintotwopiecesofrespectivelengths ��1 and ��2,so
Denotethelengthof ���� by ��
Figure1.11. Pythagoras’TheorembyProportions
(a) Showthatthetwotriangles △������ and △������ arebothsimilarto △������.
(b) Usingthesimilarityof △������ with △������,showthat
(c) Usingthesimilarityof △������ with △������,showthat
(d) NowcombinetheseequationswithEquation(1.11)toprovePythagoras’Theorem.
ThebasicproportionsherearethosethatappearinEuclid’sproofofProposition47,BookIofthe Elements,althoughhearrivesattheseviadifferentreasoning.However,inBookVI,Proposition31,Euclidpresentsageneralizationofthis theorem:drawanypolygonusingthehypotenuseasoneside;thendrawsimilar polygonsusingthelegsofthetriangle;Proposition31assertsthatthesumofthe areasofthetwopolygonsonthelegsequalsthatofthepolygononthehypotenuse. Euclid’sproofofthispropositionisessentiallytheargumentgivenabove.
(15) TheLawofCosinesforanacuteangleisessentiallygivenbyProposition13inBook IIofEuclid’s Elements[28,vol.1,p.406]:
Inacute-angledtrianglesthesquareonthesidesubtendingtheacuteangle islessthanthesquaresonthesidescontainingtheacuteanglebytwicethe rectanglecontainedbyoneofthesidesabouttheacuteangle,namelythat onwhichtheperpendicularfalls,andthestraightlinecutoffwithinbythe perpendiculartowardstheacuteangle.
Translatedintoalgebraiclanguage(seeFigure1.12,wheretheacuteangleis∠������) thissays |����|2 =|����|2+|����|2 |����||����|. ExplainwhythisisthesameastheLawofCosines.
Figure1.12. EuclidBookII,Proposition13
1.2VectorsandTheirArithmetic
Manyquantitiesoccurringinphysicshaveamagnitudeandadirection—forexample, forces,velocities,andaccelerations.Asaprototype,wewillconsider displacements. Supposearigidbodyispushed(withoutbeingrotated)sothatadistinguishedspot onitismovedfromposition �� toposition �� (Figure 1.13).Werepresentthismotion byadirectedlinesegment,orarrow,goingfrom�� to��anddenoted ⃗ ����.Notethatthis arrowencodesalltheinformationaboutthemotionofthewholebody:thatis,ifwehad distinguishedadifferentspotonthebody,initiallylocatedat��′,thenitsmotionwould bedescribedbyanarrow ⃗ ��′��′ parallelto ⃗ ���� andofthesamelength:inotherwords, theimportantcharacteristicsofthedisplacementareits direction and magnitude,but not thelocationinspaceofits initial or terminalpoints (i.e.,its tail or head).
Figure1.13. Displacement
Asecondimportantpropertyofdisplacementisthewaydifferentdisplacements combine.Ifwefirstperformadisplacementmovingourdistinguishedspotfrom �� to �� (representedbythearrow ⃗ ����)andthenperformaseconddisplacementmoving ourspotfrom �� to �� (representedbythearrow ⃗ ����),theneteffectisthesameasifwe hadpusheddirectlyfrom �� to ��.Thearrow ⃗ ���� representingthisnetdisplacementis formedbyputtingarrow ⃗ ���� withitstailattheheadof ⃗ ����anddrawingthearrowfrom thetailof ⃗ ���� totheheadof ⃗ ���� (Figure 1.14).Moregenerally,theneteffectofseveral successivedisplacementscanbefoundbyformingabrokenpathofarrowsplacedtailto-head,andforminganewarrowfromthetailofthefirstarrowtotheheadofthelast.
