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VectorAnalysis_Gibbs Wilson 10032899-2

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A scanned copy of the 1901 Yale Bicentennial Series textbook for students of mathematics and physics, with prefaces by Gibbs and Wilson. The contents list covers vector addition, dot and cross products, vector differential and integral calculus (del, divergence, curl, Gauss's and Stokes's theorems), and the linear vector function, ending with higher topics and complex vectors. The text shows no annotations by Phil; it is a book by others kept in his downloads.

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VECTORANALYSIS ATEXT-BOOKFORTHEUSEOFSTUDENTS OFMATHEMATICSANDPHYSICS FOUNDEDUPONTHELECTURES0F J.WILLARDGIBBS,PH .D .,LL .D . ProfessorofMat/2matz' ealPbyszcsinYaleUniversity BY EDWINBIDWELLWILSON,PH .D . InstructorinMat/innoticsinYaleUniversity NEWYORK:CHARLESSCRIBNER’SSONS LONDON:EDWARDARNOLD BYYALEUNIVERSITY . Published,December,1901. ska UNIVERSITYPRESSJOHNWILSON ANDSONCAMBRIDGE,U .S.A . PREFACEBYPROFESSORGIBBS SINCEtheprintingofashortpamphletontheElementsof VectorAnalysisintheyears1881—84,neverpublished,but somewhatwidelycirculatedamongthosewhowereknownto beinterestedinthesubject,thedesirehasbeenexpressed inmorethanonequarter,thatthesubstanceofthattrea tise,perhapsinfullerform,shouldbemadeaccessibleto thepublic. As,however,theyearspassedwithoutmyﬁndingthe leisuretomeetthiswant,whichseemedarealone,Iwas verygladtohaveoneofthehearersofmycourseonVector Analysisintheyear1899-1900undertakethepreparationof atext-bookonthesubject.IhavenotdesiredthatDr.Wilsonshouldaimsimply atthereproductionofmylectures,butratherthatheshould usehisownjudgmentinallrespectsfortheproductionofa text-bookinwhichthesubjectshouldbesoillustratedbyan adequatenumberofexamplesastomeetthewantsofstu dentsofgeometryandphysics. J .WILLARDGIBBS. YALEUNIVERSITY,September,1901. ENGINEERIHGl GENERALPREFACE WHENIundertooktoadaptthelecturesofProfessorGibbs onVECTORANALYSISforpublicationintheYaleBicenten nialSeries,ProfessorGibbshimselfwasalreadysofully engageduponhisworktoappearinthesameseries,Elementary PrinciplesinStatisticalMechanics,thatitwasunderstoodno materialassistanceinthecompositionofthisbookcouldbe expectedfromhim .Forthisreasonhewishedmetofeel entirelyfreetousemyowndiscretionalikeintheselection ofthetopicstobetreatedandinthemodeoftreatment.Ithasbeenmyendeavortousethefreedomthusgranted onlyinsofaraswasnecessaryforpresentinghismethodin text-bookform . Byfarthegreaterpartofthematerialusedinthefollow ingpageshasbeentakenfromthecourseoflectureson VectorAnalysisdeliveredannuallyattheUniversityby ProfessorGibbs.Someuse,however,hasbeenmadeofthe chaptersonVectorAnalysisinMr.OliverHeaviside’ sElec tromagneticTheory(ElectricianSeries,1893)andinProfessorFoppl’ slecturesonDieMaxwell’ seheTheoriederElectricitc’it (Teubner, MypreviousstudyofQuaternionshas alsobeenofgreatassistance. Thematerialthusobtainedhasbeenarrangedintheway whichseemsbestsuitedtoeasymasteryofthesubject. ThoseArts .whichitseemedbesttoincorporateinthe textbutwhichforvariousreasonsmaywellbeomittedat theﬁrstreadinghavebeenmarkedwithanasterisk Nu merousillustrativeexampleshavebeendrawnfromgeometry, mechanics,andphysics.Indeed,alargepartofthetexthas todowithapplicationsofthemethod .Theseapplications havenotbeensetapartinchaptersbythemselves,buthave GENERALPREFACE beendistributedthroughoutthebodyofthebookasfastas theanalysishasbeendevelopedsufﬁcientlyfortheiradequate treatment.Itishopedthatbythismeansthereadermaybe betterenabledtomakepracticaluseofthebook .Greatcare hasbeentakeninavoidingtheintroductionofunnecessary ideas,andinsoillustratingeachideathatisintroducedas tomakeitsnecessityevidentanditsmeaningeasytograsp. Thusthebookisnotintendedasacompleteexpositionof thetheoryofVectorAnalysis,butasatext-bookfromwhich somuchofthesubjectasmayberequiredforpracticalappli cationsmaybelearned .Henceasummary,includingalist ofthemoreimportantformulae,andanumberofexercises, havebeenplacedattheendofeachchapter,andmanyless essentialpointsinthetexthavebeenindicatedratherthan fullyworkedout,inthehopethatthereaderwillsupplythe details.Thesummarymaybefoundusefulinreviewsand forreference . ThesubjectofVectorAnalysisnaturallydividesitselfinto threedistinctparts.First,thatwhichconcernsadditionand thescalarandvectorproductsofvectors.Second,thatwhich concernsthedifferentialandintegralcalculusinitsrelations toscalarandvectorfunctions.Third,thatwhichcontains thetheoryofthelinearvectorfunction .Theﬁrstpartis anecessaryintroductiontobothotherparts.Thesecond andthirdaremutuallyindependent.Eithermaybetaken upﬁrst.Forpracticalpurposesinmathematicalphysicsthe secondmustberegardedasmoreelementarythanthethird . Butastudentnotprimarilyinterestedinphysicswouldnat urallypassfromtheﬁrstparttothethird,whichhewould probablyfindmoreattractiveandeasythanthesecond .Followingthisdivisionofthesubject,themainbodyof thebookisdividedintosixchaptersofwhichtwodealwith eachofthethreepartsintheordernamed .ChaptersI.and II.treatofaddition,subtraction,scalarmultiplication,and thescalarandvectorproductsofvectors.Theexposition hasbeenmadequiteelementary.Itcanreadilybeunder stoodbyandisespeciallysuitedforsuchreadersashavea knowledgeofonlytheelementsofTrigonometryandAna lyticGeometry .ThosewhoarewellversedinQuaternions oralliedsubjectsmayperhapsneedtoreadonlythesum maries.ChaptersIII.andIV .containthetreatmentof thosetopicsinVectorAnalysiswhich,thoughoflessvalue tothestudentsofpuremathematics,areoftheutmostimpor tancetostudentsofphysics.ChaptersV .andVI.dealwith thelinearvectorfunction .Tostudentsofphysicsthelinear vectorfunctionisofparticularimportanceinthemathemati caltreatmentofphenomenaconnectedwithnon-isotropic media;andtothestudentofpuremathematicsthispartof thebookwillprobablybethemostinterestingofall,owing tothefactthatitleadstoMultipleAlgebraortheTheory ofMatrices.Aconcludingchapter,VII.,whichcontainsthe developmentofcertainhigherpartsofthetheory,anumber ofapplications,andashortsketchofimaginaryorcomplex vectors,hasbeenadded .Inthetreatmentoftheintegralcalculus,ChapterIV ., questionsofmathematicalrigorarise.Althoughmodern theoristsaredevotingmuchtimeandthoughttorigor,and althoughtheywilldoubtlesscriticisethisportionofthebook adversely,ithasbeendeemedbesttogivebutlittleattention tothediscussionofthissubject.Andthemoresoforthe reasonthatwhateversystemofnotationbeemployedques tionsofrigorareindissolublyassociatedwiththecalculus andoccasionnonewdifﬁcultytothestudentofVector Analysis,whomustﬁrstlearnwhatthefactsareandmay postponeuntillaterthedetailedconsiderationoftherestric tionsthatareputuponthosefacts.Notwithstandingtheeffortswhichhavebeenmadeduring morethanhalfacenturytointroduceQuaternionsinto physicsthefactremainsthattheyhavenotfoundwidefavor. Ontheotherhandtherehasbeenagrowingtendencyespe ciallyinthelastdecadetowardtheadoptionofsomeformof VectorAnalysis.TheworksofHeavisideandFopplre ferredtobeforemaybecitedinevidence.ASyethowever nosystemofVectorAnalysiswhichmakesanyclaimto completenesshasbeenpublished .InfactHeavisidesays: “IaminhopesthatthechapterwhichInowﬁnishmay xii GENERALPREFACE serveasastopgaptillregularvectorialtreatisescometobe writtensuitableforphysicists,baseduponthevectorialtreat mentofvectors(ElectromagneticTheory,Vol.I.,p . ElsewhereinthesamechapterHeavisidehassetforththe claimsofvectoranalysisasagainstQuaternions,andothers haveSxpressedsimilarviews. Thekeynote,then,toanysystemofvectoranalysismust beitspracticalutility.This,Ifeelconﬁdent,wasProfessor Gibbs’ spointofviewinbuildinguphissystem .Heusesit entirelyinhiscoursesonElectricityandMagnetismandon ElectromagneticTheoryofLight.InwritingthisbookI havetriedtopresentthesubjectfromthispracticalstand point,andkeepclearlybeforethereader’smindtheques tions:Whatcombinationsorfunctionsofvectorsoccurin physicsandgeometry?Andhowmaytheseberepresented symbolicallyinthewaybestsuitedtofacileanalyticmanip ulation?Thetreatmentofthesequestionsinmodernbooks onphysicshasbeentoomuchconﬁnedtotheadditionand subtractionofvectors.Thisisscarcelyenough .Ithas beentheaimheretogivealsoanexpositionofscalarand vectorproducts,oftheOperatorv,ofdivergenceandcurl whichhavegainedsuchuniversalrecognitionsincetheap pearanceofMaxwell’sTreatiseonElectricityandMagnetism, ofslope,potential,linearvectorfunction,etc.,suchasshall beadequatefortheneedsofstudentsofphysicsatthe presentdayandadaptedtothem .IthasbeenassertedbysomethatQuaternions,Vector Analysis,andallsuchalgebrasareoflittlevalueforinvesti gatingquestionsinmathematicalphysics.Whetherthis assertionshallprovetrueornot,onemaystillmaintainthat vectorsaretomathematicalphysicswhatinvariantsareto geometry.Aseverygeometermustbethoroughlyconver santwiththeideasofinvariants,soeverystudentofphysics Shouldbeabletothinkintermsofvectors.Andthereis nowayinwhichhe,especiallyatthebeginningofhissci entiﬁcstudies,cancometosotrueanappreciationofthe importanceofvectorsandoftheideasconnectedwiththem asbyworkinginVectorAnalysisanddealindirectlwith GENERALPREFACE thevectorsthemselves.Tothosethatholdtheseviewsthe successofProfessorFoppl’ sVorlesungentiterTechnische Mechanik(fourvolumes,Teubner,1897—1900,alreadyina secondedition),inwhichthetheoryofmechanicsisdevel opedbymeansofavectoranalysis,canbebutanencour agingSign . Itakepleasureinthankingmycolleagues,Dr.M .B .Porter andProf.H .A .Bumstead,forassistingme' withthemanu script.Thegoodservicesofthelatterhavebeenparticularly valuableinarrangingChaptersIII.andIV .intheirpresent formandinsuggestingmanyoftheillustrationsusedinthe work .Iamalsounderobligationstomyfather,Mr.Edwin H .Wilson,forhelpinconnectionbothwiththeproofsand themanuscript.Finally,Iwishtoexpressmydeepindebt ednesstoProfessorGibbs.Foralthoughhehasbeenso preoccupiedastobeunabletoreadeithermanuscriptor proof,hehasalwaysbeenreadytotalkmattersoverwith me,anditishewhohasfurnishedmewithinspirationsuf ﬁcienttocarrythroughthework . EDWINBIDWELLWILSON . YALEUNIVERSITY,October,1901. PAGE PREFACEBYPROFESSORGIBBS Vii GENERALPREFACE ix CHAPTERI ADDITIONANDSCALARMULTIPLICATION ARTS. 1-3SCALARSANDVECTORS 4EQUALANDNULLVECTORS 5THEPOINTOFVIEWOFTHISCHAPTER 6—7SCALARMULTIPLICATION .THENEGATIVESIGN 8-10ADDITION .THEPARALLELOGRAMLAW 11SUBTRACTION 12LAWSGOVERNINGTHEFOREGOINGOPERATIONS 13—16COMPONENTSOFVECTORS.VECTOREQUATIONS 17THETHREEUNITVECTORSi,j,k APPLICATIONSToSUNDRYPROBLEMSINGEOMETRY . 20—22VECTORRELATIONSINDEPENDENTOFTHEORIGIN 23-24CENTERSOFGRAVITY .BARYCENTRICCOORDINATES 25THEUSEOFVECTORSTODENOTEAREAS SUMMARYOFCHAPTERIEXERCISESONCHAPTERI CHAPTERII DIRECTANDSKEWPRODUCTSOFVECTORS 2728THEDIRECT,SCALAR,0RDOTPRODUCTOFTwoVECTORS55 29-30THEDISTRIBUTIVELAWANDAPPLICATIONS 58 31—33THESKEW,VECTOR,ORCROSSPRODUCTOFTwoVECTORS60 34—35THEDISTRIBUTIVELAWANDAPPLICATIONS 63 36THETRIPLEPRODUCTA°BC 67TABLEOFCONTENTS vi CONTENTS ARTS. 37—38THESCALARTRIPLEPRODUCTA°BxCon[ABC] 39-40THEVECTORTRIPLEPRODUCTAX(BXC) 41-42PRODUCTSOFMORETHANTHREEVECTORSWITHAPPLI CATIONSToTRIGONOMETRY 43-45RECIPROCALSYSTEMSOFTHREEVECTORS 46-47SOLUTION0FSCALARANDVECTOREQUATIONSLINEARINANUNKNOWNVECTOR 48—50SYSTEMSOFFORCESACTINGONARIGIDBODY 51KINEMATICSOFARIGIDBODY 52CONDITIONSFOREQUILIBRIUMOFARIGIDBODY 53RELATIONSBETWEENTWORIGHT-HANDEDJYSTEMSOF THREEPERPENDICULARUNITVECTORS 54PROBLEMSINGEOMETRY.PLANARCOORDINATES SUMMARYOFCHAPTERIIEXERCISESONCHAPTERII CHAPTERIII THEDIFFERENTIALCALCULUSOFVECTORS 55—56DERIVATIVESANDDIFFERENTIALSOFVECTORFUNCTIONS w aging;ToASCALARVARIABLE 57CURVATUREANDTORSIONOFGAUCHECURVES 58—59KINEMATICSOFAPARTICLE .THEHODOGRAPH 60THEINSTANTANEOUSAXISOFROTATION 61INTEGRATIONWITHAPPLICATIONSToKINEMATICS 62SCALARFUNCTIONSOFPOSITIONINSPACE 63—67THEYECTORDIFFERENTIATING0PERAT0R_V 68THESGALAROPERATORAojz 69VECTORFUNCTIONSOFPosITIoNIN§PA§E 70 71INTERPRETATIONOFTHEDIVERGENCEV 72INTERPRETATIONOFTHECURLVx 73LAWSOFOPERATIONOFV,V.vx 74-76THEPARTIALAPPLICATIONOFV.EXPANSIONOFAVEC TORFUNCTIONANALOGOUSToTAYLOR’STHEOREM . 77THEDIFFERENTIATINGOPERATORSOFTHESECONDORDER 78GEOMETRICINTERPRETATIONOFLAPLACE’SOPERATOR VVAsTHEDISPERSION SUMMARYOFCHAPTERIIIEXERCISESONCHAPTERIIIPAGE 68 71 75 81 87 92 97 101 104 106 109 113 159 166 170 172 177115 120 125 131 133 136 138 147 149 150 152 155 157 CONTENTS XVII CHAPTERIV THEINTEGRALCALCULUSOFVECTORS ARTS. PAGE 79-80LINEINTEGRALS0FVECTORFUNCTIONSWITHAPPLICA TIONS 179 81GAUSS’STHEOREM 184 82 187 83CONVERSEOFSTOKES’STHEOREMWITHAPPLICATIONS 193 84TRANSFORMATIONS0FLINE,SURFACE,ANDVOLUMEIN 197 85REMARKSONMULTIPLE-VALUEDFUNCTIONS 200 86—87POTENTIAL .THEINTEGRATINGOPERATOR“POT”205 88COMMUTATIVEPROPERTYOFPOTANDV211 89REMARKSUPONTHEFOREGOING 215 90THEINTEGRATINGOPERATORS“NEW,”“LAP,”“MAX”222 91RELATIONSBETWEENTHEINTEGRATINGANDDIFFER ENTIATINGOPERATORS 228 92THEPOTENTIAL“POT”ISASOLUTIONOFPOISSON’SEQUATION 230 9394SOLENOIDALANDIRROTATIONALPARTSOFAVECTOR FUNCTION .CERTAINOPERATORSANDTHEIRINVERSE 234 95MUTUALPOTENTIALS,NEWTONIANS,LAPLACIANS,AND MAXWELLIANS 240 96CERTAINBOUNDARYVALUETHEOREMS 243 SUMMARYOFCHAPTERIV 249EXERCISES0NCHAPTERIV 255 CHAPTERV LINEARVECTORFUNCTIONS 97-98LINEARVECTORFUNCTIONSDEFINED 260 99DYADICSDEFINED 264 100ANYLINEARVECTORFUNCTIONMAYBEREPRESENTED BYADYADIC.PROPERTIESOFDYADICS 266 101THENONIONFORMOFADYADIC 269 102THEDYADORINDETERMINATEPRODUCTOFTWOVEC TORSISTHEMOSTGENERAL .FUNCTIONALPROPERTY OFTHESGALARANDVECTORPRODUCTS 271 103—104PRODUCTSOFDYADICS 276 105—107DEGREESOFNULLITYOFDYADICS 282 108THEIDEMFACTOR 288 XVIII ARTS. 109-110 111 112-114 115—116 117 118—119 120 121 122 123—124 125-126 127 128 129 130 131 132 136142 143-146 147—148 149—157 158-162CONTENTS PAGE RECIPROCALDYADICS.POWERSANDROOTSOFDYADICS290 CONJUGATEDYADICS.SELF-CONJUGATEANDANTI SELF-CONJUGATEPARTSOFADYADIC ANTI-SELF-CONJUGATEDYADICS.THEVECTORPROD UCT .QUADRANTALVERSORS REDUCTIONOFDYADICSToNORMALFORM DOUBLEMULTIPLICATIONOFDYADICS THESECONDANDTHIRDOFADYADIC CONDITIONSFORDIFFERENTDEGREES0FNULLITY NONIONFORM .DETERMINANTS INVARIANTSOFADYADIC .THEHAMILTON-CAYLEYEQUATION SUMMARYOFCHAPTERVEXERCISESONCHAPTERv294 319 321 329 CHAPTERVI ROTATIONSANDSTRAINS HOMOGENEOUSSTRAINREPRESENTEDBYADYADIC ROTATIONSABOUTAFIXEDPOINT .VERSORS THEVECTORSEMI-TANGENTOFVERSION BIQUADRANTALVERSORSANDTHEIRPRODUCTS CYCLICDYADICS RIGHTTENSORS TONICSANDCYCLOTONICS REDUCTIONOFDYADICSTOCANONICALFORMS,TONICS, CYCLOTONICS,SIMPLEANDCOMPLEXSHEARERS SUMMARYOFCHAPTERVI356 368 CHAPTERVII MISCELLANEOUSAPPLICATIONS 372 392 403 411 426QUADRICSURFACES THEPROPAGATIONOFLIGHTINCRYSTALS VARIABLEDYADICS CURVATUREOFSURFACES HARMONICVIBRATIONSANDBIVECTORS297 302 306 310 313 315 332 334 339 343 347 351 353 09C?VECTORANALYSIS Thepositiveandnegativenumbersofordinaryalgebraarethe typicalscalars.Forthisreasontheordinaryalgebraiscalled scalaralgebrawhennecessarytodistinguishitfromthevector algebraoranalysiswhichisthesubjectofthisbook . Thetypicalvectoristhedisplacementoftranslationinspace. ConsiderﬁrstapointP(Fig . LetPbedisplacedina straightlineandtakeanewpositionP’ . Thischangeofpositionisrepresentedbythe 1'linePP' .Themagnitudeofthedisplace mentisthelengthofPP’ ;thedirectionof I itisthedirectionofthelinePP’fromPto P' .Nextconsideradisplacementnotofone, butofallthepointsinSpace.Letallthe pointsmoveinstraightlinesinthesamedirectionandforthe samedistanceD .Thisisequivalenttoshiftingspaceasa rigidbodyinthatdirectionthroughthedistanceDwithout rotation .Suchadisplacementiscalledatranslation .It possessesdirectionandmagnitude.WhenSpaceundergoes atranslationT,eachpointofspaceundergoesadisplacement equaltoTinmagnitudeanddirection;andconverselyif thedisplacementPP'whichanyoneparticularpointPsuf fersinthetranslationTisknown,thenthatofanyother pointQisalsoknown:forQQ'mustbeequalandparallel toPP’ . ThetranslationTisrepresentedgeometricallyorgraphically byanarrowT(Fig .1)ofwhichthemagnitudeanddirection areequaltothoseofthetranslation .Theabsoluteposition ofthisarrowinSpaceisentirelyimmaterial.Technicallythe arrowiscalledastroke.Itstailorinitialpointisitsorigin; anditsheadorﬁnalpoint,itsterminus.Intheﬁgurethe originisdesignatedby0andtheterminusbyT .Thisgeo metricquantity,astroke,isusedasthemathematicalsymbol forallvectors,justastheordinarypositiveandnegativenum bersareusedasthesymbolsforallscalars.FIG.1. ADDITIONANDSGALARMULTIPLICATION 3 Asexamplesofscalarquantitiesmass,time,den sity,andtemperaturehavebeenmentioned .Othersaredis tance,volume,momentofinertia,work,etc.Magnitude, however,isbynomeansthesolepropertyofthesequantities. Eachimpliessomethingbesidesmagnitude.Eachhasits owndistinguishingcharacteristics,asanexampleofwhich itsdimensionsinthesensewellknowntophysicistsmay becited .Adistance3,atime3,awork3,etc.,arevery differentThemagnitude3is,however,apropertycommon tothemall—perhapstheonlyone.Ofallscalarquanti titiespurenumberisthesimplest.Itimpliesnothingbut magnitude.Itisthescalarparexcellenceandconsequently itisusedasthemathematicalsymbolforallscalars . ASexamplesofvectorquantitiesforce,displacement,velo city,andaccelerationhavebeengiven .Eachofthesehas othercharacteristicsthanthosewhichbelongtoavectorpure andSimple.Theconceptofvectorinvolvestwoideasand twoalone—magnitudeofthevectoranddirectionofthe vector.Butforceismorecomplicated .Whenitisapplied toarigidbodythelineinwhichitactsmustbetakeninto consideration;magnitudeanddirectionalonedonotsuf ﬁce.Andincaseitisappliedtoanon-rigidbodythepoint ofapplicationoftheforceisasimportantasthemagnitudeor direction .Suchisfrequentlytrueforvectorquantitiesother thanforce.Moreoverthequestionofdimensionsispresent asinthecaseofscalarquantities.Themathematicalvector, thestroke,whichistheprimaryobjectofconsiderationin thisbook,abstractsfromalldirectedquantitiestheirmagni tudeanddirectionandnothingbutthese;justasthemathe maticalscalar,purenumber,abstractsthemagnitudeand thatalone.Henceonemustbeonhisguardlestfrom analogyheattributesomepropertiestothemathematical vectorwhichdonotbelongtoit;andhemustbeevenmore carefullestheobtainerroneousresultsbyconsideringthe 4 VECTORANALYSIS vectorquantitiesofphysicsaspossessingnopropertiesother thanthoseofthemathematicalvector.Forexampleitwould neverdotoconsiderforceanditseffectsasunalteredby shiftingitparalleltoitself.Thiswarningmaynotbe necessary,yetitmaypossiblysavesomeconfusion .Inasmuchas,takeninitsentirety,avectororstroke isbutaSingleconcept,itmayappropriatelybedesignatedby oneletter.Owinghowevertothefundamentaldifference betweenscalarsandvectors,itisnecessarytodistinguish carefullytheonefromtheother.Sometimes,asinmathe maticalphysics,thedistinctionisfurnishedbythephysical interpretation .Thusifnbetheindexofrefractionit mustbescalar;m,themass,andt,thetime,arealso scalars;butf,theforce,and a,theacceleration,are vectors.When,however,thelettersareregardedmerely assymbolswithnoparticularphysicalsigniﬁcancesome typographicaldifferencemustbereliedupontodistinguish vectorsfromscalars.HenceinthisbookClarendontypeis usedforsettingupvectorsandordinarytypeforscalars. Thispermitstheuseofthesameletterdiﬁerentlyprinted torepresentthevectoranditsscalarmagnitude .1Thusif Cbetheelectriccurrentinmagnitudeanddirection,0may beusedtorepresentthemagnitudeofthatcurrent;ifgbe thevectoraccelerationduetogravity,gmaybethescalar valueofthatacceleration;ifVbethevelocityofamoving mass,vmaybethemagnitudeofthatvelocity.Theuseof Clarendonstodenotevectorsmakesitpossibletopassfrom directedquantitiestotheirscalarmagnitudesbyamere changeintheappearanceofaletterwithoutanyconfusing changeintheletteritself. DeﬁnitionTwovectorsaresaidtobeequalwhentheyhave thesamemagnitudeandthesamedirection. 1Thisconvention,however,isbynomeansinvariablyfollowed .Insome instancesitwouldprovejustasundesirableasitisconvenientinothers.Itis chieﬂyvaluableintheapplicationofvectorstophysics. ADDITIONANDSCALARMULTIPLICATION 5 TheequalityoftwovectorsAandBisdenotedbythe usualSign ThusA B . Evidentlyavectororstrokeisnotalteredbyshiftingit aboutparalleltoitselfinspace .HenceanyvectorA PP’ (Fig .1)maybedrawnfromanyassignedpoint0asorigin forthesegmentPP’maybemovedparalleltoitselfuntil thepointPfallsuponthepoint0andP’uponsomepointT . Then Inthiswayallvectorsinspacemaybereplacedbydirected segmentsradiatingfromoneﬁxedpoint0 .Equalvectors inspacewillofcoursecoincide,whenplacedwiththeirter miniatthesamepoint0 .Thus(Fig .1)APP’ ,andBQQ’ , bothfalluponT 0T .Forthenumericaldeterminationofavectorthreescalars arenecessary .Thesemaybechoseninavarietyofways.Ifr,c,6bepolarcoo‘rdinatesinSpaceanyvector1'drawn withitsoriginattheoriginofco'ordinatesmayberepresented bythethreescalarsr,a,0whichdeterminetheterminusof thevector.rN Orifx,y,2beCartesiancoo’rdinatesinspaceavectorrmay beconsideredasgivenbythedifferencesoftheco'o’rdinatesao’ , y’ ,z'ofitsterminusandthoseit,y,zofitsorigin . -zc,y’—y,z’-z). Ifinparticulartheoriginofthevectorcoincidewiththe originofcoOrdinates,thevectorwillberepresentedbythe threecoordinatesofitsterminus ry’ , Whentwovectorsareequalthethreescalarswhichrepre sentthemmustbeequalrespectivelyeachtoeach.Hence onevectorequalityimpliesthreescalarequalities. 6 VECTORANALYSIS DeﬁnitionAvectorAissaidtobeequaltozerowhenits magnitudeAiszero. SuchavectorAiscalledanullorzerovectorandiswritten equaltonaughtintheusualmanner.Thus A=0ifA=0. AllnullvectorsareregardedasequaltoeachotherWithout anyconsiderationsofdirection . Infactanullvectorfromageometricalstandpointwould berepresentedbyalinearsegmentoflengthzerothatisto say,byapoint.Itconsequentlywouldhaveawhollyinde terminatedirectionor,whatamountstothesamething,noneat all.If,however,itberegardedasthelimitapproachedbya vectorofﬁnitelength,itmightbeconsideredtohavethat directionwhichisthelimitapproachedbythedirectionofthe ﬁnitevector,whenthelengthdecreasesindeﬁnitelyandap proacheszeroasalimit.Thejustiﬁcationfordisregarding thisdirectionandlookinguponallnullvectorsasequalis thatwhentheyareadded(Art.8)toothervectorsnochange occursandwhenmultiplied(Arts.27,31)byothervectors theproductiszero. InextendingtovectorsthefundamentalOperations ofalgebraandarithmetic,namely,addition,subtraction,and multiplication,caremustbeexercisednotonlytoavoidself contradictorydeﬁnitionsbutalsotolaydownusefulones. Boththeseendsmaybeaccomplishedmostnaturallyand easilybylookingtophysics(forinthatsciencevectorscon tinuallypresentthemselves)andbyobservinghowsuch quantitiesaretreatedthere.IfthenAbeagivendisplace ment,force,orvelocity,whatistwo,three,oringeneralx timesA?What,thenegativeofA?AndifBbeanother, whatisthesumofAandB?Thatistosay,whatisthe equivalentofAandBtakentogether?Theobviousanswers tothesequestionssuggestimmediatelythedesireddeﬁnitions. ADDITIONANDSCALARMULTIPLICATION 7 ScalarMultiplication DeﬁnitionAvectorissaidtobemultipliedbya positivescalarwhenitsmagnitudeismultipliedbythatscalar anditsdirectionisleftunaltered. ThusifvbeavelocityofnineknotsEastbyNorth,times visavelocityoftwenty-oneknotswiththedirectionstill EastbyNorth .Oriffbetheforceexerteduponthescale panbyagramweight,1000timesfistheforceexertedbya kilogram.Thedirectioninbothcasesisverticallydown ward . IfAbethevectoranda:thescalartheproductofacandAis denotedasusualby a:AorAso. Itis,however,morecustomarytoplacethescalarmultiplier beforethemultiplicandA .Thismultiplicationbyascalar iscalledscalarmultiplication,anditfollowstheassociativelaw asinordinaryalgebraandarithmetic.Thisstatementisim mediatelyobviouswhenthefactistakenintoconsideration thatscalarmultiplicationdoesnotalterdirectionbutmerely multipliesthelength . Deﬁnition:Aunitvectorisonewhosemagnitudeisunity . AnyvectorAmaybelookeduponastheproductofaunit vector9.initsdirectionbythepositivescalarA,itsmagni tude. A=Aa=aA . Theunitvector9.maysimilarlybewrittenastheproductof Abyl/AorasthequotientofAandA . 1 A a—ZAA 8 VECTORANALYSIS Deﬁnition:Thenegativesign, preﬁxedtoavector reversesitsdirectionbutleavesitsmagnitudeunchanged . ForexampleifAbeadisplacementfortwofeettotheright, Aisadisplacementfortwofeettotheleft.Againifthe strokeABbeA,thestrokeBA,whichisofthesamelength asABbutwhichisinthedirectionfromBtoAinsteadof fromAtoB,willbeA .Anotherillustrationoftheuse ofthenegativesignmaybetakenfromNewton’sthirdlaw ofmotion .IfAdenotean“action,”Awilldenotethe reaction .”Thepositivesign, maybepreﬁxedtoavec tortocallparticularattentiontothefactthatthedirection hasnotbeenreversed .Thetwosignsandwhenused inconnectionwithscalarmultiplicationofvectorsfollowthe samelawsofoperationasinordinaryalgebra.Theseare symbolically +3 —A). Theinterpretationisobvious. AdditionandSubtraction Theadditionoftwovectorsorstrokesmaybetreated mostsimplybyregardingthemasdeﬁningtranslationsin space(Art. LetSbeonevectorandTtheother.LetP beapointofspace(Fig . Thetrans lationScarriesPintoP’suchthatthe linePP’isequaltoSinmagnitudeand direction .ThetransformationTwillthen carryP’intoP”thelineP’P”being paralleltoTandequaltoitinmagnitude. Fm .2. ConsequentlytheresultofSfollowedby TistocarrythepointPintothepoint P” .IfnowQbeanyotherpointinspace,SwillcarryQ intoQ’suchthatQQ’SandTwillthencarry(2’intoQ” 10 VECTORANALYSIS sequentlythepointsP,P’ ,P” ,andP’"lieattheverticesof aparallelogram .Hence P’”P”isequalandpar alleltoPP’ .Hence8 carriesP’"intoP” .Tfol« lowedbySthereforecar riesPintoP”throughP’ , WhereasSfollowedbyT carriesPintoP”through Theﬁnalresultisin eithercasethesame .Thismaybedesignatedsymbolically bywritingFIG.3. ItistobenoticedthatSPP’andTPP’arethetwosides oftheparallelogramPP’P”P’”whichhavethepointPas commonorigin;andthatR 2:PP”isthediagonaldrawn throughP .Thisleadstoanotherverycommonwayof statingthedeﬁnitionofthesumoftwovectors. Iftwovectorsbedrawnfromthesameoriginandaparallelo grambeconstructeduponthemassides,theirsumwillbethat diagonalwhichpassesthroughtheircommonorigin . Thisisthewell-known“parallelogramlaw”accordingto whichthephysicalvectorquantitiesforce,acceleration,velce ity,andangularvelocityarecompounded.Itisimportantto notethatincasethevectorsliealongthesamelinevector additionbecomesequivalenttoalgebraicscalaraddition .The lengthsofthetwovectorstobeaddedareaddedifthevectors havethesamedirection;butsubtractediftheyhaveOppo sitedirections.Ineithercasethesumhasthesamedirection asthatofthegreatervector. Afterthedeﬁnitionofthesumoftwovectorshas beenlaiddown,thesumofseveralmaybefoundbyadding togethertheﬁrsttwo,tothissumthethird,tothisthefourth, andsoonuntilallthevectorshavebeencombinedintoasin ADDITIONANDSCALARMULTIPLICATION 11 gleone.Theﬁnalresultisthesameasthatobtainedbyplacing theoriginofeachsucceedingvectorupontheterminusofthe precedingoneandthendrawingatoncethevectorfrom theoriginoftheﬁrsttotheterminusofthelast.Incase thesetwopointscoincidethevectorsformaclosedpolygon andtheirsumiszero .Interpretedgeometricallythisstates thatifanumberofdisplacementsR,S,Tooaresuchthatthe strokesR,S,Tooformthesidesofaclosedpolygontakenin order,thentheeﬁectofcarryingoutthedisplacementsisnil. Eachpointofspaceisbroughtbacktoitsstartingpoint.In terpretedinmechanicsitstatesthatifanynumberofforces actatapointandiftheyformthesidesofaclosedpolygon takeninorder,thentheresultantforceiszeroandthepoint isinequilibriumundertheactionoftheforces. Theorderofsequenceofthevectorsinasumisofnocon sequence.Thismaybeshownbyprovingthatanytwoadja centvectorsmaybeinterchangedwithoutaﬂectingtheresult. Toshow Let Then LetnowB0”Then0’B0Disaparallelogramand consequently0’D=C .Hence whichprovesthestatement.Sinceanytwoadjacentvectors maybeinterchanged,andsincethesummaybearrangedin anyorderbysuccessiveinterchangesofadjacentvectors,the orderinwhichthevectorsoccurinthesumisimmaterial. Deﬁnition:Avectorissaidtobesubtractedwhenit isaddedafterreversalofdirection .Symbolically, A BA B). Bythismeanssubtractionisreducedtoadditionandneeds 12 VECTORANALYSIS nospecialconsideration .Thereishoweveraninterestingand importantwayofrepresentingthedifferenceoftwovectors geometrically .LetA 0A,B 0B(Fig . Complete theparallelogramofwhichAandB arethesides.Thenthediagonal OC’=0isthesumA+Bofthe twovectors.Nextcompletethe parallelogramofwhichAand B OB’arethesides.Thenthedi« agonalOD Dwillbethesumof FIG ,4 , AandthenegativeofB .Butthe segmentODisparallelandequal toBA .HenceBAmaybetakenasthedifferencetothetwo vectorsAandB .ThisleadstothefollowingruleThediffer enceoftwovectorswhicharedrawnfromthesameoriginis thevectordrawnfromtheterminusofthevectortobesub tractedtotheterminusofthevectorfromwhichitissub tracted .Thusthetwodiagonalsoftheparallelogram,which isconstructeduponAandBassides,givethesumanddif ferenceofAandB . Intheforegoingparagraphsaddition,subtraction,and scalarmultiplicationofvectorshavebeendeﬁnedandinter preted.Tomakethedevelopmentofvectoralgebramathe maticallyexactandsystematicitwouldnowbecomenecessary todemonstratethatthesethreefundamentaloperationsfollow thesameformallawsasintheordinaryscalaralgebra,al thoughfromthestandpointofthephysicalandgeometrical interpretationofvectorsthismayseemsuperﬂuous.These lawsare I:m(nA) I, II: A+B=B+Aa IIIa° IIIb' III,°—A—B . ADDITIONANDSCALARMULTIPLICATION 13 Iaistheso-calledlawofassociationandcommutationof thescalarfactorsinscalarmultiplication . Ibisthelawofassociationforvectorsinvectoraddition .It statesthatinaddingvectorsparenthesesmaybeinsertedat anypointswithoutalteringtheresult. IIisthecommutativelawofvectoraddition . isthedistributivelawforscalarsinscalarmultipli cation . III,isthedistributivelawforvectorsinscalarmultipli cation . III,isthedistributivelawforthenegativesign . Theproofsoftheselawsofoperationdependuponthose propositionsinelementarygeometrywhichhavetodealwith theﬁrstpropertiesoftheparallelogramandsimilartriangles. Theywillnotbegivenhere;butitissuggestedthatthe readerworkthemoutforthesakeofﬁxingthefundamental ideasofaddition,subtraction,andscalarmultiplicationmore clearlyinmind .Theresultofthelawsmaybesummedup inthestatement: Thelawswhichgovernaddition,subtraction,andscalar multiplicationofvectorsareidenticalwiththosegoverningthese operationsinordinaryscalaralgebra. Itispreciselythisidentityofformallawswhichjustiﬁes theextensionoftheuseofthefamiliarsigns and ofarithmetictothealgebraofvectorsanditisalsothis whichensuresthecorrectnessofresultsobtainedbyoperat ingwiththosesignsintheusualmanner.Onecautiononly needbementioned .Scalarsandvectorsareentirelydifferent sortsofquantity .Forthisreasontheycanneverbeequated toeachotherexceptperhapsinthetrivialcasewhereeachis zero .Forthesamereasontheyarenottobeaddedtogether. Solongasthisisborneinmindnodifﬁcultyneedbeantici patedfromdealingwithvectorsmuchasiftheywerescalars. Thusfromequationsinwhichthevectorsenterlinearlywith 14 VECTORANALYSIS scalarcoefﬁcientsunknownvectorsmaybeeliminatedor foundbysolutioninthesamewayandwiththesamelimita tionsasinordinaryalgebra;fortheeliminationsandsolu tionsdependsolelyonthescalarcoefﬁcientsoftheequations andnotatallonwhatthevariablesrepresent.Iffor instance thenA,B,C,orBmaybeexpressedintermsoftheother three 1as D— c- l Andtwovectorequationssuchas and 2A+SB=F yieldbytheusualprocessesthesolutions A=3E—4F and B=3F—2E . ComponentsofVectors Deﬁnition:Vectorsaresaidtobecollinearwhen theyareparalleltothesameline;coplanar,whenparallel tothesameplane.Twoormorevectorstowhichnoline canbedrawnparallelaresaidtobenon-collinear.Threeor morevectorstowhichnoplanecanbedrawnparallel saidtobenon-coplanar.Obviouslyanytwovectorsare 00planar. Anyvectorbcollinearwithamaybeexpressedasthe productofaandapositiveornegativescalarwhichisthe ratioofthemagnitudeofbtothatofa.Thesignispositive whenbandahavethesamedirection;negative,whenthey haveoppositedirections.Ifthen0A a,thevectorrdrawn ADDITIONANDSGALARMULTIPLICATION 15 fromtheorigin0toanypointoftheline0Aproducedin eitherdirectionis r2:cc(1) Ifabeavariablescalarparameterthisequationmaythere foreberegardedasthe(vector)equationofallpointsinthe line0A .LetnowBbeanypointnot upontheline0Aorthatlineproduced ineitherdirection(Fig . LetOB b.Thevectorbissurely notoftheformaca.DrawthroughBFIG .5. alineparallelto0AandletBbeany pointuponit.ThevectorBBiscollinearwithaandis consequentlyexpressibleasxa .Hencethevectordrawn fromOtoBis OR=OB+BR or (2) Thisequationmayberegardedasthe(vector)equationof allthepointsinthelinewhichisparalleltoaandofwhich Bisonepoint. 14]Anyvectorrcoplanarwithtwonon-collinearvectors aandbmayberesolvedintotwocomponentsparalleltoa andbrespectively .Thisresolutionmay beaccomplishedbyconstructingthepar allelogram(Fig .6)ofwhichthesidesare paralleltoaandbandofwhichthedi agonalisr.Ofthesecomponentsoneis xa;theother,yb . ccandyarerespec tivelythescalarratios(takenwiththe prepersign)ofthelengthsofthesecomponentstothelengths ofaandb.HenceFIG .6. r=xa+yb isatypicalformforanyvector00p1anarwithaandb.If severalvectorsr1,r2,r3umaybeexpressedinthisformas 16 VECTORANALYSIS rl=xla+y1b, r2=a2a+yzb, r3=x3a+y3b. theirsumristhen Thisisthewell-knowntheoremthatthecomponentsofa sumofvectorsarethesumsofthecomponentsofthose vectors .Ifthevector1'iszeroeachofitscomponentsmust bezero.ConsequentlytheonevectorequationrOis equivalenttothetwoscalarequations t=0 . 3 Anyvector1‘inspacemayberesolvedintothree componentsparalleltoanythreegivennon-coplanarvectors. Letthevectorsbea,b, andc.Theresolution maythenbeaecom plishedbyconstructing theparallelopiped(Fig . 20 7)ofwhichtheedges areparalleltoa,b,and candofwhichthedi yagonalisr.Thispar allelopipedmaybe Fm ,7 , drawneasilybypassing threeplanesparallelre spectivelytoaandb,bandc,candathroughtheorigin0 ofthevector andasimilarsetofthreeplanesthroughits terminusB .Thesesixplaneswillthenbeparallelinpairs 18 VECTORANALYSIS Let r=r’ , r=xa+yb+za Then m=cc’ ,gzy’ ,zzz’ . For r Hence a g—y’z0,z Butthiswouldnotbetrueifa,b,and0werecoplanar.In thatcaseoneofthethreevectorscouldbeexpressedinterms oftheothertwoas c=ma+nh Then r b, r [o+nz>b=0 . Hencetheindividualcomponentsofr r’inthedirections aandb(supposeddiﬁerent)arezero. Hence Butthisbynomeansnecessitates .v,y,ztobeequalrespec tivelytox’ ,y’ ,z' .Inasimilarmannerifaandbwerecol linearitisimpossibletoinferthattheircoefﬁcientsvanish individually .Thetheoremmayperhapsbestatedasfollows: Incasetwoequalvectorsareexpressedintermsofonevector, ortwonon-collinearvectors,orthreenon-00planarvectors,the correspondingscalarcoeﬁcientsareequal.Butthisisnotno cessarilytrueifthetwovectorsbecollinear;orthethreevectors, coplanar .Thisprinciplewillbeusedintheapplications (Arts.18et TheThreeUnitVectorsi,j,k. Intheforegoingparagraphsthemethodofexpress ingvectorsintermsofthreegivennon-coplanaroneshasbeen explained.Thesimplestsetofthreesuchvectorsistherect ADDITIONANDSCALARMULTIPLICATION 19 angularsystemfamiliarinSolidCartesianGeometry.This rectangularsystemmayhoweverbeeitheroftwoverydistinct types.Inonecase(Fig .8,ﬁrstpart)theZ-axis1liesupon thatsideoftheXYplaneonwhichrotationthrougharight anglefromtheX-axistotheY-axisappearscounterclockwise orpositiveaccordingtotheconventionadoptedinTrigonome try.Thisrelationmaybestatedinanotherform.IftheX axisbedirectedtotherightandtheY-axisvertically,the Z-amswillbedirectedtowardtheobserver.OriftheX axispointtowardtheobserverandtheY-axistotheright, theZ-axiswillpointupward.Stillanothermethodofstate I 0 o y] Right-handed Left-handed FIG .8. mentiscommoninmathematicalphysicsandengineering .If aright-handedscrewbeturnedfromtheX-axistotheY axisitwilladvancealongthe(positive)Z—aseis.Suchasys temofaxesiscalledright-handed,positive,orcounterclock wise.2ItiseasytoseethattheY-axisliesuponthatsideof theZX-planeonwhichrotationfromtheZ-axistotheX axisiscounterclockwise;andtheX-axis,uponthatsideof 1BytheXY orZ-axisthepositivehalfofthataxisismeant.TheXYplanemeanstheplanewhichcontainstheX andY—axis,i.e.,theplane20 . 3Aconvenientright-handedsystemandonewhichisalwaysavailableconsists ofthethumb,ﬁrstﬁnger,andsecondﬁngeroftherighthand.Ifthethumband ﬁrstﬁngerbestretchedoutfromthepalmperpendiculartoeachother,andifthe secondﬁngerbebentovertowardthepalmatrightanglestoﬁrstﬁnger,arighthandedsystemisformedbytheﬁngerstakenintheorderthumb,ﬁrstﬁnger, secondﬁnger. 20 VECTORANALYSIS theYZ-planeonwhichrotationfromtheY-axistotheZ axisiscounterclockwise.Thusitappearsthattherelation betweenthethreeaxesisperfectlysymmetricalsolongasthe samecyclicorderXYZXYisobserved.Ifaright-handed screwisturnedfromoneaxistowardthenextitadvances alongthethird . Intheothercase(Fig .8,secondpart)theZ-axisliesupon thatsideoftheXY-planeonwhichrotationthrougharight anglefromtheX-axistotheY-axisappearsclockwiseorneg ative.TheY-axisthenliesuponthatsideoftheZX-plane onwhichrotationfromtheZ-axistotheX—axisappears clockwiseandasimilarstatementmaybemadeconcerning theX-axisinitsrelationtotheYZ—plane.Inthiscase,too, therelationbetweenthethreeaxesissymmetricalsolong asthesamecyclicorderXYZXYispreservedbutitisjust theOppositeofthatintheformercase.Ifaleft-handedscrew isturnedfromoneaxistowardthenextitadvancesalong thethird .Hencethissystemiscalledleft-handed,negative, orclockwise.1 Thetwosystemsarenotsuperposable.Theyaresym metric.Oneistheimageoitheotherasseenina mirror.IftheXandY-axesofthetwodiﬁerentsystemsbe superimposed,theZ-axeswillpointinoppositedirections. Thusonesystemmaybeobtainedfromtheotherbyreversing thedirectionofoneoftheaxes.Alittlethoughtwillshow thatiftwooftheaxesbereversedindirectionthesystemwill notbealtered,butifallthreebesoreverseditwillbe. Whichofthetwosystemsbeused,matterslittle.Butin asmuchastheformulaeofgeometryandmechanicsdiffer slightlyinthematterofsign,itisadvisabletosettleoncefor allwhichshallbeadopted .Inthisbooktheright-handedor counterclockwisesystemwillbeinvariablyemployed . 1Aleft-handedsystemmaybeformedbythelefthandjustasaright-handed onewasformedbytheright. ADDITIONANDSCALARMULTIPLICATION 21 DeﬁnitionThethreelettersi,j,kwillbereservedtode notethreevectorsoiunitlengthdrawnrespectivelyinthe directionsoftheXYandZaxesofaright-handedrectan gularsystem . Intermsofthesevectors,anyvectormaybeexpressedas (6) Thecoefﬁcientsa,g,zaretheordinaryCartesiancoordinates oftheterminusofrifitsoriginbesituatedattheoriginof coordinates.ThecomponentsofrparalleltotheXY and Z—axesarerespectively sci,yj,zk . Therotationsaboutifromjtok,aboutjfromktoi,and aboutkfromitojareallpositive.Bymeansofthesevectorsi,j,ksuchacorrespondenceis establishedbetweenvectoranalysisandtheanalysisinCar tesiancoordinatesthatitbecomespossibletopassatwill fromeitheronetotheother.Thereisnothingcontradic torybetweenthem .Onthecontraryitisoftendesirable orevennecessarytotranslatetheformulmobtainedby vectormethodsintoCartesiancoordinatesforthesakeof comparingthemwithresultsalreadyknownanditis stillmorefrequentlyconvenienttopassfromCartesian analysistovectorsbothonaccountofthebrevitythereby obtainedandbecausethevectorexpressionsshowforththe intrinsicmeaningoftheformulae . Applications Problemsinplanegeometrymayfrequentlybesolved easilybyvectormethods.Anytwonon-collinearvectorsin theplanemaybetakenasthefundamentalonesintermsof whichallothersinthatplanemaybeexpressed .Theorigin mayalsobeselectedatpleasure.Oftenitispossibleto 22 VECTORANALYSIS makesuchanadvantageouschoiceoftheoriginandfunda mentalvectorsthattheanalyticworkofsolutionismaterially simpliﬁed.Theadaptabihtyofthevectormethodisabout thesameasthatofobliqueCartesiancoordinateswithdiﬁer entscalesuponthetwoaxes. x ExampTle‘Thelinewhichjoinsonevertexofaparallelo gram’ﬁfh’ e’middlepointofanoppositesidetrisectsthediag onal(Fig . LetABC’Dbetheparallelogram,BEthelinejoiningthe vertexBtothemiddlepointBoftheside AD,Rthepointinwhichthislinecutsthe diagonalA0 .ToshowABisonethirdof AO .ChooseAasorigin,ABandADasthe twofundamentalvectorsSandT .Then A0’isthesumofSandT .Letﬂ rﬁtB .)ToshowFIG.9. 1 whereaistheratioofERtoEB—euunknownscalar. And whereyisthescalarratioofARtoAOtobeshownequal to%. Hence ;T+a(S or Hence,equatingcorrespondingcoefficients(Art. ADDITIONANDSCALARMULTIPLICATION 23 1Fromwhichy. 3 InasmuchasxisalsogthelineEBmustbetrisectedas wellasthediagonalAC . (ExamplgﬁIfthroughanypointwithinatrianglelines bedrawnparalleltothesidesthesumoftheratiosofthese Ilinestotheircorrespondingsidesis2 . LetABC’bethetriangle,Rthepointwithinit.Choose Aasorigin,ABandA0asthetwofundamentalvectorsS andT.Let (a) A mSisthefractionofABwhichiscutoffbythelinethrough RparalleltoAC.TheremainderofABmustbethefrac tion1m8.Consequentlybysimilartrianglestheratioof thelineparalleltoA0’tothelineA0’itselfis(1m). SimilarlytheratioofthehueparalleltoABtothelineAB itselfis(1 n).NextexpressBintermsofSandTSthe thirdsideofthetriangle.Evidentlyfrom(a) —S). Hence(m n)SisthefractionofABwhichiscutoffbythe linethroughRparalleltoB Consequentlybysimilartri anglestheratioofthislinetoB0’itselfis(m n).Adding thethreeratios (1 andthethgoremisproved . "Eda- Eplc Iffromanypointwithinaparallelogramlines bedrawnparalleltothesides,thediagonalsoftheparallelo gramsthusformedintersectuponthediagonalofthegiven parallelogram . LetABC’Dbeaparallelogram,Bapointwithinit,EM andLNtwolines e1respectivelytoABand M ‘C‘ 24 VECTORANALYSIS AD,thepoints B0,CDrespecti LM ofthetwoparallelogramsKRNDand“* W ' 0 at“ Ina-OW on]ChooseAasorigin,ABandE asthetwofunda mentalvectors8andT .Let RAR mS nT, andletPbethepointofintersectionofKNwithLM . Then P=AP=AK+xKM LM=(I P=AP=AL+yLM Hence and Equatingcoefﬁcients, ccm .7!n Bysolution,cc 9 Substitutingeither expressionforP, theresultis P m+n whichshowsthatPiscollinearwithA Problemsinthreedimensionalgeometrymaybe solvedinessentiallythesamemannerasthoseintwodimen sions.Inthiscasetherearethreefundamentalvectorsin termsofwhichallotherscanbeexpressed .Themethodof solutionisanalogoustothatinthesimplercase.Twon+zc(1—n) n —9 m+n—1 m m+n—l ofthesesolutionsinthe 26 VECTORANALYSIS Inlikemanner AB’:x20yaD and—B). Hence—B) and 0=1+k2(l x2=k2m, y2=k2n . 1Hence h2= 1_lPB’192—1and BB,k2 Inthesamewayitmaybeshownthat PC’_andPD’ 00’—m Addingthefourratiostheresultis 1 Example2:TOﬁndalinewhichpassesthroughagiven pointandcutstwogivenlinesinspace. LetthetwolinesbeﬁxedrespectivelybytwopointsA andB,C’andDoneach.Let0bethegivenpoint.Choose itasoriginandlet A=0A,B=OB, D=0D . AnypointPofABmaybeexpressedas —A) AnypointQOfCDmaylikewisebewritten —O). IfthepointsPandQlieinthesamelinethrough0,PandQ are .collinear.Thatis P=zQ ADDITIONANDSCALARMULTIPLICATION 27 Beforeitispossibletoequatecoefﬁcientsoneofthefour vectorsmustbeexpressedintermsOftheotherthree. Let Then P=A+x(B—A) nC Hence l—xzzyl, x=zym, Hence x:m 9 l+m 1 y— l—n’ 1—n z: l+m SubstitutinginPandQ lA+mB P_ 9 l+m nC—DQ n—l EitherOfthesemaybetakenasdeﬁningalinedrawnfromO andcuttingABand0’D . VectorRelationsindependentoftheOrigin Example1TOdividealineABinagivenratio m n(Fig . Chooseanyarbitrarypoint0as origin .Let0A Aand0B B . TOﬁndthevectorP 0Pofwhich theterminusPdividesABinthe0 FIG .10. ratiom n . P=0P=0A+m AB=A+m (B—A). m+ Thatis, (7) 28 VECTORANALYSIS ThecomponentsOfPparalleltoAandBareininverseratio tothesegmentsAPandPBintowhichthelineABis dividedbythepointP .IfitshouldsohappenthatPdivided thelineABexternally,theratioAP/PBwouldbenega tive,andthesignsofmandnwouldbeOpposite,butthe formulawouldholdwithoutchangeifthisdifferenceofsign inmandnbetakenintoaccount. Example2TOﬁndthepointofintersectionofthemedians Ofatriangle. Choosetheorigin0atrandom .LetABCbethegiven triangle.Let0A 2A,OB B,andO0G.Let C’ berespectivelythemiddlepointsOfthesidesoppositethe verticesA,B,0 .LetMbethepointofintersectionOfthe mediansandM OMthevectordrawntoit.Then (B—A) 2 and (C—B) 2 AssumingthatOhasbeenchosenoutsideoftheplaneOfthe trianglesothatA,B,Carenon-OOpIanar,correspondingcoefﬁ cientsmaybeequated. 1 l §x=1—y, l 1 sac— 2? Hencenzg Hence =1 ADDITIONANDSCALARMULTIPLICATION 29 ThevectordrawntothemedianpointOfatriangleisequal toonethirdOfthesumOfthevectorsdrawntothevertices. IntheproblemsOfwhichthesolutionhasjustbeengiven theorigincouldbechosenarbitrarilyandtheresultisin dependentOfthatchoice.Henceitisevenpossibletodisre gardtheoriginentirelyandreplacethevectorsA,B,C,etc., bytheirterminiA,B,C,etc.Thusthepointsthemselves becomethesubjectsOfanalysisandtheformulaeread PnA+mB m+n and M ThisistypicalOfawholeclassofproblemssolublebyvector methods.Infactanypurelygeometricrelationbetweenthe diﬁerentpartsOfaﬁguremustnecessarilybeindependent Oftheoriginassumedfortheanalyticdemonstration .In somecases,suchasthoseinArts.18,19,thepositionofthe originmaybespecializedwithregardtosomecrucialpoint Oftheﬁguresoastofacilitatethecomputation;butinmany othercasesthegeneralityObtainedbyleavingtheoriginun Specializedandundeterminedleadstoasymmetrywhich renderstheresultsjustaseasytocomputeandmoreeasy toremember. Theorem:Thenecessaryandsufﬁcientconditionthata vectorequationrepresentarelationindependentoftheorigin isthatthesumOfthescalarcoefﬁcientsofthevectorson onesideOfthesignOfequalityisequaltothesumOfthe coefﬁcientsOfthevectorsupontheotherside .Orifallthe termsOfavectorequationbetransposedtoonesideleaving zeroontheother,thesumofthescalarcoefﬁcientsmust bezero . Lettheequationwritteninthelatterformbe 30 VECTORANALYSIS ChangetheoriginfromOto0’byaddingaconstantvector R OO’toeachOfthevectorsA,B,C,Do Theequation thenbecomes - o 0 IfthisistobeindependentoftheoriginthecoefﬁcientofB mustvanish.Hence Thatthisconditionisfulﬁlledinthetwoexamplescited isObvious. If PnA+mB , m+n n m 1 m+u+ m+u If M 1 1 l1 Thenecessaryandsufﬁcientconditionthattwo vectorssatisfyanequation,inwhichthesumOfthescalar coefﬁcientsiszero,isthatthevectorsbeequalinmagnitude andindirection. Firstlet aA+bB=O and a+b=0. Itisofcourseassumedthatnotboththecoefﬁcientsaandb vanish .Iftheydidtheequationwouldmeannothing.Sub stitutethevalueofaObtainedfromthesecondequationinto theﬁrst. Hence A B . ADDITIONANDSCALARMULTIPLICATION 31 SecondlyifAandBareequalinmagnitudeanddirection theequation A B0 subsistsbetweenthem .ThesumOfthecoefﬁcientsiszero. Thenecessaryandsufﬁcientconditionthatthreevectors satisfyanequation,inwhichthesumofthescalarcoefﬁcients iszero,isthatwhendrawnfromacommonorigintheytermi nateinthesamestraightline.1 Firstlet aA+bB+cC=0 and Notallthecoefﬁcientsa,b,c,vanishortheequations wouldbemeaningless.Letcbeanon-vanishingcoefﬁcient. SubstitutethevalueOfaobtainedfromthesecondequation intotheﬁrst. or c(C—B). HencethevectorwhichjoinstheextremitiesofCandAis collinearwiththatwhichjoinstheextremitiesofAandB . HencethosethreepointsA,B,0'lieonaline.Secondly supposethreevectorsA OA,B OB,C00drawnfrom thesameorigin0terminateinastraighthue.Thenthe vectors AB=B—AandA0=C—A arecollinear.Hencetheequation subsists .Thesumofthecoefﬁcientsonthetwosidesis thesame. Thenecessaryandsufﬁcientconditionthatanequation, inwhichthesumofthescalarcoefﬁcientsiszero,subsist 1Vectorswhichhaveacommonoriginandterminateinonelinearecalledby Hamilton“termino-collinear.” 32 VECTORANALYSIS betweenfourvectors,isthatifdrawnfromacommon theyterminateinoneplane.l Firstlet and Letdbeanon-vanishingcoefﬁcient.Substitutethe ofaObtainedfromthelastequationintotheﬁrst. or d(D—C). ThelineADiscoplanarwithABandA Henceallfour terminiA,B,C,DofA,B,C,Dlieinoneplane.Secondly supposethattheterminiOfA,B,C,Ddolieinoneplane. ThenAD=D—A,andAB zB—Aareco planarvectors.OneOfthemmaybeexpressedintermsof theothertwo .Thisleadstotheequation l(B—A)—0, wherel,m,andnarecertainscalars.Thesumofthecoefﬁ cientsinthisequationiszero . Betweenanyﬁvevectorsthereexistsoneequationthesum Ofwhosecoefﬁcientsiszero . LetA,B,C,D,Ebetheﬁvegivenvectors.Formthe differences E_A,E_B,E_C,E—D . OneOfthesemaybeexpressedintermsOftheotherthree —orwhatamountstothesamethingtheremustexistan equationbetweenthem . l003-41)+l(E ThesumOfthecoefﬁcientsofthisequationiszero. 1Vectorswhichhaveacommonoriginandterminateinoneplanearecalled byHamiltontermino-complanar." 34 VECTORANALYSIS HoweverthepointsE,C,andAlieuponthesamestraight line .Hencetheequationwhichconnectsthevectors andAmustbesuchthatthesumOfitscoefﬁcientsiszero. Thisdeterminesa:as1 n . Hence E—nC=D—nB=(1—n)A . Byanotherrearrangementandsimilarreasoning Subtracttheﬁrstequationfromthesecond: —n)A . ThisvectorcutsBCandAG .Itmustthereforebea multipleOfFandsuchamultiplethatthesumOfthecoefﬁ cientsOftheequationswhichconnectB,C,andForG,A, andFshallbezero. Hence B+C 2Hence F i andthetheoremhasbeenproved .Theproofhascovered considerablespacebecauseeachdetailOfthereasoninghas beengiven .Inreality,however,theactualanalysishascon sistedofjustfourequationsObtainedsimplyfromtheﬁrst. Example2:TodeterminetheequationsOfthelineand plane. LetthelinebeﬁxedbytwopointsAandBuponit.LetPbeanypointOftheline.Chooseanarbitraryorigin . ThevectorsA,B,andPterminateinthesameline.Hence aA+bB+PP=O and _aA+bBTherefore 2 ab ADDITIONANDSCALARMULTIPLICATION 35 FordiﬁerentpointsPthescalarsaandbhavedifferent values.Theymaybereplacedbyxandy,whichareused moregenerallytorepresentvariables.Then _xA+3/B x+yP LetaplanebedeterminedbythreepointsA,B,and0 . LetPbeanypointoftheplane.Chooseanarbitraryorigin . ThevectorsA,B,C,andPterminateinoneplane.Hence and Therefore PaA+bB+cc a+b+c Asa,b,c,varyfordifferentpointsOftheplane,itismore customarytowriteintheirsteadx,y,z. x+y+z Example3:ThelinewhichjoinsonevertexOfacom pletequadrilateraltotheintersectionOftwodiagonals dividestheOppositesideshar monically(Fig . LetA,B,C,Dbefourvertices Ofaquadrilateral.LetABmeet CDinaﬁfthvertexE,andAD meetBCinthesixthvertexF . E LetthetwodiagonalsACand BDintersectin Toshow thatFG’intersectsABinapointE’andCDinapointE” suchthatthehuesABandCDaredividedinternallyat E'andE”inthesameratioastheyaredividedexternally byE .Thatistoshowthatthecrossratios (ABDE’)(CD.DE")—1.FIG .12 . 36 VECTORANALYSIS Choosetheoriginatrandom .ThefourvectorsA,B,C,D drawnfromittothepointsA,B,C,Dterminateinone plane.Hence and Separatetheequationsbytransposingtwoterms: aA+cC= a+c= uA+cCbB+dD Drvrde: G b+d InhkemannerF=aA+dD bB+cc a+d b+c (a+c)G cc—dDF“m (am (He or(a+o)G cC—dD E"(a) cd cd Separatetheequationsagainanddivide: aA+bB cC+dD E . a+b c+d(b) HenceEdividesABintheratioa:bandCDin c:d .Butequation(a)showsthatE”dividesCDinthe ratio c:d.HenceEandE”divideCDinternallyand externallyinthesameratio.Whichofthetwodivisionsis internalandwhichexternaldependsupontherelativesigns Ofcandd.Iftheyhavethesamesigntheinternalpoint OfdivisionisE;ifoppositesigns,itisE Inasimilarway E’andEmaybeshowntodivideABharmonically. Example4:TOdiscussgeometricnets.Byageometricnetinaplaneismeantaﬁgurecomposed ofpointsandstraighthnesObtainedinthefollowingmanner. StartwithacertainnumberOfpointsallofwhichlieinone ADDITIONANDSCALARMULTIPLICATION 37 plane.Drawallthelinesjoiningthesepointsinpairs. ThesehueswillintersecteachotherinanumberOfpoints. Nextdrawallthelineswhichconnectthesepointsinpairs. ThissecondsetOflineswilldetermineastillgreaternumber ofpointswhichmayinturnbejoinedinpairsandsoon. Theconstructionmaybekeptupindeﬁnitely .Ateachstep thenumberOfpointsandhnesintheﬁgureincreases. ProbablythemostinterestingcaseOfaplanegeometricnetis thatinwhichfourpointsaregiventocommencewith. Joiningthesetherearesixhueswhichintersectinthree pointsdifferentfromthegivenfour.Threenewlinesmay nowbedrawnintheﬁgure.Thesecutoutsixnewpoints. Fromthesemorehnesmaybeobtainedandsoon . Totreatthisnetanalyticallywritedowntheequations (c) and whichsubsistbetweenthefourvectorsdrawnfromanunde terminedorigintothefourgivenpoints.Fromtheseitis possibletoObtain aA bB cC=dD E a+b c+d’ aA+cC bB+dD F a+c b+d aA+dD bB+cC G a+d b+c bysphttingtheequationsintotwopartsanddividing.Next fourvectorssuchasA,D,E,Fmaybechosenandtheequa tionthesumOfwhosecoeﬁicientsiszeromaybedetermined . Thiswouldbe Bytreatingthisequationas(c)wastreatednewpointsmay beobtained. 38 VECTORANALYSIS —aA+dD 9 H_a+d 2a+b+c I b a+c+d KdD+(a+b)E 'c a+b+d Equationsbetweenothersetsoffourvectorsselectedfrom A,B,C,D,E,F,Gmaybefound;andfromthesemorepoints Obtained .TheprocessOfﬁndingmorepointsgoesforward indeﬁnitely.Afulleraccountofgeometricnetsmaybe foundinHamilton’sElementsofQuaternions,”BookI. Asregardsgeometricnetsinspacejustawordmaybe said .Fivepointsaregiven.Fromthesenewpointsmaybe ObtainedbyﬁndingtheintersectionsOfplanespassedthrough setsOfthreeOfthegivenpointswithhuesconnectingthe remainingpairs.Theconstructionmaythenbecarriedfor wardwiththepointsthusobtained .Theanalytictreatment issimilartothatinthecaseofplanenets.Thereare ﬁvevectorsdrawnfromanundeterminedorigintothegiven ﬁvepoints.Betweenthesevectorsthereexistsanequation thesumofwhosecoeﬁ‘icientsiszero.Thisequationmaybe separatedintopartsasbeforeandthenewpointsmaythus beObtained. If and then FaA+bB cC+dD+eE , a+b c+d+e aA+cCbB+dD+eE H 9 a+b b+d+c aretwoOfthepointsandothersmaybefoundinthesame way.NetsinspacearealsodiscussedbyHamilton,loc.cit. ADDITIONANDSCALARMULTIPLICATION 39 CentersofGravity ThecenterofgravityOfasystemofparticlesmay befoundveryeasilybyvectormethods.Thetwolawsof physicswhichwillbeassumedarethefollowing: Thecenterofgravityoftwomasses(consideredas situatedatpoints)hesonthehneconnectingthetwomasses anddividesitintotwosegmentswhichareinverselypro portionaltothemassesattheextremities. InﬁndingthecenterOfgravityoftwosystemsof masseseachsystemmaybereplacedbyasinglemassequal inmagnitudetothesumOfthemassesinthesystemand situatedatthecenterofgravityofthesystem . GiventwomassesaandbsituatedattwopointsAandB . TheircenterOfgravity0isgivenby aA+bB 9 8 G a+b wherethevectorsarereferredtoanyoriginwhatsoever. Thisfollowsimmediatelyfromlaw1andtheformula(7) fordivisionOfahneinagivenratio . ThecenterOfgravityOfthreemassesa,b,csituatedatthe threepointsA,B,Cmaybefoundbymeansoflaw2.The massesaandbmaybeconsideredasequivalenttoasingle massabsituatedatthepoint aA+bB a+b Then G=(a+ +cC a+b+c aA+bB+cCHence G ab c 40 VECTORANALYSIS EvidentlythecenterofgravityOfanynumberofmasses a,b,c,d, situatedatthepointsA,B,C,D, befoundinasimilarmanner.Theresultis w) Theorem1Thehueswhichjointhecenterofgravityofa triangletotheverticesdivideitintothreetriangleswhich areproportionaltothemassesattheOp positevertices(Fig . LetA,B,C betheverticesOfatriangleweighted withmassesa,b,c.LetCbethecen terofgravity .JoinA,B,CtoGand producethehnesuntiltheyintersect theoppositesidesinA’ ,B’ ,C’respectively.TOshowthat theareas ThelastproportionbetweenABCanda b ccomes fromcompoundingtheﬁrstthree.Itis,however,usefulin thedemonstration . ABCAA’AC +GA’b+c CBC GA’GA’CA’a ABC a+b+cFIG .13. +1. Hence 030,a BUA a+b+cInasrmrlarmanner 00A 6 CAB a+b+cand GAB c Hencetheproportionisproved. Theorem2:ThehneswhichjointhecenterofgravityOf atetrahedrontotheverticesdividethetetrahedronintofour 42 VECTORANALYSIS a,b,0maythereforebelookeduponasco'ordinatesOfthe pointsPinsideofthetriangleABC .Toeachsetthere correspondsadeﬁnitepointP,andtoeachpointPthere correspondsaninﬁnitenumberofsetsOfquantities,which howeverdonotdifferfromoneanotherexceptforafactor Ofproportionality . TOobtainthepointsPOftheplaneABCwhichheoutside ofthetriangleABConemayresorttotheconceptionof negativeweightsormasses.ThecenterOfgravityofthe masses2and1situatedatthepointsAandBrespectively wouldbeapointCdividingthehneABexternallyinthe ratio12.Thatis AnypointOfthelineABproducedmayberepresentedby asuitablesetOfmassesa,bwhichdiﬂerinsign .Similarly anypointPoftheplaneABCmayberepresentedbya suitablesetofmassesa,b,cOfwhichonewilldifferinsign fromtheothertwoifthepointPhesoutsideofthetriangle ABC .InasmuchasonlytheratiosOfa,b,andcareim portanttwoOfthequantitiesmayalwaysbetakenpositive. Theideaofemployingthemassessituatedatthevertices ascoordinatesofthecenterOfgravityisduetoMO‘biusand waspubhshedbyhiminhisbookentitled“Barycentrische Calcitl,”in1826.Thismaybefairlyregardedasthestarting pointOfmodernanalyticgeometry . TheconceptionOfnegativemasseswhichhavenoexistence innaturemaybeavoidedbyreplacingthemassesatthe verticesbytheareasOfthetrianglesCBC,GCA,and CABtowhichtheyareproportional.Thecoo‘rdinatesOf apointPwouldthenbethreenumbersproportionaltothe areasOfthethreetrianglesofwhichPisthecommonvertex; andthesidesofagiventriangleABC,thebases.Thesign oftheseareasisdeterminedbythefollowingdeﬁnition . ADDITIONANDSCALARMULTIPLICATION 43 Deﬁnition:TheareaABCOfatriangleissaidtobe positivewhentheverticesA,B,Cfolloweachotherinthe positiveorcounterclockwisedirectionuponthecirclede scribedthroughthem .Theareaissaidtobenegativewhen thepointsfollowinthenegativeorclockwisedirection. OychcpermutationOfthelettersthereforedoesnotalter thesignOfthearea. ABC=BCA=CAB . InterchangeoftwoletterswhichamountstoareversalOf thecychcorderchangesthesign . ACB=BAC:CBA z—ABC . IfPbsanypointwithinthetriangletheequation PAB+PBC+PCA=ABC musthold.ThesamewillalsoholdifPbeoutsideOfthe triangleprovidedthesignsoftheareasbetakenintocon sideration .Theareasorthreequantitiesproportionalto themmayberegardedascoordinatesofthepointP . TheextensionOftheideaof“barycentric”coordinatesto spaceisimmediate.ThefourpointsA,B,C,Dsituatedat theverticesofatetrahedronareweightedwithmassa,b,c,d respectively .ThecenterOfgravity0’isrepresentedby thesequantitiesorfourothersproportionaltothem .TO ObtainpointsoutsideOfthetetrahedronnegativemasses maybeemployed.OrinthelightOftheorem2,page40, themassesmaybereplacedbythefourtetrahedrawhich areproportionaltothem .ThentheideaOfnegativevol umestakestheplaceOfthatOfnegativeweights.Asthis ideaisofconsiderableimportancelater,abrieftreatmentOf itheremaynotbeoutofplace. Deﬁnition:ThevolumeABCDOfatetrahedronissaid tobepositivewhenthetriangleABCappearspositiveto 44 VECTORANALYSIS theeyesituatedatthepointD .Thevolumeisnegative iftheareaOfthetriangleappearnegative. TOmakethediscussionofthesignsofthevarious tetrahedraperfectlyclearitisalmostnecessarytohavea sohdmodel.Aplanedrawingisscarcelysufﬁcient.Itis difﬁculttoseefromitwhichtrianglesappearpositiveand whichnegative.Thefollowingrelationswillbeseento holdifamodelbeexamined. TheinterchangeOftwolettersinthetetrahedronABCD changesthesign. ACBD=CBAD=BACD=DBCA =ADCB=ABDC=—ABCD . ThesignofthetetrahedronforanygivenoneOfthepos sibletwenty-fourarrangementsofthelettersmaybeObtained byreducingthatarrangementtotheorderABCDby meansofanumberOfsuccessiveinterchangesoftwoletters. IfthenumberOfinterchangesiseventhesignisthesame asthatofABCD;ifodd,opposite.Thus CADB=—CABD=+ACBD z—ABCD . IfPisanypointinsideofthetetrahedronABCDthe equation ABCP—BCDP+CDAP—DABP zABCD holdsgood.ItstillistrueifPbewithoutthetetrahedron providedthesignsOfthevolumesbetakenintoconsidera tion.Theequationmaybeputintoaformmoresymmetri calandmoreeasilyrememberedbytransposingalltheterms toonenumber.Then ABCD+BCDP+ Theproportionintheorem2,page40,doesnotholdtrue ifthesignsofthetetrahedraberegarded .Itshouldread ADDITIONANDSCALARMULTIPLICATION 45 IfthepointGliesinsidethetetrahedrona,b,c,drepre sentquantitiesproportionaltothemasseswhichmustbe locatedattheverticesA,B,C,DrespectivelyifCistobethe centerOfgravity .If0liesoutsideofthetetrahedrontheymay stillberegardedasmassessomeofwhicharenegative—Or perhapsbettermerelyasfournumberswhoseratiosdetermine thepositionOfthepointC .Inthismannerasetof“bary centriccoordinatesisestabhshedforSpace. ThevectorPdrawnfromanindeterminateorigintoany pointOftheplaneABCis(page35) PxA+yB+zC xy z Comparingthiswiththeexpression aA bB cC G ab c itwillbeseenthatthequantitiesx,y,zareinrealitynothing morenorlessthanthebarycentriccoordinatesofthepointP withrespecttothetriangleABC .Inlikemannerfrom equation xA+yB+zC+wD x+y+z+w whichexpressesanyvectorPdrawnfromanindeterminate originintermsOffourgivenvectorsA,B,C,Ddrawnfrom thesameorigin,itmaybeseenbycomparisonwith a+b+c+d thatthefourquantitiesx,y,z,warepreciselythebary centriccoordinatesOfP,theterminusofP,withrespectto thetetrahedronABCD .Thusthevectormethodsinwhich theoriginisundeterminedandthemethodsofthe“Bary centricCalculusarepracticallyco-extensive. ItwasmentionedbeforeanditmaybewelltorepeathereP G 46 VECTORANALYSIS thattheoriginmaybeleftwhollyoutOfconsiderationand thevectorsreplacedbytheirtermini.Thevectorequations thenbecomepointequations xAyB 2C wD xy zw . ThisstepbringsinthepointsthemselvesastheObjectsOf alysisandleadsstillnearertotheBarycentrischeCalcul” ofMo‘biusandthe“AusdehnungslehreofGrassmann .and TheUseofVectorstodenoteAreas DeﬁnitionAnarealyinginoneplaneMNand boundedbyacontinuouscurvePQRwhichnowherecuts itselfissaidtoappearpositivefromthepoint0whenthe lettersPQRfolloweach otherinthecounterclockwise Norpositiveorder;negative, whentheyfollowinthe negativeorclockwiseorder (Fig . Itisevidentthatanarea canhavenodeterminedsign FIG .14, perse,butonlyinreference tothatdirectioninwhichits boundaryissupposedtobetracedandtosomepoint0out sideOfitsplane .FortheareaPRQisnegativerelativeto PQRandanareaviewedfromOisnegativerelativetothe sameareaviewedfromapoint0’uponthesideoftheplane oppositetoO .AcirclelyingintheXY-planeanddescribed inthepositivetrigonometricorderappearspositivefromevery pointonthatsideoftheplaneonwhichthepositiveZ-axis lies,butnegativefromallpointsonthesideuponwhich ADDITIONANDSCALARMULTIPLICATION 47 thenegativeZ-axislies.Forthisreasonthepointofview andthedirectionofdescriptionoftheboundarymustbekept clearlyinmind . AnothermethodOfstatingthedeﬁnitionisasfollows:If apersonwalkinguponaplanetracesoutaclosedcurve,the areaenclosedissaidtobepositiveifitliesuponhisleft handside,negativeifuponhisright.Itisclearthatiftwo personsbeconsideredtotraceouttogetherthesamecurveby walkinguponOppositesidesoftheplanetheareaenclosed willlieupontherighthandofoneandthelefthandofthe other.TOoneitwillconsequentlyappearpositive;tothe other,negative.ThatsideOftheplaneuponwhichthearea seemspositiveiscalledthepositiveside;thesideupon whichitappearsnegative,thenegativeside.Thisideais famihartostudentsOfelectricityandmagnetism.Ifan electriccurrentﬂowaroundaclosedplanecurvethehnesOf magneticforcethroughthecircuitpassfromthenegativeto thepositivesideoftheplane.Apositivemagneticpole placeduponthepositivesideOftheplanewillberepelledby thecircuit. Aplaneareamaybelookeduponaspossessingmorethan positiveornegativemagnitude.Itmaybeconsideredto possessdirection,namely,thedirectionofthenormaltothe positivesideOftheplaneinwhichitlies.Henceaplane areaisavectorquantity .Thefollowingtheoremsconcerning areaswhenlookeduponasvectorsareimportant. Theorem1Ifaplaneareabedenotedbyavectorwhose magnitudeisthenumericalvalueofthatareaandwhose directionisthenormaluponthepositivesideOftheplane, thentheorthogonalprojectionOfthatareauponaplane willberepresentedbythecomponentofthatvectorinthe directionnormaltotheplaneofprojection(Fig . LettheareaAlieintheplaneMN .Letitbeprojected orthogonallyupontheplaneM’N’ .LetMNandM’N’inter 48 VECTORANALYSIS sectinthelinelandletthediedralanglebetweenthese twoplanesbex.ConsiderﬁrstarectanglePQRSinMN whosesides,PQ,RSandQB,SParerespectivelyparallel andperpendiculartothelineI.Thiswillprojectintoa rectangle inM’N’ .ThesidesP’Q’andB’S’ willbeequaltoPQandRS;butthesidesQ’R’andS’P’ willbeequaltoQRandSPmultiphedbythecosineOfx, theanglebetweentheplanes.Consequentlytherectangle PQRScosx. FIG.15. Hencerectangles,ofwhichthesidesarerespectively parallelandperpendiculartol,thelineOfintersectionofthe twoplanes,projectintorectangleswhosesidesarelikewise respectivelyparallelandperpendiculartolandwhoseareais equaltotheareaoftheoriginalrectanglesmultipliedbythe cosineoftheanglebetweentheplanes. FromthisitfollowsthatanyareaAisprojectedintoan areawhichisequaltothegivenareamultipliedbythecosine oftheanglebetweentheplanes.ForanyareaAmaybedi videdupintoalargenumberOfsmallrectanglesbydrawinga seriesofhnesinMNparallelandperpendiculartothelineI. 50 VECTORANALYSIS Theorem2Thevectorwhichrepresentsaclosedpolyhedral surfaceiszero . Thismaybeprovedbymeansofcertainconsiderationsof hydrostatics.Supposethepolyhedrondrawninabodyof ﬂuidassumedtobefreefromallexternalforces,gravityin cluded .lTheﬂuidisinequihbriumunderitsowninternal pressures.Theportionoftheﬂuidboundedbytheclosed surfacemovesneitheronewaynortheother.Uponeachface ofthesurfacetheﬂuidexertsadeﬁniteforceproportional totheareaofthefaceandnormaltoit.Theresultantofall theseforcesmustbezero,astheﬂuidisinequilibrium .Hence thesumofallthevectorareasintheclosedsurfaceiszero . Theproofmaybegiveninapurelygeometricmanner. Considertheorthogonalprojectionoftheclosedsurfaceupon anyplane.ThisconsistsOfadoublearea.ThepartOfthe surfacefarthestfromtheplaneprojectsintopositivearea; thepartnearesttheplane,intonegativearea.Thusthe surfaceprojectsintoacertainportionoftheplanewhichis coveredtwice,oncewithpositiveareaandoncewithnegative. Thesecanceleachother.Hencethetotalprojectionofa closedsurfaceuponaplane(iftakenwithregardtosign)is zero .Butbytheorem1theprojectionOfanareaupona planeisequaltothecomponentOfthevectorrepresenting thatareainthedirectionperpendiculartothatplane.Hence thevectorwhichrepresentsaclosedsurfacehasnocomponent alongthehueperpendiculartotheplaneOfprojection .This, however,wasanyplanewhatsoever.Hencethevectoris zero. Thetheoremhasbeenprovedforthecaseinwhichthe closedsurfaceconsistsofplanes.Incasethatsurfacebe 1Suchastateofaffairsisrealizedtoallpracticalpurposesinthecaseofapolyhedronsuspendedintheatmosphereandconsequentlysubjectedtoatmos phericpressure.Theforceofgravityactsbutiscounterbalancedbythetension inthesuspendingstring. ADDITIONANDSCALARMULTIPLICATION 51 curveditmayberegardedasthelimitOfapolyhedralsurface whosenumberOffacesincreaseswithouthmit.Hencethe vectorwhichrepresentsanyclosedsurfacepolyhedralor curvediszero.Ifthesurfacebenotclosedbutbecurvedit mayberepresentedbyavectorjustasifitwerepolyhedral. ThatvectoristhehmitIapproachedbythevectorwhich representsthatpolyhedralsurfaceOfwhichthecurvedsurface isthehmitwhenthenumberOffacesbecomesindeﬁnitely great SUMMARYOFCHAPTERI Avectorisaquantityconsideredaspossessingmagnitude anddirection .Equalvectorspossessthesamemagnitude andthesamedirection .Avectorisnotalteredbyshiftingit paralleltoitself.Anullorzerovectorisonewhosemag nitudeiszero.TOmultiplyavectorbyapositivescalar multiplyitslengthbythatscalarandleaveitsdirection unchanged.TOmultiplyavectorbyanegativescalarmul tiplyitslengthbythatscalarandreverseitsdirection . Vectorsaddaccordingtotheparallelogramlaw .Tosubtract avectorreverseitsdirectionandadd .Addition,subtrac tion,andmultiphcationofvectorsbyascalarfollowthesame lawsasaddition,subtraction,andmultiphcationinordinary algebra.Avectormayberesolvedintothreecomponents paralleltoanythreenon-OOplanarvectors.Thisresolution canbeaccomphshedinonlyoneway . Thecomponentsofequalvectors,paralleltothreegiven non-coplanarvectors,areequal,andconverselyifthecom ponentsareequalthevectorsareequal.Thethreeunit vectorsi,j,kformaright-handedrectangularsystem .In 1Thislimitexistsandisunique.Itisindependentofthemethodinwhich thepolyhedralsurfaceapproachesthecurvedsurface. 52 VECTORANALYSIS termsofthemanyvectormaybeexpressedbymeansOfthe Cartesiancoordinatesx,y,z . (6) Applications.Thepointwhichdividesahneinagiven ratiom:nisgivenbytheformula nA+mB m+n(7) Thenecessaryandsufﬁcientconditionthatavectorequation representarelationindependentOftheoriginisthatthesum ofthescalarcoeﬂicientsintheequationbezero.Between anyfourvectorsthereexistsanequationwithscalarcoeﬂi cients.IfthesumOfthecoeﬂicientsiszerothevectorsare termino-coplanar.Ifanequationthesumofwhosescalar coefﬁcientsiszeroexistsbetweenthreevectorstheyare termino-collinear.ThecenterofgravityOfanumberof massesa,b,c situatedattheterminiOfthevectors A,B,Cooosupposedtobedrawnfromacommonoriginis givenbytheformula GaA+bB+cC+ (9) Avectormaybeusedtodenoteanarea.Iftheareais planethemagnitudeOfthevectorisequaltothemagnitude ofthearea,andthedirectionOfthevectoristhedirectionof thenormaluponthepositivesideoftheplane.Thevector representingaclosedsurfaceiszero. EXERCISESONCHAPTERI 1.DemonstratethelawsstatedinArt.12 . 2.Atrianglemaybeconstructedwhosesidesareparallel andequaltothemediansofanygiventriangle. ADDITIONANDSCALARMULTIPLICATION 53' 3 .Thesixpointsinwhichthethreediagonalsofacom pletequadrangle1meetthepairsOfOppositesideshethree bythreeuponfourstraighthnes. 4 .Iftwotrianglesaresosituatedinspacethatthethree pointsOfintersectionOfcorrespondingsidesheonaline,then thehuesjoiningthecorrespondingverticespassthrougha commonpointandconversely . 5 .Givenaquadrilateralinspace.Findthemiddlepoint ofthehnewhichjoinsthemiddlepointsofthediagonals. Findthemiddlepointofthehuewhichjoinsthemiddle pointsOftwooppositesides.Showthatthesetwopointsare thesameandcoincidewiththecenterofgravityofasystem ofequalmassesplacedattheverticesofthequadrilateral. 6.IftwoOppositesidesOfaquadrilateralinspacebe dividedproportionallyandiftwoquadrilateralsbeformedby joiningthetwopointsOfdivision,thenthecentersOfgravity Ofthesetwoquadrilateralsheonahnewiththecenterof gravityOftheoriginalquadrilateral.Bythecenterofgravity ismeantthecenterOfgravityoffourequalmassesplacedat thevertices.Canthistheorembegenerahzedtothecase wherethemassesarenotequal? 7 .ThebisectorsOftheanglesOfatrianglemeetina point. 8 .IftheedgesOfahexahedronmeetfourbyfourinthree points,thefourdiagonalsOfthehexahedronmeetinapoint. Inthespecialcaseinwhichthehexahedronisaparallelopiped thethreepointsareataninﬁnitedistance . 9 .Provethatthethreestraightlinesthroughthemiddle pointsofthesidesofanyfaceofatetrahedron,eachparallel tothestraightlineconnectingaﬁxedpointPwiththemid dlepointOftheOppositeedgeOfthetetrahedron,meetina 1Acompletequadrangleconsistsofthesixstraighthneswhichmaybepassed throughfourpointsnothreeofwhicharecollinear.Thediagonalsarethelines whichjointhepointsofintersectionofpairsofsides. 