Arepresentationofaphysical(orgeometric)quantitywiththesecharacteristicsis sometimescalleda vectorialrepresentation.Withrespecttovelocities,the“parallelogramofvelocities”appearsinthe Mechanica,aworkincorrectlyattributedto,but contemporarywith,Aristotle(384-322BC)[25,vol.I,p.344],andisdiscussedatsome lengthinthe Mechanics byHeronofAlexandria(ca. 75AD)[25,vol.II,p.348].The
Figure1.14. CombiningDisplacements
vectorialnatureofsomephysicalquantities,suchasvelocity,accelerationandforce, waswellunderstoodandusedbyIsaacNewton(1642-1727)inthePrincipia[40,Corollary1,Book1(p.417)].Inthelateeighteenthandearlynineteenthcentury,Paolo Frisi(1728-1784),LeonardEuler(1707-1783),JosephLouisLagrange(1736-1813),and othersrealizedthatotherphysicalquantities,associatedwithrotationofarigidbody (torque,angularvelocity,momentofaforce),couldalsobeusefullygivenvectorial representations;thiswasdevelopedfurtherbyLouisPoinsot(1777-1859),SiméonDenisPoisson(1781-1840),andJacquesBinet(1786-1856).Ataboutthesametime,variousgeometricquantities(e.g.,areasofsurfacesinspace)weregivenvectorialrepresentationsbyGaetanoGiorgini(1795-1874),SimonLhuilier(1750-1840),JeanHachette(1769-1834),LazareCarnot(1753-1823)),MichelChasles(1793-1880)andlater byHermannGrassmann(1809-1877)andGiuseppePeano(1858-1932).Intheearly nineteenthcentury,vectorialrepresentationsofcomplexnumbers(andtheirextension,quaternions)wereformulatedbyseveralresearchers;theterm vector wascoined byWilliamRowanHamilton(1805-1865)in1853.Finally,extensiveuseofvectorial propertiesofelectromagneticforceswasmadebyJamesClerkMaxwell(1831-1879) andOliverHeaviside(1850-1925)inthelatenineteenthcentury.However,ageneral theoryofvectorswasonlyformulatedintheverylatenineteenthcentury;thefirstelementaryexpositionwasgivenbyEdwinBidwellWilson(1879-1964)in1901[55],based onlecturesbytheAmericanmathematicalphysicistJosiahWillardGibbs(1839-1903)4 [18].
Byageometricvectorinℝ3(orℝ2)wewillmeanan“arrow”whichcanbemoved toanyposition,provideditsdirectionandlengtharemaintained.5 Wewilldenote vectorswithalettersurmountedbyanarrow,likethis:⃗�� . 6 Wedefinetwooperations onvectors.The sum oftwovectorsisformedbymoving sothatits“tail”coincides inpositionwiththe“head”of⃗�� ,thenformingthevector ⃗��+⃗�� whosetailcoincides withthatof⃗�� andwhoseheadcoincideswiththatof (Figure 1.15).Ifinsteadwe place withitstailatthepositionpreviouslyoccupiedbythetailof⃗�� andthenmove
4IlearnedmuchofthisfromSandroCaparrini[6–8].Thisnarrativediffersfromthestandardone, givenbyMichaelCrowe[10]
5Thismobilityissometimesexpressedbysayingitisa freevector.
6Forexample,allofthearrowsinFigure 1.13 representthevector ⃗ ����.
Figure1.15. Sumoftwovectors
⃗�� sothatitstailcoincideswiththeheadof ,weform ⃗��+⃗�� ,anditisclearthatthese twoconfigurationsformaparallelogramwithdiagonal ⃗��+⃗��=⃗��+⃗��.
Thisisthe commutativeproperty ofvectoraddition.
Asecondoperationis scaling or multiplicationofavectorbyanumber.We naturallydefine(positiveinteger)multiplesofavector: 1⃗��=⃗��,2⃗��=⃗��+⃗��,3⃗��=⃗��+⃗��+ ⃗��=2⃗��+⃗�� ,andsoon.Thenwecandefinerationalmultiplesby⃗��= �� �� ⃗��⇔��⃗��=��⃗�� .