54 VECTORANALYSIS pointEandthatthispointissuchthatPEpasses andisbisectedbythecenterofgravityofthetetrah 10 .Showthatwithoutexceptionthereexists0 equationwithscalarcoefﬁcientsbetweenanyfO vectorsA,B,C,D . 11.Discusstheconditionsimposeduponthree, ﬁvevectorsiftheysatisfytwoequationsthesum0 eﬂicientsineachofwhichiszero. CHAPTERII DIRECTANDSKEWPRODUCTSOFVECTORS ProductsofTwoVectors THEOperationsOfaddition,subtraction,andscalar multiphcationhavebeendeﬁnedforvectorsintheway suggestedbyphysicsandhavebeenemployedinafew apphcations.Itnowbecomesnecessarytointroducetwo newcombinationsOfvectors.Thesewillbecalledproducts becausetheyObeythefundamentallawOfproducts;i.e.,the distributivelawwhichstatesthattheproductOfAintothe sumofBandCisequaltothesumOftheproductsofAinto BandAinto0 . Deﬁnition:ThedirectproductOftwovectorsAandBis thescalarquantityObtainedbymultiplyingtheproductof themagnitudesOfthevectorsbythecosineOftheanglebe tweenthem . Thedirectproductisdenotedbywritingthetwovectors withadotbetweenthemas AoB . ThisisreadAdotBandthereforemayOftenbecalledthe dotproductinsteadOfthedirectproduct.Itisalsocalled thescalarproductowingtothefactthatitsvalueissca lar.IfAbethemagnitudeOfAandBthatofB,thenby deﬁnition A-B=ABcos (1) Obviouslythedirectproductfollowsthecommutativelaw AoBBoA. (2) 56 VECTORANALYSIS Ifeithervectorbemultipliedbyascalartheproductis multiphedbythatscalar.Thatis (xA)-B). IncasethetwovectorsAandBarecollineartheanglebe tweenthembecomeszerooronehundredandeightydegrees anditscosineisthereforeequaltounitywiththepositiveor negativesign .Hencethescalarproductoftwoparallel vectorsisnumericallyequaltotheproductoftheirlengths. ThesignoftheproductispositivewhenthedirectionsOfthe vectorsarethesame,negativewhentheyareOpposite .The productOfavectorbyitselfisthereforeequaltothesquare ofitslength AAA2(3) Consequentlyiftheproductofavectorbyitselfvanishthe vectorisanullvector. IncasethetwovectorsAandBareperpendicularthe anglebetweenthembecomesplusorminusninetydegrees andthecosinevanishes.HencetheproductAoBvanishes. ConverselyifthescalarproductAoBvanishes,then ABcos(A,B):0. HenceeitherAorBorcos(A,B)iszero,andeitherthe vectorsareperpendicularoroneofthemisnull.Thusthe conditionfortheperpendicularityoftwovectors,neitherof whichvanishes,isAoB0 . Thescalarproductsofthethreefundamentalunit vectorsi,j,kareevidently (4) Ifmoregenerallyaandbareanytwounitvectorsthe product aob 008(a,b). 58 VECTORANALYSIS Thescalarordirectproductfollowsthedistributive lawofmultiphcation .Thatis (A+R)-C=A-C+B-C . (6) ThismaybeprovedbymeansOfprojections.LetCbeequal toitsmagnitudeCmultiphedbyaunitvector0initsdirec tion .Toshow -(Cc) or -c. A-cistheprojectionOfAupon0;Boc,thatOfBupon0; (A B)oc,thatofABupon0.Buttheprojectionofthe sumA Bisequaltothesumoftheprojections.Hence therelation(6)isproved .Byanimmediategenerahzation (A+B+ +Bor+B-e+ Thescalarproductmaybeusedjustastheproductinordi naryalgebra.Ithasnopeculiardifﬁculties. IftwovectorsAandBareexpressedintermsofthe threeunitvectorsi,j,kas and then AoB= o(Bli+B2j+Bak) _A1B1ioi+A1B2ioj+A1B3i-k +A2B1joi+A2B2joj+A2B3jok +A3Blk-j+A3B2k-j+A3B3kok.BymeansOf(4)thisreducesto A(7) IfinparticularAandBareunitvectors,theircomponents A,,A2,A3andBl,B2,B,3arethedirectioncosinesofthe huesAandBreferredtoX,Y,Z . DIRECTANDSKEWPRODUCTSOFVECTORS59 AIcos(A,X),A2cos(A,Y),A8cos(A,Z), B1cos(B,X),B2cos(B,Y),B3cos(B,Z). MoreoverAoBisthecosineoftheincludedangle.Hence theequationbecomes cos(A,B)=cos(A,X)cos(B,Z)cos(A,Y)cos(B,Y) cos(A,Z)cos(B,Z). IncaseAandBareperpendicularthisreducestothewell knownrelation 0=cos(A,X)cos(B,X)cos(A,Y)cos(B,Y) cos(A,Z)cos(B,Z) betweenthedirectioncosinesofthe lineAandthehneB . IfAandBaretwosides0A andOBofatriangle0AB,thethird sideABisCBA(Fig. CoC=(B—A)=B-B+A-A—2A-Bcos or 02=A2+B2FIG .17. Thatis,thesquareofonesideOfatriangleisequaltothe sumofthesquaresoftheothertwosidesdiminishedbytwice theirproducttimesthecosineoftheanglebetweenthem . Or,thesquareOfonesideOfatriangleisequaltothesumOf thesquaresoftheothertwosidesdiminishedbytwicethe projectionofeitherofthosesidesupontheother,thetheorem sometimesknownasthegenerahzedPythagoreantheorem . IfAandBaretwosidesofaparallelogram,CA B andDA Barethediagonals.Then D-D=(A—B)o(A—B)=AoA—2A-B+BoB, 0o0+D-D=2(A-A+B-B)a or 60 VECTORANALYSIS Thatis,thesumofthesquaresOfthediagonalsofaparallelo gramisequaltotwicethesumofthesquaresoftwosides. Inlikemanneralso C-C—D-D=4AoB or 02—D2=4ABcos Thatis,thedifferenceOfthesquaresofthediagonalsofa parallelogramisequaltofourtimestheproductOfoneofthe sidesbytheprojectionoftheotheruponit. IfAisanyvectorexpressedintermsOfi,j,kas then 4422+A82(8) ButifAbeexpressedintermsOfanythreenon-OOplanarunit vectorsa,b,cas A=aa+bb+ca Aoh +2bcb-c+2cacos 2cacos(c,a). ThisformulaisanalogoustotheoneinCartesiangeometry whichgivesthedistancebetweentwopointsreferredto Obhqueaxes.Ifthepointsbexl,y,,z],andx2,ya,22the distancesquaredis D2(x2w021902(zszr)2 2(x2mi)(312003(X:Y) 2(592 (22z1)cos(KZ) 2(z2—zl)(x2x1)cos(Z,X). Deﬁnition:TheskewproductofthevectorAinto thevectorBisthevectorquantity0whosedirectionisthe normaluponthatsideoftheplaneOfAandBonwhich DIRECTANDSKEWPRODUCTS0FVECTORS 61 rotationfromAtoBthroughanangleoflessthanone hundredandeightydegreesappearspositiveorcounter clockwise;andwhosemagnitudeisobtainedbymultiplying theproductofthemagnitudesofAandBbythesineofthe anglefromAtoB . ThedirectionofAxBmayalsobedeﬁnedasthatin whichanordinaryright-handed screwadvancesasitturnssoasc:Ax}; tocarryAtowardB(Fig . Theskewproductisdenotedby acrossasthedirectproductwas byadot.Itiswritten En}.la, C=AXE andreadAcrossB .Forthisreasonitisoftencalledthecross product.Morefrequently,however,itiscalledthevectorprod uct,owingtothefactthatitisavectorquantityandincon trastwiththedirectorscalarproductwhosevalueisscalar. Thevectorproductisbydeﬁnition (9) whenAandBarethemagnitudesofAandBrespectivelyand wherecisaunitvectorinthedirectionofC .IncaseAand BareunitvectorstheskewproductAxBreducestothe unitvectorcmultipliedbythesineoftheanglefromAtoB . ObviouslyalsoifeithervectorAorBismultipliedbyascalar a:theirproductismultipliedbythatscalar. (23A)XB=AX(xB)=zC . IfAandBareparalleltheanglebetweenthemiseitherzero oronehundredandeightydegrees.Ineithercasethesine vanishesandconsequentlythevectorproductAxBisanull vector.AndconverselyifAXBiszero ABsin 62 VECTORANALYSIS HenceAorBorsin(A,B)iszero .Thustheconditionfor parallelismoftwovectorsneitherofwhichvanishesisAXB 0 .Asacorollarythevectorproductofanyvectorinto itselfvanishes. Thevectorproductoftwovectorswillappearwher everthesineoftheincludedangleisofimportance,justas thescalarproductdidinthecaseofthecosine.Thetwoprod uctsareinacertainsensecomplementary .Theyhavebeen denotedbythetwocommonsignsofmultiplication,thedot andthecross.Invectoranalysistheyoccupytheplaceheld bythetrigonometricfunctionsofscalaranalysis.Theyare atthesametimeamenabletoalgebraictreatment,aswillbe seenlater.Atpresentafewusesofthevectorproductmay becited. IfAandB(Fig .18)arethetwoadjacentsidesofaparallel ogramthevectorproduct C=AXB=ABsin(A,B)c representstheareaofthatparallelograminmagnitudeand direction(Art. ThisgeometricrepresentationofAXB isofsuchcommonoccurrenceandimportancethatitmight wellbetakenasthedeﬁnitionoftheproduct.Fromitthe trigonometricdeﬁnitionfollowsatonce.Thevectorproduct appearsinmechanicsinconnectionwithcouples.IfAand Aaretwoforcesformingacouple,themomentofthe coupleisAXBprovidedonlythatBisavectordrawnfrom anypointofAtoanypointof—A .Theproductmakesits appearanceagaininconsideringthevelocitiesoftheindivid ualparticlesofabodywhichisrotatingwithanangular' ve locitygiveninmagnitudeanddirectionbyA .IfRbethe radiusvectordrawnfromanypointoftheaxisofrotationA theproductAXB.willgivethevelocityoftheextremityof B(Art. Thisvelocityisperpendicularaliketotheaxis ofrotationandtotheradiusvectorR . DIRECTANDSKEWPRODUCTS0FVECTORS63 ThevectorproductsAXBandBXAarenotthe same.Theyareinfactthenegativesofeachother.Forif rotationfromAtoBappearpositiveononesideoftheplane ofAand° B,rotationfromBtoAwillappearpositiveonthe other.HenceAXBisthenormaltotheplaneofAandB uponthatsideoppositetotheoneuponwhichBXAisthe normal.ThemagnitudesofAXBandBXAarethesame. Hence AXB BXA (10) Thefactorsinavectorproductcanbeinterchangedifandonly ifthe oftheproductbereversed . Thisistheﬁrstinstanceinwhichthelawsofoperationin vectoranalysisdifferessentiallyfromthoseofscalaranaly sis.Itmaybethatatﬁrstthischangeofsignwhichmust accompanytheinterchangeoffactorsinavectorproductwill giverisetosomedifﬁcultyandconfusion .Changessimilarto thisare,however,veryfamiliar.Noonewouldthinkofinter changingtheorderofacandyintheexpressionsin(a: withoutpreﬁxingthenegativesigntotheresult.Thus sin(y—x)=—sin(x—y), althoughthesignisnotrequiredforthecaseofthecosine. cos(y—y). AgainifthecyclicorderofthelettersAB0'intheareaofa trianglebechanged,theareawillbechangedinsign(Art. ABO=—AOB Inthesamemannerthisreversalofsign,whichoccurs whentheorderofthefactorsinavectorproductisreversed, willappearafteralittlepracticeandacquaintancejustas naturalandconvenientasitisnecessary . Thedistributivelawofmultiplicationholdsinthe caseofvectorproductsjustasinordinaryalgebra—except 64 VECTORANALYSIS thattheorderofthefactorsmustbecarefullymaintained (11) O Averysimpleproofmaybegivenbymakinguseoftheideas developedinArt.26.SupposethatC isnotcoplanarwithAandB .LetA andBbetwosidesofatriangletaken inorder.Then(A+B)willbethe thirdside(Fig . Formtheprism ofwhichthistriangleisthebaseand ofwhich0istheslantheightoredge. FIG.19. Theareasofthelateralfacesofthis prismare Axc,Bxc, Theareasofthebasesare inxB)and—3 2-(AxB). Butthesumofallthefacesoftheprismiszero;forthe prismisaclosedsurface.Hence AXC+BX‘ C A><C+BXC or (11) Therelationisthereforeprovedincase0isnon-coplanar withAandB .ShouldCbecoplanarwithAandB,chooseD, anyvectoroutofthatplane.ThenGDalsowilllieoutof thatplane.Henceby(11) SincethethreevectorsineachsetA,C,D,andB,C,D,and A B,C,Dwillbenon-coplanarifDisproperlychosen,the productsmaybeexpanded . 66 VECTORANALYSIS Thismaybewrittenintheformofadeterminantas ij k AXB AIA2A3 Theformulasforthesineandcosineofthesumordif ferenceoftwoanglesfollowimmediatelyfromthedotand crossproducts.Letaandbbetwounitvectorslyinginthe ij-plane.Ifxbetheanglethatamakeswithi,andythe anglebmakeswithi,then a—x), a Hence cos(y If b’=cosyi—sinyj, Hence—sinysinx. aXb=k(sinycosa—sinxcosy). Hence sin(y—x)=sinycosa—sinxcosy. Hencesin(y+a)=sinycosx+sinxcos IfI,m,nandl' ,m’ ,n'arethedirectioncosinesoftwo unitvectorsaanda'referredtoX,Y,Z,then ace/=cos ashasalreadybeenshowninArt.29.Thefamiliarformula forthesquareofthesineoftheanglebetweenaanda'may befound. DIRECTANDSKEWPRODUCTS0FVECTORS 67 aXa'=sin(a,a')e=(mu'(nl'—7b'l)j (lm’—l’m)k, whereeisaunitvectorperpendiculartoaanda' (aXa’)o(aXa’)sin2(a,a')eoesin2(a, sin2n’m'n)2+(nl' n’l)2+(lm'l’rn)2 Thisleadstoaneasywayofestablishingtheusefulidentity (mn'rn’n)2(nl’n’l)2(lm'l’m)2 (12m2n2)(l'2m’2n’z)(W mm'rim/)2 . ProductsofMorethanTwoVectors Uptothispointnothinghasbeensaidconcerning productsinwhichthenumberofvectorsisgreaterthan two .Ifthreevectorsarecombinedintoaproducttheresult iscalledatripleproduct.Nexttothesimpleproducts ABandAXBthetripleproductsarethemostimportant. Allhigherproductsmaybereducedtothem. Thesimplesttripleproductisformedbymultiplyingthe scalarproductoftwovectorsAandBintoathirdCas (AB)C . Thisinrealitydoesnotdifferessentiallyfromscalarmulti plication(Art. Thescalarinthiscasemerelyhappensto bethescalarproductofthetwovectorsAandB .Moreover inasmuchastwovectorscannotstandsidebysideinthe formofaproductasBCwithouteitheradotoracrossto unitethem,theparenthesisin(AB)Cissuperﬂuous .The expressionAoB0 cannotbeinterpretedinanyotherway1thanastheproduct ofthevector0bythescalarAB . 1Later(Chap .V .)theproductBC,wherenosigneitherdotorcrossoccurs, willbedeﬁned .Butitwillbeseentherethat(AoB)CandA-(BC)areidentical andconsequentlynoambiguitycanarisefromtheomissionoftheparenthesis. 68 VECTORANALYSIS Thesecondtripleproductisthescalarproductof twovectors,ofwhichoneisitselfavectorproduct,as A-(BXC)or(AXB)-C . Thissortofproducthasascalarvalueandconsequentlyis oftencalledthescalartripleprod uct.Itspropertiesareperhapsmost easilydeducedfromitscommonest geometricalinterpretation .LetA,B, andCbeanythreevectorsdrawn fromthesameorigin(Fig . ThenBXCistheareaofthe__par allelogramofwhichBandCaretwoadjacentsides.The seal“A-(BXC)v(14)A FIG .20 . willthereforebethevolumeoftheparallelopipedofwhich BXCisthebaseandAtheslantheightoredge.SeeArt.28 . ThisvolumevispositiveifAandBXClieuponthesame sideoftheBC-plane;butnegativeiftheylieonopposite sides.InotherwordsifA,B,0formaright-handedor positivesystemofthreevectorsthescalarAo(BXC)isposi tive;butiftheyformaleft-handedornegativesystem,it isnegative. IncaseA,B,andCarecoplanarthisvolumewillbe neitherpositivenornegativebutzero.Andconverselyif thevolumeiszerothethreeedgesA,B,Coftheparallelo« pipedmustlieinoneplane .Hencethenecessaryandsufﬁ cientconditionforthecoplanarityofthreevectorsA,B,0none ofwhichvanishesisAo(BXC)0 .Asacorollarythescalar tripleproductofthreevectorsofwhichtwoareequalor collinearmustvanish;foranytwovectorsarecoplanar. ThetwoproductsA-(BXC)and(AXB)-Careequaltothe samevolumevoftheparallelopipedwhoseconcurrentedges areA,B,C.Thesignofthevolumeisthesameinboth DIRECTANDSKEWPRODUCTSOFVECTORS 69 Thisequalitymaybestatedasaruleofoperation .Thedot andthecrossinascalartripleproductmaybeinterchanged withoutalteringthevalueoftheproduct. ItmayalsobeseenthatthevectorsA,B,Cmaybeper mutedcycliclywithoutalteringtheproduct. Ao(BXC)Bo(CXA)C (15) Foreachoftheexpressionsgivesthevolumeofthesame parallelopipedandthatvolumewillhaveineachcasethe samesign,becauseifAisuponthepositivesideoftheBC plane,BwillbeonthepositivesideoftheCA-planeand0 uponthepositivesideoftheAB—plane.Thetripleproduct maythereforehaveanyoneofsixequivalentforms A-(BXC)B-(CxA)E»(AxB) (AxB)oC(BXC)-A(CXA)-B . Ifhoweverthecyclicorderofthelettersischangedthe productwillchangesign . A-(BXC) (16) Thismaybeseenfromtheﬁgureorfromthefactthat BXC CXB . Hence:Ascalartripleproductisnotalteredbyinterchanging thedotorthecrossorbypermutingcycliclytheorderofthe vectors,butitisreversedinsignifthecyclicorderbechanged . Awordisnecessaryuponthesubjectofparentheses inthistripleproduct.Cantheybeomittedwithoutem biguity?Theycan .Theexpression AoBXC canhaveonlytheoneinterpretation A Fortheexpression(A«B)XCismeaningless.Itisimpos sibletoformtheskewproductofascalarABandavector 70 VECTORANALYSIS C .HenceasthereisonlyonewayinwhichA-BXCmay beinterpreted,noconfusioncanarisefromomittingthe parentheses .Furthermoreowingtothefactthatthereare sixscalartripleproductsofA,B,andCwhichhavethesame valueandareconsequentlygenerallynotworthdistinguish ingtheonefromanother,itisoftenconvenienttousethe symbol [ABC] todenoteanyoneofthesixequalproducts. [ABC]A-BxcB-CxA C-AxB AxB-CBXC-A CxA-B then [ABC] [ACB]. Thescalartripleproductsofthethreeunitvectorsi,j,1: allvanishexceptthetwowhichcontainthethreedifferent vectors. [ijk]=(17) HenceifthreevectorsA,B,Cbeexpressedintermsofi,j,k as Czali+02j+03k9 then B203+B102A3+ A2B318 —A1B302—B103A2—01A382. Thismaybeobtainedbyactuallyperformingthemultiplica tionswhichareindicatedinthetripleproduct.Theresult maybewrittenintheformofadeterminant.1 AlA2A3 [ABo]R,E,E, 010203 1Thisistheformulagiveninsolidanalyticgeometryforthevolumeofa tetrahedrononeofwhoseverticesisattheorigin.Foramoregeneralformula seeexercises. DIRECTANDSKEWPRODUCTS0FVECTORS71 IfmoregenerallyA,B,Careexpressedintermsofanythree non-coplanarvectorsa,b,cwhicharenotnecessarilyunit vectors, A=ala+a2b+a30 C=cla+c2b+c3c wherea1,a2,as;b1,b2,b3;andcl,c3arecertaincon stants,then (19)—a1b302—b103a2—ela3b2)[abc]. “1“2“s 610203 Thethirdtypeoftripleproductisthevectorproduct oftwovectorsofwhichoneisitselfavectorproduct.Such are AX(BXC)and(AXB)XC. ThevectorAX(BXC)isperpendiculartoAandto(EXO). But(BXC)isperpendiculartotheplaneofBandC .Hence AX(BXC),beingperpendicularto(BXC)mustlieinthe planeofBandCandthustaketheform AX(BXC)aLByc, wherea:andyaretwoscalars.Inlikemanneralsothe vector(AXB)XC,beingperpendicularto(AXB)mustlie intheplaneofAandB .Henceitwillbeoftheform (AXB)XC:mA nB wheremandvaretwoscalars.Fromthisitisevidentthat ingeneral (AXB)XCisnotequaltoAX(BXC). Theparenthesesthereforecannotberemovedorinter changed .Itisessentialtoknowwhichcrossproductis 72 VECTORANALYSIS formedﬁrstandwhichsecond.Thisproductistermedthe vectortripleproductincontrasttothescalartripleproduct. Thevectortripleproductmaybeusedtoexpressthatcom ponentofavectorBwhichisperpendiculartoagivenvector A .Thisgeometricuseoftheproductisvaluablenotonlyin itselfbutforthelightitsheds Ax3 uponthepropertiesoftheproduct. LetA(Fig .21)beagiven~vector andBanothervectorwhosecom ponentsparallelandperpendicular AtoAaretobefound .Letthe componentsofBparallelandper Ax(Axn) pendiculartoAbeB’andB"re FIG .21.spectively.DrawAandBfroma commonorigin .TheproductAXB isperpendiculartotheplaneofAandB .Theproduct AX(AXB)liesintheplaneofAandB .Itisfurthermore perpendiculartoA.HenceitiscollinearwithB” .An examinationoftheﬁgurewillshowthatthedirectionof AX(AXB)isoppositetothatofB” .Hence AX(AXB)cB" , wherecissomescalarconstant. Now AX(AXB)='AZBsin(A,B)b” but—cB"=—cin(A,R)b" , ifb’beaunitvectorinthedirectionofB" Hence 0A2A-A . AX(AxB)Hence B" .2 AA(20) ThecomponentofBperpendiculartoAhasbeenexpressed intermsofthevectortripleproductofA,A,andB .The componentB’paralleltoAwasfoundinArt.28tobe 74 VECTORANALYSIS Substitutingthesevaluesin AX(BXC)A-CBA-BC . (24) TherelationisthereforeprovedforanythreevectorsA,B,C. Anothermethodofgivingthedemonstrationisasfollows. Itwasshownthatthevectortripleproduct. AX(BXC)was oftheform AX(BXC)xByC . SinceAX(AXC)isperpendiculartoA,thedirectproductof itbyAiszero .Hence A oBoC0 and a:yA-C-AoB . Hence AX(BXC):1n(A-CBABC), wherenisascalarconstant.Itremainstoshown 1. MultiplybyB . AX(BXC)oB n(A~CBoBAoBGB). Thescalartripleproductallowsaninterchangeofdotand cross.Hence A-(BXC)XB iftheorderofthefactors(EXO)andBbeinverted . A-[BX(DXC)]A{BCBBB0] —B-CAoBB-BA00 . Hencen1and AX(BXC)A-CBABC .(24) FromthethroelettersA,B,Cbydiﬁerentarrangements, fouralliedproductsineachofwhichBandCareincludedin parenthesesmaybeformed .Theseare AX(BXC),AX(CXB), (BXC)XA . Asavectorproductchangesitssignwhenevertheorderof twofactorsisinterchanged,theaboveproductsevidently satisfytheequations AX(BXC)AX(CXB) DIRECTANDSKEWPRODUCTS0FVECTORS75 Theexpansionforavectortripleproductinwhichthe parenthesiscomesﬁrstmaythereforebeobtaineddirectly fromthatalreadyfoundwhentheparenthesiscomeslast. CX(AXB)C-BA C-AB . Theformulasthenbecome AX(BXC)A-CBAB0 (24) and A0B0-3A (24) Thesereductionformulaeareofsuchconstantoccurrenceand greatimportancethattheyshouldbecommittedtomemory. Theircontentmaybestatedinthefollowingrule.Toexpand avectortripleproductﬁrstmultiplytheexteriorfactorintothe remotertermintheparenthesistoformascalarcoeﬂicientfor thenearerone,thenmultiplytheexteriorfactorintothenearer termintheparenthesistoformascalarcoeﬁcientforthe remoterone,andsubtractthisresultfromtheﬁrst. Asfarasthepracticalapplicationsofvectoranalysis areconcerned,onecangenerallygetalongwithoutany formulaemorecomplicatedthanthatforthevectortriple product.Butitisfrequentlymoreconvenienttohaveat handotherreductionformulasofwhichallmaybederived simplybymakinguseoftheexpansionforthetripleproduct AX(BXC)andoftherulesofoperationwiththetriplepro ductA-BXC . Toreduceascalarproductoftwovectorseachofwhich isitselfavectorproductoftwovectors,as (AxB) Letthisberegardedasascalartripleproductofthethree vectorsA,B,andCXD—thus AXB(CXD) Interchangethedotandthecross. 76 VECTORANALYSIS AxB-(CXD)A-Bx(CXD) BX(CXD)BB0B-0 Hence (AXB)-(CXD)2ACB-DABBC .(25) Thismaybewrittenindeterminantalform . A-D D 25(Axis(oxM IfAandDbecalledtheextremesBandCthemeans3A andCtheantecedents;BandDtheconsequentsinthis productaccordingtothefamiliarusageinproportions,then theexpansionmaybestatedinwords .Thescalarproduct oftwovectorproductsisequaltothe(scalar)pfoductofthe antecedentstimesthe(scalar)productoftheconsequents diminishedbythe(scalar)productofthemeanstimesthe (scalar)productoftheextremes. Toreduceavectorproductoftwovectorseachofwhich isitselfavectorproductoftwovectors,as LetCXD E .Theproductbecomes A-EBBEA . SubstitutingthevalueofEbackintotheequation: (AoCxD)B(BoCxD)A (26)LetFAXB .Theproductthenbecomes ED0F-CD (AxB-D)C(AxBoC)D . Byequatingthesetwoequivalentresultsandtransposing allthetermstoonesideoftheequation, [BCD]A(27) Thisisanequationwithscalarcoefﬁcientsbetweenthefour vectorsA,B,C,D .Thereisingeneralonlyonesuchequa DIRECTANDSKEWPRODUCTSOFVECTORS77 tion,becauseanyoneofthevectorscanbeexpressedinonly onewayintermsoftheotherthree:thusthescalarcoefﬁ cientsofthatequationwhichexistsbetweenfourvectorsare foundtobenothingbutthefourscalartripleproductsof thosevectorstakenthreeatatime.Theequationmayalso bewrittenintheform c . Moreexamplesofreductionformula,ofwhichsomeare important,aregivenamongtheexercisesattheendofthe chapter.Inviewoftheseitbecomesfairlyobviousthat thecombinationofanynumberofvectorsconnectedin anylegitimatewaybydotsandcrossesortheproductofany numberofsuchcombinationscanbeultimatelyreducedto asumoftermseachofwhichcontainsonlyonecrossatmost. Theproofofthistheoremdependssolelyuponanalyzingthe possiblecombinationsofvectorsandshowingthattheyall fallunderthereductionformulainsuchawaythatthe crossesmayberemovedtwoatatimeuntilnotmorethan oneremains. Theformuladevelopedintheforegoingarticlehave interestinggeometricinterpretations.Theyalsoaﬁorda simplemeansofdeducingtheformulaofSphericalTrigo nometry.Thesedonotoccurinthevectoranalysisproper. Theirplaceistakenbythetwoquadrupleproducts, (AXB)-(CXD)M:B-D BA:AB (25) and [ACD]B[BOD]A [ABD]c[ABC]D,(26) whicharenowtobeinterpreted . Letaunitsphere(Fig .22)begiven .Letthevectors A,B,C,Dbeunitvectorsdrawnfromacommonorigin,the centreofthesphere,andterminatinginthesurfaceofthe SphereatthepointsA,B,C,D .Thegreatcirculararcs 78 VECTORANALYSIS AB,A etc.,givetheanglesbetweenthevectorsAandB, AandC,etc.ThepointsA,B,C,Ddetermineaquadrilateral uponthesphere.A0andBDareone pairofoppositesides;ADandBC,the other.ABandCDarethediagonals. (AXE)A-CB~DA-DB-C AXBsin(A,B),CXD sin(C,D). TheanglebetweenAXBandCXDisthe FIG .22.anglebetweenthenormalstotheAB andCD-planes.Thisisthesameas theanglebetweentheplanesthemselves.Letitbedenoted byso.Then sin(A,B)sin(C,D)cosx. Theangles(A,B),(C,B)maybereplacedbythegreat circulararcsAB,CDwhichmeasurethem .Then (AXB)-(CXD)sinABsinCDcos Act}BoD—AoDBee2cosACcosBD cosADcosBC . Hence sinABsin0Dcosx=cosA0cosBD—cosADcosBC’ . Inwords:Theproductofthecosinesoftwooppositesides ofasphericalquadrilaterallesstheproductofthecosinesof theothertwoOppositesidesisequaltotheproductofthe sinesofthediagonalsmultipliedbythe cosineoftheanglebetweenthem .This theoremiscreditedtoGauss. LetA,B,C(Fig .23)beasphericaltri angle,thesidesofwhicharearcsofgreat circles.Letthesidesbedenotedbya,b,c respectively .LetA,B,Cbetheunitvectors drawnfromthecenterofthespheretothepointsA,B, Furthermoreletpa,p,,p,bethegreatcirculararcsdroppedFIG .23. DIRECTANDSKEWPRODUCTSOFVECTORS 79 perpendicularlyfromtheverticesA,B,0tothesidesa,b,c. Interprettheformula (AXB)-(CXA)AOBoA B-CA-A . (AXB)sin(A,B)sinc,(OXA)sin(C,A)sinb. Then (AXB)-(CXA)sincsinbcosx, wherexistheanglebetweenAXBandOXA .This angleisequaltotheanglebetweentheplaneofA,Bandthe planeofC,A .Itis,however,nottheinteriorangleAwhich isoneoftheanglesofthetriangle:butitistheexterior angle180°A,asanexaminationoftheﬁgurewillshow . Hence sincsinbcos(180°A) sin0sinbcosA ACB-ABoCAA cosbcosccosa1. Byequatingtheresultsandtransposing, cosa:cosbcosc-sinbsinccosA cosb=‘ cosccosa sincsinacosB cosc=cosacosb—srnasrnbcos Thelasttwomaybeobtainedbycyclicpermutationofthe lettersorfromtheidentities B-ACoBC-A, C-BA-CB-C . Nextinterprettheidentity inthespecial casesinwhichoneofthevectorsisrepeated . [ABC]A . Letthethreevectorsa,b,cbeunitvectorsinthedirectionof BXC,CXA,AXBrespectively.Then AXB=csinc, AXC=—bsinb csin0sinbAsincsinbsinA [ABC](AXB)-C 0-0sinccos(90°-p,)sinc [ABC]A=sincsinp,~ 80 VECTORANALYSIS Byequatingtheresultsandcancellingthecommonfactor, sinp,sinbsinA sinpasin0sinB sinp,sinasin Thelasttwomaybeobtainedbycyclicpermutation letters.Theformulagivethesinesofthealtitudesofthe triangleintermsofthesinesoftheangleandsides. write [ABC]A [BCA]B [GAB]0. Hence sincsinbsinA.[ABC] sinasinesinB:[BOA] sinbsinasin [GAB]. Theexpressions[ABC],[BCA],[CAB]areequal. theresultsinpairsandtheformula sinbsinA=sinasinB smcsrns inbsinO’ sinasinOzzsincsinA 82 VECTORANALYSIS wherea,b,carethreescalarconstantstobedetermined. MultiplybyobXc. t) zx b 0 at 0bboc cc-c or Inlikemannerbymultiplyingtheequationby ocXaand oaXbthecoefﬁcientsband6maybefound. [rca]=b[bca] [rab]=c[cab] [rbc]Hence [abc]a [bca]. [cab]0°(28) Thedenominatorsareallequal.Hencethisgivesthe equation [abc]r[era]b[rab]c=0 whichmustexistbetweenthefourvectorsr,a,b,c. Theequationmayalsobewritten roc a+roc roaxb [abc] [abc] [abc] DIRECTANDSKEPVPRODUCTSOFVECTORS 83 whicharefoundbydividingthethreevectorproductsbXc, cXa,aXbofthreenon-coplanarvectorsa,b,cbythescalar product[abc]iscalledthereciprocalsystemtoa,b,c. Thewordnon-coplanarisimportant.Ifa,b,0wereco planarthescalartripleproduct[abc]wouldvanishand consequentlythefractions bXc cXa aXb [abc][abc][abc] wouldallbecomemeaningless.Threecoplanarvectorshave noreciprocalsystem .Thismustbecarefullyremembered . Hereafterwhenthetermreciprocalsystemisused,itwillbe understoodthatthethreevectorsa,b,carenotcoplanar. Thesystemofthreevectorsreciprocaltosystema,b,0 willbedenotedbyprimesasa’ ,b’ ,c’ aXb (29) [abc] [abc] [abc] Theexpressionfor1‘reducesthentotheverysimpleform r ros’a rob’b r-c’c. (30) Thevectorrmaybeexpressedintermsofthereciprocal systema’ ,b’ ,0’insteadofintermsofa,b,c.Intheﬁrst placeitisnecessarytonotethatifa,b,carenon-coplanar, a’b’ ,c’whicharethenormalstotheplanesofbandc, canda,aandbmustalsobenon-COplanar.Hencermay beexpressedintermsofthembymeansofproperscalar coefﬁcientsas,y,z. or Multiplysuccessivelybyos,ob,oc.Thisgives [ab0]’ azr-a y=rob z=roc Hence r r-aa’robb’r-cc’ . (31) 84 VECTORANALYSIS 44]Ifa’ ,b’ ,c’bethesystemreciprocaltoa,b,cthe scalarproductofanyvectorofthereciprocalsystemintothe correspondingvectorofthegivensystemisunity;but theproductoftwonon-correspondingvectorsiszero. a’ oab’ ob c’ oc1 (32) a’ ob a’ oc:2b’ oab’ oc c’ oa c’ ob0 . Thismaybeseenmosteasilybyexpressinga’b’ ,c’in termsofthemselvesaccordingtotheformula(31) r roaa’robb’r-cc’ . Hence a’a’ oaa’a’ obb’a’ o' cc’ b'b’ oaa’b’ obb’b’ occ’ o’c’ oaa’c’ obb’c’ ecc’ . Sincea’ ,b’ ,c'arenon-coplanarthecorresponding' coefﬁ cientsonthetwosidesofeachofthesethreeequationsmust beequal.Hencefromtheﬁrst 1 a’ oaO a’ ob0a’c. Fromthesecond0Va1b’ obOb’ oc. Fromthethird 0 c’ oaO c’ ob1 c’ oo. Thisprovestherelations.Theymayalsobeproved directlyfromthedeﬁnitionsofa’ ,b’ ,c’ c_c-a[boa] ' 0 O aa [abc]3 [abc][abc]_1 c c-b 0’ 0b 0 a [abc]b [abc][abc]0 andsoforth. Converselyiftwosetsofthreevectorseach,sayA,B,C, anda,b,c,satisfytherelations Aoa=Boh'20-01 A03A00BoaB007008Cob=0 86 VECTORANALYSIS TheoremIfa’ ,b’ ,c’anda,b,cbereciprocalsystems thescalartripleproducts[a’b’c’]and[abc]arenumerical reciprocals.Thatis [abc]=1 (33) bXc cXaaXb [abc][abc][abc] [c cxaaXb]. [cc aXb] But [abc]o. Hence[c cxaaXb][abc]c-aXb[abc]2 . 1 1 Hence [abc13[abc]2_ [abc] Bymeansofthisrelationbetween[a’b’c’]and[abc]it ispossibletoproveanimportantreductionformula, P-APoBac Q-Aanac (34) BABBas whichreplacesthetwoscalartripleproductsbyasumof ninetermseachofwhichistheproductofthreedirectpro ducts.Thusthetwocrosseswhichoccurinthetwoscalar productsareremoved.TogivetheproofletP,Q,Rbe expressedas P=PoAA’P-BB'PoCC’ Q:Q-AA’+ B’QC0' B=RoAA’BoBB’c' , P-APoBP-C Then [PQB]=onasac RoAR-BR-C Bilt1 [ABCJ' DIRECTANDSKEWPRODUCTS0FVECTORS 87 BABBPoC Hence [PQR][ABC] Q-AQoBQoC RoARoBR-C ThesystemOfthreeunitvectorsi,j,kisitsownreciprocal system .jXk i kXii, —=iaJ, [mg— 1.ixj [iiklzj’k (35) Forthisreasontheprimesi’ ,j’ ,k’arenotneededtodenote asystemofvectorsreciprocaltoi,j,k .Theprimeswill thereforebeusedinthefuturetodenoteanothersetofrect angularaxesi,j,kjustasX’ ,Y’ ,Z’areusedtodenotea setofaxesdiﬁerentfromX,Y,Z . Theonlysystemsofthreevectorswhicharetheirownreciprocals aretheright-handedandleft-handedsystemsofthreeunit vectors.Thatisthesystemi,j,kandthesystemi,j,—k . LetA,B,Cbeasetofvectorswhichisitsownreciprocal. Thenby(32) AA BoB 0-01. Hencethevectorsareallunitvectors. ABA4)0. HenceAisperpendiculartoBandC . BA 2B-020. HenceBisperpendiculartoAandC. C-A CoB0 . HenceCisperpendiculartoAandB . HenceA,B,Cmustbeasystemlikei,j,korlikei,j,k . Ascalarequationoftheﬁrstdegreeinavectorris anequationineachtermofwhichroccursnotmorethan once .Thevalueofeachtermmustbescalar.Asanexam pleofsuchanequationthefollowingmaybegiven. aa-e cfordO, 88 VECTORANALYSIS wherea,b,c,d,e,fareknownvectors;anda,b,c,d,known scalars .Obviouslyanyscalarequationoftheﬁrstdegreein anunknownvectorrmaybereducedtotheform r-A=a whereAisaknownvector;anda,aknownscalar.Toao complishthisresultinthecaseofthegivenequationproceed asfollows . aaXborb(cxd)xeor cford0 {aaXbb(c)xc cf}or d. Inmorecomplicatedformsitmaybenecessarytomakeuse ofvariousreductionformulabeforetheequationcanbemade totakethedesiredform, r-A a . Asavectorhasthreedegreesoffreedomitisclearthatone scalarequationisinsuﬁ‘icienttodetermineavector.Three scalarequationsarenecessary. Thegeometricinterpretationoftheequa tion r-A a (36) isinteresting .Letrbeavariablevector (Fig.24)drawnfromaﬁxedorigin .Let Abeaﬁxedvectordrawnfromthesame origin .Theequationthenbecomes 7‘ACOS a,FIG .24. or rcos(r,A)E, ifrbethemagnitudeofr;andAthatofA.Theexpression rcos(r,A) istheprojectionofruponA .Theequationthereforestates thattheprojectionofruponacertainﬁxedvectorAmust 90 VECTORANALYSIS Fromfourscalarequations r-A a roBb (39) r-C c r-Dd thevectorrmaybeentirelyeliminated .Toaccomplishthis solvethreeoftheequationsandsubstitutethevalueinthe fourth . rzaAI+bBI+CC’ aA’ oD bB’ oD cC’ oDzd or (40) Avectorequationoftheﬁrstdegreeinanunknown vectorisanequationeachtermofwhichisavectorquantity containingtheunknownvectornotmorethanonce.Such anequationis DE-r nrF 0, whereA,B,C,D,E,Fareknownvectors,naknownscalar, andrtheunknownvector.Onesuchequationmayingen eralbesolvedforr.Thatistosay,onevectorequationisin generalsufﬁcienttodeterminetheunknownvectorwhichis containedinittotheﬁrstdegree. Themethodofsolvingavectorequationistomultiplyit withadotsuccessivelybythreearbitraryknownnon-coplanar vectors.Thusthreescalarequationsareobtained .These maybesolvedbythemethodsoftheforegoingarticle.Inthe ﬁrstplacelettheequationbe whereA,B,C,D,a,b,careknownvectors.Noscalarcoefﬁ cientearewrittenintheterms,fortheymaybeincorporatedin thevectors.MultiplytheequationsuccessivelybyA’ ,B’ ,C’ . ItisunderstoodofcoursethatA,B,Carenon-coplanar. DIRECTANDSKEIVPRODUCTSOFVECTORS91 aorD-A’ b-rD-B’ cor2DC’ But ra’a-rb’borc’cor. Hence rD-A’a’D-B’b’D-C’c’ . ThesolutionisthereforeaccomplishedincaseA,B,Carenon Coplanaranda,b,0alsonon-coplanar.Thespecialcasesin whicheitherofthesesetsofthreevectorsiscoplanarwillnot bediscussedhere. Themostgeneralvectorequationoftheﬁrstdegreeinan unknownvectorrcontainstermsofthetypes Aaor,nr,EXr,D . Thatisitwillcontaintermswhichconsistofaknown vectormultipliedbythescalarproductofanotherknownvec torandtheunknownvector;termswhicharescalarmulti plesoftheunknownvector;termswhicharethevector productofaknownandtheunknownvector;andconstant terms.ThetermsofthetypeAaormayalwaysbereduced tothreeinnumber.Forthevectorsa,b,c,-whichare multipliedintormayallbeexpressedintermsofthreenon coplanarvectors .Hencealltheproductsa-r,b-r,cor, maybeexpressedintermsofthree .Thesumofalltermsof thetypeAaorthereforereducestoanexpressionofthree terms,as AaorBb-rCcor. ThetermsofthetypesnrandExrmayalsobeexpressed inthisform . -r+nc’c-r ExrExa’aorEXb’b-r+EXc’cor. Addingallthesetermstogetherthewholeequationreduces totheform 92 VECTORANALYSIS Thishasalreadybeensolvedas rKoL’a’K-m’b’KoN’c’ . Thesolutionisintermsofthreenon-coplanarvectorsa’b’ , Theseformthesystemreciprocaltoa,b,cintermsofwhich theproductscontainingtheunknownvectorrwereexpressed. SUNDRYAPPLICATIONSorPnonuors ApplicationstoMechanics Inthemechanicsofarigidbodyaforceisnota vectorinthesenseunderstoodinthisbook .SeeArt.3 . Aforcehasmagnitudeanddirection;butithasalsoaline ofapplication .Twoforceswhicharealikeinmagnitude anddirection,butwhichlieupondifferentlinesinthebody donotproducethesameeﬁect.Neverthelessvectorsare sufﬁcientlylikeforcestobeusefulintreatingthem . Ifanumberofforcesf1,f2,f3,“actonabodyatthe samepoint0,thesumoftheforcesaddedasvectorsiscalled theresultantR . Inthesamewayiff1,f2,f3 donotactatthesamepoint thetermresultantisstillappliedtothesumoftheseforces addedjustasiftheywerevectors. (41) aforcedoesnotdifferfromavector. DefinitionThemomentofaforcefaboutthepoint0is equaltotheproductoftheforcebytheperpendiculardis tancefromOtothelineofactionoftheforce .Themoment howeverisbestlookeduponasavectorquantity.Itsmag nitudeisasdeﬁnedabove.Itsdirectionisusuallytakento 94 VECTORANALYSIS ThisisthemagnitudeofthemomentMo{f}.Thedirection ofdisthesameasthedirectionofthemoment.Hence therelationisproved. MO2:d . Thesumofthemomentsabout0ofanumberofforces f1,f2,actingatthesamepointPisequaltothemoment oftheresultantRoftheforcesactingatthatpoint.Forlet (1bethevectorfromOtoP .Then Thetotalmomentabout0’ofanynumberofforcesf1,f2,o actingonarigidbodyisequaltothetotalmomentofthose forcesabout0increasedbythemomentabout0’ofthe resultantBoconsideredasactingat0 . Mo'{fraf2,m}MOifo129W}Mo'{303°(44) Let d2, bevectorsdrawnfromOtoanypointin f1,f2,orespectively.Let(1dz’ ,obethevectorsdrawn from0’tothesamepointsinf1,f2,ooorespectively .Letc bethevectorfromOto Then Mo{f1,f2,:2dl 1(’lzxf2 Mo'{f1,f2,o(ll’Xfldz’s o (d1c)Xf1c)Xf2 But cisthevectordrawnfromO’to0 .Hence—cXf, isthemomentabout0’ofaforceequalinmagnitudeand parallelindirectiontof1butsituatedat0 .Hence DIRECTANDSKEWPRODUCTSOFVECTORS95 —cx(f1+f2ocXRoMy{R0}. HenceM01{f1,f2,.o M0{f1,f2, My{Rage Thetheoremisthereforeproved. TheresultantRisofcoursethesameatallpoints.The subscriptOisattachedmerelytoshowatwhatpointitis supposedtoactwhenthemomentabout0’istaken .For thepointofapplicationofBaffectsthevalueofthatmoment. Thescalarproductofthetotalmomentandtheresultant isthesamenomatteraboutwhatpointthemomentbetaken . Inotherwordstheproductofthetotalmoment,theresult ant,andthecosineoftheanglebetweenthemisinvariant forallpointsofspace. R'MO'{flafza=R'M0{f1’f2’ where0’andOareanytwopointsinspace .Thisimportant relationfollowsimmediatelyfromtheequation M0'{f19f2,ZM0{flaf29 MO’ ForB-MOI§f1,f2,=R-Mo{f1,f2,- o+R-M0,{B0}. ButthemomentofB.isperpendiculartoB.nomatterwhat thepoint0ofapplicationbe.Hence R°M0I{B0}0 andtherelationisproved .Thevariationinthetotal momentduetoavariationofthepointaboutwhichthe momentistakenisalwaysperpendiculartotheresultant. Apoint0’maybefoundsuchthatthetotalmoment aboutitisparalleltotheresultant.Theconditionfor parallelismis RXM0'{f1,f2, 0 m o,if“f2,m;=Rxmo{f1,f2, +RXM0I 96 VECTORANALYSIS where0isanypointchosenatrandom.ReplaceMolfﬁo} byitsvalueandforbrevityomittowritethe£1,f2,oointhe braces Then m aRXMORX(cXB)0. Theproblemistosolvethisequationforc. RXMOBB0B-cB0 . NowBisaknownquantity .MOisalsosupposedtobe known.Letcbechosenintheplanethrough0perpen diculartoR .ThenBooOandtheequationreducesto RXMO:3ROB.0 RXMO R-R Ifcbechosenequaltothisvectorthetotalmomentabout thepoint whichisatavectordistancefrom0equalto0, willbeparalleltoR .Moreover,sincethescalarproductof thetotalmomentandtheresultantisconstantandsincethe resultantitselfisconstantitisclearthatinthecasewhere theyareparallelthenumericalvalueofthetotalmoment willbeaminimum . Thetotalmomentisunchangedbydisplacingthepoint aboutwhichitistakeninthedirectionoftheresultant. f2, f2, If000’isparalleltoB,cXBvanishesandthemoment about0’isequaltothatabout0 .Henceitispossibleto ﬁndnotmerelyonepoint0’aboutwhichthetotalmoment isparalleltotheresultant;butthetotalmomentaboutany pointinthelinedrawnthrough0’paralleltoB.isparallel toB .Furthermorethesolutionfoundinequationforcis theonlyonewhichexistsintheplaneperpendiculartoB. unlesstheresultantB.vanishes.Theresultsthathavebeen obtainedmaybesummedupasfollows: 98 VECTORANALYSIS theaxis .Thevelocityofanypointinitscircleisequal totheproductoftheangularvelocityandtheradiusofthe circle.Itisthereforeequaltotheproductoftheangular velocityandtheperpendiculardis tancefromthepointtotheaxis. Thedirectionofthevelocityis perpendiculartotheaxisandto theradiusofthecircledescribed bythepoint. Leta(Fig .25)beavectordrawn alongtheaxis'ofrotationinthat directioninwhicharight-handed screwwouldadvanceifturnedin FIG .25. thedirectioninwhichthebodyis rotating .Letthemagnitudeofa bea,theangularvelocity.Thevectoramaybetakento representtherotationofthebody .Letrbearadiusvector drawnfromanypointoftheaxisofrotationtoapointinthe body .Thevectorproduct axr=arsin(a,r) isequalinmagnitudeanddirectiontothevelocityvofthe terminusofr.Foritsdirectionisperpendiculartoaandr anditsmagnitudeistheproductofaandtheperpendicular distancersin(a,r)fromthepointtothelinea.Thatis v an . (45) Ifthebodyberotatingsimultaneouslyaboutseveralaxes a], a3 whichpassthroughthesamepointasinthe caseofthegyroscope,thevelocitiesduetothevarious rotationsare v2a,,Xr.z v8asxr3 DIRECTANDSKEWPRODUCTSOFVECTORS99 wherer1,r2,r8,ooaretheradiivectoresdrawnfrompoints ontheaxisa1,a2,a3,ootothesamepointofthebody .Let thevectorsr1,r2,r3, bedrawnfromthecommonpointof intersectionoftheaxes.Then 1'=r=1“ l 2 8and Thisshowsthatthebodymovesasifrotatingwiththe angularvelocitywhichisthevectorsumoftheangular velocitiesa1,a2,33,ooThistheoremissometimesknown astheparallelogramlawofangularvelocities. Itwillbeshownlater(Art)60thatthemotionofany rigidbodyonepointofwhichisﬁxedisateachinstantof timearotationaboutsomeaxisdrawnthroughthatpoint.Thisaxisiscalledtheinstantaneousaxisofrotation .The axisisnotthesameforalltime,butconstantlychangesits position .Themotionofarigidbodyonepointofwhichis ﬁxedisthereforerepresentedby v aXr (45) whereaistheinstantaneousangularvelocity;andr,the radiusvectordrawnfromtheﬁxedpointtoanypointofthe body . Themostgeneralmotionofarigidbodynopointofwhich isﬁxedmaybetreatedasfollows.Chooseanarbitrary point0 .Atanyinstantthispointwillhaveavelocityv0. Relativetothepoint0thebodywillhaveamotionofrotation aboutsomeaxisdrawnthrough0 .Hencethevelocityvof anypointofthebodymayberepresentedbythesumof V0thevelocityofOandaxtthevelocityofthatpoint relativeto0 . v voaxr. (46) 100 VECTORANALYSIS Incasevoisparalleltoa,thebodymovesaroundaand alongitsimultaneously.Thisispreciselythemotionofa screwadvancingalonga.IncaseV0isperpendiculartoa,it ispossibletoﬁndapoint,givenbythevectorr,suchthat itsvelocityiszero .Thatis exr v0. Thismaybedoneasfollows.Multiplybyxa. (axr)Xa voxs or asr a-ra voxa . Letrbechosenperpendiculartoa.Thena-riszeroand aoar=—voxs r=—voxs as Thepointr,thusdetermined,hasthepropertythatitsveloc ityiszero .Ifalinebedrawnthroughthispointparallelto a,themotionofthebodyisoneofinstantaneousrotation aboutthisnewaxis. Incasev0isneitherparallelnorperpendiculartoaitmay beresolvedintotwocomponents whicharerespectivelyparallelandperpendiculartoa v vo’v,”aXr Apointmaynowbefoundsuchthat v,”axr. Letthedifferentpointsofthebodyreferredtothispointbe denotedbyr’ .Thentheequationbecomes v vo’am“ . Themotionhereexpressedconsistsofrotationaboutanaxis aandtranslationalongthataxis.Itisthereforeseenthat themostgeneralmotionofarigidbodyisatanyinstant 102 VECTORANALYSIS Asf’isparalleltoathescalarproduct[adfvanishes. a-d aod” . Ontheotherhandtheworkdonebyf”isequaltothework doneby1‘duringthedisplacement.Forf’beingparallelto aisperpendiculartoitslineofaction.Ifhbethecommon vectorperpendicularfromthelineatotheforcefthework donebyfduringarotationofangularvelocityafortime tisapproximately W :hf”at aohxf”t. Thevector(1drawnfromanypointofatoanypointoffmay bebrokenupintothreecomponentsofwhichoneish,another isparalleltoa,andthethirdisparalleltof” .Inthescalar tripleproduct[adfonlythatcomponentofdwhichis perpendicularaliketoaandf”hasanyeﬁect.Hence W :atf”t ad’t’ad t. Ifarigidbodyuponwhichtheforcesf1,f2, actbedis placedbyanangularvelocity9.foraninﬁnitesimaltimet andif uobethevectorsdrawnfromanypoint0of atoanypointsoff1,f2,oorespectively,thentheworkdone bytheforcesf1,f2, willbeapproximMely W :(aolflaodzxf2ooo)t a-(dIXf1(12sot a'Mo £29 t Ifthebodybeinequilibriumthisworkmustbezero. Hence a-Moif},f2,n o}t:O . Thescalarproductoftheangularvelocity9.andthetotal momentoftheforcesf1,f2, aboutanypoint0mustbe zero.Asamaybeanyvectorwhatsoeverthemomentitself mustvanish . Mo{InI“,oo DIRECTANDSKEWPRODUCTSOFVECTORS103 Thenecessaryconditionsthatarigidbodyheinequilib riumundertheactionofasystemofforcesisthattheresult antofthoseforcesandthetotalmomentaboutanypointin spaceshallvanish . Converselyiftheresultantofasystemofforcesandthe momentofthoseforcesaboutanyoneparticularpointinspace vanishsimultaneously,thebodywillbeinequilibrium . IfR 0,thenforanydisplacementoftranslationD D-R=0 . andthetotalworkdoneiszero,whenthebodysuffersany displacementoftranslation . LetMO{f1,f2,oobezeroforagivenpoint0 .Thenfor anyotherpoint0’ MO’{fp{2:’3MOlfp£29'1'MO’{BO} ButbyhypothesisRisalsozero.Hence Mol{f1,f2, =0. Hence aoMo:{fvf2,.3t0 whereaisanyvectorwhatsoever.Butthisexpressionis equaltotheworkdonebytheforceswhenthebodyisrotated foratimetwithanangularvelocityaaboutthelinea passingthroughthepoint Thisworkiszero. Anydisplacementofarigidbodymayberegardedasa translationthroughadistanceDcombinedwitharotation foratimetwithangularvelocityaaboutasuitablelineain space.Ithasbeenprovedthatthetotalworkdonebythe forcesduringthisdisplacementiszero .Hencetheforces mustbeinequilibrium .Thetheoremisproved. 104 VECTORANALYSIS ApplicationstoGeometry Relationsbetweentworight-handedsystemsofthree mutuallyperpendicularunitvectors .Leti,j,kandi’ ,j’ ,k’ betwosuchsystems.Theyformtheirownreciprocalsystems. Hence r=r.ii+r-jj+r-kk 47and r roi’i’r-j’j’r-k’k’ . Fromthis ’i’=i’-ii+i’ ojj i’-kk=a1i+a2j+a3k .l’jukk=bri+s+bak +c3k . Thescalarsa1,a2,a3;bl,b2,b3;cl,02,c3arerespectivelythe directioncosinesofi’ ;j’ ;k’withrespecttoi,j,k. Thatis a1cos(i’ ,i)a2cos(i’ ,j)a3cos(i’ ,k) b1cos(j’ ,i)b22cos j)b32cos(j’ ,k)(48) c1cos(k’ ,i)c2cos(k’ ,j)oscos(k’ ,k). Inthesamemanner i ioi'i'i.j'j’ a1i’+b,j'c,k' jjoirirj.jrjr+joklkl zazil+62jr+02kl kk.i'i'k~j’j'kok’k’a3i’baj’csk’ i’ oi’1 alza22agz i'-i’1 b,2(49)kukl1 012022032 andii 1 a;bec,2 andj’ ok’0b1c1b202b303 (50) 106 VECTORANALYSIS theidirectionbuttoliewhollyinthejk-planeandfrom itsformupontheleftitisseentolieinthej’k’-plane. Henceitmustbethelineofintersectionofthosetwoplanes. ItsmagnitudeisW or1/W Thisgivesthe scalarrelations azz(132blz0121 alz . Themagnitude1 alzisthesquareofthesineoftheangle betweenthevectorsiandi’ .Hencethevector brkl_01j’=asj—“2k (53) isthelineofintersectionofthej’k’andjk-planes,and itsmagnitudeisthesineoftheanglebetweentheplanes. Eightothersimilarvectorsmaybefound,eachofwhichgives oneoftheninelinesofintersectionofthetwosetsofmu tuallyorthogonalplanes.Themagnitudeofthevectorisin eachcasethesineoftheanglebetweentheplanes. VariousexamplesinPlaneandSolidGeometrymay besolvedbymeansofproducts. Example1Theperpendicularsfromtheverticesofatrian gletotheoppositesidesmeetinapoint.LetAB0’bethe triangle.LettheperpendicularsfromAtoBCandfromB to0Ameetinthepoint0 .Toshow00isperpendicular toAB .Choose0asoriginandletOA A,OB=B,and 000 .Then BC’:C—B,CA=A—C,AB zB—A . Byhypothesis A-(CB)0 and B-(A C)0. Subtract; A)0, whichprovesthetheorem . Example2:Toﬁndthevectorequationofalinedrawn throughthepointBparalleltoagivenvectorA . DIRECTANDSKEWPRODUCTSOFVECTORS107 Let0betheoriginandBthevectorOB .LetBbethera dinsvectorfromOtoanypointoftherequiredline.Then B.BisparalleltoA .Hencethevectorproductﬁanishes. A—B=0 . X(R A Thisisthedesiredequation .Itisavectorequationinthe unknownvectorB .Theequationofaplanewasseen(page 88)tobeascalarequationsuchas 3°C c intheunknownvectorB . Thepointofintersectionofalineandaplanemaybe foundatonce.Theequationsare AX(E B)0 B4! c AxB AXB (AXB)XC(AXB)XC A-CB C-BA(AXB)XC A-CR cA(AXB)XC Hence(AXB)XC cA ThesolutionevidentlyfailswhenA4)0.Inthiscasehow everthelineisparalleltotheplaneandthereisnosolution; or,ifitliesintheplane,thereareaninﬁnitenumberofsolu tions. Example3Theintroductionofvectorstorepresentplanes. Heretoforevectorshavebeenusedtodenoteplaneareasof deﬁniteextent.Thedirectionofthevectorwasnormalto theplaneandthemagnitudewasequaltotheareatobere presented .Butitispossibletousevectorstodenotenota planeareabuttheentireplaneitself,justasavectorrepresents apoint.Theresultisanalogoustotheplanecoordinatesof analyticgeometry .Let0beanassumedorigin .LetMNbe aplaneinspace.TheplaneMNistobedenotedbyavector 108 VECTORANALYSIS whosedirectionisthedirectionoftheperpendiculardropped upontheplanefromtheorigin0andwhosemagnitudeisthe reciprocalofthelengthofthatperpendicular.Thusthenearer aplaneistotheoriginthelongerwillbethevectorwhich representsit. Ifrbeanyradiusvectordrawnfromtheorigintoapoint intheplaneandifpbethevectorwhichdenotestheplane, then rep21 istheequationoftheplane.For toprcos(r,p)p. Nowp,thelengthofpisthereciprocaloftheperpendicular distancefromOtotheplane.Ontheotherhandrcos(r,p) isthatperpendiculardistance.Henceropmustbeunity . Ifrandpbeexpressedintermsofi,j,k r=xi+yj+2k p=ui+vj+wk Hence Thequantitiesu,v,warethereciprocalsoftheinterceptsof theplanepupontheaxes. Therelationbetweenrandpissymmetrical.Itisarela tionofduality .Ifintheequation r-p:1 rberegardedasvariable,theequationrepresentsaplanep whichisthelocusofallpointsgivenbyr.Ifhoweverpbe regardedasvariableandrasconstant,theequationrepre sentsapointrthroughwhichalltheplanesppass.The developmentoftheideaofdualitywillnotbecarriedout. Itisfamiliartoallstudentsofgeometry.Theuseofvec torstodenoteplaneswillscarcelybealludedtoagainuntil ChapterVII. 110 VECTORANALYSIS vanishesisthattheirvectorproductvanishes.Thecom mutativelawsdonothold . AXB: BXA(10) ixi=i=k O ixj jxi=k(12) i kxji l iXk=1 Thescalartripleproductofthreevectors[ABC]isequal tothevolumeoftheparallelopipedofwhichA,B,Carethree edgeswhichmeetinapoint . [ABC]Ao0BoCxACoAxB AxBoCBxCoACXAoB [ABC] [A03].(my DIRECTANDSKEWPRODUCTS0FVECTORS111 IfthecomponentofBperpendiculartoAbeB” , AX(AXB) A-A AX(BXC)A-CBABC (AXB)XCA-CB C-BA (AXB) A4)B1)A-DB-c [A B[BCD]A [ABD]C—[ABC]D . (26) TheequationwhichsubsistsbetweenfourvectorsA,B,C,D is [BOD]A[CDA]B[DAB]0[ABC]D=0.(27)B" Applicationofformulaofvectoranalysistoobtainthefor mulaofPlaneandSphericalTrigonometry . Thesystemofvectorsa’ ,b’ ,c’issaidtobereciprocaltothe systemofthreenon-coplanarvectorsa,b,c bxc cxa axb [abc],b, [abc],(Bl(29) Avectorrmaybeexpressedintermsofasetofvectorsand itsreciprocalintwosimilarwayswhen a’ r r.a’a r.h’b r.c’c (30) or r roaa,’rubb’roc (31) Thenecessaryandsufﬁcientconditionsthatthetwosystemsof non-COplanarvectorsa,b,canda’ ,b’ ,c’bereciprocalsisthat b' ob 1 (32) a' ob b' ocbl oa Clea O' ob0 . Ifa’ ,b’ ,0’formasystemreciprocaltoa,b,c;thena,b,0will formasystemreciprocaltoa’ ,b' ,c’ . 1 [abc](20) (24) (25) 112 VECTORANALYSIS PoAP-BP-C [resume]:conQ-Bac (34) B-AB-BB4: Thesystemi,j,kisitsownreciprocalandifconverselya systembeitsownreciprocalitmustbearightorlefthanded systemofthreemutuallyperpendicularunitvectors .Appli cationofthetheoryofreciprocalsystemstothesolutionof scalarandvectorequationsoftheﬁrstdegreeinanunknown vector.Thevectorequationofaplaneis r-A a . (36) ApplicationsofthemethodsdevelopedinChapterIL,tothe treatmentofasystemofforcesactingonarigidbodyandin particulartothereductionofanysystemofforcestoasingle forceandacoupleofwhichtheplaneisperpendiculartothat force.Applicationofthemethodstothetreatmentof instantaneousmotionofarigidbodyobtaining v=vo+aXr (46) wherevisthevelocityofanypoint,v0atranslationalveloc ityinthedirectiona,andathevectorangularvelocityofro tation .Furtherapplicationofthemethodstoobtainthe conditionsforequilibriumbymakinguseoftheprincipleof virtualvelocities.Applicationsofthemethodtoobtain therelationswhichexistbetweentheninedirectioncosines oftheanglesbetweentwosystemsofmutuallyorthogonal axes.Applicationtospecialproblemsingeometryincluding theformunderwhichplanecoordinatesmaketheirappear anceinvectoranalysisandthemethodbywhichplanes(as 114v‘VECTORAIVALYSIS 12 .Showbyvectormethodsthattheformulaforthevol umeofatetrahedronwhosefourverticesare yrs21)(x2,92’zz)(x3,l/arzs) 9024) zr1 221 6x3y8es1 “34341 13 . useofformula(34)ofthetextshowthatH8N toe [abo]=abo n1l ml1 wherea,b,carethelengthsofa,b,0respectivelyandwhere l:cos(b,c),m cos(c,a),ncos(a,b). 14Determinetheperpendicular(asavectorquantity) whichisdroppedfromtheoriginuponaplanedeterminedby theterminiofthevectorsa,b,c.Usethemethodofsolution giveninArt.46. 15 .Showthatthevolumeofatetrahedronisequaltoone sixthoftheproductoftwooppositeedgesbytheperpendicu lardistancebetweenthemandthesineoftheincludedangle. 16 .Ifalineisdrawnineachfaceplaneofanytriedralangle throughthevertexandperpendiculartothethirdedge,the threelinesthusobtainedlieinaplane. CHAPTERIII THEDIFFERENTIALCALCULUSOFVECTORS DiferentiationofFunctionsOfOneScalarVariable IFavectorvariesandchangesfromrtor’theincre mentofrwillbethedifferencebetweenI’andrandwillbe denotedasusualbyAr. Ar=r’—r, (1) whereArmustbeavectorquantity.IfthevariableIbe unrestrictedtheincrementArisofcoursealsounrestricted: itmayhaveanymagnitudeandanydirection .If,however, thevectorrberegardedasafunction(avectorfunction)of asinglescalarvariabletthevalueofArwillbecompletely determinedwhenthetwovaluestandt’oft,whichgivethe twovaluesrandr’ ,areknown . Toobtainaclearerconceptionofthequantitiesinvolved itwillbeadvantageoustothinkofthevectorrasdrawn fromaﬁxedorigin0(Fig . When theindependentvariabletchangesits valuethevectorrwillchange,andast possessesonedegreeoffreedomrwill varyinsuchawaythatitsterminus describesacurveinspace.rwillbe theradiusvectorofonepointPof thecurve;r’ ,ofaneighboringpointP’Ar bethe chordPP’ofthecurve.Theratio Ar At411' 0! FIG. 116 VECTORANALYSIS willbeavectorcollinearwiththechordPP’butmagniﬁed intheratio1:At.WhenAtapproacheszeroP’willap proachP,thechordPP’willapproachthetangentatP,and thevector Ar dr Atwillapproach 67; whichisavectortangenttothecurveatPdirectedinthat senseinwhichthevariabletincreasesalongthecurve . Ifrbeexpressedintermsofi,j,kas r=rli+r2j+r3k thecomponentsr1,r2,r8willbefunctionsofthescalart. r’=(r2+Ar2)j(r3+Ar3)k Ar=r’—r=Arli+Ar2j+Ar3k ArArl AtAtl+m anddrq11i]dt_ dt dtJ+ dtk °(2)Arz,A 1+At Hencethecomponentsoftheﬁrstderivativeofrwithre specttotaretheﬁrstderivativeswithrespecttotofthe componentsofr.Thesameistrueforthesecondandhigher derivatives. dzrdzrlidzr2 dc?dc?dt2J+ dtz1" 2r d"rd“dnr2 d” 118 VECTORANALYSIS A(aob)_AbAa obAa-Ab At At+At At HenceinthelimitwhenAt=0, d dbda 0 0 0h 3 dt<a a dt+dt d4(aXb)=aX xb dt cc . (5) d X +3 X[c]. (6) l Thelastthreeoftheseformulamaybedemonstratedexactly astheﬁrstwas. Theformalprocessofdifferentiationinvectoranalysis differsinnowayfromthatinscalaranalysisexceptinthis onepointinwhichvectoranalysisalwaysdiffersfromscalar analysis,namely:Theorderofthefactorsinavectorproduct THEDIFFERENTIALCALCULUSOFVECTORS119 cannotbechangedwithoutchangingthesignoftheproduct. Henceofthetwoformula d6;' bf' Xa\ d t‘Xp;7 dand dt(aXb)= Xb+aX theﬁrstisevidentlyincorrect,butthesecondcorrect.In otherwords,scalardifferentiationmusttakeplacewithout alteringtheorderofthefactorsofavectorproduct.The factorsmustbedifferentiatedinsitu .Thisofcoursewasto beexpected . Incasethevectorsdependuponmorethanonevariable theresultsarepracticallythesame.Inplaceoftotalderiva tiveswithrespecttothescalarvariables,partialderivatives occur.Supposeaandbaretwovectorswhichdependon threescalarvariablesx,y,z.Thescalarproductaobwill dependuponthesethreevariables,anditwillhavethree partialderivativesoftheﬁrstorder. 9 ax(a-b)_ .b+a 9 ay(aob)= ob+ao(7) 9 as ob Ob 92(a 92+3 Thesecondpartialderivativesareformedinthesameway . 929a 9b a? a? 120 VECTORANALYSIS Oftenitismoreconvenienttousenotthederivativesbut thedifferentials.Thisisparticularlytruewhendealirigwith ﬁrstdifferentials.Theformula (4)become d(aob)=da-b+aodb, (3) andsoforth .Asanillustrationconsiderthefollowing example.Ifrbeaunitvector ror=1 . Thelocusoftheterminusofrisasphericalsurfaceofunit radiusdescribedabouttheorigin .rdependsupontwovari ables.Differentiatetheequation. (dr)or+r :0. Hence rodr=0. Hencetheincrementdrofaunitvectorisperpendicularto thevector.Thiscanbeseengeometrically.Ifrtracesa spherethevariationdrmustbeateachpointinthetangent planeandhenceperpendiculartor. Vectormethodsmaybeemployedadvantageously inthediscussionofcurvatureandtorsionofcurves .Letr denotetheradiusvectorofacurve wherefissomevectorfunctionofthescalart.Inmostappli cationsinphysicsandmechanicstrepresentsthetime.Let sbethelengthofarcmeasuredfromsomedeﬁnitepointof thecurveasorigin .TheincrementAristhechordofthecurve.HenceArAsisapproximatelyequalinmagnitudetounityandapproachesunityasitslimitwhenA3becomes infinitesimal. 122 VECTORANALYSIS tot=coc=non=1 and Differentiatingtheﬁrstset andthesecond todc+dtoc=c Butdtisparalleltocandconsequentlyperpendicularton . nodt=o . Hence dnot=0. Theincrementofnisperpendicularto Buttheincrement ofnisalsoperpendicularton .Itisthereforeparalleltoc. AsthetortuosityisT du/ds,itisparalleltodnandhence toc. ThetortuosityTis d ddrdar 1 T:_ dsdsx dsz (11) Tdzr xdzr 1 +dr xd3r 1 as?0mmdsda3vC.c drd2rd 1 +dsx d82dc.c Theﬁrsttermofthisexpressionvanishes.Tmoreoverhas beenseentobeparalleltoC r/d32 .Consequentlythe magnitudeofTisthescalarproductofTbytheunitvec torcinthedirectionofC .Itisdesirablehowevertohave thetortuositypositivewhenthenormal11appearstoturnin thepositiveorcounterclockwisedirectionifviewedfrom thatsideoftheno-planeuponwhichtorthepositivepart ofthecurvelies.Withthisconventiond11appearstomove inthedirection—cwhenthetortuosityispositive,thatis,11 turnsawayfromc.Thescalarvalueofthetortuositywill thereforebegivenby—coT . THEDIFFERENTIALCALCULUS0FVECTORS123 cT_ cdr xd3r1cdr xdzrd 1 dsds3 q/c.cd8dszds But0isparalleltothevectoral2r/d32 .Hence 0dr xdzr 0 d8 (Ts— 2 IAnd0isaunitvectorinthedirection0.Hence 0 d2r 1 c: dzrdr (1%1HenceT:—c-T= dszo dsx dsgcoc(12) drdzr d d2 Or T3 3 (13) dzrdzr dszo ds2 Thetortuositymaybeobtainedbyanothermethodwhich issomewhatshorterifnotquitesostraightforward. t-c=con=n-t=0 . Hence dtoc=—dc-t doenz—dnoc dn-t=—dton.Nowdtisparalleltoc;henceperpendicularton .Hence dton0.Hencednt0.Butdnisperpendicularto11. Henced11mustbeparalleltoc.Thetortuosityisthemag nitudeofdnd3takenhoweverwiththenegativesign becausednappearsclockwisefromthepositivedirectionof thecurve.HencethescalartortuosityTmaybegivenby (14) T=tXOO 124 VECTORANALYSIS d dc'C°Ci? d0 txc —txco d8v0-C—txc'-C But dc txc o d0t0x d3 T 0°C drdr2d3r Td8d82ds3 dzrdzr(13) d32d82 InCartesiancoordinatesthisbecomes da:dydz d3 d3d8 dzxdzydzz d32Ill— 32d“ ? d3x d33de3d33 Thosewhowouldpursuethestudyoftwisted surfacesinSpacefurtherfromthestandpointofvectorswill ﬁndthebook“ApplicationdelaMéthodeVectorielledcGrass manna‘laGéome’trieInﬁmItém' male”1byFEHBextremely 1Paris,Cari-éetNaud,1899 . 126 VECTORANALYSIS introducedbyNewton.Itwillalsobeconvenienttodenote theunittangenttothecurvebyt.Theequationsbecome dr V=r= E- t (16) v='vt. (17) Theaccelerationistherateofchangeofvelocity.It isavectorquantity.LetitbedenotedbyA . deﬁnition LIMAvdv AAtioAtdtv dv ddr dzr a‘nd v dtdtdt t2 Differentiatetheexpression v vt. dvdWt do dt“an:513— 217“a? de ﬁcits dtdtz dtdtds zcv dtdsdt whereCisthe(vector)curvatureofthecurveandvisthe speedinthecurve.Substitutingthesevaluesintheequation theresult18 a ta ” 2 Theaccelerationofaparticlemovinginacurvehasthere forebeenbrokenupintotwocomponentsofwhichoneisparal leltothetangenttandofwhichtheotherisparalleltothe curvatureC,thatis,perpendiculartothetangent.Thatthis resolutionhasbeenaccomplishedwouldbeunimportantwere THEDIFFERENTIALCALCULUSOFVECTORS127 itnotfortheremarkablefactwhichitbringstolight.The componentoftheaccelerationparalleltothetangentisequal inmagnitudetotherateofchangeofspeed.Itisentirely independentpfwhatsortofharmthe Fparticleisdescribing. Itwouldbethesameiftheparticledescribedarightline withthesamespeedasitdescribesthecurve.Ontheother handthecomponentoftheaccelerationnormaltothetangent isequalinmagnitudetotheproductofthesquareofthe Speedoftheparticleandthecurvatureofthecurve.The sharperthecurve,thegreaterthiscomponent.Thegreater thespeedoftheparticle,thegreaterthecomponent.Butthe rateofchangeofspeedinpathhasnoeffectatallonthis normalcomponentoftheacceleration . Ifrbeexpressedintermsofi,j,kas mx+yy+zz Fromtheseformulaethediﬁerencebetween therateof changeofspeed,andA therateofchangeofvelocity, isapparent.Justwhenthisdiﬁerenceﬁrstbecameclearly recognizedwouldbehardtosay.Butcertainitisthat Newtonmusthavehaditinmindwhenhestatedhissecond lawofmotion .Therateofchangeofvelocityisproportional totheimpressedforce;butrateofchangeofspeedisnot. ThehodographwasintroducedbyHamiltonasan aidtothestudyofthecurvilinearmotionofaparticle. Withanyassumedoriginthevectorvelocityiislaidoff. Thelocusofitsterminusisthehodograph .Inotherwords, theradiusvectorinthehodographgivesthevelocityofthe 128 VECTORANALYSIS particleinmagnitudeanddirectionatanyinstant.Itis possibletoproceedonestepfurther‘andconstructthehodo graphofthehodograph .Thisisdonebylaying0Ethe vectoraccelerationA' rfromanassumedoriginThe radiusvectorinthehodographofthehodographtherefore givestheaccelerationateachinstant. Example1 Letaparticlerevolveinacircle(Fig.29) A ofradius7'withauniform vangularvelocity The speedoftheparticlewillthen beequalto vzan Letrbetheradiusvector drawntotheparticle.The velocityvisperpendiculartorandtoa.ItisFIG.29. i=v=aXL Thevectorvisalwaysperpendicularandofconstantmagni tude.Thehodographisthereforeacircleofradiusa ar. Theradiusvectori'inthiscircleisjustninetydegreesin advanceoftheradiusvectorrinitscircle,anditcouse quentlydescribesthecirclewiththesameangularvelocity aTheaccelerationAwhichistherateofchangeofvis alwaysperpendiculartovandequalinmagnitudeto off A=av=a27n TheaccelerationAmaybegivenbytheformula ii:A axv aora aoar. Butasaisperpendiculartotheplaneinwhichrlies,aor0 . HenceizA=—aoar=—a2r. Theaccelerationduetotheuniformmotionofaparticlein acircleisdirectedtowardthecentreandisequalinmagni tudetothesquareoftheangularvelocitymultipliedbythe radiusofthecircle. 130 VECTORANALYSIS Perhapsitwouldbewelltogoalittlemorecarefullyinto thisquestion .Ifrbetheradiusvectoroftheparticlein itspathatoneinstant,theradiusvectoratthenextinstant isrAr.TheareaofthevectorofwhichrandrArare theboundingradiiisapproximatelyequaltotheareaofthe triangleenclosedbyr,rAr,andthechordAr.This areais l1 -rxr+2rxAr._ 2rxAr. Therateofdescriptionofareabytheradiusvectoris consequently A,(wr._ LIM-kAr) LIM 1 Ar 1. Ari-02At Ate- .OErXA‘ t Letiandi,betwovaluesofthevelocityattwopoints PandPowhichareneartogether.Theaccelerationii oatP0isthelimitof " o.Thequantitya:ap proachesunitywhenAtapproacheszero.Thequantity3]approacheszerowhenAtapproacheszero. THEDIFFERENTIALCALCULUS0FVECTORS131 Hence A rxr Buteachofthethreetermsupontheright-handsideisan inﬁnitesimalofthesecondorder.Hencetheratesofdescrip tionofareaatPandPodiﬂerbyaninﬁnitesimalofthe secondorderwithrespecttothetime.Thisistrueforany pointofthecurve.Hencetheratesmustbeexactlyequal atallpoints.Thisprovesthetheorem . Themotionofarigidbodyonepointofwhichis ﬁxedisatanyinstantarotationaboutaninstantaneousaxis passingthroughtheﬁxedpoint. Leti,j,kbethreeaxesﬁxedinthebodybutmovingin space.Lettheradiusvectorrbedrawnfromtheﬁxedpoint toanypointofthebody .Then drzxdi+ydj+zdh But dr=(dr Substitutingthevaluesofdroi,drj,drokobtainedfrom thesecondequation drz(wiodi+yiodj+ziodk)i +(xjodi+yjodj+zjodk)i +(xk But Henceiodj+jodi=0orj-di=—iodi j-dk+k-dj=0orkodj=—jodh k-di+iodk=0oriodk=—kodi. Moreover i-izj Hence iodi=j-di=k-dh=o. 132 VECTORANALYSIS Substitutingthesevaluesintheexpressionfordr. dr=(zi.dk—ci~di)i+(xi-di—zk~di)i (yk-dj—xiodk)k. Thisisavectorproduct. -dkj+j Let djdk, di Thenr—c —i —E—axt _ dt_ Thisshowsthattheinstantaneousmotionofthebodyisone ofrotationwiththeangularvelocityaaboutthelinea. Thisangularvelocitychangesfrominstanttoinstant.The proofofthistheoremﬁllsthelacunaintheworkinArt.51. Twoinﬁnitesimalrotationsmaybeaddedlikevectors. Leta1anda2betwoangularvelocities.Thedisplacements duetothemare dlr=alxrdt, d2r a2xrdt. Ifrbedisplacedbya,itbecomes r+dlr=r+ale t. Ifitthenbedisplacedbya2,itbecomes Hence(cit)! Iftheinﬁnitesimals(dt)2oforderhigherthantheﬁrstbe neglected, dr=al><rdt+a2 xrdt, whichprovesthetheorem .Ifbothsidesbedividedbydt 134 VECTORANALYSIS whereCissomeconstantvector.Toaccomplishtheintegra tioninanyparticularcasemaybeamatterofsomedifﬁculty justasitisinthecaseofordinaryintegrationofscalars. Example1 Integratetheequationofmotionofa projectile. Theequationofmotionissimply iig, whichexpressesthefactthattheaccelerationisalwaysver ticallydownwardandduetogravity . wherebisaconstantofintegration.Itisevidentlythe velocityatthetimetO. cisanotherconstantofintegration.Itisthepositionvector ofthepointattimeiz0.Thepathwhichisgivenbythis lastequationisaparabola.Thatthisissomaybeseenby expressingitintermsofxandyandeliminatingt. Example9Therateofdescriptionofareaswhena ticlemovesunderacentralaccelerationisconstant f Sincetheaccelerationisparalleltotheradius, rxizo. But rxii—d dl dFor dHence O andrxin=c’ whichprovesthestatement. THEDIFFERENTIALCALCULUSOFVECTORS135 Example3Integratetheequationofmotionforaparticle movingwithanaccelerationtowardthecentreandequalto aconstantmultipleoftheinversesquareofthedistance fromthecentre. 02Given r— 3r. 7' Then 1'x' 1‘0 . Hence rxi0. equationstogetherwithx . " xc—1—1 r ozT3{r-rr r-rr}. Hence Eachsideofthisequalityisaperfectdiﬁerential. e=df 7‘ Integrate.Then whereeIisthevectorconstantofintegration .eisitsmagni. tudeandIaunitvectorinitsdirection.Multiplytheequa tionbyro roixO ror 2+eroI. c 9' Butroi-xc rXi'oOC-O 136 VECTORANALYSIS Letpc. 20andCOSu 003(r,I)‘ 0 Or rP= 1+ecosu Thisistheequationoftheellipseofwhicheistheeccentri city.ThevectorIisdrawninthedirectionofthemajor Thelengthofthisaxisis P a 1 e2 Itispossibletocarrytheintegrationfurtherandobtain thetime.Sofarmerelythepathhasbeenfound. ScalarFunctionsofPositioninSpace.TheOperatorV AfunctionV(x,y,2)whichtakesonadeﬁnitescalar valueforeachsetofcoordinatesx,y,zinspaceiscalleda scalarfunctionofpositioninspace.Suchafunction,forex ample,is V(x,y,z)=x2+y2+22=r2 . Thisfunctiongivesthesquareofthedistanceofthepoint (x,y,z)fromtheorigin.ThefunctionVwillbesupposedto beingeneralcontinuousandsingle-valued.Inphysicsscalar functionsofpositionareofconstantoccurrence.Inthe theoryofheatthetemperatureTatanypointofabodyisa scalarfunctionofthepositionofthatpoint.Inmechanics andtheoriesofattractionthepotentialistheall-important function.This,too,isascalarfunctionofposition . IfascalarfunctionVbesetequaltoaconstant,theequa tion V(x,y,z)c. (20) deﬁnesasurfaceinspacesuchthatateverypointofitthe functionVhasthesamevaluec.IncaseVbethetempera 138 VECTORANALYSIS Thevectorsumwhichistheresultantrateofincrease ofVisdenotedbyVV. 9V.aV' 9Vk VV i ax+Jay+ az VVrepresentsadirectedrateofchangeofV adirected orvectorderivativeofV,sotospeak .ForthisreasonVV willbecalledthederivativeofV;andV,theprimitiveof VV.ThetermsgradientandslopeofVarealsousedfor VV .ItiscustomarytoregardVasanoperatorwhichobtains avectorVVfromascalarfunctionVofpositioninspace. 9 9 k 21!VV ax+19y+ e 9 9OV 1 9x+19y+k 9 ThissymbolicoperatorVwasintroducedbySirW .R . Hamiltonandisnowinuniversalemployment.There seems,however,tobenouniversallyrecognizedname1forit, althoughowingtothefrequentoccurrenceofthesymbol somenameisapracticalnecessity .Ithasbeenfoundby experiencethatthemonosyllabledelissoshortandeasyto pronouncethatevenincomplicatedformulainwhichVoccurs anumberoftimesnoinconveniencetotheSpeakerorhearer arisesfromtherepetition .VVisreadsimplyasdelV” AlthoughthisoperatorVhasbeendeﬁnedas .9 9 9 1SomeusethetermNablaowingtoitsfanciedresemblancetoanAssyrian harp.OthershavenoteditslikenesstoaninvertedAandhaveconsequently coinedthenonetooeuphoniousnameAtledbyinvertingtheorderofthelettersin thewordDelta.FlipplinhisEinﬂhrungindieMaxwell’scheTheoriederEleo tricitc‘itavoidsanyspecialdesignationandreferstothesymbolas“dieOperation V .”Howthisistobereadisnotdivulged.Indeed,forprintingnoparticular nameisnecessary,butforlecturingandpurposesofinstructionsomethingisre quired—somethingtoothatdoesnotconfusethespeakerorhearerevenwhen oftenrepeated. THEDIFFERENTIALCALCULUSOFVECTORS139 sothatitappearstodependuponthechoiceoftheaxes,it isinrealityindependentofthem .Thiswouldbesurmised fromtheinterpretationofVasthemagnitudeanddirection ofthemostrapidincreaseofV .Todemonstratetheinde pendencetakeanothersetofaxes,i’ ,j’ ,k’andanewsetof Variablesx’ ,y’ ,z’referredtothem .ThenVreferredtothis systemis V’=i’a +j’a +k’—a 9x’c?y’9z’ Bymakinguseoftheformula and Art53,page 104,fortransformationofaxesfromi,j,ktoi’ ,j’ ,k’andby actuallycarryingoutthedifferentiationsandﬁnallyby takingintoaccounttheidentities(49)and V’may actuallybetransformedintoV . V’=V . Thedetailsoftheproofareomittedhere,becauseanother shortermethodofdemonstrationistobegiven. Considertwosurfaces(Fig .30) V(x,y,z):0 and uponwhichVisconstantandwhicharemoreoverinﬁnitely neartogether.Letx,y,2beagivenpointuponthesurface V:0.Letrdenotethera dinsvectordrawntothisA” pointfromanyﬁxedorigin . V Thenanypointnearbyin t theneighboringsurfaceV cdcmayberepresented c bytheradiusvectorrdr. TheactualincreaseofVfrom theﬁrstsurfacetothesecond isaﬁxedquantitydo.Therateofincreaseisavariable 140 VECTORANALYSIS quantityanddependsuponthedirectiondrwhichisfol lowedwhenpassingfromonesurfacetotheother. directiondr.Letnbeaunitnormaltothe surfacesanddnthesegmentofthatnormalintercepted betweenthesurfaces,11dnwillthenbetheleastvaluefor dr.Thequotientd A c a (in willthereforebeamaximumwhendrisparalleltonand equalinmagnitudeofdn .Theexpression n (23) isthereforeavectorofwhichthedirectionisthedirectionof mostrapidincreaseofVandofwhichthemagnitudeisthe rateofthatincrease.Thisvectorisentirelyindependentof theaxesX,Y,Z .LetdcbereplacedbyitsequaldVwhich istheincrementofVinpassingfromtheﬁrstsurfacetothe second .ThenletVVbedeﬁnedagainas M\g VV=—n . Fromthisdeﬁnition,VViscertainlythevectorwhich givesthedirectionofmostrapidincreaseofVandtherate inthatdirection.MoreoverVVisindependentoftheaxes. dVVV‘ -no dr dndr (25) nisaunitnormal.Hencenodristheprojectionofdron nandmustbeequaltotheperpendiculardistancednbetween thesurfaces. 142 VECTORANALYSISA: Moreoverthisequationdeﬁnesdydx.InaSimilarmanner itispossibletolaydownthefollowingdeﬁnition. Definition:ThederivativeVVofascalarfunctionof positioninspaceshallsatisfytheequation drVV dV forallvaluesofdr. Thisdeﬁnitioniscertainlythemostnaturalandimportant fromtheoreticalconsiderations.Butforpracticalpurposes eitherofthedeﬁnitionsbeforegivenseemstobebetter. Theyaremoretangible.TherealSigniﬁcanceofthislast deﬁnitioncannotbeappreciateduntilthesubjectoflinear vectorfunctionshasbeentreated.SeeChapterVII. ThecomputationofthederivativeVofafunctionismost frequentlycarriedonbymeansoftheordinary differentiation. Let r 9x 93/ 9x x .7/Vr=1 rm5W +kz 1Hence Vr(ix+13;kz) and VrI Thederivativeofrisaunitvectorinthedirectionofr. Thisisevidentlythedirectionofmostrapidincreaseofr andtherateofthatincrease. THEDIFFERENTIALCALCULUS0FVECTORS143 1 1 1 a:3;V 7.1 (x2 J(x2ya kz H VI 1 ix kz) ence 9'(332+312+22‘)i13’ 1 r —rr and V 'r(r-r)ar3Ir. Thederivativeofl/risavectorWhosedirectionisthat r,andWhosemagnitudeisequaltothereciprocalofthe Theproofislefttothereader. Example4Lety,z)log “3 +53’+01: x2+y2x2+yz 1 (ix wa+gg Ifrdenotethevectordrawnfromtheorigintothepoint (x,y,z)ofspace,thefunctionVmaybewrittenas and ix+jy=r—kk;r. r—kkorHence IOr_(boa r—kha (r—kkor)-(r—kkor) 144 VECTORANALYSIS ThereisanothermethodofcomputingVwhich upontheidentity dr-VV=dV. Let dr-r r (117 —dl‘ o =droVVVror H VV encevhr r LetV:roa,whereaisaconstantvector. dV=droa=droVV . Hence VV=a. Example3:LetV:(rxa)(rxb),wherea.andb constantvectors. V=rorgob—pa, CM‘Ard‘alb.