Finally,todefinemultiplicationbyanarbitrary(positive)realnumber,suppose ���� ���� →ℓ isasequenceofrationalsconvergingtotherealnumber ℓ.Foranyfixedvector⃗�� ,if wedrawarrowsrepresentingthevectors(����/����)⃗�� withalltheirtailsatafixedposition, thentheheadswillformaconvergentsequenceofpointsalongaline,whoselimitis thepositionfortheheadof ℓ⃗��.Alternatively,ifwepickaunitoflength,thenforany vector⃗�� andanypositiverealnumber ��,thevector ��⃗�� hasthesamedirectionas⃗�� ,and itslengthisthatof⃗�� multipliedby ��.Forthisreason,werefertorealnumbers(ina vectorcontext)as scalars.
If ⃗��=⃗��+⃗�� thenitisnaturaltowrite ⃗��=⃗��−⃗�� andfromthis(Figure 1.16)itis naturaltodefinethenegative−⃗�� ofavector asthevectorobtainedbyinterchanging theheadandtailof .Thisallowsustoalsodefinemultiplicationofavector⃗�� byany negative realnumber ��=−|��| as (−|��|)⃗��≔|��|(−⃗��)
—thatis,wereversethedirectionof⃗�� and“scale”by |��|. -
Figure1.16. Differenceofvectors
Chapter1.CoordinatesandVectors
Additionofvectors(andofscalars)andmultiplicationofvectorsbyscalarshave manyformalsimilaritieswithadditionandmultiplicationofnumbers.Welistthemajorones(thefirstofwhichhasalreadybeennotedabove):
• Additionofvectorsis
commutative: ⃗��+⃗��=⃗��+⃗�� ,and associative: ⃗��+(⃗��+⃗��)=(⃗��+⃗��)+⃗�� .
• Multiplicationofvectorsbyscalars distributesovervectorsums: ��(⃗��+⃗��)=��⃗��+��⃗��,and distributesoverscalarsums: (��+��)⃗��=��⃗��+��⃗��.
WewillexploresomeofthesepropertiesfurtherinExercise 3
Theinterpretationofdisplacementsasvectorsgivesusanalternativewaytorepresentvectors.Ifweknowthatanarrowhasitstailattheorigin(wecallthisstandard position),thenthevectoritrepresentsisentirelydeterminedbythecoordinatesof itshead.Thisgivesusanaturalcorrespondencebetween vectors⃗�� in ℝ3 (or ℝ2)and points ��∈ℝ3 (resp ℝ2):the positionvector ofthepoint �� isthevector ⃗ ����;itrepresentsthatdisplacementof ℝ3 whichmovestheoriginto��.Weshallmakeextensive useofthecorrespondencebetweenvectorsandpoints,oftendenotingapointbyits positionvector ⃗��∈ℝ 3,orspecifyingavectorbythecoordinates (��,��,��) ofitshead whenrepresentedinstandardposition.Wereferto ��, �� and �� asthe components or entries of⃗�� ,andsometimeswrite ⃗��=(��,��,��) .Vectorarithmeticisveryeasyto calculateinthisrepresentation:if ⃗��=(△��,△��,△��) ,thenthedisplacementrepresentedby movestheoriginto (△��,△��,△��);thesumofthisand ⃗��=(��,��,��) is thedisplacementtakingtheoriginfirstto (��,��,��) andthento ⃗��+⃗��=(��+△��,��+△��,��+△��); thatis, weaddvectorscomponentwise
Similarly,if �� isanyscalarand ⃗��=(��,��,��) ,then ��⃗��=(����,����,����)∶
ascalarmultipliesallentriesofthevector
Thisrepresentationpointsoutthepresenceofanexceptionalvector—the zero vector 0≔(0,0,0) whichistheresultofeithermultiplyinganarbitraryvectorbythescalarzero(0⃗��=0) orofsubtractinganarbitraryvectorfromitself(⃗��−⃗��= 0).Asa point, 0 corresponds totheorigin �� itself.Asan arrow,itstailandheadareatthesameposition.Asa displacement,itcorrespondstonotmovingatall.Noteinparticularthatthezerovector doesnothaveawell-defineddirection—afeaturewhichwillbeimportanttoremember inthefuture.Fromaformal,algebraicpointofview,thezero vector playstherole for vector additionthatisplayedbythe number zeroforadditionof numbers:itisan additiveidentityelement,whichmeansthataddingittoanyvectorgivesbackthat vector:
⃗��+ 0=⃗��=0+⃗��.