‘ bdrah arn’nuI‘P.e (17’2droraobdroarobdrobr-adr0VV Hence VV=2raoh arobbr-a VV=(raoh ar-b)(raohbr-a) aX(rxb). WhichofthesetwomethodsforcomputingVshallbe appliedinaparticularcasedependsentirelyupontheir relativeeaseofexecutioninthatcase.Thelattermethodis independentofthecoordinateaxesandmaythereforebe preferred .ItisalsoshorterincasethefunctionVcanbe expressedeasilyintermsofr.ButwhenVcannotbeso expressedtheformermethodhastoberesortedto. ThegreatimportanceoftheoperatorVinmathe maticalphysicsmaybeseenfromafewillustrations.Sup P059T(w, 2)bethetemperatureatthepointx,y,zofa 146 VECTORANALYSIS Potentialinelectricityormagnetismisthepotentialenergy perunitchargeorpole;andpotentialinattractionproblems ispotentialenergyperunitmasstaken,however,withthe negativesign . i‘67.]Itisoftenconvenienttotreatanoperatorasa quantityprovideditobeysthesameformallawsasthat quantity.Considerforexamplethepartialdiﬁerentiators 9x93/92 Asfarascombinationsoftheseareconcerned,theformallaws arepreciselywhattheywouldbeifinsteadofdifferentiators threetruescalars a,b,c weregiven .Forinstance thecommutativelaw 99 99 9a:9y9y9:c theassociativelaw 9 92 9x93/a— za(bc) andthedistributivelaw 9 9 99 99 9xay+ 9x9y+ axaz holdforthediﬁerentiatorsjustasforscalars.Ofcoursesuch formulaeas THEDIFFERENTIALCALCULUS0FVECTORS147 Inthesamewayagreatadvantagemaybeobtainedby lookingupon 9 9 9Vz. 1 9x+19y+ 92 asavector.Itisnotatruevector,forthecoefﬁcients 9 9 9 9x9y9z arenottruescalars.Itisavectordifferentiatorandof courseanoperandisalwaysimpliedwithit.Asfarasformal operationsareconcerneditbehaveslikeavector.For instance V(a Vv, V0012)(Va)7) aCV'v), eVaV(ca), ifaandvareanytwoscalarfunctionsofthescalarvariables x,y,zandifcbeascalarindependentofthevariableswith regardtowhichthediﬁerentiationsareperformed. IfArepresentanyvectortheformalcombination AVis A— z(27) provided A=A1i+Agj+A3k . ThisoperatorAVisascalardiﬁerentiator.When toascalarfunctionV(x,y,2)itgivesascalar. (A.V)V:AI (28) SupposeforconveniencethatAisaunitvector 9V 9V 9V (80V)V=a1- a— a-+a2— a az 148 VECTORANALYSIS wherea1,a2,asarethedirectioncosinesofthelineareferred totheaxesX,Y,Z .Consequently(aV)Vappearsasthe well-knowndirectionalderivativeofVinthedirectiona. Thisisoftenwritten 9V 9V 9V 9V 92 It_expressesthemagnitudeoftherateofincreaseofVin thedirectiona.Intheparticularcasewherethisdirectionis thenormal11toasurfaceofconstantvalueofV,thisrelation becomesthenormalderivative. 9V 9V 9V 9V aw+n293/(29)9n ifn1,n2,nabethedirectioncosinesofthenormal. TheoperatoraVappliedtoascalarfunctionofposition Vyieldsthesameresultasthedirectproductofaandthe vectorVV. (a oV)V=ao(VV). (30) Forthisreasoneitheroperationmaybedenotedsimplyby aoVV withoutparenthesesandnoambiguitycanresultfromthe omission .Thetwodifferentforms(a oV)Vanda(VV) mayhoweverbeinterpretedinanimportanttheorem. (8vV)Kisthe_directionaLderivativaofVin~thedirection a.—0nthaotherhanda(VV)isthenomponentnf .VV_in thedirection8.Hence:ThedirectionalderivativeofVin anydirectionisequaltothecomponentofthederivative VVinthatdirection .IfVdenotegravitationalpotentialthe theorembecomes:Thedirectionalderivativeofthepotential inanydirectiongivesthecomponentoftheforceperunit massinthatdirection .IncaseVbeelectricormagnetic potentialadiﬁerenceofsignmustbeobserved. 150 VECTORANALYSIS Thismaybewrittenintheform 9V 9V2.9V1 3(a-V)V 981+ 981+ 93 Hence(a oV)Visthedirectionalderivativeofthevector functionVinthedirectiona.Itispossibletowrite (aeV)V aoVV withoutparentheses.Forthemeaningofthevectorsymbol a.a (31y deﬁned;Hencefromthepresentstandpointtheexpression aVVcanhavebuttheoneinterpretationgiventoitby (aoV)V . AlthoughtheOperationVVhasnotbeendeﬁnedand cannotbeatpresent,1twoformalcombinationsofthevector OperatorVandavectorfunctionVmaybetreated .These arethe(formal)scalarproductandthe(formal)vectorprod uotofVintoV .Theyare vovz —+j—+k.v (32) and VxV=—+j—+k XV . (33) V0VisreaddeldotV;andVXV,delcrossV . O O 9a aThediffeetito rnars axa?!az,berngscalar0perators,pass bythedotandthecross.Thatis 9V 9V 9VV-V=ro 9x+j9y 9z(32) 9V 9V 9Vv =ix(33) THEDIFFERENTIALCALCULUSOFVECTORS151 9vw9V1.i9Vg.C9V3 L L 92:9xl+ 9c’+ 9xk' 9V9VI.9V2.917 334 ‘1 ‘9V9Vli+9V2j+9V3bLtdK 9292 92 azJxlﬁ i'4 . t c) j :2 1: Hence (32 Moreover ix .ix kX O I! HenceVxV 1 +5az 9x(33) 9V,9Vl+k 9a: 93/ Thismaybewrittenintheformofadeterminant 1J k v =i .3 .9 152 VECTORANALYSIS Itistobeunderstoodthattheoperatorsaretobeappliedto thefunctionsV1,V2,V8whenexpandingthedeterminant . Fromsomestandpointsobjectionsmaybebroughtforward againsttreatingVasasymbolicvectorandintroducingVv andVxVrespectivelyasthesymbolicscalarandmom; peM ofljntoLTheseobjectionsmaybeavoidedby simplylayingdownthedeﬁnitionthatthesymbolsVoand VX,whichmaybelookeduponasentirelynewoperators quitedistinctfromV,shallbe 0 o 0 k. 2' VV l axial ay‘i" 9z(3) av av av . Butforpracticalpurposesandforrememberingformqit seemsbyallmeansadvisabletoregard 9m 9y 92 asasymbolicvectordifferentiator.Thissymbolobeysthe samelawsasavectorjustinsofarasthediﬁerentiators 999 37,E,E7obeythesamelawsasordinaryscalarquantities . ThatthetwofunctionsVaVandVXVhavevery importantphysicalmeaningsinconnectionwiththevector functionVmaybeeasilyrecognized .Bythestraight forwardproofindicatedinArt.63itwasseenthatthe 154 VECTORANALYSIS 9V mdxdydz V Thetotalﬂuxoutwardfromthecubethroughthesetwo facesisthereforethealgebraicsumofthesequantities.This issimply6 9V 9dxdydz: 99:da:dydz. Inlikemannertheﬂuxesthroughtheotherpairsoffacesof thecubeare joggdxdydzandkeg—d dydz. 9z Thetotalﬂuxoutfromthecubeistherefore av av ' ax+"ay Thisisthenetquantityofﬂuidwhichleavesthecubeper unittime.Thequotientofthisbythevolumeda:dydzof thecubegivestherateofdiminutionofdensity .Thisis av av avW1N2W3 9x+199+ko 9z9x 93/92' BecauseVoVthusrepresentsthediminutionofdensity ortherateatwhichmatterisleavingapointperunitvolume perunittime,itiscalledthedivergence.Maxwellemployed theterm convergencetodenotetherateatwhichﬂuidap proachesapointperunitvolumeperunittime.Thisisthe negativeofthedivergence.Incasetheﬂuidisincompressﬂile, asmuchmattermustleavethecubeasentersit.Thetotal changeofcontentsmustthereforebezero.Forthisreason thecharacteristicdiﬁerentialequationwhichanyincompres sibleﬂuidmustsatisfyis V0V0+k dxdydz. Vov=io THEDIFFERENTIALCALCULUSOFVECTORS155 whereVistheﬂuxoftheﬂuid.Thisequationisoften knownasthehydrodynamicequation .Itissatisﬁedbyany ﬂowofwater,sincewaterispracticallyincompressible.The greatimportanceoftheequationforworkinelectricityisdue tothefactthataccordingtoMaxwell’shypothesiselectricdis placementobeysthesamelawsasanincompressibleﬂuid.If thenDbetheelectricdisplacement, divD=VoD=0. TotheoperatorVXMaxwellgavethenamecurl. Thisnomenclaturehasbecomewidelyaccepted. VxV=cue . ThecurlofavectorfunctionVisitselfavectorfunction ofpositioninspace.Asthename closely connectedwiththeangularvelocityorspinoftheﬂuxat eachpoint.Buttheinterpretationofthecurlisneitherso easilyObtainednorsosimpleasthatofthedivergence. ConsiderasbeforethatVrepresentstheﬂuxofaﬂuid. Takeatadeﬁniteinstantaninﬁnitesimalsphereaboutany point(cc, Atthenextinstantwhathasbecomeofthe Sphere?Intheﬁrstplaceitmayhavemovedoﬂasawhole inacertaindirectionbyanamountdr.Inotherwordsit mayhaveatranslationalvelocityofdr/dt.Inadditionto thisitmayhaveundergonesuchadeformationthatitisno longerasphere.Itmayhavebeensubjectedtoastrainby virtueofwhichitbecomesslightlyellipsoidalinshape. Finallyitmayhavebeenrotatedasawholeaboutsome axisthroughanangledw .Thatistosay,itmayhavean angularvelocitythemagnitudeofwhichisdw/dt.An inﬁnitesimalspherethereforemayhaveanyoneofthree distincttypesofmotionorallofthemcombined.Err} ,a translationwithdeﬁnitevelocity .Secofriji,astrainwiththree deﬁniteratesofelongationalongtheaxesofanellipsoid. 156 VECTORANALYSIS Thﬁd,anangularvelocityaboutadeﬁniteaxis.Itisthis thirdtypeofmotionwhichisgivenbythecurl.Infact, thecurloftheﬂuxVisavectorWhichhasateachpointof spacethedirectionoftheinstantaneousaxisofrotationat thatpointandamagnitudeequaltotwicetheinstantaneous angularvelocityaboutthataxis. Theanalyticdiscussionofthemotionofaﬂuidpresents moredifﬁcultiesthanitisnecessarytointroduceintreating thecurl.Themotionofarigidbodyissufﬁcientlycomplex togiveanadequateideaoftheoperation .Itwasseen(Art. 51)thatthevelocity' oftheparticlesofarigidbodyatany instantisgivenbytheformula5.w v=vo+axr. curlv=v =VXvo+VX(axt).Let a=a1i+a2j+a3k expandVX(aXr)formallyasifitwerethevectortriple productofV,a,andr.Then V0isaconstantvector.HencethetermVXvovanishes. _9x999z__V°r — z—3. Asaisaconstantvectoritmaybeplacedupontheotherside ofthedifferentialoperator,Voa aV . 9 9 ‘l-Vrz alﬁ+azﬁ+a8$rzali+a21+a8k=a. w v =3a—a=2a Thereforeinthecaseofthemotion-ofarigidbodythecurl ofthelinearvelocityatanypointisequaltotwicethe angularvelocityinmagnitudeandindirection. 158 VECTORANALYSIS Vac(Va)1) Vic-v:(Va)-v Vaxv=(Vn)xv. IfVistobeappliedtomorethantheonetermwhichfollows it,thetermstowhichitisappliedareenclosedinaparen thesisasupontheleft-handsideoftheaboveequations. Theproofsoftheformulaemaybegivenmostnaturally byexpandingtheexpressionsintermsofthreeassumedunit vectorsi,j,k.Thesign2ofsummationwillbefoundcon venient.BymeansofittheOperatorsV,v.,Axtakethe form Thesummationextendsoverac,y,z. Todemonstrate VX 9 9vVX(uV) v+a VX —v (av)21X ax+E1X n ew 9n 9v 97vXV+2u1X a— x Hence Todemonstrate V(nov)=v-Vu+u(VXV). THEDIFFERENTIALCALCULUS0FVECTORS159 9x Now 911 911 911 vx(VXu)=vx21x a— x= p awl v.15?” 911.9112v. a— x1_vx or ZV 9x 9v.InhkemannerZu- a— i1_ux(VXV)+u-VV. HenceV(u-v)=v-Vu+u-Vv +vX(VXu)+uX(VXv). Theotherformulaearedemonstratedinasimilarmanner. Thenotation1 V ‘0u willbeusedtodenotethatinapplyingtheOperatorVtothe product(11ov),thequantityuistoberegardedasconstant. Thatis,theoperationViscarriedoutonlypartiallyupon theproduct(ugv).IngeneralifVistobecarriedout partiallyuponanynumberoffunctionswhichoccurafter itinaparenthesis,thosefunctionswhichareconstantforthe differentiationsarewrittenaftertheparenthesisassubscripts. v=v1i+v2j+vsln 1ThisideaandnotationofapartialVsotospeakmaybeavoidedbymeans oftheformula41.Butacertainamountofcompactnessandsimplicityislostthereby.TheideaofV(nov).issurelynomorecomplicatedthanuVvor vX(VXu). 160 VECTORANALYSIS then QuaV(uov)=21 721+ 93:222+ ax123 9112 9723 Hence V(u -v)v. (45) Thisformulacorrespondstothefollowingoneinthenota tionofdiﬂerentials d(acv)=d(aov)u+d(uov)v or Theformulae(35) givenabove(Art.73)maybe writteninthefollowingmanner,asisobviousfromanalogy withthecorrespondingformulaeindifferentials: (37y 162 VECTORANALYSIS formallyasifV,n,vwereallrealvectors.Then 11X(VXv)=v—:rv. Thesecondtermiscapableofinterpretationasitstands. Theﬁrstterm,however,isnot.TheoperatorVhasnothing uponwhichtooperate.Itthereforemustbetransposedso thatitshallhavenovasanoperand.Butnbeingoutside oftheparenthesisinnX(VXv)isconstantforthedifferen tiations.Hence novV=V(uov)u and—u-Vv. (46) Ifubeaunitvector,saya,theformula ,U .aV«av’ ao =V(a (47) expressesthefactthatthedirectionalderivativea-Vvofa vectorfunctionvinthedirectionaisequaltothederivative oftheprojectionofthevectorvinthatdirectionplusthe vectorproductofthecurlofvintothedirectiona. Considerthevaluesofvattwoneighboringpoints. and v(x+dx,y+dy,z+dz) Let v=i1i+v2j+v3k dv=dv1i+dozj+dogh. Butby do1=droVO1 dv2=dro2 dv8=dro3. Hence dv=dr Hence dv=dr-Vv. By dVV(dl‘V)“(VXv)Xdr.(48) THEDIFFERENTIALCALCULUSOFVECTORS163 Orifvodenotethevalueofvatthepoint(x,y,z)andvthe valueataneighboringpoint v=vo+V(dr (49) Thisexpressionofvintermsofitsvaluev0atagivenpoint, thedels,andthedisplacementdrisanalogoustotheexpan sionofascalarfunctorofonevariablebyTaylor’stheorem, fCC)=f(xo)+f’dx Thederivativeof(rov)whenvisconstantisequaltov. Thatis For -Vr VXr=O . Hence V(r-v)v=v. Inlikemannerifinsteadoftheﬁnitevectorr,aninﬁnitesimal vectordrbesubstituted,theresultstillis V(dr By(47)v=vo+V(dr V(dr Hence V(drcv)—v. Substituting: (50) Thisgivesanotherformof(49)whichissometimesmore convenientItisalsoslightlymoresymmetrical. 164 VECTORANALYSIS Consideramovingﬂuid.Letv(z,y,z,t)bethe velocityoftheﬂuidatthepoint(x,y,2)atthetimet.Sur roundapoint yo,zo)withasmallsphere. drdr c2 . Ateachpointofthisspherethevelocityis v2v0dreVv. Intheincrementoftime8tthepointsofthisspherewillhave movedthedistance (vo+dro)St. Thepointatthecenterwillhavemovedthedistance v,8t. Thedistancebetweenthecenterandthepointsthatwere uponthesphereofradiusdratthecommencementofthe interval8thasbecomeattheendofthatinterval8t dr’=dr+dr-Vv8t. Toﬁndthelocusoftheextremityofdr'itisnecessaryto eliminatedrfromtheequations 02=drodn Theﬁrstequationmaybesolvedfordrbythemethodof Art.47,page90,andthesolutionsubstitutedintothesecond. Theresultwillshowthattheinﬁnitesimalsphere dr-dr=c2 hasbeentransformedintoanellipsoidbythemotionofthe ﬂuidduringthetime8t. Amoredeﬁniteaccountofthechangethathastakenplace maybeobtainedbymakinguseofequation(50) 166 VECTORANALYSIS spectivelyparalleltothem .Thentheexpressionabove becomessimmy Thepointwhosecoordinatesreferredtothecenterofthe inﬁnitesimalsphereare doc,dy,dz isthereforeendowedwiththis ,velocity.Inthetime8tit willhavemovedtoanewposition 922 d 1—8td28,d3 ac + 92:3]t z1+ az8t Thetotalityofthepointsuponthesphere dr-dr=dx2+dy2dz2=c2 goesoverintothetotalityofpointsupontheellipsoidof whichtheequationis $2yazz_62 972311 M1+ 923 Thestatementsmadebefore(Art.72)concerningthethree typesofmotionwhichaninﬁnitesimalsphereofﬂuidmay possesshavethereforenowbeendemonstrated . ThesymbolicoperatorVmaybeappliedseveraltimes insuccession .Thiswillcorrespondinageneralwayto formingderivativesofanorderhigherthantheﬁrst.The expressionsfoundbythusrepeatingVwillallbeindepend entoftheaxesbecauseVitselfis.Therearesixofthese delsofthesecondorder. LetV(x,y,2)beascalarfunctionofpositioninspace. ThederivativeVVisavectorfunctionandhencehasacurl andadivergence.Therefore V°VVs VXVV THEDIFFERENTIALCALCULUSOFVECTORS167 arethetwoderivativesofthesecondorderwhichmaybe obtainedfromV . VXVV=cueV . (52) ThesecondexpressionVXVVvanishesidentically.Thatis, thederivativeofanyscalarfunctionVpossessesnocurl.This maybeseenbyexpandingVXVVintermsofi,j, All thetermscancelout.Later(Art.83)itwillbeshowncon verselythatifavectorfunctionWpossessesnocurl,i.e.if WisthederivativeofsomescalarfunctionV . TheﬁrstexpressionVoVVwhenexpandedintermsof i,j,kbecomes 92V92V92VI 929292 Symbohcally, V-V 6Mcz+ ayﬁ+ TheoperatorVoVisthereforethewell-knownoperatorofLaplace.Laplace’sEquation WWWVV 9x,M,M,0 (53) becomesinthenotationhereemployed VVV0. WhenappliedtoascalarfunctionVtheOperatorVVyields ascalarfunctionwhichis,moreover,thedivergenceofthe derivative. LetTbethetemperatureinabody.Letcbethecon ductivity,pthedensity,andkthespeciﬁcheat.The ﬂowfis f=—eVT . 168 VECTORANALYSIS Therateatwhichheatisleavingapointperunitvolumeper unittimeisVof.Theincrementoftemperatureis dT= —1 —Vofdt. pk dT c —V-VT . dtpk ThisisFourier’sequationfortherateofchangeoftempera LetVbeavectorfunction,andVI,V2,17 3itsthreecom ponents.TheoperatorVeVofLaplacemaybeappliedtoV . IfavectorfunctionVsatisﬁesLaplace’sEquation,eachof itsthreescalarcomponentsdoes.Otherdelsofthesecond ordermaybeobtainedbyconsideringthedivergenceandcurl ofV .ThedivergenceVoVhasaderivative VVoV=VdivV . (55) ThecurlVXVhasinturnadivergenceandacurl, and V-VxV,VXVXV . V-VXV=divcue (56) and VXVXV curlcurlV . (57) OftheseexpressionsVoVXVvanishesidentically.Thatis, thedivergenceOfthecurlOfanyvectoriszero.Thismaybe seenbyexpandingVeVXVintermsofi,j,k .Later(Art. 83)itwillbeshownconverselythatifthedivergenceofa vectorfunctionWvanishesidentically,i.e.if V WisthecurlofsomevectorfunctionV . 170 VECTORANALYSIS ThegeometricinterpretationofV-Vuisinteresting. Itdependsuponageometricinterpretationofthesecond derivativeofascalarfunctionuoftheonescalarvariablecc. Letu,bethevalueofuatthepoint92,Letitberequired toﬁndthesecondderivativeofuwithrespecttoa:atthe pointso.Letx1andas,betwopointsequidistantfrom .130. Thatis,let xz—mo=aco—xl=a. “1“2 2 “3' istheratioofthedifferencebetweentheaverageofuatthe pointsx,andx2andthevalueofuatnotothesquareofthe distanceofthepointsx1,232fromdo.Thatup Then “ll-uzb1a §dx2 -O a2 iseasilyprovedbyTaylor’stheorem . Letubeascalarfunctionofposition1nspace.Choose threemutuallyorthogonallinesi,j,kandevaluatethe expressions 92a ,92a ,92a 9x29y2922 Letx2andT,betwopointsonthelineiatadistanceafrom ao;x4andas,twopointsonjatthesamedistanceafrom (co;cc,andas,twopointsonkatthesamedistanceafromwe. THEDIFFERENTIALCALCULUSOFVECTORS171 ifs 1927“LIM2_u° 2922ai0€12 132%92%_1 6 LIM6 a-0 a2 AsVandVareindependentoftheparticularaxeschosen, thisexpressionmaybeevaluatedforadifferentsetofaxes, thenforstilladiﬂerentone,etc.Byaddingtogetherall theseresults u1+u2+ 6nterms 1 _Lm 6” 6Vv“ a0 a2—uo Letnbecomeinﬁniteandatthesametimeletthediﬂerent setsofaxespointineverydirectionissuingfrom(so.The fraction u1+u26nterms 6n thenapproachestheaveragevalueofituponthesurfaceofa sphereofradiusasurroundingthepointdo.Denotethis byna. I _LIMuct—no 6vvu a .-0 a2 VeVuisequaltosixtimesthelimitapproachedbytheratio oftheexcessofuonthesurfaceofasphereabovethevalue atthecentertothesquareoftheradiusofthesphere.The samereasoningheldincaseuisavectorfunction . IfubethetemperatureofabodyV-Vu(exceptfora constantfactorwhichdependsuponthematerialofthe 172 VECTORANALYSIS body)isequaltotherateofincreaseoftemperature(Art. IfV-Vuispositivetheaveragetemperatureupona smallsphereisgreaterthanthetemperatureatthecenter. Thecenterofthesphereisgrowingwarmer.Inthecase ofasteadyﬂowthetemperatureatthecentermustremain constant.Evidentlythereforetheconditionforasteady ﬂowis VoVuzo . Thatis,thetemperatureisasolutionofLaplace’sEquation. MaxwellgavethenameconcentrationtoVoVitwhether ubeascalarorvectorfunction .ConsequentlyVVumay becalledthedispersionofthefunctionitwhetheritbescalar orvector.Thedispersionisproportionaltotheexcessof theaveragevalueofthefunctiononaninﬁnitesimalsurface abovethevalueatthecenter.Incaseitisavectorfunction theaverageisavectoraverage.Theadditionsinitare vectoradditions. SUMMARYOFCHAPTER111 Ifavectorrisafunctionofascalartthederivativeof rwithrespecttotisavectorquantitywhosedirectionis thatofthetangenttothecurvedescribedbytheterminus ofrandwhosemagnitudeisequaltotherateofadvanceof thatterminusalongthecurveperunitchangeoft.The derivativesofthecomponentsofavectorarethecomponents ofthederivatives. d"rd”rl.d“r2.d"r3 dt" (2) Acombinationofvectorsorofvectorsandscalarsmaybe differentiatedjustasinordinaryscalaranalysisexceptthat thediﬂ‘ erentiationsmustbeperformedinsitu. 174 VECTORANALYSIS Ifrdenotethepositionofamovingparticle,tthetime, vthevelocity,Atheacceleration, v i'(15) vj:s (16) dvdzr A—V— R— Zz- d— Z— zzl' . (18) whichoneisparalleltothetangentanddependsuponthe rateofchangeofthescalarvelocityvoftheparticleinits path,andofwhichtheotherisperpendiculartothetangent anddependsuponthevelocityoftheparticleandthecurva tureofthepath . (19) Applicationstothehodograph,inparticularmotionina circle,parabola,orunderacentralacceleration .Application totheproofofthetheoremthatthemotionofarigidbody onepointofwhichisﬁxedisaninstantaneousrotationabout axisthroughtheﬁxedpoint. Integrationwithrespecttoascalarismerelytheinverse ofdiﬂerentiation .Applicationtoﬁndingthepathsdueto givenaccelerations. TheoperatorVappliedtoascalarfunctionofpositionin spacegivesavectorwhosedirectionisthatofmostrapid increaseofthatfunctionandwhosemagnitudeisequalto therateofthatincreaseperunitchangeofpositioninthat direction VV=i—+j—+k— z, (21) V=i‘9 +5 (22) THEDIFFERENTIALCALCULUSOFVECTORS175 TheoperatorVisinvariantoftheaxesi,j,k .Itmaybe deﬁnedbytheequation dV VV= CT7;11, (24) VV-drd . (25y ComputationofthederivativeVVbytwomethodsdepend inguponequations(21)and Illustrationoftheoc currenceofVinmathematicalphysics. Vmaybelookeduponasaﬁctitiousvector,avector differentiator.Itobeystheformallawsofvectorsjustin sofarasthescalardifferentiatorsof9/9cc,99y,99zobey theformallawsofscalarquantities 9V 9V 9V IfabeaunitvectoraVVisthedirectionalderivativeofV inthedirectiona. a.VV=(a-V) (30) IfVisavectorfunctionaVVisthedirectionalderivative ofthatvectorfunctioninthedirectiona. V-V=i —g, (32y v.v=:17 ; +1: 176 VECTORANALYSIS ProofthatVeVisthedivergenceofVandVxV, ofV . V°V=d1VV, VXV=cue . V -u+Vov, Vo(uv)=Vu-v+uVov, V(uov)=v-Vn+r v+vx(VXu) +11X(VXv),(41) V -VXu—u-VXV,(42) VX(nxv)=voVu-vVou—r v+uVcv.(43) Introductionofthepartialdel,V(no inwhichthedif ferentiationsareperformeduponthehypothesisthatuis constant. -r v. (46) Ifabeaunitvectorthedirectionalderivative a (47) Theexpansionofanyvectorfunctionvintheneighborhood ofapointyo, atwhichittakesonthevalueofv(,is v=vo+V(dr Xdr.(49) or (50) Applicationtohydrodynamics. Thedelsofthesecondorderaresixinnumber.(35) (36) (37) (38) (39) (40) 178 VECTORANALYSIS 3.Obtaintheaccelerationsofamovingparticleparallel andperpendiculartothetangenttothepathandreducethe resultstotheusualform . 4 .Ifr,0beasystemofpolarcoordinatesinspace, whereristhedistanceofapointfromtheorigin,cf)the meridianalangle,and0thepolarangleobtaintheexpressions forthecomponentsofthevelocityandaccelerationalongthe radiusvector,ameridian,andaparalleloflatitude.Reduce theseexpressionstotheordinaryformintermsof .v,y,z. 5 .ShowthedirectmethodsuggestedinArt.63thatthe operatorVisindependentoftheaxes. 6 .BythesecondmethodgivenforcomputingVﬁnd thederivativeVofatripleproduct[ab0]eachtermofwhich isafunctionofy,zincase b=(r-a)e, c=rXf, whered,“ e,fareconstantvectors. 1 17.ComputeV-Vt enVisr2 ,r, or rzr 8 .ComputeVoVV,VVoV,andVXVXt enVis equaltorandwhenVisequalto£39andshowthatinthese casestheformula(58)holds. 9.ExpandVXVVandVoVXVintermsofi,j,kand showthattheyvanish(Art. 10.Showbyexpandingintermsofi,j,kthat VXVXV=VVoV—V-VV . 11.ProveAoV(V-W)=VAoVW+WAoVV. and (VXV)XW=VX CHAPTERIV THEINTEGRALCALCULUSOFVECTORS LetW(ac,y,z)beavectorfunctionofpositionin space.LetCbeanycurveinspace,andrtheradiusvector fromsomeﬁxedorigintothepointsofthecurve. Dividethecurveintoinﬁnitesimalelementsdr.Fromthe sumofthescalarproductoftheseelementsdrandthevalue ofthefunctionWatsomepointoftheelement thus 2Wdr. Thelimitofthissumwhentheelementsdrbecomeinﬁnite innumber,eachapproachingzero,iscalledthelineintegralof WalongthecurveCandiswritten fW-dr. 0 and few.(1) Thedeﬁnitionofthelineintegralthereforecoincideswith thedeﬁnitionusuallygiven.Itishowevernecessaryto specifyinwhichdirectiontheradiusvectorrissupposedto describethecurveduringtheintegration .Fortheelements drhaveoppositesignswhenthecurveisdescribedinoppo 180 VECTORANALYSIS sitedirections.Ifonemethodofdescriptionbedenotedby CandtheotherbyC, fWodr= Wodr. -O IncasethecurveCisaclosedcurveboundingaportionof surfacethecurvewillalwaysberegardedasdescribedin suchadirectionthattheenclosedareaappearspositive (Art. If1‘denotetheforcewhichmaybesupposedtovaryfrom pointtopointalongthecurveC,theworkdonebytheforce whenitspointofapplicationismovedfromtheinitialpoint roofthecurveCtoitsﬁnalpointristhelineintegral ff-dr:r f-dr. 0’ to Theorem:ThelineintegralofthederivativeVVofa scalarfunctionV(a:,y,z)alonganycurvefromthepoint r0tothepointrisequaltothedifferencebetweenthevalues ofthefunctionV(x,y,2)atthepointrandatthepointto. Thatis, fr VVd:V(r)V(r,)z)v 1' Bydeﬁnition dr'-VV=dV V:V0.)V(ro)V673:y’z)V(3’30!yo,zo)° Theorem:ThelineintegralofthederivativeVVofa singlevaluedscalarfunctionofpositionVtakenarounda closedcurvevanishes. 182 VECTORANALYSIS Thevalueoftheintegralisthereforeascalarfunctionof thepositionofthepointrwhosecoordinatesarey,z. odr Lettheintegralbetakenbetweentwopointsinﬁnitelynear together. WdrdV(x,y,z). Butbydeﬁnition VV~dr=dV Hence W VV . Thetheoremisthereforedemonstrated . Letfbetheforcewhichactsuponaunitmassnear thesurfaceoftheearthundertheinﬂuenceofgravity.Let asystemofaxesi,j,kbechosensothatkisvertical.Then f=—gk . Theworkdonebytheforcewhenitspointofapplication movesfromthepositionrotothepositionris r r w=ffodr=f—gkodr=_9012, r r Hence w=_g(z Theforcefissaidtobederivablefromaforce-functionV whenthereexistsascalarfunctionofpositionVsuchthat theforceisequalateachpointofthederivativeVV. EvidentlyifVisoneforce-function,anothermaybeobtained byaddingtoVanyarbitraryconstant.Intheaboveex ampletheforce-functionis Ormoresimply Vgz, Theforceis f_ THEINTEGRALCALCULUSOFVECTORS 183 Thenecessaryandsuﬁicientconditionthataforce-function V(x,y,z)exist,isthattheworkdonebytheforcewhenits pointofapplicationmovesaroundaclosedcircuitbezero. Theworkdonebytheforceis ws-dr. Ifthisintegralvanisheswhentakenaroundeveryclosed contourf=VV=Vw . Andconverselyif fVV theintegralvanishes.Theforce-functionandtheworkdone differonlybyaconstant. V:w const. Incasethereisfrictionnoforce-functioncanexist.Forthe workdonebyfrictionwhenaparticleismovedaroundina closedcircuitisneverzero. TheforceofattractionexertedbyaﬁxedmassMupon aunitmassisdirectedtowardtheﬁxedmassandispropor tionaltotheinversesquareofthedistancebetweenthe [1138888. M f=—c—r. ThisisthelawofuniversalgravitationasstatedbyNewton. Itiseasytoseethatthisforceisderivablefromaforce functionV .Choosetheoriginofcoordinatesatthecenter oftheattractingmassM .Thentheworkdoneis But rodrzrdr, w=-cd zr=—OM1 r 07 7.ro 184 VECTORANALYSIS Byaproperchoiceofunitstheconstantcmaybemade equaltounity.Theforce-functionVmaythereforebe chosenas V IftherehadbeenseveralattractingbodiesM1,M2,M3, theforce-functionwouldhavebeen V 71 wherer1,r2,r3, arethedistancesoftheattractedunit massfromtheattractingmassesM1,M2,M8ooo Thelawoftheconservationofmechanicalenergyrequires thattheworkdonebytheforceswhenapointismoved aroundaclosedcurveshallbezero.Thisisontheassump tionthatnoneofthemechanicalenergyhasbeenconverted intootherformsofenergyduringthemotion .Thelawof conservationofenergythereforerequirestheforcestobe derivablefromaforce-function .Converselyifaforce functionexiststheworkdonebytheforceswhenapointis carriedaroundaclosedcurveiszeroandconsequentlythere isnolossofenergy .Amechanicalsystemforwhichaforce functionexistsiscalledaconservativesystem .Fromthe examplejustcitedaboveitisclearthatbodiesmovingunder thelawofuniversalgravitationformaconservativesystem atleastsolongastheydonotcollide. LetW(as,y,z)beanyvectorfunctionofpositionin space.LetSbeanysurface.Dividethissurfaceintoin ﬁnitesimalelements.Theseelementsmayberegardedas planeandmayberepresentedbyinﬁnitesimalvectorsof whichthedirectionisateachpointthedirectionofthe normaltothesurfaceatthatpointandofwhichthemagni tudeisequaltothemagnitudeoftheareaoftheinﬁnitesimal 186 VECTORANALYSIS givestheamountofthatsubstancewhichispassingthrough thesurfaceperunittime.Itwasseenbefore(Art.71)that therateatwhichmatterwasleavingapointperunit volumeperunittimewasVof.Thetotalamountofmat terwhichleavesaclosedspaceboundedbyasurfaceSper unittimeistheordinarytripleintegral fffVofdv. (6) Hencetheveryimportantrelationconnectingasurfacein tegralofaﬂuxtakenoveraclosedsurfaceandthevolume integralofthedivergenceoftheﬂuxtakenoverthespace enclosedbythesurface Vofd‘v. 7"w Writtenoutinthenotationoftheordinarycalculusthis becomes ff[Xdydzd dxd dy] axaY+as 9%93/ 92dacdydz (s) whereX,Y,Zarethethreecomponentsoftheﬂuxf.The theoremisperhapsstillmorefamiliarwheneachofthethree componentsistreatedseparately. ffd dysfgfdxdydz. (s)' ThisisknownasCauss’ sTheorem .Itstatesthatthesurface integral(takenoveraclosedsurface)oftheproductofa functionXandthecosineoftheanglewhichtheexterior normaltothatsurfacemakeswiththeX-axisisequalto thevolumeintegralofthepartialderivativeofthatfunction THEINTEGRALCALCULUSOFVECTORS 187 withrespecttosotakenthroughoutthevolumeenclosedby thatsurface. IfthesurfaceSbethesurfaceboundinganinﬁnitesimal sphereorcube fff-da=Vofdv S Wheredvisthevolumeofthatsphereorcube.Hence Vof= dl vff-da. (9) Thisequationmaybetakenasadeﬁnitionofthedivergence Vof.Thedivergenceofavectorfunctionfisequaltothe limitapproachedbythesurfaceintegralof1‘takenoverasur faceboundinganinﬁnitesimalbodydividedbythatvolume whenthevolumeapproacheszeroasitslimit.Thatis Sid f-da. (10) Fromthisdeﬁnitionwhichisevidentlyindependentofthe axesallthepropertiesofthedivergencemaybededuced.In ordertomakeuseofthisdeﬁnitionitisnecessarytodevelop atleasttheelementsoftheintegralcalculusofvectorsbefore thedifferentiatingoperatorscanbetreated.Thisdeﬁnition ofV-fconsequentlyisinterestingmorefromatheoretical thanfromapracticalstandpoint. TheoremThesurfaceintegralofthecurlofavector functionisequaltothelineintegralofthatvectorfunction takenaroundtheclosedcurveboundingthatsurface. XW-da=fW-dr. (11) 0 ThisisthecelebratedtheoremofStokes.Onaccountofits greatimportanceinallbranchesofmathematicalphysicsa numberofdifferentproofswillbegiven . 188 VECTORANALYSIS FirstProof:Considerasmalltriangle193uponthesurface S(Fig. LetthevalueofWatthevertex1beW0. Thenby Chap .III.,thevalueatanyneighboringpointis 8r wherethesymbol8rhasbeenintroducedforthesakeofdis tinguishingitfromdrwhichistobeusedastheelementof integration .TheintegralofWtakenaroundthe 123is FIG .32. 1 1LWodr—éfAWO-dr-f- ELVe y-dr X8r-dr. Theﬁrstterm A2A vanishesbecausetheintegralofdraroundaclosedﬁgure,in thiscaseasmalltriangle,iszero.Thesecondterm gLvrydr vanishesbyvirtueof(3)page180.Hence 190 VECTORANALYSIS Thesecondmember2VXWoda 8 isthesurfaceintegralofthecurlofW . ZVXW-da VXWoda. 8 Inaddingtogetherthelineintegralswhichoccurintheﬁrst memberitisnecessarytonoticethatallthesidesoftheele mentarytrianglesexceptthosewhichliealongthebounding curveofthesurfacearetracedtwiceinoppositedirections. Henceallthetermsinthesum odr whicharisefromthosesidesofthetriangleslyingwithinthe surfaceScancelout,leavinginthesumonlytheterms whicharisefromthosesideswhichmakeupthebounding curveofthesurface.Hencethesumreducestothelinein tegralofWalongthecurvewhichboundsthesurfaceS . nWodr= .dr. Hence xWoda odr. (11) Fro .33.SecondProof:LetCbeanyclosed contourdrawnuponthesurfaceS (Fig . ItwillbeassumedthatC iscontinuousanddoesnotcutitself. LetC’beanothersuchcontournear toC .Considerthevariation8which takesplaceinthelineintegralofW inpassingfromthecontourCtothe contourC’ . THEINTEGRALCALCULUSOFVECTORS 191 8fWodr=Wodrodr, 0/ odr=f8(Wodr -8dr+f8Wodr. But d(W-8r)=dW-8r+Wod8r and 8dr=d8r. Henceo8dr=fW-d8r=fd(W-8r)dW-8r. Theexpressiond(Wo8r)isbyitsformaperfectdiﬂerential. Thevalueoftheintegralofthatexpressionwillthereforebe thedifferencebetweenthevaluesofWodrattheendandat thebeginningofthepathofintegration .Inthiscasethe integralistakenaroundtheclosedcontourC .Hence HencefW-8dr—do8r, and 8fWodr=f8Wodr—d-8r, sfw-dr=f{8Wodr—dW-8r 9W 9W 9WBut dW— awdx+aydy+azdz, 9W or dW—aw iodr+aw jodr+ k-dr,9m 9y9z 9W, 9W 9W 1-8r+ jo8r+92k-8r. 192 VECTORANALYSIS Substitutingthesevalues 9W 9W 0. 080 0 dr1 r dx8rid similartermsinyand2. Butby(25)page111 9W 9W 9W rx axo(9e r)axodr1o9r axo9riodr. Hence 8fW-dr=fix:Z-8rxdr similartermsinyandz or 8fWodr=fVXW-8rxdr. InFig.33itwillbeseenthatdristheelementofare alongthecurveCand8risthedistancefromthecurveCto thecurve Hence8rXdrisequaltotheareaofanele mentaryparallelogramincludedbetweenCand0’uponthe surfaceS .Thatis 8e r=da, 8fW-drzsXWoda. LetthecurveCstartingatapoint0inSexpanduntilit coincideswiththecontourboundingS .Thelineintegral fW-dr willvaryfromthevalue0atthepoint0tothevalue fW-dr O 194 VECTORANALYSIS AdeﬁnitionofVXWwhichisindependentoftheaxes i,j,kmaybeobtainedbyapplyingStokes’stheoremtoanin ﬁnitesimalplanearea.ConsiderapointP .Passaplane throughPanddrawinit,concentricwithP,asmallcircleof areada. VXWoda Wodr. (13) WhendahasthesamedirectionasVXWthevalueofthe lineintegralwillbeamaximum,forthecosineoftheangle betweenVXWanddawillbeequaltounity.Forthis valueofda, wdr (13) HencethecurlVXW ofavectorfunctionWhasateach pointofspacethedirectionofthenormaltothatplanein whichthelineintegralofWtakenaboutasmallcirclecon centricwiththepointinquestionisamaximum .Themag nitudeofthecurlatthepointisequaltothemagnitudeof thatlineintegralofmaximumvaluedividedbytheareaof thecircleaboutwhichitistaken .Thisdeﬁnitionlikethe onegiveninArt.81forthedivergenceisinterestingmore fromtheoreticalthanfrompracticalconsiderations. Stokes’stheoremorratheritsconversemaybeusedtode duceMaxwell’sequationsoftheelectro-magneticﬁeldina simplemanner.LetEbetheelectricforce,Bthemagnetic induction,Hthemagneticforce,andCtheﬂuxofelectricity perunitareaperunittime(i.e.thecurrentdensity). Itisafactlearnedfromexperimentthatthetotalelectro motiveforcearoundaclosedcircuitisequaltothenegative oftherateofchangeoftotalmagneticinductionthrough thecircuit.Thetotalelectromotiveforceisthelineintegral oftheelectricforcetakenaroundthecircuit.Thatis fE-dr. O THEINTEGRALCALCULUSOFVECTORS 195 Thetotalmagneticinductionthroughthecircuitisthesur faceintegraloithemagneticinductionBtakenoverasurface boundedbythecircuit.Thatis Experimentthereforeshowsthat dLE-dr: dtf SR-da, orodrs—Boda. o 8 HencebytheconverseofStokes’stheorem Vs—awﬂE=—n Itisalsoafactofexperimentthattheworkdoneincarry ingaunitpositivemagneticpolearoundaclosedcircuitis equalto47:timesthetotalelectricﬂuxthroughthecircuit. Theworkdoneincarryingaunitpolearoundacircuitis thelineintegralofHaroundthecircuit.Thatis Hodn Thetotalﬂuxofelectricitythroughthecircuitisthe surfaceintegralof0takenoverasurfaceboundedbythe circuit.Thatis fro-do» Experimentthereforeteachesthat Hodr=47r Coda. 196 VECTORANALYSIS BytheconverseofStokes’stheorem VXH=47rC. Withaproperinterpretationofthecurrent0,asthedis placementcurrentinadditiontotheconductioncurrent, aninterpretationdependingupononeofMaxwell’sprimary hypotheses,thisrelationandtheprecedingonearethefunda mentalequationsofMaxwell’stheory,intheformusedby HeavisideandHertz . ThetheoremsofStokesandGaussmaybeusedto stratetheidentities V divcue=O. VXVV:0, cueV=0. AccordingtoGauss’stheorem ffV-Vd v: VXW-‘da. AccordingtoStokes’stheorem ffswva:Wodr. Hence V-VXd Wodr. Applythistoaninﬁnitesimalsphere.Thesurfacebounding thesphereisclosed.Henceitsboundingcurvereducestoa point;andtheintegralaroundit,tozero. V-VXd =fW-dr=0, o VOVXW=0. 198 VECTORANALYSIS thoseconnectedwith“integrationbypartsinordinary calculus.Theyareobtainedbyintegratingbothsidesofthe formula,page161,fordiﬂerentiating. First fV(uv)-dr=faq-dr= vVu-dr. 0 Henceqv vVuodr.(14) Theexpression [uv]: representsthediﬂerencebetweenthevalueof(uv)atr,the endofthepath,andthevalueatto,thebeginningofthepath. Ifthepathbeclosed qvodr=—vuodr. O 0 Second Hence V X0dcdoffsu v afouv rffsuVXvda,(15) 01‘ THEINTEGRALCALCULUS0FVECTORS 199 -da. s s Hence Vuv-da=qv-dr=—foVu-dr,(16) O 0 Fourth V Vo(uv)do: uV-vdv+ Vuovdv. Hence uV-vdv=ff8uvoda—fffVuovdv,(17) Vu-vd'vsuv-da—ffq-vdv, —Vuovxv. Vo(Vuxv)= ffVo(Vuxv)dv=—fffVu-v dv . Henceffsvuxvoda=—fffVu-v dv .(18) Inalltheseformulaewhichcontainatripleintegralthe surfaceSistheclosedsurfaceboundingthebodythroughout whichtheintegrationisperformed . Examplesofintegrationbypartslikethoseabovecanbe multipliedalmostWithoutlimit.Onlyonemorewillbe givenhere.ItisknownasGreen’ sTheoremandisperhaps themostimportantofall.Ifuand'vareanytwoscalar functionsofposition, 200 VECTORANALYSIS V.(uvv)=Vu.Vv+avov'v, V°(vvu)=Vu-V'v+vvovu. Vu°Vv=V-uV-Vv=v-(vvu)—vvovu, fffVu-l v= -(q)dv—ffqodv, V-(vVu)dv V-vade. Hence ffVuov'vd'v:fuvvoda uV.Vvdv, fvu-da—ffvoVudv.(19) By ,subtractingtheseequalitiestheformula (20) fftV-v'u—vvu)oda. isobtained.Byexpandingtheexpressionintermsofi,j,k theordinaryformofGreen’stheoremmaybeobtained.A furthergeneralizationduetoThomson(LordKelvin)isthe following Vr vdvzfq voda uVo[wVv]dv, ffv u-da (21) wherewisathirdscalarfunctionofposition . Theelementofvolumedohasnothingtodowiththescalar function12intheseequationsorinthosethatgobefore.The useof'vinthesetwodifferentsensescanhardlygiveriseto anymisunderstanding. Intheprecedingarticlesthescalarandvectorfunc tionswhichhavebeensubjecttotreatmenthavebeensup 202 VECTORANALYSIS thevaluealongonepathbeingthenegativeofthevalue alongtheother.Theintegralaroundthecirclewhichisa closedcurvedoesnotvanish,butisequalto:l:272 ItmightseemthereforetheresultsofArt.79werefalse andthatconsequentlytheentirebottomoftheworkwhich followsfellout.Thishoweverisnotso.Thediﬂicultyis thatthefunction V:tan1'7! a: isnotsingle-valued .Atthepoint forinstance ,the functionVtakesonnotonlythevalue butawholeseriesofvalues I+k7r, wherekisanypositiveornegativeinteger .Furthermoreat theorigin,whichwasincludedbetweenthetwosemicircular pathsofintegration,thefunctionVbecomeswhollyinde terminateandfailstopossessaderivative.Itwillbeseen thereforethattheoriginisapeculiarorsingularpointofthe functionV .Ifthetwopathsofintegrationfrom1,O)to hadnotincludedtheoriginthevaluesoftheintegral wouldnothavediffered.Inotherwordsthevalueofthe integralaroundaclosedcurvewhichdoesnotincludethe onymvanishesasitshould . vitiatestheresultsobtained,letitbeconsideredasmarkedbyanimpassablebarrier.Anyclosedcurve0whichdoes THEINTEGRALCALCULUS0FVECTORS 203 mayshrinkuptonothingwithoutabreakinitscontinuity; but0canonlyshrinkdownandﬁtcloserandcloserabout theorigin .Itcannotbeshrunkdowntonothing.Itmust alwaysremainencirclingtheorigin.ThecurveCissaidto bereducibleC,irreducible.IncaseofthefunctionV,then, itistruethattheintegraltakenaroundanyreduciblecircuit Cvanishes;buttheintegralaroundanyirreduciblecircuit0 doesnotvanish . SupposenextthatVisanyfunctionwhatsoever.Letall thepointsatwhichVfailstobecontinuousortohavecon tinuousﬁrstpartialderivativesbemarkedasimpassable barriers.Thenanycircuit0'whichcontainswithinitno suchpointmaybeshrunkuptonothingandissaidtobe reducible;butacircuitwhichcontainsoneormoresuch pointscannotbesoshrunkupwithoutbreakingitscontinuity anditissaidtobeirreducible.Thetheoremmaythenbe stated:ThelineintegralofthederivativeVVofanyfunction VuanishesaroundanyreduciblecircuitC .Itmayormaynot vanisharoundanirreduciblecircuit.Incaseoneirreducible circuitCmaybedistortedsoastocoincidewithanother irreduciblecircuit0withoutpassingthroughanyofthe singularpointsofVandwithoutbreakingitscontinuity, thetwocircuitsaresaidtobereconcilableandthevaluesof thelineintegralofVVaboutthemarethesame. Aregionsuchthatanyclosedcurve0’withinitmaybe shrunkuptonothingwithoutpassingthroughanysingular pointofVandwithoutbreakingitscontinuity,thatis,a regioneveryclosedcurveinwhichisreducible,issaidtobe acyclic.Allotherregionsarecyclic. Bymeansofasimpledeviceanycyclicregionmayberen deredacyclic.Consider,forinstance,theregion(Fig.34)en closedbetweenthesurfaceofacylinderandthesurfaceofa cubewhichcontainsthecylinderandwhosebasescoincide withthoseofthecylinder.Sucharegionisrealizedinaroom 204 VECTORANALYSIS inwhichacolumnreachesfromtheﬂoortotheceiling .It isevidentthatthisregioniscyclic.Acircuitwhichpasses aroundthecolumnisirreducible.Itcannotbecontractedto nothingwithoutbreakingitscontinuity.If nowadiaphragmbeinsertedreachingfrom thesurfaceofthecylinderorcolumntothe surfaceofthecubetheregionthusformed boundedbythesurfaceofthecylinder,the surfaceofthecube,andthetwosidesofthe diaphragmisacyclic.Owingtotheinser tionofthediaphragmitisnolongerpossible todrawacircuitwhichshallpasscompletelyaroundthecyl inder—thediaphragmpreventsit.Henceeveryclosedcir cuitwhichmaybedrawnintheregionisreducibleandthe regionisacyclic. Inlikemanneranyregionmayberenderedacyclicby insertingasufﬁcientnumberofdiaphragms.Thebounding surfacesofthenewregionconsistoftheboundingsurfacesof thegivencyclicregionandthetwofacesofeachdiaphragm . Inacyclicregionsorregionsrenderedacyclicbythefore goingdevicealltheresultscontainedinArts .79etseq. holdtrue.Forcyclicregionstheymayormaynothold true.Toenterfurtherintothesequestionsatthispointis unnecessary.Indeed,evenasmuchdiscussionashasbeen giventhemalreadymaybesuperﬂuous.Fortheyareques tionswhichdonotconcernvectormethodsanymorethanthe correspondingCartesianones.Theybelongproperlytothe subjectofintegrationitself,ratherthantotheparticular notationwhichmaybeemployedinconnectionwithitand whichistheprimaryobjectofexpositionhere.Inthis respectthesequestionsaresimilartoquestionsofrigor.FIG .34. 206 VECTORANALYSIS Thelengthofrl2isthenr12andwillbeassumedtobe positive. (22 Considerthetripleintegralr12 17052922)I(5'71,31vzl)dwadysdz? Theintegrationisperformedwithrespecttothevariables x2,y2,z2thatis,withrespecttothebodyofwhichV representsthedensity(Fig . During theintegrationthepoint(x1,yl,zl)re mainsﬁxed.TheintegralIhasadeﬁnite valueateachdeﬁnitepoint(x1,y1, Itisafunctionofthatpoint.Thein terpretationofthisintegralIiseasy,if thefunctionVberegardedasthedensityofmatterinspace. Theelementofmassdmat(x2,y2,zz)is dmV(x2,y2,za)datadyzdz2Vdu. TheintegralIisthereforethesumoftheelementsofmass inabody,eachdividedbyitsdistancefromaﬁxedpoint (xv3hrz1)°FIG .35. dm Thisiswhatistermedthepotentialatthepoint(x1,yl,zl) duetothebodywhosedensityis 22) ThelimitsofintegrationintheintegralImaybelookedat ineitheroftwoways.Intheﬁrstplacetheymaybe regardedascoincidentwiththelimitsofthebodyofwhich Visthedensity .Thisindeedmightseemthemostnatural setoflimits.OntheotherhandtheintegralImaybe THEINTEGRALCALCULUSOFVECTORS 207 regardedastakenoverallspace.Thevalueoftheintegral isthesameinbothcases.Forwhenthelimitsareinﬁnite thefunctionVvanishesidenticallyateverypoint(x2,y2,22) situatedoutsideofthebodyandhencedoesnotaugment thevalueoftheintegralatall.Itisfoundmostconvenient toconsiderthelimitsasinﬁniteandtheintegralasextended overallspace.Thissavesthetroubleofwritinginspecial limitsforeachparticularcase.ThefunctionVofitselfthen practicallydeterminesthelimitsowingtoitsvanishingiden ticallyatallpointsunoccupiedbymatter. TheOperationofﬁndingthepotentialisofsuch frequentoccurrencethataspecialsymbol,Pot,isusedforit. MH”m a;a22>a.a.dz.<22) 12 ThesymbolisreadthepotentialofV .”Thepotential, PotV,isafunctionnotofthevariablesx2,y2,22with regardtowhichtheintegrationisperformedbutofthepoint (221,y,,21)whichisﬁxedduringtheintegration .These variablesenterintheexpressionforr12.ThefunctionV andPotVthereforehavedifferentsetsofvariables. ItmaybenecessarytonotethatalthoughVhashitherto beenregardedasthedensityofmatterinspace,suchan interpretationforVisentirelytoorestrictedforconvenience. Wheneveritbecomesnecessarytoformtheintegral 170132,92’zz) _ff5 2rfff rm fmdo,2) ofanyscalarfunctionV,nomatterwhatVrepresents,that integraliscalledthepotentialofV .Thereasonforcalling suchanintegralthepotentialevenincasesinwhichithas noconnectionwithphysicalpotentialisthatitisformed accordingtothesameformallawasthetruepotentialand 208 VECTORANALYSIS byvirtueofthatformationhascertainsimplerulesofOpera tionwhichothertypesofintegralsdonotpossess. Pursuanttothisideathepotentialofavectorfunction W 22) Inthiscasetheintegralisthesumofvectorquantities andisconsequentlyitselfavector.Thusthepotentialofa vectorfunctionWisavectorfunction,justasthepotential ofascalarfunctionVwasseentobeascalarfunctionofposi tioninspace.IfWberesolvedintoitsthreecomponents w 22)ix(902’32)+5Y929zz) 92’22) PotW=iPotX+jPotY+kPotZ .(24) ThepotentialofavectorfunctionWisequaltothevector sumofthepotentialsofitsthreecomponentsI,Y,Z . ThepotentialofascalarfunctionVexistsatapoint (901,y,,zv)whenandonlywhentheintegral PotV 23dog, takenoverallspaceconvergestoadeﬁnitevalue.If, forinstance,VwereeverywhereconstantinSpacethein tegralwouldbecomegreaterandgreaterwithoutlimitas thelimitsofintegrationwereextendedfartherandfarther outintospace.Evidentlythereforeifthepotentialistoexist Vmustapproachzeroasitslimitasthepoint recedesindeﬁnitely.Afewimportantsufﬁcientconditions fortheconvergenceofthepotentialmaybeobtainedby transformingtopolarcoordinates.Let 210 VECTORANALYSIS IfthefunctionVremainﬁniteorifitbecomeinfiniteso weaklythattheproduct Vr remainsfinitewhenrapproacheszero,thentheintegralconverges asfarasregionsneartotheoriginareconcerned.Forlet Vr<K r=R r: ffesmddrd6ale<jfdrd6as r=0 r: r=R ffKdrdoas: wKR r=0 Hencethetripleintegraltakenoverallspaceinsideasphere ofradiusR(whereRisnowsupposedtobeasmallquantity) islessthan47rKRandconsequentlyconvergesasfaras regionsneartotheoriginwhichisthepoint(c,yl,21)are concerned . Ifatanypoint(x2,y2,z2)notcoincidentwiththeorigin, i.e.thepoint(x1,yl,zl),thefunctionVbecomesinﬁniteso weaklythattheproductofthevalueofVatapointnearto (x2,y2,zz)bythesquareofthedistanceofthatpointfrom (x2,y2,Z2)remainsfiniteasthatdistanceapproacheszero,then theintegralconvergesasfarasregionsneartothepoint(x2,y2,Z?) areconcerned .Theproofofthisstatementislikethosegiven before.Thesethreeconditionsfortheconvergenceofthe integralPotVaresufﬁcient.Theyarebynomeansneces sary.Theintegralmayconvergewhentheydonothold . Itishoweverindispensabletoknowwhetherornotanintegral underdiscussionconverges.Unlessthetestsgivenabove showtheconvergence,morestringentonesmustberesorted to.Such,however,willnotbediscussedhere.Theybelong tothetheoryofintegrationingeneralratherthantothe THEINTEGRALCALCULUSOFVECTORS 211 theoryoftheintegratingoperatorPot.Thediscussionof theconvergenceofthepotentialofavectorfunctionWre ducesatoncetothatofitsthreecomponentswhicharescalar functionsandmaybetreatedasabove. Thepotentialisafunctionofthevariablesx1,yl,zl whichareconstantwithrespecttotheintegration .Letthe valueofthepotentialatthepoint(x1,yl,21)bedenotedby [POtVlzu3h:21. Theﬁrstpartialderivativeofthepotentialwithrespecttoa, istherefore aPOtV V]31A319y”’1 V]zl9yl’‘1 9x1Ano,iO A331 Thevalueofthislimitmaybedeterminedbyasimple device(Fig . Consider thepotentialatthepoint (xlA901,l/vzl) duetoacertainbodyT .This isthesameasthepotentialat thepoint (x1,y1,21)FIG.36. duetothesamebodyTdisplacedinthenegativedirectionby theamountAa1.ForinﬁndingthepotentialatapointP duetoabodyTtheabsolutepositionsinspaceofthebody TandthepointPareimmaterial.Itisonlytheirpositions relativetoeachotherwhichdeterminesthevalueofthepoten tial.Ifbothbodyandpointbetranslatedbythesame amountinthesamedirectionthevalueofthepotentialisun changed .ButnowifTbedisplacedinthenegativedirection bytheamountAac,thevalueofVateachpointofspaceis changedfrom 17032,22)to7002A502,dz, whereAx2A$1. 212 VECTORANALYSIS Hence [PotV Ail,y”8,[Pot17962A232’y2’z2)]‘1 HenceLIMVs , A561i0 A231 Aalio Itwillbefoundconvenienttointroducethelimitsof integration.Lettheportionofspaceoriginallyﬁlledbythe bodyTbedenotedbyM;andlettheportionﬁlledbythe bodyafteritstranslationinthenegativedirectionthrough thedistanceAa,bedenotedbyM’ .TheregionsMandM' overlap .LettheregioncommontobothbeIf;andletthe remainderofMbem;theremainderofM’ ,m' .Then M=M+m, Pot—_fff V(x2+srysrzz)dv d,vo ’122 2 Vx 2PMV032,3/2vea)C3,ya 2)d”: 12 Vff(3325%22)dvz+V6132y222)dvz I! Hence(25)becomes,whenAx,isreplacedbyitsequalAan, iAsallthefollowingpotentialsareforthepoint:1,yl,21thebracketand indiceshavebeendropped. 214 VECTORANALYSIS ThenifitbeassumedthattheregionTisﬁniteandthatV vanishesuponthesurfaceboundingT LIM A552,zz) széoA$2 LIMff17052: si’0m r12A232(102—0° Consequentlytheexpressionforthederivativeofthepoten tialreducestomerelydv2=0 QPotV 19V 9V 9x19x2 ThepartialderivativeofthepotentialofascalarfunctionV isequaltothepotentialofthepartialderivativeofV . ThederivativeVofthepotentialofVisequaltothepotential ofthederivativeVV . VPotV:PotVV: (27) Thisstatementfollowsimmediatelyfromtheformer.As theVupontheleft-handsideappliestothesetofvari ablesxl,yl,zl,itmaybewrittenVI.Inlikemannerthe Vupontheright-handsidemaybewrittenV2tocallatten tiontothefactthatitappliestothevariables:22,pg,zzofV . Then VlPotV:PotV2V . TodemonstratethisidentityVmaybeexpandedintermsof i,j,k. iQPotV +.9PotV QPotV 9331J93119zl 9V 9V taV 9x28%52. THEINTEGRALCALCULUS0FVECTORS 215 Asi,j,kareconstantvectorstheymaybeplacedunder thesignofintegrationandthetermsmaybecollected .Then bymeansof(26) V1PotV=POtVzV . ThecurlVxanddivergenceVoofthepotentialofavector fit/notionWareequalrespectivelytothepotentialofthecurland divergenceofthatfunction . 01' curlPotW Potcue(28) and VIPotw:Pot;V2w, 9 0r divPotw Potdivw , Theserelationsmaybeprovedinamanneranalogoustothe above.Itisevenpossibletogofurtherandformthedels ofhigherorder V0VPotV:PotVoVV, (30) V-VPotW=PotV-VW, (31) VV-PotW=PotVV-W, (32) VxVxPotW=PotVXVXW . (33) Thedelsupontheleftmighthaveasubscript1attachedto showthatthedifferentiationsareperformedwithrespectto thevariables:cl,p1,21,andforasimilarreasonthedelsupon therightmighthavebeenwrittenwithasubscript2 .The resultsofthisarticlemaybesummedupasfollows: Theorem:ThediferentiatingoperatorVandtheintegrating OperatorPotarecommutative. Intheforegoingworkithasbeenassumedthatthe regionTwasﬁniteandthatthefunctionVwaseverywhere ﬁniteandcontinuousinsideoftheregionTandmoreover decreasedsoastoapproachzerocontinuouslyatthesurface boundingthatregion .Theserestrictionsareinconvenient 216 VECTORANALYSIS andmayberemovedby useofasurfaceintegral. Thederivativeofthepotentialwasobtained(page213)in essentiallytheform QPotV 19V dv29x1 Ir12Jazz LIM 1 V(ag+A séosvr12dv LIM 1 V002,yz,22) AxaéOAxar12dv2. LetdabeadirectedelementofthesurfaceSboundingthe regionM .Theelementofvolumedv2intheregionm'is thereforeequalto dv2=Ax2ioda. lo V032A“729 2Hence dv, Theelementofvolumedv2intheregionmisequalto dva=—A:c2ioda. 1 m ye.,z.)m V052»$92922)ida. Consequently 9PtV 19—Zdv,+V iuta.(34) 218 VECTORANALYSIS thispointwithasmallSphereofradiusR .LetSdenotethe surfaceofthisSphereandMalltheregionTnotincluded withinthesphere.Then V 9PotV 19V dvz+ ioda. BytheconditionsimposeduponV VrK ffgmza Eded¢=27rK . 8 ConsequentlywhenthesphereofradiusRbecomessmaller andsmallerthesurfaceintegralmayormaynotbecomezero. Moreoverthevolumeintegral 19V dvz r129332 mayormaynotapproachalimitwhenRbecomessmaller andsmaller.Hencetheequation 9PotVPot9V 9x19332 hasnotalwaysadeﬁnitemeaningatapointoftheregion TatwhichVbecomesinﬁniteinsuchamannerthatthe productVrremainsﬁnite. If,however,Vremainsﬁniteatthepointinquestionso thattheproductVrapproacheszero,theconstantKiszero andthesurfaceintegralbecomessmallerandsmallerasR approacheszero.Moreoverthevolumeintegral 1 9V dv29x2 THEINTEGRALCALCULUSOFVECTORS 219 approachesadeﬁnitelimitasRbecomesinﬁnitesimal.Con sequentlytheequation 9PotVPot67V 9x19wz holdsintheneighborhoodofallisolatedpointsatwhichV remainsﬁniteeventhoughitbediscontinuous. SupposethatVbecomesinﬁniteatsomesinglepoint (x2,y2,22)notcoincidentwith(xi,gl, Accordingtothe conditionslaiduponV W2K. wherelisthedistanceofthepoint(x2,ya,22)fromapoint neartoit.Thenthesurfaceintegral V ioda neednotbecomezeroandconsequentlytheequation neednotholdforanypointgl,zl)oftheregion.But ifVbecomesinﬁniteatg2,z2insuchamannerthat Vl<K, thenthesurfaceintegralwillapproachzeroasitslimitand theequationwillhold . FinallysupposethefunctionVremainsﬁniteuponthe surfaceSboundingtheregionT,butdoesnotvanishthere. InthiscasethereexistsasurfaceofdiscontinuitiesofVI WithinthissurfaceVisﬁnite;without,itiszero.The V i-da 220 VECTORANALYSIS doesnotvanishingeneral.Hencetheequation 9PotV 9V 9x1"‘POt ex, cannothold . Similarreasoningmaybeappliedtoeachofthethree partialderivativeswithrespectto:61,gl, Bycombining theresultsitisseenthatingeneral v,PotV:PotV,V+ da.(35) LetVbeanyfunctioninspace,andletitbegranted ’that PotVexists.SurroundeachpointofspaceatwhichV ceasestobeﬁnitebyasmallsphere.Letthesurfaceofthe spherebedenotedbyS .Drawinspaceallthosesurfaces whicharesurfacesofdiscontinuityofV.Letthesesur facesalsobedenotedbyS .Thentheformula(35)holds wherethesurfaceintegralistakenoverallthesurfaces whichhavebeendesignatedbyS .Iftheinten taken overallthesesurfacesvanisheswhentheradiiofthespheres abovementionedbecomeinﬁnitesimal,then VIPotV:PotV2V. Thisformula VIPotV:PotV2V: willsurelyholdatapoint(x1,yl,21)ifVremainsalwaysﬁniteorbecomesinﬁniteatapoint(x2,y2,z,)sothatthe productVIremainsﬁmite,andifVpossessesnosurfacesof discontinuity,andiffurthermoretheproductVr3remainsfi/nite asrbecomesinfinite.lInothercasesspecialtestsmustbe appliedtoascertainwhethertheformula canbeused orthemorecomplicatedone(35)mustberesortedto . iForextensionsandmodiﬁcationsofthistheorem,eeeexercises. 222 VECTORANALYSIS Theirregularitieswhichmayarisearethrownintotheinter pretation,notintotheanalyticappearanceoftheformula . ThisistheessenceofProfessorGibbs’ smethodoftreatment. Theﬁrstpartialderivativesofthepotentialmayalso beobtainedbydifferentiatingunderthesignofintegration.1 dxdydzz 2 2 (36) Pa a 9otV l:(ac2y222) 2dx2dy2dzaax1V 1 9x1(37) InlikemannerforavectorfunctionW QPOLW (932—5171)W(332’y2922)dd) 93:2 1(38) 9PotV (az—xl)V Or 8061dv2 OPotW 2—al)W 38' and 9x173 12dv2 Pt 9Pt vPotV ziQPotV +j9oV +k0V 9501921 10132dva. 7' 127' 127’ 12 But iota +5(yz—y1)+k(552 1Ifanattemptweremadetoobtainthesecondpartialderivativesinthesame manner,itwouldbeBeenthatthevolumeintegralsnolongerconverged.fff fff‘x THEINTEGRALCALCULUSOFVECTORS 223 Hence VPotV: dv (39) 12 Inlikemanner VxPotW dv2, (40) r1oWand VPotW:3 ”dv2 (41) Thesethreeintegralsobtainedfromthepotential differentiatingOperatorsareofgreatimportanceinmathe meticalphysics.Eachhasitsowninterpretation .Couse quentlyalthoughobtainedsosimplyfromthepotentialeach isgivenaseparatename.Moreoverinasmuchasthese integralsmayexistevenwhenthepotentialisdivergent, theymustbeconsideredindependentofit.Theyareto belookeduponasthreenewintegratingoperatorsdeﬁned eachuponitsownmeritsasthepotentialwasdeﬁned . Let,therefore, r12XW(392,dxzdyzdzzLapW(43) r-W x2, ,z12( 3‘7”(44) 7' 12 Ifthepotentialexists,then VPotV:NewV VXPotW LapW (45) VPotWMaxW . TheﬁrstiswrittenNewVandread“TheNewtonianofV .”fff fff =fff 224 VECTORANALYSIS ThereasonforcallingthisintegraltheNewtonianisthatif Vrepresentthedensityofabodytheintegralgivestheforce ofattractionatthepoint(x1,g1,sl)duetothebody.This willbeprovedlater.ThesecondiswrittenLapW and read“theLaplacianofW .”Thisintegralwasusedtoa considerableextentbyLaplace.Itisoffrequentoccurrence inelectricityandmagnetism .IfWrepresentthecurrent 0inspacetheLaplacianof0givesthemagneticforceatthe point(xl,pl, duetothecurrent.Thethirdiswritten MaxWandreadtheMaxwellianofW .”Thisintegralwas usedbyMaxwell.It,too,occursfrequentlyinelectricity andmagnetism .ForinstanceifWrepresenttheintensity ofmagnetizationI,theMaxwellianofIgivesthemagnetic potentialatthepoint(x1,p1,21)duetothemagnetization . ToshowthattheNewtoniangivestheforceofattraction accordingtothelawoftheinversesquareofthedistance. Letdmzbeanyelementofmasssituatedatthepoint (x2,g2, Theforceat(x1,pl,21)duetodmisequalto dma inmagnitudeandhasthedirectionofthevectorr1,fromthe P0int(“3p21)tothepoint(a2,g2, Hencetheforceis Integratingovertheentirebody,oroverallspaceaccording totheconventionhereadopted,thetotalforceis dmz1'ldv=NewV r3 12r3 123 whereVdenotesthedensityofmatter. 226 VECTORANALYSIS Integratingoverallspace,thetotalmagneticforceactingat thepointg1,21)uponaunitpositivepoleis fffT3 12 Thisintegralmaybeexpandedintermsofi,j,k.Let W(x2,zz)iX(w2’ +jI7(5’32a 52)kZ(x23 rlaxi)i+zr)k ° Thei,j,kcomponentsofLapWarerespectively 191)Z(zz21)Y ioLapW= 3 j'LaPW= 3dv,I koLaPW=fff(x2—x1)Y T' 3 12 Intermsofthepotential(ifoneexists)thismaybewritt QPotY 991931 _OPotXaPotZJoLapW az19x1 k .Lapw_9P0tYOPotX 9931 ToshowthatifIbetheintensityofmagnetizationatthe point(x2,v2,zz),thatis,ifIbeavectorwhosemagnitudeis equaltothemagneticmomentperunitvolumeandwhose THEINTEGRALCALCULUS0FVECTORS 227 directionisthedirectionofmagnetizationoftheelementatv2 fromsouthpoletonorthpole,thentheMaxwellianofIisthe magneticpotentialduetothedistributionofmagnetization. Themagneticmomentoftheelementofvolumedv2isIdv2. Thepotentialat(al,yl,sl)duetothiselementisequaltoits magneticmomentdividedbythesquareofthedistancerm andmultipliedbythecosineoftheanglebetweenthedirec tionofmagnetizationIandthevectorr12.Thepotentialis therefore 3 Integrating,thetotalmagneticpotentialisseentobe 3dvz=MaxI. Thisintegralmayalsobewrittenoutintermsofx,y,2. Let 1002’92’zz)i14(5’32’ 22)+jB(mzayzrzz)k 92a32) r12°1=(x2" 'xr)A(ya (32"z1)0~ Ifinsteadof$1,pl,21thevariablesx,g,z;andinsteadof x2,pg,22thevariables5,n,Cbeused1theexpressiontakes ontheformgivenbyMaxwell. MaxI=ffA(E AccordingtothenotationemployedfortheLaplacian Maxw=fff T8 12 1Maxwell:ElectricityandMagnetism,Vol.II.p.9. 228 VECTORANALYSIS TheMaxwellianofavectorfunctionisascalarquantity . Itmaybewrittenintermsofthepotential(ifitexists)as X 9PotY9PotMaxWa (921Z . Thisformofexpressionismuchusedinordinarytreatises uponmathematicalphysics. TheNewtonian,Laplacian,andMaxwellian,however,should notbeassociatedindissolublywiththeparticularphysical interpretationsgiventothemabove.Theyshouldbelooked uponasintegratingoperatorswhichmaybeapplied,asthe potentialis,toanyfunctionsofpositioninspace.TheNew tonianisappliedtoascalarfunctionandyieldsavector function .TheLaplacianisappliedtoavectorfunction andyieldsafunctionofthesamesort.TheMaxwellian isappliedtoavectorfunctionandyieldsascalarfunction . Moreover,theseintegralsshouldnotbelookeduponasthe derivativesofthepotential.Ifthepotentialexiststhey areitsderivatives.Buttheyfrequentlyexistwhenthe potentialfailstoconverge. LetVandWbesuchfunctionsthattheirpotentials existandhave1ngeneraldeﬁnitevalues.Thenby(27)and (29) VoVPotV:VoPotVV=PotVoVV Butby(45)VPotV:NewV, and VoPotVV=MaxVV Hence V-VPotV zV-Nes MaxVV PotV-VV(46) By(27)and(29)VVPotW=VPotVoW :PotVVoW . Butby'(45) VoPotWMaxW, andby(45) VPotVoW=NewV-W . 230 VECTORANALYSIS Poisson’ sEquation LetVbeanyfunctioninspacesuchthatthepotential PotV hasingeneraladeﬁnitevalue.Then V-VPotV:—4vrV, (52) «92PotV 92PotV <92PotV 4V or9x12Oylz92127r ThisequationisknownasPoisson’sEquation . Theintegralwhichhasbeendeﬁnedasthepotentialisa solutionofPoisson’sEquation.Theproofisasfollows. V VIPotV:NewV:{1:olv2 VI~1 Vdv27' 12 oV ThesubscriptsIand2havebeenattachedtodesignate clearlywhatarevariableswithrespecttowhichthediﬁerem tiationsareperformed. 1VI'VIPOtVzvloNeWV=fffV1 Vs'vz. 1 1But VI V2 1 1 1andV2VV2=V2oV2V+VV2~V2 THEINTEGRALCALCULUSOFVECTORS 231 1 1V2V:VV2V2—V2VV2 1 1 1orVIoV2V=VV2-V2—+V2oVV1 1 1VIoV2Vdvz: V2-V2—dv2TI? 1But Thatistosay satisﬁesLaplace’sEquation.Andby(8) ffv2.VV,. dvz:ffVV11 oda. 8 1 V2Vdv2(53) 8 Thesurfaceintegralistakenoverthesurfacewhichbounds theregionofintegrationofthevolumeintegral.Thisis takenoverallspace.”Hencethesurfaceintegralmustbe takenoverasphereofradiusR,alargequantity,andRmust beallowedtoincreasewithoutlimitAtthepoint(x1,yvel), however,theintegrandofthesurfaceintegralbecomesin ﬁniteowingtothepresenceoftheterm 1 232 VECTORANALYSIS HencethesurfaceSmustincludenotonlythesurfaceofthe SphereofradiusR,butalsothesurfaceofaSphereofradius R’ ,asmallquantity,surroundingthepoint(331,yl,21)andR’ mustbeallowedtoapproachzeroasitslimit. AsithasbeenassumedthatthepotentialofVexists,itis assumedthattheconditionsgiven(Art.87)fortheexistence ofthepotentialhold .Thatis VraK,whenrislarge VrK,whenrissmall. Introducepolarcoordinateswiththeoriginatthepoint (x1,y,, Thenr12becomessimplyr 1 1 r 12 12 ThenforthelargesphereofradiusR 1 moda:%r2sin6d6as. V1 Hencethesurfaceintegraloverthatsphereapproacheszero asitslimit.For 1 K 2am HencewhenRbecomesinﬁnitethesurfaceintegraloverthe largesphereapproacheszeroasitslimit. ForthesmallSphere 1odar rzsinddddp. r12r3V1 Hencetheintegraloverthatspherebecomes fVsin0d0as. 234 VECTORANALYSIS Theorem:IfVandWaresuchfunctionsofpositioninspace thattheirpotentialsexistingeneral,thenforallpointsatwhich V andW areﬁniteandcontinuousthosepotentialssatisfy Poisson’ sEquation, V-VPotV:-47rV, (52) V-VPotW :—4rrW . Themodiﬁcationsinthistheoremwhicharetobemadeat pointsatwhichVandWbecomediscontinuouswillnotbe takenuphere. Itwasseen(46)Art.91that VeVPotV:VoNewV=MaxVV Hence VNewV=—471'V (53) or MaxVV 4vrV . Inasimilarmanneritwasseen(51)Art.91that V-VPotW:V~ MaxW—VXLapW —NewV-W—LapVXW . Hence VMaxW VXLapW—47rW,(54) or NeoW—LapVXW=—47rW . ByvirtueofthisequalityWisdividedintotwoparts . 1 1W—ELapVXW— 4wNeoW .(55) Let W=W1+W2, 1 1whereW1— 4— wLapVXw: Z;Lapcue (56) 1 1andW2= —NewV-W_ Newde .(57)471'471' THEINTEGRALCALCULUSOFVECTORS 235 Equation(55)statesthatanyvectorfunctionWmultiplied by471'isequaltothedifferenceoftheLaplacianofitscurl andtheNewtonianofitsdivergence.Furthermore I 1 Vowl—V~LapVXW— 4w 4WV-VXLal. Butthedivergenceofthecurlofavectorfunctioniszero. Hence V-W1divW1:0 (58) 1 1 VXW2=— 4WVXN8WV0W2= 4WVXVMas. Butthecurlofthederivativeofascalarfunctioniszero. Hence VXW2curlW20 . (59) ConsequentlyanyvectorfunctionWwhichhasapotential maybedividedintotwopartsofwhichonehasnodivergence andofwhichtheotherhasnocurl.ThisdivisionofWinto twosuchpartsisunique. Incaseavectorfunctionhasnopotentialbutbothitscurl anddivergencepossesspotentials,thevectorfunctionmaybe dividedintothreepartsofwhichtheﬁrsthasnodivergence thesecond,nocurl;thethird,neitherdivergencenorcurl. 1 1ILetW— 4— 7rLapVXW 47rNeoW+W3.(55) Asbefore 1 1 V-LapVXW Vo PotW=0 471'471' 1—1and VXNeoW VXVPotV-W :O. 471' 471' Thedivergenceoftheﬁrstpartandthecurlofthesecond partofWarethereforezero. 236 VECTORANALYSIS 1 VXLapVXW—iVXVXPotVXW 471' 471' —1—V-VPotW . 47; 471' 1 VVoPotVXW=—1 VPotVoVXW=O, 471' 471' for VoVXW:O. Hence 4;V0VP0tVXW=VXW . Hence 41 7VXLapVXW:VXW=VXW1. ThecurlofWisequaltothecurloftheﬁrstpart 1LaVXW 471'P intowhichWisdivided .Henceasthesecondparthasno curl,thethirdpartcanhavenone.Moreover 1 4WVNeoW:V-W:V-Wl. ThusthedivergenceofWisequaltothedivergenceof thesecondpart ﬁl NeoW . 471' intowhichWisdivided .Henceastheﬁrstparthasno divergencethethirdcanhavenone.Consequentlythethird partW8hasneithercurlnordivergence.Thisprovesthe statement. BymeansofArt.96itmaybeseenthatanyfunctionW3 whichpossessesneithercurlnordivergence,musteither 238 VECTORANALYSIS WithrespecttoascalarfunctionVtheoperators Voor—divand1New,471' andalso MaxandV areinverseoperators. NewV:V (63) 1and MaxVV=V . 471' WithrespecttoasolenoidalfunctionW1theoperators 1 4Potand VXorcurlcurl areinverseoperators.Thatis WithrespecttoanirrotationalfunctionW2theoperators 1Potand—VV0 471' areinverseoperators.Thatis 11— EPotVV-W2:—VVo 47rPotW2=Wzo(65) WithrespecttoanyscalarorvectorfunctionV,Wthe Operators 1 —Potand—VoV471' areinverseoperators.Thatis THEINTEGRALCALCULUSOFVECTORS 239 1 1 471'471' 1 1and—PotV-VW:—VoV 471'471' WithrespecttoasolenoidalfunctionW1thediferentiating operatorsofthesecondorder VeVandVXVX areequivalent (67) WithrespecttoanirrotationalfunctionW2thediferentiat ingoperatorsofthesecondorder VoVandVV areequivalent.Thatis V-VW2:VV-WZ. (68) Byintegratingtheequations 4n'V VNewV and 47rW=VXLapW—VMaxW bymeansofthepotentialintegralPot 47rPotV: PotVNewV:MaxNewV(69) 47rPotW :PotVXLapW—PotVMaxW 471'PotW LapLapW NewMaxW .(70) Henceforscalarfunctionsandirrotationalvectorfunctions 1NewMax 471' isanoperatorwhichisequivalenttoPot.Forsolenoidalvector functionstheoperator1LapLap 240 VECTORANALYSIS givesthepotential.ForanyvectorfunctiontheﬁrstOperator givesthepotentialoftheirrotationalpart;thesecond,the potentialofthesolenoidalpart. Thereareanumberofdoublevolumeintegralswhich areofsuchfrequentoccurrenceinmathematicalphysicsas tomeritapassingmention,althoughthetheoryofthemwill notbedevelopedtoanyconsiderableextent.Thesedouble integralsareallscalarquantities.Theyarenotscalarfunc tionsofpositioninSpace.TheyhavebutaSinglevalue. Theintegrationsintheexpressionsmaybeconsideredfor convenienceasextendedoverallSpace.Thefunctionsby vanishingidenticallyoutsideofcertainﬁnitelimitsdeter mineforallpracticalpurposesthelimitsofintegrationin casetheyareﬁnite.GiventwoscalarfunctionsU,VofpositioninSpace. Themutualpotentialorpotentialproduct,asitmaybecalled, ofthetwofunctionsisthesextupleintegral Pot(U,V) dvldo,. 71 Oneoftheintegrationsmaybeperformed PotVdv, ffV(:c2,y,,22)PotUdvz. (72) InaSimilarmannerthemutualpotentialorpotentialproduct oftwovectorfunctionsW’ ,W”is I! Potzl)°W dvldc, Thisisalsoascalarquantity.Oneintegrationmaybecar riedout 242 VECTORANALYSIS Oneintegrationyields Max(W,V) (78) By(53)Art.93. 47rUPotV:(VNewU)PotV . V.[NewUPotV]=(VoNewU)PotV+(NewU)-VPotV. (VoNewU)PotV:—Vo[NewUPotV]+NewUNewV Integrate wffPotVdv:—fffVo[NewUPotVJdv +ffNeo Ned v. 47rPot(U,V):fffNewUNed v _ffPotVNeo da. (79) S ThesurfaceintegralistobetakenovertheentiresurfaceS boundingtheregionofintegrationofthevolumeintegral. AsthisregionofintegrationisallSpace,”thesurfaceSmay belookeduponasthesurfaceofalargesphereofradiusR . IfthefunctionsUandVvanishidenticallyforallpointsout sideofcertainﬁnitelimits,thesurfaceintegralmustvanish. Hence By(54)Art.93, 471'W”oPotW’VXLapW”PotW' VMaxW"oPotW' . THEINTEGRALCALCULUS0FVECTORS 243 ButVo[LapW”XPotW’]PotW’oVXLapW” LapW”VXPotW’ , andVo[MaxW”PotW’]PotW’VMaxW” MaxW”VPotW’ . HenceVXLapW”PotW’:Vo[LapW”XPotW’] LapW”LapW’ , and VMaxW”oPotW’:Vo[MaxW”PotW’] MaxW”MaxW’ Hencesubstituting: 471'W”oPotW’:LapW’LapW’MaxW’MaxW” Vo[LapW”XPotW’] V[MaxW”Pot Integrating: 471'Pot(W’ ,W”) ffPotW’XLapW”da MaxW”PotW’ oda. 8 IfnowW’andW”existonlyinﬁniteSpacethesesurface integralstakenoveralargesphereofradiusRmustvanish andthen 471'Pot:fffLapW’LapW”dv ffMaxwIMaxW”dv. Thereareanumberofusefultheoremsofafunction theoreticnaturewhichmayperhapsbementionedhereowingffLapW’ oLapW”dv ffMaxW’MaxW”dv(80) -ff. 244 VECTORANALYSIS totheirintimateconnectionwiththeintegralcalculusof vectors .Theproofsofthemwillinsomeinstancesbegiven andinsomenot.Thetheoremsareoftenusefulinpractical applicationsofvectoranalysistophysicsaswellasinpurely mathematicalwork. Theorem:IfV(:c,y,z)beascalarfunctionofposition inspacewhichpossessesingeneraladeﬁnitederivativeVV andifinanyportionofspace,ﬁniteorinﬁnitebutnecessarily continuous ,thatderivativevanishes,thenthefunctionVis constantthroughoutthatportionofSpace. Given VV:0. ToShow V:const. Chooseaﬁxedpoint(901,y,, intheregion.By(2)page 180 ButfVV.d1 Hence V(.v,y,z)V(zc1,y,,21) const. TheoremIfV(ac,y,2)beascalarfunctionofposition inspacewhichpossessesingeneraladeﬁnitederivativeVV; ifthedivergenceofthatderivativeexistsandiszerothrough outanyregionofSpace,1ﬁniteorinﬁnitebutnecessarily continuous;andiffurthermorethederivativeVVvanishes ateverypointofanyﬁnitevolumeorofanyﬁniteportionof surfaceinthatregionorboundingit,thenthederivative vanishesthroughoutallthatregionandthefunctionVre ducestoaconstantbytheprecedingtheorem. 1Thetermthroughoutanyregionofspacemustberegardedasincludingthe boundanesoftheregionaswellastheregionitself. 246 VECTORANALYSIS anextensionofthereasoningVisseentobeconstant throughouttheentireregionT . TheoremIfV(x,y,2)beascalarfunctionofpositionin spacepossessingingeneraladerivativeVVandifthrough outacertainregion1TofSpace,ﬁniteorinﬁnite,continuous ordiscontinuous,thedivergenceVoVVofthatderivative existsandiszero,andiffurthermorethefunctionVpossesses aconstantvaluecinallthesurfacesboundingtheregion andV(a:,y,z)approaches0asalimitwhenthepoint(cc,y,z) recedestoinﬁnity,thenthroughouttheentireregionTthe functionVhasthesameconstantvaluecandthederivative VVvanishes. Theproofdoesnotdifferessentiallyfromtheonegiven inthecaseofthelasttheorem .Thetheoremmaybegen eralizedasfollows: Theorem:IfV(x,y,z)beanyscalarfunctionofposition inspacepossessingingeneraladerivativeVV;ifU(x,y,e) beanyotherscalarfunctionofpositionwhichiseitherposi tiveornegativethroughoutandupontheboundariesofa regionT,ﬁniteorinﬁnite,continuousordiscontinuous;if thedivergenceV[UVV]oftheproductofUandVV existsandiszerothroughoutandupontheboundariesofT andatinﬁnity;andiffurthermoreVbeconstantandequal tocuponalltheboundariesofTandatinﬁnity;thenthe functionVisconstantthroughouttheentireregionTand isequaltoc. Theorem:IfV(x,y,2)beanyscalarfunctionofposition inspacepossessingingeneraladerivativeVV;ifthrough outanyregionTofspace,ﬁniteorinﬁnite,continuousor discontinuous,thedivergenceVeVVofthisderivativeexists andiszero;andifinalltheboundingsurfacesoftheregion TthenormalcomponentofthederivativeVVvanishesand atinﬁnitedistancesinT(ifsuchtherebe)theproduct 1Theregionincludesitsboundaries. THEINTEGRALCALCULUS0FVECTORS 247 r29V/6)rvanishes,whererdenotesthedistancemeasured fromanyﬁxedorigin;thenthroughouttheentireregionT thederivativeVVvanishesandineachcontinuousportion ofTVisconstant,althoughfordiﬁerentcontinuousportions thisconstantmaynotbethesame. Thistheoremmaybegeneralizedastheprecedingone wasbythesubstitutionoftherelationV UVV)0for V-VV=OandUrZOV/9r20forr29V/9r:0 . AScorollariesoftheforegoingtheoremsthefollowing statementsmaybemade.Thelanguageisnotsoprecise asinthetheoremsthemselves,butwillperhapsbeunder stoodwhentheyareborneinmind. IfVU=VV,thenUandVdiﬁeratmostbya constant. IfV-VU:V-VVandifVU=VVinanyﬁnite portionofsurfaceS,thenVUVVatallpointsandU diﬁersfromVonlybyaconstantatmost. IfV-VU:VoVVandifU:Vinallthebounding surfacesoftheregionandatinﬁnity(iftheregionextend thereto),thenatallpointsUandVareequal. IfVVUVVVandifinalltheboundingsurfaces oftheregionthenormalcomponentsofVUandVVare equalandifatinﬁnitedistancesr2(9U(Jr—9V/9r)is zero,thenVUandVVareequalatallpointsoftheregion andUdiﬁersfromVonlybyaconstant. TheoremIfW’andW”aretwovectorfunctionsofposition inSpacewhichingeneralpossesscurlsanddivergences;if foranyregionT,ﬁniteorinﬁnitebutnecessarilycontinuous, thecurlofW’isequaltothecurlofW”andthedivergence ofW’isequaltothedivergenceofW”andifmoreover thetwofunctionsW’andW”areequaltoeachotherat everypointofanyﬁnitevolumeinTorofanyﬁnitesurface inTorboundingit;thenW’isequaltoW”ateverypoint oftheregionT . 248 VECTORANALYSIS SinceVXW’:VXW” ,VX(W’Avec torfunctionwhosecurlvanishesisequaltothederivative1 ofascalarfunctionV(page LetVV=W’—W” . ThenVoVV:0owingtotheequalityofthedivergences. Thetheoremthereforebecomesacorollaryofaprecedingone. TheoremIfW’andW”aretwovectorfunctionsofposi tionwhichingeneralpossessdeﬁnitecurlsanddivergences ifthroughoutanyaperiphractic2regionT,ﬁnitebutnot necessarilycontinuous,thecurlofW’isequaltothecurlof W”andthedivergenceofW’isequaltothedivergenceof W”andiffurthermoreinalltheboundingsurfacesofthe regionTthetangentialcomponentsW’andW”areequal; thenW’isequaltoW”throughouttheaperiphracticregionT . Theorem:IfW’andW”aretwovectorfunctionsofposi tioninspacewhichingeneralpossessdeﬁnitecurlsand divergences;ifthroughoutanyacyclicregionT,ﬁnitebutnot necessarilycontinuous,thecurlofW’isequaltothecurl W”andthedivergenceofW’isequaltothedivergenceof W”;andifinalltheboundingsurfacesoftheregionTthe dlcomponentsofW’andW”areequal;thenthefunc tionsW’andW”areequalthroughouttheregionacyclicT . Theproofsofthesetwotheoremsarecarriedoutbymeans ofthedevicesuggestedbefore. Theorem:IfW’andW”aretwovectorfunctionssuch thatVoVW’andVVW”haveingeneraldeﬁnitevalues inacertainregionT,ﬁniteorinﬁnite,continuousordiscon tinuous;andifinalltheboundingsurfacesoftheregion andatinﬁnitythefunctionsW’andW”areequal;thenW’ isequaltoW”throughouttheentireregionT . Theproofisgivenbytreatingseparatelythethreecom ponentsofW’andW” . 