Bythinkingofvectorsin ℝ3 astriplesofnumbers,wecanrecovertheentriesofa vectorgeometrically:if ⃗��=(��,��,��) thenwecanwrite
⃗��=(��,0,0)+(0,��,0)+(0,0,��)=��(1,0,0)+��(0,1,0)+��(0,0,1).
1.2.VectorsandTheirArithmetic 15
Thismeansthatanyvectorin ℝ3 canbeexpressedasasumofscalarmultiples(or linearcombination)ofthreespecificvectors,knownasthe standardbasis for ℝ3 (seeFigure 1.17),anddenoted ⃗��=(1,0,0),⃗��=(0,1,0), ��=(0,0,1).
Wehavejustseenthateveryvector ⃗��∈ℝ 3 canbeexpressedas ⃗��=(��,��,��)=��⃗��+��⃗��+�� ��, where ��, ��,and �� arethecoordinatesof⃗�� .
Figure1.17. TheStandardBasisfor ℝ3
Weshallfinditconvenienttomovefreelybetweenthecoordinatenotation ⃗��= (��,��,��) andthe“arrow”notation ⃗��=��⃗��+��⃗��+�� ��;generally,weadoptcoordinate notationwhen⃗�� isregardedasapositionvector,and“arrow”notationwhenwewant topictureitasanarrowinspace.
Webeganbythinkingofavector⃗�� in ℝ3 asdeterminedbyitsmagnitudeandits direction,andhaveendedupthinkingofitasatripleofnumbers.Tocomefullcircle,werecallthatthevector ⃗��=(��,��,��) hasasitsstandardrepresentationthearrow ⃗ ���� fromtheorigin �� tothepoint �� withcoordinates (��,��,��);thusitsmagnitude(or length,denoted ‖⃗��‖ or | | ⃗�� | | )isgivenbythedistanceformula ‖⃗��‖=√��2+��2+��2 . Whenwewanttospecifythe direction of⃗�� ,we“point,”usingasourstandardrepresentationthe unitvector—thatis,thevectoroflength 1—inthedirectionof⃗�� .From thescalingpropertyofmultiplicationbyrealnumbers,weseethattheunitvectorin thedirectionofa(nonzero7)vector⃗�� (⃗��≠ 0)is u( ⃗�� )= 1
Inparticular,thestandardbasisvectors⃗�� ,⃗�� ,and�� areunitvectorsinthedirection(s)of the(positive)coordinateaxes.
7Avectoris nonzero ifitisnotequaltothezerovector: some ofitsentriescanbezero,but notall of them.
Chapter1.CoordinatesandVectors
Two(nonzero)vectorspointin thesame directionpreciselyiftheirrespectiveunit vectorsarethesame: 1 ‖��‖ ⃗��= 1 ‖��‖ ,or
⃗��=��⃗��,⃗��= 1 ��⃗��,
wherethe(positive)scalar �� is ��= ‖��‖ ‖��‖.Similarly,thetwovectorspointin opposite directionsifthetwounitvectorsare negatives ofeachother,or ⃗��=��⃗�� (resp ⃗��= 1 �� ⃗�� ),
wherethe negative scalar �� is ��=−‖��‖ ‖��‖ .Weshallrefertotwovectorsas parallel if theypointinthesameoroppositedirections,thatis,ifeachisa nonzero (positiveor negative)multipleoftheother.
Wecansummarizethisby
Remark1.2.1. Fortwononzerovectors ⃗��=(�� 1,��1,��1) and ⃗��=(�� 2,��2,��2),thefollowingareequivalent:
•⃗�� and areparallel(i.e.,theypointinthesameoroppositedirections);
• ⃗��=��⃗�� forsomenonzeroscalar ��;
• ⃗��=�� ′⃗�� forsomenonzeroscalar ��′;
•
=�� forsomenonzeroscalar�� (whereifoneoftheentriesof iszero, soisthecorrespondingentryof⃗�� ,andthecorrespondingratioisomittedfromthese equalities);
• ��2 ��1 = ��2 ��1 = ��2 ��1 =��′ forsomenonzeroscalar��′ (whereifoneoftheentriesof iszero, soisthecorrespondingentryof⃗�� ,andthecorrespondingratioisomittedfromthese equalities).