1TheregionTmayhavetobemadeacyclicbytheinsertionofdiaphragms. 3Aregionwhichencloseswithinitselfanotherregionissaidtobeperiphracv ac.Ifitenclosesnoregionitisaperiphractic. 250 VECTORANALYSIS oror + d” or9x67y —ff[Xdydz+d dx+d dy], (8) S ifX,Y,ZbethethreecomponentsofthevectorfunctionW . Stokes’sTheorem:Thesurfaceintegralofthecurlofa vectorfunctiontakenoveranysurfaceisequaltotheline integralofthefunctiontakenaroundthelineboundingthe surface.Andconverselyifthesurfaceintegralofavector functionUtakenoveranysurfaceisequaltothelineintegral ofafunctionWtakenaroundtheboundary,thenUisthe curlofW . ffVXW-da:Wodr, (11) S andifffSU-da:fW (12) 0 ApplicationofthetheoremofStokestodeducingthe equationsoftheelectro-magneticﬁeldfromtwoexperimental factsduetoFaraday .ApplicationofthetheoremsofStokes andGausstotheproofthatthedivergenceofthecurlof avectorfunctioniszeroandthecurlofthederivativeof ascalarfunctioniszero . Formulaanalogoustointegrationbyparts qv—vuodr, (14) ffSVuxvoda:ﬂuvodr—ffSuVXv ffsVuXVv —LvVu-dr,(16) dv: qodaqo dv Vu-da—ffv-Vudv,(19) Vu)dv: (q—vVu)-da(20) VPotV:PotVV; VXPotW:PotVXW, V-PotW:PotV-W, V-VPotV=PotV-VV, 252 VECTORANALYSIS V-VPotW :PotV-VW, (31) VV-PotW:PotVV-W, (32) VXVXPotW:PotVXVXW . (33) TheintegratingoperatorPotandthedifferentiatingoperator Varecommutative. Thethreeadditionalintegratingoperatorsknownasthe Newtonian,theLaplacian,andtheMaxwellian . NewV=fff12 m(42) LapW (43) MaxW :fff12(44) Ifthepotentialexiststheseintegralsarerelatedtoitasfol lows: VPotV:NewV, VXPotW LapW; (45) VPotWMaxW . TheinterpretationofthephysicalmeaningoftheNewtonian ontheassumptionthatVisthedensityofanattracting body,oftheLaplacianontheassumptionthatWiselectric ﬂux,oftheMaxwellianontheassumptionthatWisthe intensityofmagnetization .Theexpressionoftheseintegrals ortheircomponentsintermsofac,y,zformula and VoNewV:MaxVV, (46) VMaxW NewVoW, (47) VXLapW:LapVXW, (48) 254 VECTORANALYSIS —V1 NewV:V 1471' (63) MaxVV=V . 471' (64) 71' 1 1 PotVV-W2:—V-V wPotW2:W2.(65)471' 4 1 1 PotV-VV=—V-V PotV:V 471'471' 1 1(66) PotV-VW :—VoV PotW=W . 471'471' (67) VoVW2:VVoW2 (68) 471'PotV:MaxNewV (69) 471'PotW LapLapW NewMaxW .(70) MutualpotentialsNewtonians,Laplacians,andMaxwellians maybeformed .Theyaresextupleintegrals.Theintegra tionscannotallbeperformedimmediately;buttheﬁrstthree maybe .Formula(71)to(80)inclusivedealwiththeseinte grals.Thechaptercloseswiththeenunciationofanumber oftheoremsofafunction-theoreticnature.Bymeansof thesetheoremscertainfactsconcerningfunctionsmaybe inferredfromtheconditionsthatsatisfyLaplace’sequation andhavecertainboundaryconditions. Amongtheexercisesnumber6isworthyofespecialatten tion.Theworkdoneinthetexthasforthemostpartassumed thatthepotentialexists.ButmanyOftheformulaeconnecting Newtonians,Laplacians,andMaxwelliansholdwhenthepoten tialdoesnotexist.ThesearetakenupinExercise6referredto. THEINTEGRALCALCULUSOFVECTORS 255 EXERCISESonCHAPTERIV IfVisascalarfunctionofpositioninspacetheline integral Vdr isavectorquantity .Showthat Vdr: Thatisthelineintegralofascalarfunctionarounda closedcurveisequaltotheskewsurfaceintegralofthederiv ativeofthefunctiontakenoveranysurfaceSpannedinto thecontourofthecurve.ShowfurtherthatifVisconstant theintegralaroundanyclosedcurveiszeroandconversely iftheintegralaroundanyclosedcurveiszerothefunctionV isconstant. Hint:Insteadoftreatingtheintegralasitstandsmultiply it(withadot)byanarbitraryconstantunitvectorandthus reduceittothelineintegralofavectorfunction. 2 .IfWisavectorfunctionthelineintegral H=fWXdr c isavectorquantity.Itmaybecalledtheskewlineintegral ofthefunctionW .Ifcisanyconstantvector,Showthatif theintegralbetakenaroundaclosedcurve Hoc:fj;(cV-W—c-VW)-da:coLWXdr, 1TheﬁrstfourexercisesaretakenfromFOppl’ SEinfiihrungindieMax well’ scheTheoriederElectricitatwheretheyareworkedout. 256 VECTORANALYSIS andHoc:cjfﬂVoWda—ILV(Woda) LWc[c Incasetheintegralistakenoveraplanecurveandthe surfaceSistheportionofplaneincludedbythecurve H :fL[VoWda Showthattheintegraltakenoveraplanecurvevanishes whenWisconstantandconverselyiftheintegraloverany planecurvevanishesWmustbeconstant. 3.ThesurfaceintegralofascalarfunctionVis Thisisavectorquantity .Showthatthesurfaceintegral ofVtakenoveranyclosedsurfaceisequaltothevolume integralofVVtakenthroughoutthevolumeboundedby thatsurface.Thatis ffsVda:—fffVVdv. Henceconcludethatthesurfaceintegraloveraclosedsur facevanishesifVbeconstantandconverselyifthesurface integraloveranyclosedsurfacevanishesthefunctionVmust beconstant. 4 .IfWbeavectorfunction,thesurfaceintegral T=ffdaXW S maybecalledtheshewsurfaceintegral.Itisavector quantity.Showthattheskewsurfaceintegralofavector 258 VECTORANALYSIS Byexercise(3)fffV2(p12V)dv2=ffP12Vda: ItcanbeShownthatifVissuchafunctionthatNewV exists,thenthissurfaceintegraltakenoveralargeSphereof radiusRandasmallsphereofradiusR’approacheszero whenRbecomesindeﬁnitelygreat;andR’ ,indeﬁnitely small.Hence ﬂamem. or NewV:PotVV (85) ProveinaSimilarmannerthat LapW PotVXW, (86) MaxW PotVoW . (87) Bymeansof (87)itispossibletoprove VXLapW=LapVXW, V-NewV:MaxVV, VMaxW=NeoW . Thenprove —fffpmvovwczv, and VMaxW=fffp12VV-d2. HenceVXLapW—VMaxW= Hence VXLapW VMaxW 47rW . (88) 7.AnintegralusedbyHelmholtzis H(V)=fffrmVdvz, THEINTEGRALCALCULUSOFVECTORS 259 orifWbeavectorfunction (90) ShowthattheintegralconvergesifVdiminishessorapidly Vr5K whenrbecomesindeﬁnitelygreat. VH(V):H(VV):New(r2V), V-H(W):H(VoW):Max(rZW), VX:H(VXW):Lap(rZW), VoVH(V):H(VoVV):Max(rZVV):2PotV V-VH(W):H(V 21 7‘PotPotV: H(W)21 7PotPotw . (97) (98) 8.GiveaproofofGauss’sTheoremwhichdoesnotdepend uponthephysicalinterpretationofafunctionastheﬂuxofa ﬂuid .ThereasoningisSimilartothatemployedinArt.51 andintheﬁrstproofofStokes’sTheorem . 9.ShowthatthedivisionofWintotwoparts,page235, isunique. 10.Treat,inamanneranalogoustothatuponpage220, thecaseinwhichVhascurvesofdiscontinuities.(91) (92) (93) (94) (95) CHAPTERV LINEARVECTORFUNCTIONS AFTERthedeﬁnitionsofproductshadbeenlaiddown andapplied,twopathsofadvancewereopen.Onewas differentialandintegralcalculus;theother,higheralgebra inthesenseofthetheoryoflinearhomogeneoussubstitutions. Thetreatmentoftheﬁrstofthesetopicsledtonewideas andnewsymbolstothederivative,divergence,curl,scalar andvectorpotential,thatis,toV,VVX,andPotwiththe auxiliaries,theNewtonian,theLaplacian,andtheMaxwellian . Thetreatmentofthesecondtopicwilllikewiseintroduce noveltybothinconceptandinnotation thelinearvector function,thedyad,andthedyadicwiththeirappropriate symbolization . Thesimplestexampleofalinearvectorfunctionisthe productofascalarconstantandavector.Thevectorr’ r’:cr (1) isalinearfunctionofr.Amoregenerallinearfunction maybeobtainedbyconsideringthecomponentsofrindivid ually.Leti,j,kbeasystemofaxes.Thecomponentsof rare ior,jor,k-r. Leteachofthesebemultipliedbyascalarconstantwhich maybedifferentforthedifferentcomponents. cli.r’Czj'r’ 03k°ro 262 VECTORANALYSIS Definition:Avectorr’issaidtobealinearvectorfunc tionofanothervectorrwhenthecomponentsof1"along threenon-OOpIanarvectorsareexpressiblelinearlywithscalar coefﬁcientsintermsofthecomponentsofralongthosesame vectors. If where and endif (3) z’:a3x+bsy+csz, thenr’isalinearfunctionofr.(Theconstantsdl,c1, etc.,havenoconnectionwiththecomponentsofa,b,0par alleltoi,j,k .)Anotherdeﬁnitionhoweverisfoundtobe moreconvenientandfromittheforegoingmaybededuced. Deﬁnition:Acontinuousvectorfunctionofavectoris saidtobealinearvectorfunctionwhenthefunctionofthe sumofanytwovectorsisthesumofthefunctionsofthose vectors.Thatis,thefunctionfislinearif f(r1=f(r1)+f(ra) (4) Theorem IfabeanypositiveornegativeScalarandiff bealinearfunction,thenthefunctionofatimes1'isatimes thefunctionofr. f(ar)af(r). (5) f(a1r1“2%“are‘l‘ a1f(r1)azf(r2)+ Theproofofthistheoremwhichappearsmoreorless obviousisatriﬂelong.Itdependsuponmakingrepeated useofrelation f(r r)=f(r)+f(r)2m).Andhence LINEARVECTORFUNCTIONS 263 Hence f(2r)2f(r). Inlikemanner f(nr)nf(r) wherenisanypositiveinteger. Letmbeanyotherpositiveinteger.Thenbytherelation justobtained f(r)=fm =mf andi—f(r). 7LHence n =f—1‘ Thatis,equation(5)hasbeenprovedincasetheconstanta isarationalpositivenumber. Toshowtherelationfornegativenumbersnotethat fCO)=f(00)2f Hence f(0)0 . But f(0)=f(r—r)=f(r r))=f(r)+f(—t)o Hence f(r) f r). Toprove(5)forincommensurablevaluesoftheconstant a,itbecomesnecessarytomakeuseofthecontinuityofthe functionf.Thatis f(wr)f (w9 Let:13approachtheincommensurablenumberabypassing throughasuiteofcommensurablevalues.Then f(er)xx(r). HenceLIMf(xr)af(r) {13 . 264 VECTORANALYSIS LIM {via(xr)_ar. Hence f(ar)af(0 whichprovesthetheorem . Theorem:Alinearvectorfunctionf(r)isentirelydeter. minedwhenitsvalueforthreenon-coplanarvectorsa,b,care known . Let l=f(a), m=f(b), I=f Since1'isanyvectorwhatsoever,itmaybeexpressedas r:xa+yb+za Hence InArt.97aparticularcaseofalinearfunctionwas expressedas r’ Forthesakeofbrevityandtosaverepeatingthevectorr whichoccursineachofthesetermsinthesamewaythis maybewritteninthesymbolicform Inlikemannerifa1,a2,a3oocbeanygivenvectors,andb1,b2, 113,oooanothersetequalinnumber,theexpression r’:alblor+a2b2-r+a3b3-r+n o(6) isalinearvectorfunctionofr;forowingtothedistributive characterofthescalarproductthisfunctionofrsatisﬁes relation Forthesakeofbrevityr’maybewrittensym bolicallyintheform 266 VECTORANALYSIS multipliedintorbydirectorScalarmultiplication .The orderofthefactorsOandrisimportant.Thedirect productofrintoOis toO=ro(elbl+azb2a3b8 _roa1b1+r-a2b2+roa3b3+n o(9) EvidentlythevectorsQ;rﬁnd-Mareingeneraldiﬂerent. Deﬁnition:WhenthedyadicOismultipliedinto1‘asOor, Oissaidtobeaprefactortor.WhenrismultipliedinOas roO,Oissaidtobeapostfactortor. AdyadicOusedeitherasaprefactororasapostfactortoa vectorrdeterminesalinearvectorfunctionofr.Thetwolinear vectorfunctionsthusobtainedareingeneraldifferentfrom oneanother.Theyarecalledconjugatelinearvectorfunc tions.Thetwodyadics ¢ zalb1+a2b2+a3b3 eachofwhichmaybeobtainedfromtheotherbyinter changingtheantecedentsandconsequents,arecalledconju gatedyadics.Thefactthatonedyadicistheconjugateof anotherisdenotedbyaffixingasubscriptCtoeither. Theorem:Adyadicusedasapostfactorgivesthesame resultasitsconjugateusedasaprefactor.Thatis T:Toot. Deﬁnition:AnytwodyadicsOandTaresaidto beequal when OorTor forallvaluesofr, orwhen roO roT forallvaluesofr, (10) orwhen soO or:soTorforallvaluesofsandr. LINEARVECTORFUNCTIONS 267 Thethirdrelationisequivalenttotheﬁrst.For,ifthe vectorsO-randTorareequal,thescalarproductsofany vector3intothemmustbeequal.Andconverselyifthe scalarproductofanyandeveryvector3intothevectorsOor andTorareequal,thenthosevectorsmustbeequal.In likemanneritmaybeShownthatthethirdrelationisequiva lenttothesecond .Henceallthreeareequivalent. Theorem:AdyadicOiscompletelydeterminedwhenthe values$03,wob,$.c, wherea,b,careanythreenon-coplanarvectors,areknown. Thisfollowsimmediatelyfromthefactthatadyadicdeﬁnes alinearvectorfunction .If r xaybzc, O-r: :xOoa+yOob+zO-c, consequentlytwodyadicsOandTareequalprovidedequa tions(10)holdforthreenon-coplanarvectorsrandthree non-coplanarvectors8. TheoremAnylinearvectorfunctionfmayberepresented byadyadicOtobeusedasaprefactorandbyadyadicT, whichistheconjugateofO,tobeusedasapostfactor. Thelinearvectorfunctioniscompletelydeterminedwhen itsvaluesforthreenon-coplanarvectors(sayi,j,k)are known(page Let f6) a,re)=11»N!)c. ThenthelinearfunctionfisequivalenttothedyadicO O aibj ck, tobeusedasapostfactorandtothedyadicT T: tobeusedasaprefactor. f(r)Oor:rO0. 268 VECTORANALYSIS Thestudyoflinearvectorfunctionsthereforeisidentical withthestudyofdyadics. Deﬁnition:Adyadabissaidtobemultipliedbyascalar awhentheantecedentortheconsequentismultipliedby thatscalar,orwhenaisdistributedinanymannerbetween theantecedentandtheconsequent.Ifa a’a" a(ab)(aa)b a(ab)(a’a) AdyadicOissaidtobemultipliedbythescalarawhen eachofitsdyadsismultipliedbythatscalar.Theproduct iswritten aO orOa. ThedyadicaOappliedtoavector1‘eitherasaprefactoror asapostfactoryieldsavectorequaltoatimesthevector obtainedbyapplyingOtor thatis -r). Theorem:Thecombinationofvectorsinadyadisdistrib utive.Thetis be and a(b+c):abcﬁrac.(11) Thisfollowsimmediatelyfromthedeﬁnitionofequalityof dyadics For and Henceitfollowsthatadyadwhichconsistsoftwofactors, eachofwhichisthesumofanumberofvectors,maybe multipliedoutaccordingtothelawofordinaryalgebra -exceptthattheorderofthefactorsinthedyadsmustbe maintained. form,theScalarcoefﬁcientsofthecorrespondingdadsbe equal. Ifthecoefficientsbeequal,thenobviously O-r:Tor foranyvalueofrandthedyadicsby(10)mustbe3qua1, Conversely,ifthedyadicsOandTareequal,thenby(0) s-Oor:soTor forallvaluesofsandr.Letsandreachtakeonthevlues i,j,k .Then’14) io¢0i=ieWoLiewojzzieWej, jo¢oi2je¢ei, jewekzjomk k.QuizkeWei,k0$o].sWei,k0wekzkeWk. Butthesequantitiesarepreciselytheninecoefﬁcientsinthe expansionofthedyadicsOandT .Hencethecorresponding coefﬁcientsareequalandthetheoremisproved .1Th1; analyticstatementoftheequalityoftwodyadicscansome timesbeusedtogreateradvantagethanthemorefundamental deﬁnition(10)basedupontheconceptionofthedyadicas deﬁningalinearvectorfunction. 5TheoremAdyadicOmaybeexpressedasthesumofnine dyadsofwhichtheantecedentsareanythreegivennon coplanarvectors,a,b,candtheconsequentsanythreegiven non-coplanarvectors1,m,11. Everyantecedentmaybeexpressedintermsofa,b,c; andeveryconsequent,intermsofl,m,n.Thedyadicmay thenbereducedtotheform O:allal+a12em+a13an a21bl a22bm G23bn (15) a3131+aazcm+aagon . C 1Asacorollaryofthetheoremitisevidentthattheninedyads(12)areindependent.Noneofthemmaybeexpressedlinearlyintermsoftheothers. LINEARVECTORFUNCTIONS 271 ThisexpressionofOismoregeneralthanthatgivenin Itreducestothatexpressionwheneachsetofvectors a,b,candl,m,ncoincideswithi,j,k . Theorem:AnydyadicOmaybereducedtothesumof threedyadsofwhicheithertheantecedentsortheconsequents, butnotboth,maybearbitrarilychosenprovidedtheybenon coplanar. LetitberequiredtoexpressOasthesumofthreedyads ofwhicha,b,caretheconsequents.Let1,m,nbeanyother threenon-coplanarvectors.Omaythenbeexpressedasin Hence 0 “ssm‘l‘ or O:aA+bB+00. (16) Inlikemannerif-itberequiredtoexpressOasthesumof threedyadsofwhichthethreenon-COplanarvectors1,m,nare theconsequents where L alla a21b a81c, M=a123+a22b+a320, Theexpressions forOareunique.Twoequal dyadicswhichhavethesamethreenon-OOplanarante cedents,a,b,c,havethesameconsequentsA,B,C these howeverneednotbenon-coplanar.Andtwoequaldyadics whichhavethesamethreenon-COplanarconsequents1,m,n, havethesamethreeantecedents. Deﬁnition:Thesymbolicproductformedbythejuxta positionoitwovectorsa,bwithouttheinterventionofadot oracrossiscalledtheindeterminateproductofthetwovectors aandb. 272 VECTORANALYSIS Thereasonforthetermindeterminateisthis.Thetwo productsaohandaxbhavedeﬁnitemeanings.Oneisa certainscalar,theotheracertainvector.Ontheotherhand theproductabisneithervectornorscalar—itispurely Symbolicandacquiresadeterminatephysicalmeaningonly whenusedasanoperator.Theproductabdoesnotobey thecommutativelaw.Itdoeshoweverobeythedistributive law(11)andtheassociativelawasfarasscalarmultiplication isconcerned(Art. TheoremTheindeterminateproductaboftwovectorsis themostgeneralproductinwhichscalarmultiplicationis associative. Themostgeneralproductconceivableoughttohavethe propertythatwhentheproductisknownthetwofactorsare alsoknown .Certainlynoproductcouldbemoregeneral. Inasmuchasscalarmultiplicationistobeassociative,thatis a(ab)(aa)b a(ab)(a’a) itwillbeimpossibletocompletelydeterminethevectorsa andbwhentheirproductabisgiven.Anyscalarfactor maybetransferredfromonevectortotheother.Apartfrom thispossibletransferenceofascalarfactor,thevectorscom posingtheproductareknownwhentheproductisknown .In otherwords Theorem:Ifthetwoindeterminateproductsabanda’b’ areequal,thevectorsaanda’ ,bandb’mustbecollinearand theproductofthelengthsofaandb(takingintoaccountthe positiveornegativeSignaccordingasaandbhaverespec tivelyequaloroppositedirectionstoa’andb’)isequaltothe productofthelengthsofa’andb’ . Let 274 VECTORANALYSIS Theindeterminateproductabimposesﬁveconditionsupon thevectorsaandb.Thedirectionsofaandbareﬁxedand likewisetheproductoftheirlengths .Thescalarproduct aob ,beingascalarquantity,imposesonlyoneconditionupon aandb .ThevectorproductaXb,beingavectorquantity, imposesthreeconditions.Thenormaltotheplaneofaand bisﬁxedandalsotheareaoftheparallelogramofwhichthey aretheSide.Thenineindeterminateproducts(12)ofi,j,k intothemselvesareindependent.Theninescalarproducts arenotindependent.Onlytwoofthemaredifferent. i and ioj:joi:jok:k Theninevectorproductsarenotindependenteither;for andi=—i ,i:—k ,kXi:—iXk. ThetwoproductsaobandaXbobtainedrespectivelyfrom theindeterminateproductbyinsertingadotandacrossbe tweenthefactorsarefunctionsoftheindeterminateproduct. Thatistosay,whenabisgiven,aobandaXbaredetermined. Fortheseproductsdependsolelyuponthedirectionsofaandb andupontheproductofthelengthofaandb,allofwhich areknownwhenabisknown .Thatis if abza’b’ , a-b:a’(17) ItdoesnotholdconverselythatifaobandaXbareknown abisﬁxed;fortakentogetherabandaXbimposeuponthe vectorsonlyfourconditions,whereasabimposesﬁve.Hence abappearsnotonlyasthemostgeneralproductbutasthe mostfundamentalproduct.Theothersaremerelyfunctions ofit.Theirfunctionalnatureisbroughtoutclearlybythe notationofthedotandthecross. LINEARVECTORFUNCTIONS 275 Deﬁnition:AscalarknownasthescalarofOmaybeob tainedbyinsertingadotbetweentheantecedentandconse quentofeachdyadinadyadic.Thisscalarwillbedenoted byasulscriptSattachedtoO.1 If (18) Inlikemannera.vectorknownasthevectorOfOmaybe obtainedbyinsertingacrossbetweentheantecedentandcon sequentofeachdyadinO .Thisvectorwillbedenotedby attachingasubscriptcrosstoO . Ox—a1Xb1+a2s +a8Xb3+ (19) IfObeexpandedinnonionformintermsofi,j,k, “22“as, (20) ¢x=(a23 Or -Ooj+koO-k, Ox:(j-Ook—k-Ooj)i+(koOoi—icO-k)j +(i-O-j—j-Ooi)k . Inequations(20)and(21)thescalarandvectorofOare expressedintermsofthecoefﬁcientsofOwhenexpanded inthenonionform .HenceifOandTaretwoequal dyadics,thescalarofOisequaltothescalarofTandthe vectorofOisequaltothevectorofT . If O T,OS:T5andOxTX. (22) FromthisitappearsthatO,andO)(arefunctionsofO uniquelydeterminedwhenOisgiven .Theymaysometimes beobtainedmoreconvenientlyfrom(20)and(21)thanfrom (18)and andsometimesnot. 1Asubscriptdotmightbeusedforthescalarof(bifitweresufﬁcientlydistinct andfreefromliabilitytomisinterpretation . 276 VECTORANALYSIS ProductsOfDyadics Ingivingthedeﬁnitionsandprovingthetheorems concerningproductsofdyadics,thedyadismadetheunder lyingprinciple.Whatistrueforthedyadistrueforthe dyadicingeneralowingtothefactthatdyadsanddyadics obeythedistributivelawofmultiplication . Deﬁnition:Thedirectproductofthedyadabintothe dyadcdiswritten (ab)(cd) andisbydeﬁnitionequaltothedyad(boc)all (ah)-c)d:bocsail (23) Thatis,theantecedentoftheﬁrstandtheconsequentofthe seconddyadaretakenfortheantecedentandconsequent respectivelyoftheproductandthewholeismultipliedby thescalarproductoftheconsequentoftheﬁrstandthe antecedentofthesecond. Thusthetwovectorswhichstandtogetherintheproduct (ab).(cd) aremultipliedastheystand .Theothertwoarelefttoform anewdyad .Thedirectproductoftwodyadicsmaybe deﬁnedastheformalexpansion(accordingtothedistributive law)oftheproductintoasumofproductsofdyads.Thus f1 and (oldl -c3d3+-n -a2b2-cld1+a2b2.02d2+a2b2.03d3+ +a3b3 iTheparenthesesmaybeomittedineachofthesethreeexpressions. 278 VECTORANALYSIS O.(T+ O.T+O.T' and (25) Henceingeneraltheproduct (49+ 971+ maybeeXpandedformallyaccordingtothedistributivelaw . TheoremTheproductofthreedyadicsO,T,.Qisassocia tive.Thetis (26) andconsequentlyeitherproductmaybewrittenwithout parentheses,as (Dgr.9 . Theproofconsistsinthedemonstrationofthetheoremfor threedyadsab,ed,eftakenrespectivelyfromthethree dyadicsO,T,.9. (abocd)-ef:(boc)adoef:(boo)(doe)af, abo(cdoef):(doe)aboct:(doe)(b-c)at. TheproofmayalsobegivenbyconsideringO,T,and .Q aSOperators {(OoT)o.Q§or:(Oo Let .Qor:r’ ot:(OoT)-r’:O Let {(OT)o.Q§or: Again (To9)or:T -r:O-[T Hence{(O. forallvaluesofr.Consequently (O . LINEARVECTORFUNCTIONS 279 Thetheoremmaybeextendedbymathematicalinduction tothecaseofanynumberofdyadics.Thedirectproduct ofanynumberofdyadicsisassociative.Parenthesesmay beinsertedoromittedatpleasurewithoutalteringtheresult. ItwasShownabove(24)that Hencetheproductoftwodyadicsandavectorisassociative. Thetheoremistrueincasethevectorprecedesthedyadics andalsowhenthenumberofdyadicsisgreaterthantwo . Butthetheoremisuntruewhenthevectoroccursbetween thedyadics.Theproductofadyadic,avector,andanother dyadicisnotassociative. (27) LetabbeadyadofO,andedadyadofT . abo(rocd) Hence (abot)oca:6abo(r-cd). Theresultsofthisarticlemaybesummedupasfollows: Theorem:Thedirectproductofanynumberofdyadics orofanynumberofdyadicswithavectorfactorateither endoratbothendsobeysthedistributiveandassociative lawsofmultiplication parenthesesmaybeinsertedor omittedatpleasure.Butthedirectproductofanynumber ofdyadicswithavectorfactoratsomeotherpositionthanat eitherendisnotassociative—parenthesesarenecessaryto givetheexpressionadeﬁnitemeaning . Lateritwillbeseenthatbymakinguseoftheconjugate dyadicsavectorfactorwhichoccursbetweenotherdyadics maybeplacedattheendandhencetheproductmaybe madetoassumeaforminwhichitisassociative. 280 VECTORANALYSIS Deﬁnition:Theskewproductsofadyadabinto a.vectorrandofavectorrintoadyadabaredeﬁned respectivelybytheequations abXrabXr), (28)rx(ab)(rXa)b . Theskewproductofadyadandavectorateitherendisa dyad.Theobviousextensiontodyadicsis -)xr ._a1blxr Thwrem:Thedirectproductofanynumberofdyadics multipliedateitherendoratbothendsbyavectorwhether themultiplicationbeperformedwithacrossoradotis associative.Butincasethevectoroccursatanyother positionthantheendtheproductisnotassociative.Thatis, (rxO)-T:rX(O-T):rXO-T, -s):rXOos, (29) but To(rXO)¢(Tor)xT . Furthermoretheexpressions sorxOandOXr.s canhavenoothermeaningthan Oxr(30) 282 VECTORANALYSIS DegreesofNullityOfDyadics ItwasShown(Art.101)thatadyadiccouldalways bereducedtoasumofthreetermsatmost,andthisreduction canbeaccomplishedinonlyonewaywhentheantecedents ortheconsequentsareSpeciﬁed .Inparticularcasesitmay bepossibletoreducethedyadicfurthertoasumoftwo termsortoasingletermortozero.Thuslet O=al+bm+cn If1,m,narecoplanaroneofthethreemaybeexpressed intermsoftheothertwoas l:xm+yn . Then (ay+c)n. Thedyadichasbeenreducedtotwoterms.If1,m,nwere allcollinearthedyadicwouldreducetoaSingletermandif theyallvanishedthedyadicwouldvanish . Theorem:IfadyadicObeexpressedasthesumofthree terms O:al+bm+on Ofwhichtheantecedentsa,b,careknowntobenon-coplanar, thenthedyadicmaybereducedtothesumoftwodyads whenandonlywhentheconsequentsareCoplanar. Theproofoftheﬁrstpartofthetheoremhasjustbeen given .Toprovethesecondpartsupposethatthedyadic couldbereducedtoasumoftwoterms O:dp+eq andthattheconsequentsl,m,nofOwerenon-coplanar. Thissuppositionleadstoacontradiction .Forletm’ ,n' bethesystemreciprocalto1,m,11.Thatis, a uxl lI_ o [lmn]’m n I [lmn]’[lmn] LINEARVECTORFUNCTIONS 283 Thevectors m’ ,n’existandarenon-Coplanarbecause 1,m,nhavebeenassumedtobenon-coplanar.Anyvectorr maybeexpressedintermsofthemas But lol’:m and -n:n Hence O BygivingtorasuitablevaluethevectorOormaybemade equaltoanyvectorinspace. But ThisshowsthatOormustbecoplanarwithdande.Hence Oorcantakeononlythosevectorvalueswhichlieinthe planeofdand6.Thustheassumptionthatl,m,narenon coplanarleadstoacontradiction .Hence1,m,11mustbe coplanarandthetheoremisproved . Theorem:IfadyadicObeexpressedasthesumofthree terms ofwhichtheantecedentsa,b,careknowntobenon-coplanar, thedyadicOcanbereducedtoasingledyadwhenandonly whentheconsequents1,m,narecollinear. Theproofoftheﬁrstpartwasgivenabove.Toprove thesecondpartsupposeOcouldbeexpressedas T zdp. Let T=alxp+bmxp+cnxn 284 VECTORANALYSIS FromthesecondequationitisevidentthatTused a postfactorforanyvector wherea’ ,b’ ,c’isthereciprocalsystemtoa,b,0gives ro Fromtheﬁrstexpression Hence x1Xp+m p+a p mustbezeroforeveryvalueofr,thatis,foreveryvalueofx, y,2.Hence lxp:m mXp:mnXp=d Hence1,m,andnareallparalleltopandthetheoremhas beendemonstrated . Ifthethreeconsequents1,m,11hadbeenknowntobenon Coplanarinsteadofthethreeantecedents,thestatementof thetheoremswouldhavetobealteredbyinterchangingthe wordsantecedentandconsequentthroughout.Thereisafur thertheoremdealingwiththecaseinwhichbothantecedents andconsequentsofOarecoplanar.ThenOisreducibleto thesumoftwodyads. Deﬁnition:Adyadicwhichcannotbereducedto thesumoffewerthanthreedyadsissaidtobecomplete.A dyadicwhichmaybereducedtothesumoftwodyads,but cannotbereducedtoasingledyadissaidtobeplanar.In casetheplaneoftheantecedentsandtheplaneofthecon sequentscoincidewhenthedyadicisexpressedasthesumof twodyads,thedyadicissaidtobeuniplanar.Adyadic whichmaybereducedtoaSingledyadissaidtobelinear. Incasetheantecedentandconsequentofthatdyadarecol 286 VECTORANALYSIS sequentofO;butnoothervalues.ThedyadicOusedasa prefactorreducesanyvectorrtothelineoftheantecedent ofO.Inparticularanyvectorsperpendiculartothecon sequentoiOarereducedtozero .ThedyadicOusedasa postfactorreducesanyvectorrtothelineoftheconsequent ofO.Inparticularanyvectorsperpendiculartotheante cedentofOarethusreducedtozero. IfOisazerodyadicthevectors8andtarebothzerono matterwhatthevalueOfrmaybe. Deﬁnition:AplanardyadicissaidtopossessonedegreeOf nullity.AlineardyadicissaidtopossesstwodegreesOf nullity.Azerodyadicissaidtopossessthreedegreesofnul lityorcompletenullity. Theorem:Thedirectproductoftwocompletedyadics iscomplete;ofacompletedyadicandaplanardyadic, planar;ofacompletedyadicandalineardyadic,linear. Theorem:Theproductoftwoplanardyadicsisplanar exceptwhentheplaneoftheconsequentoftheﬁrstdyadic intheproductisperpendiculartotheplaneoftheantece dentoftheseconddyadic.Inthiscasetheproductreduces toalineardyadic—andonlyinthiscase. Let O:a1b,32b2, .9:ToT . Thevector3 TortakesonallvaluesintheplaneofO1andc2 8 x01yoz. Thevector3’Oo3takesonthevalues s’:O-s:x(blocl)a1+y(blocz)a1 s’:(bl-c2»a1+{x(112-00+ LINEARVECTORFUNCTIONS 287 Let s’x’aly’a2, Where ‘3’“3(b1ci)9(b1c2), and9’37(b201)3’(b2 Theseequationsmayalwaysbesolvedforxandywhen anydesiredvalues22'andy’aregiven—thatis,when3’has anydesiredvalueintheplaneofa1andaz—unlessthe determinant b1°1b102 b261b2‘32 Butby Chap . thisismerelytheproduct (b.xb.)(c.xc.)0. ThevectorblXb2isperpendiculartotheplaneofthecon sequentsofO;and01X02,totheplaneoftheantecedentsof T .Theirscalarproductvanisheswhenandonlywhenthe vectorsareperpendicular thatis,whentheplanesareper pendicular.Consequently8’maytakeonanyvalueinthe planeofa1anda2andOoTisthereforeaplanardyadic unlesstheplanesofb1andb2,01andc2areperpendicular. Ifhoweverb1andb2,01andc2areperpendiculars’cantake ononlyvaluesinacertainlineoftheplaneofa1anda2,and henceOoTislinear.Thetheoremisthereforeproved . Theorem:Theproductoftwolineardyadicsislinear exceptwhentheconsequentoftheﬁrstfactorisperpen diculartotheantecedentofthesecond .Inthiscasethe productiszero—andonlyinthiscase. Theorem:Theproductofaplanardyadicintoalinearis linearexceptwhentheplaneoftheconsequentsofthe planardyadicisperpendiculartotheantecedentofthelinear dyadic .Inthiscasetheproductiszero—andonlyinthis case. Theorem:Theproductofalineardyadicintoaplanar dyadicislinearexceptwhentheconsequentofthelinear 288 VECTORANALYSIS dyadicisperpendiculartotheplaneoftheantecedentsof theplanardyadic.Inthiscasetheproductiszero—and onlyinthiscase. Itisimmediatelyevidentthatinthecasesmentionedthe productsdoreducetozero.Itisnotquitesoapparentthat theycanreducetozeroinonlythosecases.Theproofsare similartotheonegivenaboveinthecaseoftwoplanar dyadics.Theyarelefttothereader.Theproofofthe ﬁrsttheoremstated,page286,isalsolefttothereader. TheIdemfactor;1ReciprocalsandConjugatesofDyadics Deﬁnition:Ifadyadicappliedasaprefactororas apostfactortoanyvectoralwaysyieldsthatvectorthe dyadicissaidtobeanidemfactor.Thatis if Oorrforallvaluesofr, orif roO rforallvaluesofr, thenOisanidemfactor.ThecapitalIisusedasthesym bolforanidemfactor.Theidemfactorisacompletedyadic. FortherecanbenodirectioninwhichIorvanishes. TheoremWhenexpressedinnonionformtheidemfactoris (33) Henceallidemfactorsareequal. Toprovethattheidemfactortakestheform(33)itis merelynecessarytoapplytheidemfactorItothevectors i,j,1:respectively .Let I=auii+alzij+alsik 'i‘a21ji+a22jj+azsjk a31ki+a82kja33kk. 1InthetheoryofdyadicstheidemfactorIplaysamileanalogoustounityin ordinaryalgebra.Thenotationisintendedtosuggestthisanalogy. 290 VECTORANALYSIS IntheﬁrstplaceSinceOistheidemfactor,itisacomplete dyadic.Hencetheantecedentsa,b,carenon-coplanarand possessasetofreciprocalsa’b’ , Let Byhypothesis roO r. Then r forallvaluesofr,thatis,forallvaluesofx,y,2.Hencethe correspondingcoefﬁcientsmustbeequal.Thatis, l:a’ ,m=b’ ,n:o’ . Theorem:IfOandTbeanytwodyadics,andiftheproduct OoTisequaltotheidemfactor;1thentheproductToO, whenthefactorsaretakeninthereversedorder,isalso equaltotheidemfactor. Let OoT:I. Toshow ToO:I. ro(OT):r-I:r, ro(OoT)-O:r-O, Thisrelationholdsforallvaluesofr.AsOiscompleteroO musttakeonalldesiredvalues.Hencebydeﬁnition T-O:I. Iftheproductoftwodyadicsisanidemfactor,thatproduct maybetakenineitherorder. Deﬁnition:Whentwodyadicsaresorelatedthat theirproductisequaltotheidemfactor,theyaresaidtobe 1Thisnecessitatesboththedyadics<1»and\IItobecomplete.Fortheproduct oftwoincompletedyadicsisincompleteandhencecouldnotbeequaltothe idemfactor. LINEARVECTORFUNCTIONS 291 reciprocals.lThenotationusedforreciprocalsinordinary algebraisemployedtodenotereciprocaldyadics.Thatis , I if—1(35) Theorem:Reciprocalsofthesameorequaldyadicsare equal. Let andTbetwogivenequaldyadics,60-1and theirreciprocalsasdeﬁnedabove.Byhypothesis W, (Dod9*1—I, and ToW‘l—I. Toshow —I=ToW‘l . As—1 40-1.0 ¢oW4 , IoQ—l—w‘l—IoW4: Hence Va . Thereciprocalof(Disthedyadicwhoseantecedentsarethe reciprocalsystemtotheconsequentsof(Dandwhoseconse quentsarethereciprocalsystemtotheantecedentsofa Ifacompletedyadic(Pbewrittenintheform a1bm on, itsreciprocalis—1’a’m’b'n'c' . (36) For o(l’a'u’h' n’o')=aa’bb’ Theorem:Ifthedirectproductsofacompletedyadic intotwodyadics97and .9areequalasdyadicsthen1”and .Q 1Anincompletedyadichasno(ﬁnite)reciprocal. 292 VECTORANALYSIS areequal.Iftheproductofa-dyadic intotwovectors 1'and3(whetherthemultiplicationbeperformedwithadot oracross)areequal,thenthevectors1'andsareequal. Thatis, if ¢oT=CD-.Q,thenT: .Q, endif thenr=s, (37) andif thenr=s. Thismaybeseenbymultiplyingeachoftheequations throughbythereciprocalofa), ¢_lo¢oT:W=¢-lo¢ w’I-Q Toreducethelastequationproceedasfollows.Lettbe anyvector, t-IXr=t-Ixs, toI17. Hence Astisanyvector,risequalto8. Equations(37)givewhatisequivalenttothelawof celationforcompletedyadics.Completedyadicsmaybe canceledfromeitherendofanexpressionjustasifthey werescalarquantities.Thecancelationofanincomplete dyadicisnotadmissible.Itcorrespondstothecancelation ofazerofactorinordinaryalgebra. TheoremThereciprocaloftheproductofany numberofdyadicsisequaltotheproductofthereciprocals takenintheoppositeorder. Itwillbesufﬁcienttogivetheproofforthecaseinwhich theproductconsistsoftwodyadics.Toshow 294 VECTORANALYSIS Geometricallythetransformation wet isareﬂectionofspaceinthejk-plane.Thistransformation replaceseachﬁgurebyasymmetricalﬁgure,symmetrically situatedupontheoppositesideofthejk-plane .Thetrans formationissometimescalledperversion .Theidemfactor hasalsoadoublyinﬁnitesystemofsquarerootsoftheform 2?iijjkk. Geometricallythetransformation r’=T-r isareﬂectioninthei-axis.Thistransformationreplaceseach ﬁgurebyitsequalrotatedaboutthei-axisthroughanangle of Theidemfactorthuspossessesnotonlytwosquare roots;butinadditiontwodoublyinﬁnitesystemsofsquare roots;anditwillbeseen(Art.129)thatthesearebyno meansall. Theconjugateofadyadichasbeendeﬁned(Art.99) asthedyadicobtainedbyinterchangingtheantecedentsand consequentsofagivendyadicandthenotationofasubscript 0'hasbeenemployed.Theequation hasbeendemonstrated.Thefollowingtheoremsconcerning conjugatesareuseful. Theorem:Theconjugateofthesumordifferenceoftwo dyadicsisequaltothesumordifferenceoftheconjugates, (QIt weiwe. Theorem:Theconjugateofaproductofdyadicsisequal totheproductoftheconjugatestakenintheoppositeorder. LINEARVECTORFUNCTIONS 295 Itwillbesufﬁcienttodemonstratethetheoremincase theproductcontainstwofactors.Toshow (w‘T70:T‘ O'we: (40) «0no-rnomsoar-eon r-T zwoor, (roO)T:To-(r-T)=TOoOgor. Hence (4’oT)0=Too(00. TheoremTheconjugateofthepowerofadyadicisthe poweroftheconjugateofthedyadic. (41) Thisisacorollaryoftheforegoingtheorem .Theexpression T2.maybeinterpretedineitheroftwoequalways. Theorem:Theconjugateofthereciprocalofadyadicis equaltothereciprocaloftheconjugateofthedyadic. (1761(42) For (T-l)co T Theidemfactorisitsownconjugateasmaybeseenfrom thenonionform . I=ii+ii+kk (¢C)—l$0 o (¢c)—l$0: 0wo. Hence (TOD—1 Theexpression maythereforebeinterpretedineither oftwoequivalentways—asthereciprocaloftheconjugate orastheconjugateofthereciprocal. Deﬁnition:Ifadyadicisequaltoitsconjugate,itissaid tobeseéf-conj' ugate.Ifitisequaltothenegativeofitscon 296 VECTORANALYSIS jugate,itissaidtobeanti-selfconjngate.Forself-conjugate dyadics. Foranti-self-conjugatedyadics roT z—Tor,T=—Te. TheoremAnydyadicmaybedividedinoneand wayintotwopartsofwhichoneISself-conjugateandthe otheranti-self-conjugate. For—Ta). (43) and—¢CO=(Iv—w . Hencethepart%(TTo)isself-conjugate;andthe“TTo),anti-self-conjugate.Thusthedivisionhasbeen accomplishedinoneway.Let %(d)o0)o' andgmoc)o" . T=T'+T" . SupposeitwerepossibletodecomposeTinanotherway intoaself-conjugateandananti-self—conjugatepart.Let then T:(T,+9)(T’I—Q). Where(o' .Q)(o'9. Henceif(T’ .Q)isself-conjugate,.Qisself-conjugate. (T" .9)(T" .Q)0=T’I a.Qo= 90‘ Henceif(T" .Q)isanti-self-conjugate .Qisanti-self conjugate. 298 VECTORANALYSIS é(¢ TxXr, 1ér-(T—Tc)=— §rxTx. (44) Theorem:Anyantiself-conjugatedyadicT”possessesone degreeofnullity.Itisauniplanardyadictheplaneof whoseconsequentsandantecedentsisperpendiculartoTX” , thevectorofT. Thistheoremfollowsasacorollaryfromequations Theorem:AnydyadicTmaybebrokenupintotwoparts ofwhichoneisself-conjugateandtheotherequivalentto minusonehalfthevectorofTusedincrossmultiplication. 