Thevaluesof��(resp.��′)arethesamewherevertheyappearabove,and��′ isthereciprocal of ��.
�� (hencealso ��′)is positive preciselyif⃗�� and pointinthe same direction,and negative preciselyiftheypointin opposite directions.
Twovectorsare linearlydependent if,whenwepicturethemasarrowsfrom acommoninitialpoint,thetwoheadsandthecommontailfallonacommonline. Algebraically,thismeansthatoneofthemisascalarmultipleoftheother.ThisterminologywillbeextendedinExercise7—butformorethantwovectors,thecondition ismorecomplicated.Vectorswhicharenot linearlydependentarelinearlyindependent.Remark 1.2.1 saysthattwo nonzero vectorsarelinearly dependent preciselyif theyare parallel
Exercisesfor§ 1.2
Practiceproblems:
(1) Ineachpart,youaregiventwovectors,⃗�� and .Find (i) ⃗��+⃗�� ; (ii) ⃗��−⃗�� ; (iii) 2⃗��; (iv) 3⃗��−2⃗��; (v)thelengthof⃗�� , ‖ ‖ ⃗�� ‖ ‖; (vi)theunitvector inthedirectionof⃗�� :
(a) ⃗��=(3,4) , ⃗��=(−1,2)
(b) ⃗��=(1,2,−2) , ⃗��=(2,−1,3)
(c) ⃗��=2⃗��−2⃗��− ��, ⃗��=3⃗��+⃗��−2 ��
(2) Ineachcasebelow,decidewhetherthegivenvectorsarelinearlydependentor linearlyindependent.
(a) (1,2), (2,4)
(b) (1,2), (2,1)
(c) (−1,2), (3,−6) (d) (−1,2), (2,1)
(e) (2,−2,6), (−3,3,9) (f) (−1,1,3), (3,−3,−9)
(g) ⃗��+⃗��+ ��, 2⃗��−2⃗��+2�� (h) 2⃗��−4⃗��+2��, −⃗��+2⃗��−��
Theoryproblems:
(3) (a) Wehaveseenthatthecommutativepropertyofvectoradditioncanbeinterpretedviathe“parallelogramrule”(Figure A.5).Giveasimilarpictorialinterpretationoftheassociativeproperty.
(b) Givegeometricargumentsforthetwodistributivepropertiesofvectorarithmetic.
(c) Showthatif ��⃗��=0 theneither ��=0 or ⃗��= 0.(Hint: Whatdoyouknow abouttherelationbetweenlengthsfor⃗�� and ��⃗��?)
(d) Showthatifavector⃗�� satisfies��⃗��=��⃗��,where��≠��aretwospecific,distinct scalars,then ⃗��= 0
(e) Showthatvectorsubtractionis not associative.
(4) Polarnotationforvectors:
(a) Showthatanyplanarvector oflength 1 canbewrittenintheform ⃗��=(cos��,sin��), where �� isthe(counterclockwise)anglebetween andthepositive ��-axis.
(b) Concludethateverynonzeroplanarvector⃗�� canbeexpressedin polarform ⃗��=‖⃗��‖(cos��,sin��), where �� isthe(counterclockwise)anglebetween⃗�� andthepositive ��-axis.
(5) (a) Showthatif⃗�� and aretwolinearlyindependentvectorsintheplane,then everyvectorintheplanecanbeexpressedasalinearcombinationof⃗�� and . (Hint: Theindependenceassumptionmeanstheypointalongnon-parallel lines.Givenapoint�� intheplane,considertheparallelogramwiththeorigin and �� asoppositevertices,andwithedgesparallelto⃗�� and .Usethisto constructthelinearcombination.)
(b) Nowsuppose ,⃗�� and are three nonzerovectorsin ℝ3.If⃗�� and arelinearlyindependent,showthateveryvectorlyingintheplanethatcontainsthe twolinesthroughtheoriginparallelto⃗�� and canbeexpressedasalinear combinationof⃗�� and .Nowshowthatif doesnotlieinthisplane,then everyvectorinℝ3 canbeexpressedasalinearcombinationof ,⃗�� and .The twostatementsabovearesummarizedbysayingthat⃗�� and (resp. ,⃗�� and ) spanℝ2 (resp. ℝ3).
Challengeproblem:
(6) Show(usingvectormethods)thatthelinesegmentjoiningthemidpointsoftwo sidesofatriangleisparalleltoandhashalfthelengthofthethirdside.
(7) Givenacollection{ ⃗�� 1,⃗��2,…,⃗����}ofvectors,considertheequation(intheunknown coefficients ��1,...,����)
1⃗�� 1+��2⃗��2+⋯+��
Chapter1.CoordinatesandVectors thatis,anexpressionforthezerovectorasalinearcombinationofthegivenvectors.Ofcourse,regardlessofthevectors⃗�� ��,onesolutionofthisis
��1 =��2 =⋯=0; thecombinationcomingfromthissolutioniscalledthe trivialcombination of thegivenvectors.Thecollection { ⃗�� 1,⃗��2,…,⃗����} is linearlydependent ifthereexistssome nontrivial combinationofthesevectors—thatis,asolutionofEquation(1.12)with atleastone nonzerocoefficient.Itislinearlyindependentifitis notlinearlydependent—thatis,iftheonlysolutionofEquation(1.12)isthetrivial one.
(a) Showthatanycollectionofvectorswhichincludesthezerovectorislinearly dependent.
(b) Showthatacollectionof two nonzerovectors { ⃗�� 1,⃗��2} in ℝ3 islinearlyindependentpreciselyif(instandardposition)theypointalongnon-parallellines.
(c) Showthatacollectionof three positionvectorsin ℝ3 islinearlydependent preciselyifatleastoneofthemcanbeexpressedasalinearcombinationof theothertwo.
(d) Showthatacollectionofthreepositionvectorsin ℝ3 islinearly independent preciselyifthecorrespondingpointsdetermineaplaneinspacethatdoesnot passthroughtheorigin.
(e) Showthatanycollectionof fourormore vectorsin ℝ3 islinearly dependent (Hint: Useeitherpart(a)ofthisproblemorpart(b)ofExercise 5.)
1.3LinesinSpace
ParametrizationofLines. Anequationoftheform
����+����=��, where��,��,and��areconstantswithatleastoneof��and��nonzero,iscalleda“linear” equationbecauseifweinterpret��and��astherectangularcoordinatesofapointinthe plane,theresultinglocusisaline(atleastprovided ��, �� and �� arenotallzero).Via straightforwardalgebraicmanipulation,(if ��≠0)8 wecanrewritethisasthe slopeinterceptformula
��=����+��, (1.13) wherethe slope �� isthetangentoftheanglethelinemakeswiththehorizontaland the��-intercept�� istheordinate(signedheight)ofitsintersectionwiththe��-axis.We canthinkofthisformulaasatwo-stepdeterminationofaline:theslopedetermines adirection,andtheinterceptpicksoutaparticularlinefromthefamilyof(parallel) linesthathavethatslope.
Thelocusinspaceofa“linear”equationinthethreerectangularcoordinates ��, �� and ��, ����+����+����=��,isa plane,notaline,butwecanconstructa vector equationforalineanalogousinspirittothepoint-slopeformula(1.13).Adirection in 3-spacecannotbedeterminedbyasinglenumber,butitisnaturallyspecifiedbya nonzerovector,sothethree-dimensionalanalogueoftheslopeofalineisa direction vector⃗��=��⃗��+��⃗��+�� �� towhichitisparallel.9 Giventhedirection⃗�� ,wecanspecifya
8��=0 meanswehave ��=��,averticalline.
9Notethatadirectionvectorneednotbeaunitvector.
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