1Tor=T’ or— 2-TXX1‘ , orsymbolically ToT’o T)(X. (45) Anyvectorcusedinvectormultiplicationdeﬁnesa linearvectorfunction.For Henceitmustbepossibletorepresenttheoperatorcxasa dyadic.Thisdyadicwillbeuniplanarwithplaneofits antecedentsandconsequentsperpendicularto0,sothatit willreduceallvectorsparalleltoctozero.Thedyadicmay befoundasfollows By(3l) -I} -r =I-(cxI) -r. Hence and rxc=r(46) Thismaybestatedinwords. LINEARVECTORFUNCTIONS 299 Theorem:Thevectorcusedinvectormultiplicationwith avectorrisequaltothedyadicIXcorcXIusedindirect multiplicationwithr.If0precedesrthedyadicsaretobe usedasprefactors;ifcfollowsr,aspostfactors.Thedyadics IXcandcXIareanti-self-conjugate. Incasethevector0isaunitvectortheapplicationofthe operator0Xtoanyvectorrinaplaneperpendiculartocis equivalenttoturningrthroughapositiverightangleabout theaxisc .ThedyadiccXIorIXcwherecisaunitvector thereforeturnsanyvectorrperpendiculartocthrougha rightangleaboutthelinecasanaxis.Ifrwereavector lyingoutofaplaneperpendiculartoctheeﬁectofthedyadic IXcorcXIwouldbetoannihilatethatcomponentofrwhich isparalleltocandturnthatcomponentofrwhichisperpen dicularto6througharightangleabout0asaxis. Ifthedyadicbeappliedtwicethevectorsperpendicularto rarerotatedthroughtworightangles.Theyarereversedin direction .Ifitbeappliedthreetimestheyareturnedthrough threerightangles.ApplyingtheOperatorIXcorcXIfour timesbringsavectorperpendicularto0backtoitsoriginal position.Thepowersofthedyadicaretherefore xI)2=—(I—cc), —Ixc—cXI, —cc,(47) ItthusappearsthatthedyadicIXcorcXIobeysthesame lawasfarasitspowersareconcernedasthescalarimaginary v 1inalgebra. ThedyadicIXcorcXIisaquadrantalversoronlyfor vectorsperpendiculartoc.Forvectorsparalleltocitacts asanannihilator.Toavoidthiseﬁectandobtainatrue 800 VECTORANALYSIS quadrantalversorforallvectorsrinspaceitismerelyneces sarytoaddthedyadcctothedyadicIXcorcXI. If X2=—I, X3=—X, (48) X4=I, X‘=X . ThedyadicXthereforeappearsasafourthrootofthe idemfactor.ThequadrantalversorXisanalogoustothe imaginaryV 1ofascalaralgebra.ThedyadicXiscom pleteandconsistsoftwopartsofwhichIXcisanti-self conjugate;andcc,self—conjugate. Ifi,j,karethreeperpendicularunitvectors IXi=iXI=kj—jk, IXj=jXI=ik—ki, (49) k =kXI=ji—ij, asmaybeseenbymultiplyingtheidemfactor Iii51‘k1: intoi,j,andksuccessively .Theseexpressionsrepresent quadrantalversorsabouttheaxisi,j,1!respectivelycombined withannihilatorsalongthoseaxes.Theyareequivalent, whenusedindirectmultiplication,toiX,jX,kXrespectively, —(ji—ij), Theexpression(IXk)4isanidemfactorfortheplaneofiand butanannihilatorforthedirection Inasimilarman nerthedyadkkisanidemfactorforthedirection butan 302 VECTORANALYSIS ReductionofDyadicstoNormalForm LetTbeanycompletedyadicandletrbeaunit vector.Thenthevectorr' r’=Tor isalinearfunctionofr.Whenrtakesonallvaluescousis tentwithitsbeingaunitvectorthatis,whentheterminus ofrdescribesthesurfaceofaunitsphere,—thevectorr' variescontinuouslyanditsterminusdescribesasurface.This surfaceisclosed.Itisfactanellipsoid .1 TheoremItisalwayspossibletoreduceacompletedyadic toasumofthreetermsofwhichtheantecedentsamong themselvesandtheconsequentsamongthemselvesaremutu allyperpendicular.ThisiscalledthenormalformofT . T=ai’ibj'jok’h . Todemonstratethetheoremconsiderthesurfacedescribed by r':Tor. Asthisisaclosedsurfacetheremustbesomedirectionofr whichmakesr'amaximumoratanyrategivesr’asgreat avalueasitispossibleforr'totakeon .Letthisdirection ofrbecalledi,andletthecorrespondingdirectionofr’ thedirectioninwhichr'takesonavalueatleastasgreatas any—becalleda.Considernextallthevaluesofrwhich lieinaplaneperpendiculartoi.Thecorrespondingvalues ofI’lieinaplaneowingtoafactthatTorisalinearvector 1Thismaybeprovedasfollows: -r-1 Hence torzl:{0(éc-lo¢ Byexpressing‘lrinnonionform,theequation1"A?or’:1isseentobeofthesecond degree.Hencer’describesaquadricsurface.Theonlyclosedquadricsurface istheellipsoid. LINEARVECTORFUNCTIONS 303 function.Ofthesevaluesofr'onemustbeatleastasgreat asanyother.Callthisbandletthecorrespondingdirection ofrbecalledj.Finallychoosekperpendiculartoiandj uponthepositivesideofplaneofiandj.Letcbethe valueofr’whichcorrespondstork.SincethedyadicT changesi,j,1:intoa,b,citmaybeexpressedintheform T aibj ck . Itremainstoshowthatthevectorsa,b,casdetermined abovearemutuallyperpendicular. +bj+ck)-r, +bj+ck)odr, r' odr'=r' oai-dr+r’ obj-dr+r’ ock-dr. Whenrisparalleltoi,r’isamaximumandhencemustbe perpendiculartodr’ .Sincerisaunitvectordrisalways perpendiculartor.Hencewhenrisparalleltoi r’ objedr+r' aokodr=0 . Iffurtherdrisperpendiculartoj,r’ ccvanishes,andif drisperpendiculartok,r’ obvanishes.Hencewhenris paralleltoi,r’isperpendiculartobothband0.Butwhen risparalleltoi,r’isparalleltoa.Hencea.isperpendicular toband0.Considernexttheplaneofjandkandthe planeofband0.Letrbeanyvectorintheplaneofjandk. r’odr’zr’obj-dr+r’ cckodr. Whenrtakesthevaluej,r’isamaximuminthisplaneand henceisperpendiculartodr' .Sincerisaunitvectoritis 304 VECTORANALYSIS perpendiculartodr.Hencewhenrisparalleltoj,dr isperpendiculartoj,and k-dr. Hencer’ociszero.Butwhenrisparalleltoj,r’takesthe valueb.Consequentlybisperpendiculartoc. Ithasthereforebeenshownthataisperpendiculartoband c,andthatbisperpendicularto0.Consequentlythethree antecedentsofTaremutuallyperpendicular.Theymaybe denotedbyi’ ,j’ ,k’ .ThenthedyadicTtakestheform T=ai’i+bj’j+ck'k, (52) wherea,b,carescalarconstantspositiveornegative. Theorem:ThecompletedyadicTmayalwaysbe reducedtoasumofthreedyadswhoseantecedentsand whoseconsequentsformaright-handedrectangularsystem ofunitvectorsandwhosescalarcoefﬁcientsareeitherall positiveorallnegative. o::t(ai’i+bj’j (53) Theproofofthetheoremdependsuponthestatements madeonpage20thatifoneorthreevectorsofaright-handed systembereversedtheresultingsystemisleft-handed,but iftwobereversedthesystemremainsright-handed .Ifthen oneofthecoefﬁcientsin(52)isnegative,thedirectionsofthe othertwoaxesmaybereversed .Thenallthecoefﬁcients arenegative.Iftwoofthecoefﬁcientsin(52)arenegative, thedirectionsofthetwovectorstowhichtheybelongmay bereversedandthenthecoefﬁcientsinTareallpositive. Henceinanycasethereductiontotheforminwhichall thecoefﬁcientsarepositiveorallarenegativehasbeen performed. Asalimitingcasebetweenthatinwhichthecoefﬁcients areallpositiveandthatinwhichtheyareallnegativecomes 306 VECTORANALYSIS ToTo:azi’i’s’j’czk’k’ , T aziisjczkk. Since TTo, T.To:To.T:T2 . I=ii+jjkk=i’i’+j’jk’k’ , T2azI(b2a2)j’j’(02a2)k’k’ , (Tz Tzazlz(b2a2)jj(oz—a2)kk, (Tz—a21)-i=0. Ifiandi’werenotparallel(T2a2I)wouldannihilate twovectorsiandi’andhenceeveryvectorintheirplane. (T2a2I)wouldthereforepossesstwodegreesofnullity andbelinear.Butitisapparentthatifa,b,carediﬁerent thisdyadicisnotlinear.Itisplanar.Henceiandi’must beparallel.Inlikemanneritmaybeshownthatjandj’ , kandk’areparallel.ThedyadicTthereforetakestheform a:oii+bjj+ckk wherea,b,carepositiveornegativescalarconstants. DoubleMultiplication1 Deﬁnition:Thedoubledotproductoftwodyadsis thescalarquantityobtainedbymultiplyingthescalarproduct oftheantecedentsbythescalarproductoftheconsequents. Theproductisdenotedbyinsertingtwodotsbetweenthe dyads° ab:cd=a-cb-d. (56) Thisproductevidentlyobeysthecommutativelaw 1TheresearchesofProfessorGibbsuponDoubleMultiplicationarehereprintedfortheﬁrsttime. LINEARVECTORFUNCTIONS 307 andthedistributivelawbothwithregardtothedyadsand withregardtothevectorsinthedyads.Thedoubledot productoftwodyadicsisobtainedbymultiplyingtheprod uctoutformallyaccordingtothedistributivelawintothe sumofanumberofdoubledotproductsofdyads. If and T:T=(a1blazb2cab3czdz cede _alblmld1a1b1:c2d2alblzc3d3+ agbzzcldl32b2:02d2 agbgzcldla3b3:02d2aabazcad8 Deﬁnition:Thedoublecrossproductoftwodyadsisthe dyadofwhichtheantecedentisthevectorproductofthe antecedentsofthetwodyadsandofwhichtheconsequentis thevectorproductoftheconsequentofthetwodyads.The productisdenotedbyinsertingtwocrossesbetweenthe dyads ab§0d=aXcbXd. (57) Thisproductalsoevidentlyobeysthecommutativelawbl‘d3+ bz'aa+ bs‘d3+ 308 VECTORANALYSIS andthedistributivelawbothwithregardtothedyadsand withregardtothevectorsofwhichthedyadsarecomposed. Thedoublecrossproductoftwodyadicsisthereforedeﬁned astheformalexpansionoftheproductaccordingtothe distributivelawintoasum ofdoublecrossproductsof dyads. If and w: +c3d3+ (57y +a3b8§cldl+a8 blxdl+alxc2bIXdz‘l'alxcg +a2><clbzxdl+agxc2b2Xd2+a2Xc3bzxd3+m baxd1+a3xezbaxd1+aa><c3 Theorem:Thedoubledotanddoublecrossproductsof twodyadicsobeythecommutativeanddistributivelawsof multiplication .Butthedoubleproductsofmorethantwo dyadics(whenevertheyhaveanymeaning)donotobeythe associativelaw . TT T:T rzr=rzr we (rmwieurm. Thetheoremissufﬁcientlyevidentwithoutdemonstration. 310 VECTORANALYSIS thefactorsisreversedeachscalartripleproductchanges sign.Theirproductthereforeisnotaltered. AdyadicTmaybemultipliedbyitselfwithdouble cross.Let T=al+bm+cn axa1X1+aXblxm+aXc n +bXal +bXbm +bxcmxn +c nXl+c nXm+cXca . Theproductsinthemaindiagonalvanish .Theothersare equalinpairs.Hence T§T=2(c a +cxanXl+aXbl).(60) Ifa,b,candl,m,narenon—coplanarthismaybewritten 2wiw: [abc][lmn](a’l’b’m’ TheproductT:5TisaspeciesofpowerofT .Itmaybere gardedasasquareofToThenotationT2willbeemployed torepresentthisproductafterthescalarfactor2hasbeen strickenout. ago 2 ThetripleproductofadyadicTexpressedasthesumof threedyadswithitselftwicerepeatedis TﬁTzT=3T2:T(c a +cxa T2:T=(bxca +cxa nXl+aXbl) Inexpandingthisproducteveryterminwhichaletteris repeatedvanishes.Forascalartripleproductofthreevec LINEARVECTORFUNCTIONS 311 torstwoofwhichareequaliszero.Hencetheproduct reducestothreetermsonly T2:T:[boa][mnl][cab][nlm][abc][lmn] or T2:T:3[abc][lmn] T:T:T:6[abc][lmn]. Thetripleproductofadyadicbyitselftwicerepeatedis equaltosixtimesthescalartripleproductofitsantecedents multipliedbythescalartripleproductofitsconsequents. Theproductisaspeciesofcube.ItwillbedenotedbyT3 afterthescalarfactor6hasbeenstrickenout. o, [abc][lmn]. (62) IfT2becalledthesecondofT;andT3,thethirdof T,thefollowingtheoremsmaybestatedconcerningthe secondsandthirdsofconjugates,reciprocals,andproducts. TheoremThesecondoftheconjugateofadyadicisequal totheconjugateofthesecondofthatdyadic .Thethirdof theconjugateisequaltothethirdofthedyadic. we2(0698‘ Theorem:Thesecondandthirdofthereciprocalofa dyadicareequalrespectivelytothereciprocalsofthesecond andthird.(63) —02(CDT-1Q22?-1 (64)-93-1Qa—l Let T=al+bm+cn T’l(36) IIIbII II $2a m+cn (6O), [abc][lmn] 312 VECTORANALYSIS [abc][lmn](lamb no) (0H)2[:blic’lfE17313] But [abc]1and[1’m’n’][lmn]1. Hence (T2)—1(T“1)2T24 . $3[MN][lmn], 1 1 [abc][lmn], (T"1)3[a’b’c’] Hence (T’l)3Ts‘l . Theorem:Thesecondandthirdofaproductareequal respectivelytotheproductofthesecondsandtheproductof thethirds. “0 ° we:(65) Chooseanythreenon-coplanarvectors1,m,nasconsequents ofTandletm’ ,n’betheantecedentsofT . T (T-T)2=c eXf+cxafXd+aXbdxc, T2=bxc' a +cxa nXl+aXbl, T2=m’Xn’eXf+n’Xl’fXd+l’Xm’dxe. HenceTo 2=c eXf+c fXd+aXbdXe. Hence (ToT)3:[abc][def] 314 VECTORANALYSIS T2=bxcmxn+c nxl+aXbl, T3[ab0][lmn]. Theorem:Thenecessaryandsufﬁcientconditionthata dyadicTbecompleteisthatthethirdofTbedifferentfrom zero . Foritwasshown(Art.106)thatboththeantecedentsand theconsequentsofacompletedyadicarenon-coplanar. HencethetwoscalartripleproductswhichoccurinT3 cannotvanish . Theorem:Thenecessaryandsufﬁcientconditionthata dyadicTbeplanaristhatthethirdofTshallvanishbutthe secondofTshallnotvanish . Itwasshown(Art.106)thatifadyadicTbeplanaritscon sequentsl,m,11mustbeplanarandconverselyiftheoonss quentsbe00planarthedyadicisplanar.Henceforaplanar dyadicT8mustvanish.ButT2cannotvanish .Sincea, b,ehavebeenassumednon-coplanar,thevectorsbXc,cXa, aXbarenon-coplanar.HenceifT2vanisheseachofthe vectorsmXn,nX1,lXmvanishes—thatis,l,m,narecol linear.ButthisisimpossiblesincethedyadicTisplanar andnotlinear. Theorem:Thenecessaryandsufﬁcientconditionthata non-vanishingdyadicbelinearisthatthesecondofT,and consequentlythethirdofT,vanishes. ForifTbelineartheconsequents1,m,n,arecollinear. Hencetheirvectorproductsvanishandtheconsequentsof T2vanish.IfconverselyT2vanishes,eachofitsconsequents mustbezeroandhencetheseconsequentsofTarecollinear. Thevanishingofthethird,unaccompaniedbythevanish ingofthesecondofadyadic,impliesonedegreeofnullity. Thevanishingofthesecondimpliestwodegreesofnullity . LINEARVECTORFUNCTIONS 315 Thevanishingofthedyadicitselfiscompletenullity.The resultsmaybeputintabularform . T8:50,Tiscomplete. T80,T2atO,Tisplanar.(69) T3=O,T2=O,T1:O,Tislinear. Itfollowsimmediatelythatthethirdofanyanti-self—conjugate dyadicvanishes;buttheseconddoesnot.Foranysuch dyadicisplanarbutcannotbelinear. NonionForm .Determinants.1InvariantsofaDyadic IfTbeexpressedinnonionform T=a11ii+a12ij+a13ik (13) 'l’“ziji‘l'“2253“zejk a31ki a82kja33kk . TheconjugateofThasthesamescalarcoefﬁcientsasT,but theyarearrangedsymmetricallywithrespecttothemain diagonal.Thus T0:a11ii a21ijaslik, a12ji+“sail (70) alaki d2ajk a33kk . ThesecondofTmaybecomputed.Take,forinstance,one term.LetitberequiredtoﬁndthecoefﬁcientofijinT2. WhattermsinTcanyieldadoublecrossproductequalto ijThevectorproductoftheantecedentsmustbeiand thevectorproductoftheconsequentsmustbej.Hencethe antecedentsmustbejandk;andtheconsequents,kandi. Thesetermsare a21J1§a33kk“21“3341 oxo00“311” xazslk“31 1Theresultsholdonlyfordeterminantsofthethirdorder.Theextensionto determinantsofhigherordersisthroughMultipleAlgebra. 316 VECTORANALYSIS HencetheterminijinT2is “23“21 Thisistheﬁrstminorofaninthedeterminant 11“12“13 “21“22“23 “31“32“33 Thisminoristakenwiththenegativesign .Thatis,the coefﬁcientofijinT2iswhatistermedthecofactorofthe coefﬁcientofijinthedeterminant.Thecofactorismerely theﬁrstminortakenwiththepositiveornegativesign accordingasthesumofthesubscriptsofthetermwhose ﬁrstminorisunderconsiderationisevenorodd .Thecc efﬁcientofanydyadinT2iseasilyseentobethecofactorof thecorrespondingterminT.Thecofactorsaredenoted generallybylargeletters. a a A1122 23 18thecofactorofan.“32“33 a a A1221 2318thecofactorofan.“31“33 a a A32u 12isthecofactorofam.“21“23 WiththisnotationthesecondofTbecomes +A21li+A2255+A23kk (71) Ag1kiA32kjAB3kk . ThevalueofthethirdofTmaybeobtainedbywritingT asthesumofthreedyads w(“11i+“21j+“31k)i+ “225 “235“213k 318 VECTORANALYSIS IfthedeterminantbedenotedbyD .ik IfTisaseconddyadicgiveninnonionformas suii+buijblsik, I’21li'i'bzzllbzalk’ 332mbagkk, theproductToTofthetwodyadicsmayreadilybefound byactuallyperformingthemultiplication ww bu“12521“13531)ii 512“12622 “13532)ij 513“12I’23“13533)ik “11“22“21“23“12“22622 “23I’13“22623“23I’33”k bu“32621“33 ki 512“32622 “33532)kl 512“32623“33633)kk w:w“11bu“12612“13513 “21I’21“22I’22“23“23 (75) “31“32532“33I933 Sincethethirdordeterminantofaproductisequaltothe productofthedeterminants,thelawofmultiplicationof determinantsfollowsfrom(65)and LINEARVECTORFUNCTIONS 319 “11“12“13I’11I’12I’13“11I)11“12“21'l'“13“31 “21“22“23621“22I’23“21bu“22“23531 “31“32“33531532“33“31I’11“32521“33531 “11512“12622“13532“11513“12523“13533’ “21“12“22622“23632“21613“22I’23“23533’(76) “31“12“32“22“33632“31Z’I3“32623“33 Therulemaybestatedinwords.Tomultiplytwodeter minantsformthedeterminantofwhichtheelementinthe mthrowandnthcolumnisthesumoftheproductsofthe elementsinthemthrowoftheﬁrstdeterminantandnth columnofthesecond. If T2=c a +c nXl+aXbl . Then c aXb][a nXll] Hence IT2(T2)8[ab[lmn]2T3“ . 2 Hence A11A12A13“11“12“13 l¢2| A21A22A23“21“22“23 (77) A31A32A33“31“32“33 Thedeterminantofthecofactorsofagivendeterminantof thethirdorderisequaltothesquareofthegivendeterminant. AdyadicThasthreescalarinvariantsthatis threescalarquantitieswhichareindependentoftheformin whichTisexpressedTheseare (93,“a, thescalarofT,thescalarofthesecondofT,andthethird ordeterminantofT .IfTbeexpressedinnonionformthese quantitiesare 320 VECTORANALYSIS $8“11“22“23 SA11A22A83 (78) “11“12“13 $3:“21“22“23 “31“32“33 Nomatterintermsofwhatright-handedrectangularsystem oftheseunitvectorsTmaybeexpressedthesequantitiesare thesame.ThescalarofTisthesumofthethreecoefﬁcients inthemaindiagonal.ThescalarofthesecondofTisthe sumoftheﬁrstminorsorcofactorsofthetermsinthe maindiagonalThethirdofTisthedeterminantofthe coefﬁcients.Thesethreeinvariantsarebyfarthemost importantthatadyadicTpossesses. Theorem:Anydyadicsatisﬁesacubicequationofwhich thethreeinvariantsTS,T23,T8arethecoefﬁcients. By(68)(o (e (o “11“12“13 (w—“Ih:“12“22—33“23 “13“32“33“90 Hence(T—wI)3=T3T3—x3 asmaybeseenbyactuallyperforming'theexpansion . (T—TI)2-(T—c)0=Ts—xT25+ss—x" Thisequationisanidentityholdingforallvaluesofthe scalarac.Itthereforeholds,ifinplaceofthescalarx,the dyadicTwhichdependsuponninescalarsbesubstituted. Thatis (TT-l)ao(TT-I)3=IT3TT25+T2T5T3 . Butthetermsupontheleftareidenticallyzero .Hence Tz—T3T2+T23T—T81=O. (79) 322 VECTORANALYSIS Twodyadicsareequalwhentheyareequalasoperators uponallvectorsoruponthreenon—coplanarvectors.That is,when TorTorforallvaluesorforthreenon coplanarvaluesofr, (10) or roT roTforallvaluesorforthreenon coplanarvaluesofr, or sTor soTorforallvaluesorforthreenon coplanarvaluesofrand3. Anylinearvectorfunctionmayberepresentedbyadyadic. Dyadsobeythedistributivelawofmultiplicationwith regardtothetwovectorscomposingthedyad Multiplicationbyascalarisassociative.Invirtueofthese twolaws3dyadicmaybeexpandedintoasumofnineterms bymeansofthefundamentaldyads, ii,ij,ik, itiiiks (12) ki,kj,kk, as li+“2255“235k: (13) _aslki+a32kjasgkk . Iftwodyadicsareequalthecorrespondingcoefﬁcientsin theirexpansionsintononionformareequalandconversely. LINEARVECTORFUNCTIONS 323 Anydyadicmaybeexpressedasthesumofthreedyadsof whichtheantecedentsortheconsequentsareanythree givennon-coplanarvectors.Thisexpressionofthedyadicis unique. Thesymbolicproductabknownasadyadisthemost generalproductoftwovectorsinwhichmultiplicationbya scalarisassociative.Itiscalledtheindeterminateproduct. Theproductimposesﬁveconditionsuponthevectorsaand b .Theirdirectionsandtheproductoftheirlengthsare determinedbytheproduct.Thescalarandvectorproducts arefunctionsoftheindeterminateproduct.Ascalarand avectormaybeobtainedfromanydyadicbyinsertingadot andacrossbetweenthevectorsineachdyad.Thisscalar andvectorarefunctionsofthedyadic. (18) (19) T5=ioToi+joToj+koTok (20) -Toi—ioTok)j +(ioT-j—j-Toi)k (21) Thedirectproductoftwodyadsisthedyadwhoseante cedentandconsequentarerespectivelytheantecedentofthe ﬁrstdyadandtheconsequentofthesecondmultipliedby thescalarproductoftheconsequentoftheﬁrstdyadand theantecedentofthesecond. (ab)o(cd)(boc) (23) Thedirectproductoftwodyadicsistheformalexpansion, accordingtothedistributivelaw,oftheproductintothe 324 VECTORANALYSIS sumofproductsofdyads.Directmultiplicationofdyadics orofdyadicsandavectorateitherendoratbothendsobeys thedistributiveandassociativelawsofmultiplication .Con sequentlysuchexpressionsas wowor,ﬂo¢0¢,SOTOTOI‘ ,¢°w°g maybewrittenwithoutparentheses;forparenthesesmay beinsertedatpleasurewithoutalteringthevalueofthe product.Incasethevectoroccursatotherpositionsthan attheendtheproductisnolongerassociative. Theskewproductofadyadandavectormaybedeﬁned bytheequation (ab)Xt=abXr, (28) Theskewproductofadyadicandavectorisequaltothe formalexpansionofthatproductintoasumofproductsof dyadsandthatvector.Thestatementmadeconcerningthe associativelawfordirectproductsholdswhenthevectoris connectedwiththedyadicsinskewmultiplication .The expressions rXT-T,ToTXr,rXTos, r-TXs,rxTXs(29) maybewrittenwithoutparenthesesandparenthesesmaybe insertedatpleasurewithoutalteringthevalueoftheproduct. Moreover -T, -T . Buttheparenthesescannotbeomitted . Thenecessaryandsuﬁicientconditionthatadyadicmay bereducedtothesumoftwodyadsortoasingledyador tozeroisthat,whenexpressedasthesum ofthree dyadsofwhichtheantecedents(orconsequents)areknown 326 VECTORANALYSIS Iftheproductoftwocompletedyadicsisequaltotheidem factorthedyadicsarecommutativeandeitheriscalled thereciprocaloftheother.Acompletedyadicmaybe canceledfromeitherendofaproductofdyadicsandvectors asinordinaryalgebra;forthecancelationisequivalentto multiplicationbythereciprocalofthatdyadic.Incomplete dyadicspossessnoreciprocals.Theycorrespondtozeroin ordinaryalgebra.Thereciprocalofaproductisequaltothe productofthereciprocalstakenininverseorder. (a.313-1r—I(0-1 . (33) Theconjugateofadyadicisthedyadicobtainedbyinter changingtheorderoftheantecedentsandconsequents.The conjugateofaproductisequaltotheproductofthecon jugatestakenintheOppositeorder. (40) Theconjugateofthereciprocalisequaltothereciprocalof theconjugate.Adyadicmaybedividedinoneandonly onewayintothesumoftwopartsofwhichoneisself conjugateandtheotheranti-self—conjugate. —a>. (43) Anyanti-self-conjugatedyadicortheanti-self-conjugate partofanydyadic,usedindirectmultiplication,isequivalent tominusone-halfthevectorofthatdyadicusedinskew multiplication . ;(T—Tc)orz—nXr, ér (44) Adyadicoftheform3XIorIXcisanti-self-conjugateand usedindirectmultiplicationisequivalenttothevector0 usedinskewmultiplication. LINEARVECTORFUNCTIONS 327 Also c =(IXc)ar, (46) ThedyadiccXIorIXc,wherecisaunitvectorisaquad rantalversorforvecto1sperpendiculartocandanannihilator forvectorsparalleltoc.ThedyadicIXc ccisatrue quadrantalversorforallvectors.Thepowersofthesedyadics behavelikethepowersoftheimaginaryunitVi— l,asmay beseenfromthegeometricinterpretation .Appliedtothe unitvectorsi,j,k IXi=iXI:kj—jk,etc. (49) Thevector3Xbinskewmultiplicationisequivalentto (aXb)XIindirectmultiplication . —ab(50) (axb)xr=(ba—ab)-r rX(aXb)=r-(ba—ab). (51) Acompletedyadicmaybereducedtoasumofthree dyadsofwhichtheantecedentsamongthemselvesandthe consequentsamongthemselveseachformaright-handed rectangularsystemofthreeunitvectorsandofwhichthe scalarcoefﬁcientsareallpositiveorallnegative. (53) Thisiscalledthenormalformofthedyadic.Anincom pletedyadicmaybereducedtothisformbutoneormoreof thecoefﬁcientsarezero.Thereductionisuniqueincase theconstantsa,b,carediﬁerent.Incasetheyarenot differentthereductionmaybeaccomplishedinmorethan oneway .Anyself-conjugatedyadicmaybereducedto thenormalform (55) inwhichtheconstantsa,b,carenotnecessarilypositive. 328 VECTORANALYSIS Thedoubledotanddoublecrossmultiplicationofdyads isdeﬁnedbytheequations ab:cd=aocb od, (56) abicd=axcbXd . (57) Thedoubledotanddoublecrossmultiplicationofdyadics isobtainedbyexpandingtheproductformally,accordingto thedistributivelaw,intoasumofproductsofdyads.The doubledotanddoublecrossmultiplicationofdyadicsiscom mutativebutnotassociative. One-halfthedoublecrossproductofadyadicTbyitself iscalledthesecondofT .If T2:%TXT=c a +c nXl+aXbl .(61) One-thirdofthedoubledotproductofthesecondofTandT iscalledthethirdofTandisequaltotheproductofthe scalartripleproductoftheantecedentsofTandthescalar tripleproductoftheconsequentofT . T3=éT:T:[abc][lmn]. Thesecondoftheconjugateistheconjugateofthesecond . Thethirdoftheconjugateisequaltothethirdofthe originaldyadic.Thesecondandthirdofthereciprocalare thereciprocalsofthesecondandthirdofthesecondand thirdofadyadic.Thesecondandthirdofaproductarethe productsofthesecondsandthirds. (¢O)2 (¢O)Bwe, (63) (W52 -1 : (64) (ToT32$2oT2 (07 330 VECTORANALYSIS 6 .ProvethestatementsmadeinArt.106andthecon verseofthestatements. 7.ShowthatifQiscompleteandifTo.9T .Qthen TandTareequal.Givetheproofbymeansoftheory developedpriortoArt.109. 8 .Definition:TwodyadicssuchthatToT:ToTthat istosay,twodyadicsthatarecommutative—aresaidtobe homologous.Showthatifanynumberofdyadicsarehomoge neoustooneanother,anyotherdyadicswhichmaybeobtained fromthembyaddition,subtraction,anddirectmultiplication arehomologoustoeachotherandtothegivendyadics.Show alsothatthereciprocalsofhomologousdyadicsarehomolo gous.JustifythestatementthatifToT orT'IoT, whichareequal,becalledthequotientofTbyT,thenthe rulesgoverningaddition,subtraction,multiplicationand divisionofhomologousdyadicsareidenticalwiththerules governingtheseOperationsinordinaryalgebraitbeing understoodthatincompletedyadicsareanalogoustozero, andtheidemfactor,tounity .Hencethealgebraandhigher analysisofhomologousdyadicsispracticallyidenticalwith thatofscalarquantities. 9.Showthat(Ixc)oT=cxTand(cXI)oT=cXT . 10.Showthatwhetherornota,b,cbecoplanar and 11.Ifa,b,carecoplanarusetheaboverelationtoprove thelawofsinesforthetriangleandtoobtaintherelation withscalarcoefﬁcientswhichexistsbetweenthreecoplanar vectors.Thismaybedonebymultiplyingtheequationbya unitnormaltotheplaneofa,b,andc. 12 .Whatistheconditionwhichmustsubsistbetweenthe coefﬁcientsintheexpansionofadyadicintononionformif LINEARVECTORFUNCTIONS 331 thedyadicbeself-conjugate?What,ifthedyadicbeanti self-conjugate 13 .ProvethestatementsmadeinArt.116concerningthe numberofwaysinwhichadyadicmaybereducedtoits normalform . 14.Thenecessaryandsufﬁcientconditionthatananti self-conjugatedyadicTbezeroisthatthevectorofthe dyadicshallbezero. 15.ShowthatifTbeanydyadictheproductToTc.is self-conjugate. 16.ShowhowtomakeuseoftherelationTx0to demonstratethattheantecedentsandconsequentsofaself conjugatedyadicarethesame(Art. 17.Showthat T2T2T,2T and 18.ShowthatifthedoubledotproductTTofadyadic byitselfvanishes,thedyadicvanishes.Henceobtainthe conditionforalineardyadicintheformT2T2O. 19.Showthat -TZ-f. 20 .Showthat(T+ T3+T2:T+T:T2+T3. 21.Showthatthescalarofaproductofdyadicsisun changedbycyclicpermutationofthedyadics.Thatis (T913924522mgr—(roe CHAPTERVI ROTATIONSANDSTRAINS INtheforegoingchaptertheanalyticaltheoryof dyadicshasbeendealtwithandbroughttoastateof completenesswhichisnearlyﬁnalforpracticalpurposes. Thereare,however,anumberofnewquestionswhichpresent themselvesandsomeoldquestionswhichpresentthemselves underanewformwhenthedyadicisappliedtophysics orgeometry.Moreoveritwasforthesakeoftheapplica tionsofdyadicsthatthetheoryofthemwasdeveloped .Itis thentheobjectofthepresentchaptertosupplyanextended applicationofdyadicstothetheoryofrotationsandstrains andtodevelop,asfarasmayappearnecessary,thefurther analyticaltheoryofdyadics. ThatthedyadicTmaybeusedtodenoteatransformation ofspacehasalreadybeenmentioned .Aknowledgeofthe precisenatureofthistransformation,however,wasnotneeded atthetime.Considerrasdrawnfromaﬁxedorigin,andr’ asdrawnfromthesameorigin .Letnow r’=T-r. Thisequationthereforemayberegardedasdeﬁningatrans formationofthepointsPofspacesituatedattheterminusof rintothepointP’ ,situatedattheterminusofr’ .Theorigin remainsﬁxed .Pointsintheﬁniteregionsofspaceremainin theﬁniteregionsofspace.Anypointuponaline r:b+xa becomesapoint r’=Tob+xT-a . 334 VECTORANALYSIS Itisimportanttonoticethatthevector8denotingaplane areaisnottransformedintothesamevector3’asitwould beifitdenotedaline.Thisisevidentfromthefactthatin thelattercaseTactson2whereasintheformercaseT2acts upon3. ToshowthatvolumesaremagniﬁedintheratioofT3to unitychooseanythreevectors6,e,fwhichdeterminethe volumeofaparallelopiped[de ExpressTwiththevec torswhichformthereciprocalsystemtod,e,fasconsequents. T=aw+bd+cﬂ ThedyadicTchangesd,e,fintoa,b,c(whicharedifferent fromthea,b,0aboveunless(i,e,fareequaltol’ ,m’ , Hencethevolume[def]ischangedintothevolume[ab T32[abc] [def]. Hence [ab3][def]T3. Theratioofthevolume[abc]to[def]isasT3istounity. Butthevectors(1,e,fwereanythreevectorswhichdeter mineaparallelopiped .Henceallvolumesarechangedby theactionofTinthesameratioandthisratioisasT3isto1. RotationsaboutaFixedPoint.Versors Theorem:Thenecessaryandsufﬁcientconditionthat adyadicrepresentarotationaboutsomeaxisisthatitbe reducibletotheform (1) wherei’ ,j’ ,k’andi,j,karetworight-handedrectangular systemsofunitvectors.Let r=xi+yj+zk Tor=xi’+ ROTATIONSANDSTRAINS 335 HenceifTisreducibletothegivenformthevectorsi,j,k arechangedintothevectors j’ ,k’andanyvectorris changedfromitspositionrelativetoi,j,kintothesameposi tionrelativetoi’ ,j’ ,k’ .Hencebythetransformationno changeofshapeiseffected .Thestrainreducestoarotation whichcarriesi,j,kintoi’j’k’ .Converselysupposethe bodysuﬁersnochangeofshape—thatis,supposeitsubjected toarotation .Thevectorsi,j,1:mustbecarriedintoanother right-handedrectangularsystemofunitvectors.Letthese bei’ ,j’ ,k’ .ThedyadicTmaythereforebereducedtothe form +k’k. Deﬁnition:Adyadicwhichisreducibletotheform i'ij'jk’k andwhichconsequentlyrepresentsarotationiscalleda versor. Theorem:Theconjugateandreciprocalofaversorare equal,andconverselyiftheconjugateandreciprocalofa dyadicareequalthedyadicreducestoaversororaversor multipliedbythenegativesign . Let w. T-l—TC. Hencetheﬁrstpartofthetheoremisproved.Toprovethe secondpartlet T aibj ck, T If T4:Tc,ToTc: Hence aa+bb+cc=L 336 VECTORANALYSIS Hence(Art.108)theantecedentsa,b,candtheconsequents a,b,0mustbereciprocalsystems.Hence(page87)they mustbeeitheraright-handedoraleft-handedrectangular systemofunitvectors.Theleft-handedsystemmaybe changedtoaright-handedonebypreﬁxingthenegative signtoeachvector.Then ' +k’k J .J . or T= Thethirdordeterminantofaversorisevidentlyequalto unity;thatoftheversorwithanegativesign,tominusone. Hencethecriterionforaversormaybestatedintheform T0T0=L T3=lTl=L (2) OrinasmuchasthedeterminantofTisplusorminusone ifToT0:I,itisonlynecessarytostatethatif ToT0=I, Tisaversor. Therearetwogeometricinterpretationsofthetransforma tionduetoadyadicTsuchthat ToTczlT3=lTl=—1 (3) T= ThetransformationduetoTisoneofrotationcombinedwith reﬂectionintheorigin.Thedyadici’i+j’j+k’kcausesa rotationaboutadeﬁniteaxisitisaversor.Thenegative signthenreversesthedirectionofeveryvectorinspaceand replaceseachﬁgurebyaﬁguresymmetricaltoitwithrespect totheorigin .Byreversingthedirectionsofi’andj’the systemi’ ,j' ,k’stillremainsright-handedandrectangular, butthedyadictakestheform —k’k, or 338 VECTORANALYSIS —jk).(4) .ij+kk=I‘—iio kj—jkzIXi. Hence T=ii+cosg(I—ii)+singIXi. (5) Ifmoregenerallyinplaceofthei-axisanyaxisdenoted bytheunitvectorabetakenastheaxisofrotationandifas beforetheangleofrotationaboutthataxisbedenotedbyq, thedyadicTwhichaccomplishestherotationis T:aa+cosg(I (6) Toshowthatthisdyadicactuallydoesaccomplishthe rotationapplyittoavectorr.Thedyadaaisanidemfactor forallvectorsparalleltoa;butanannihilatorforvectors perpendiculartoa.ThedyadicI—aaisanidemfactor forallvectorsintheplaneperpendiculartoa;butan annihilatorforallvectorsparalleltoa .ThedyadicIXa isaquadrantalversor(Art.113)forvectorsperpendicular toa;butanannihilatorforvectorsparalleltoa.Ifthen rbeparalleltoa wer=aa°r=ro HenceTleavesunchangedallvectors(orcomponentsof vectors)whichareparalleltoa.Ifrisperpendiculartoa Tor=cosgr+singaXL Hencethevectorrhasbeenrotatedinitsplanethroughthe angleg.Ifrwereanyvectorinspaceitscomponentparallel toasuﬁersnochange;butitscomponentperpendiculartoa isrotatedaboutathroughanangleofgdegrees.Thewhole vectoristhereforerotatedaboutathroughthatangle. Letabegivenintermsofi,j,kas 20 0.0 o aa—a111+a1a211+a1a31k ROTATIONSANDSTRAINS 339 jk +a3a1ki+agazkja32kk, Ixa=0ii—a31j+azik, +a35i+015—a.jk. T{alz(1 gala,3(1-cosq)-dasing}ij +{ala3(l—cosq)-a2singjik {a2a1(1 008Mi +{a2a8(1—cosg)—a1sing}jk +{a3a1(l—cosg)—a2sing}ki {a3a2(1 {agz(1+cosg)+cosg}kk .(7) IfTbewrittenasinequation(4)thevectorofT andthescalarofTmaybefound. T5=1+2cosg. TheaxisofrotationiisseentohavethedirectionofTX, thenegativeofthevectorofT .Thisistrueingeneral. Thedirectionoftheaxisofrotationofanyversoristhe negativeofthevectorofT .Theproofofthisstatement dependsontheinvariantpropertyofTX.AnyversorT maybereducedtotheform(4)bytakingthedirectionofi 340 VECTORANALYSIS coincidentwiththedirectionoftheaxisofrotation .After thisreductionhasbeenmadethedirectionoftheaxisisseen tobethenegativeofTX.ButTxisnotalteredbythe reductionofTtoanyparticularform—noristheaxisof rotationalteredbysuchareduction .Hencethedirectionof theaxisofrotationisalwayscoincidentwithTX,thedirco tionofthenegativeofthevectorofT . Thetangentofone-halftheangleofversiongis sing x/TxoTx(8)1+cosg 1+T5tang Thetangentofone-halftheangleofversionistherefore determinedwhenthevaluesofT,(andT5areknown .The vectorTxandthescalarT3,whichareinvariantsofT,deter minecompletelytheversorT .LetQ.beavectordrawn inthedirectionoftheaxisofrotation .Letthemagnitude ofQ.beequaltothetangentofone-halftheangle9of version . 42x 1+T; Thevector0.determinestheversorTcompletely .Q.willbe calledthevectorsemi-tangentofversion. By(6)aversorTwasexpressedintermsofaunitvector paralleltotheaxisofrotation .Q T=aa+cosq(I—aa)+singIXa . HenceifQ.bethevectorsemi-tangentofversion Q.Q +cosg +singl><Q Q-0Vacs ThereisamorecompactexpressionforaversorTinterms ofthevectorsemi-tangentofversion.Letcbeanyvectorin space.TheversionrepresentedbyQ.carries(10) c—QXcintoc+QXc. 342 VECTORANALYSIS and (I+IXQ)-(I Multiplybyc —IxQ)‘1-(c Hencethedyadic —IXQ)‘l carriesthevector0QX0intothevector0QXenomatter whatthevalueOf0.HencethedyadicTdeterminesthe versionduetothevectorsemi-tangentofversionQ . ThedyadicI+IXQcarriesthevectorc—QX0into —Qxc—QX(Qxc) (I+IXQ)-(c Hencethedyadic I+IXQ 1+QoQ carriesthevector0QX0intothevector0,ifcbeperpen diculartoQashasbeensupposed.Consequentlythedyadic=(I-1 (I+IXQ)2 1+Q°Q producesarotationofallvectorsintheplaneperpendicular toQ.If,however,itbeappliedtoavectorxQparalleltoQ theresultisnotequalto Q 1+O-Q I+QoQ 1+g.o, ROTATIONSANDSTRAINS 343 ToobviatethisdifﬁcultythedyadQQ.whichisanannihilator forallvectorsperpendiculartoQ,maybeaddedtothenu merator.TheversorTmaythenbewritten 1+Q-Q(10) (1+IXQ) (IXQ) -QQ—Q-QI . Hencesubstituting: (1—Q 1+Q-QT wIII Thismaybeexpandedinnonionform .Let (l+a2—bz —az+b2 —2a)jk(11) +(2ao—a3—62+02)kk 1+a2+bz+cz Ifaisaunitvectoradyadicoftheform T=2aa—I (12) isabiquadrantalversor.Thatis,thedyadicTturnsthe pointsofspaceabouttheaxisathroughtworightangles. Thismaybeseenbysetting9equalto71inthegeneral expressionforaversorT_ T:aa+cosg(I—aa)+singIXa, oritmaybeseendirectlyfromgeometricalconsiderations . ThedyadicTleavesavectorparalleltoaunchangedbutre verseseveryvectorperpendiculartoaindirection. Theorem:Theproductoftwobiquadrantalversorsisa versortheaxisofwhichisperpendiculartotheaxesofthe 344 VECTORANALYSIS biquadrantalversorsandtheangleofwhichistwicethe anglefromtheaxisofthesecondtotheaxisoftheﬁrst. Letaandbbetheaxesoftwobiquadrantalversors.The product .0:(2bb—I)o(2aa—I) iscertainlyaversor;fortheproductofanytwoversors isaversor.Considerthecommonperpendiculartoaandb . ThebiquadrantalversorZaa—Ireversesthisperpendicular indirection .(2bb—I)againreversesitindirectionandcon sequentlybringsitbacktoitsoriginalposition .Hencethe product9leavesthecommonperpendiculartoaandbun changed . .9isthereforearotationaboutthislineasaxis. .Q-a=(2bb—I)o(2aa—I)oa=2bboa—a. Thecosineoftheanglefromato .9.ais a-Q-a:2boaboa—aoa=2(b-a)2—1=cos HencetheangleoftheversorQisequaltotwicetheangle fromatob. Theorem:Converselyanygivenversormaybeexpressed astheproductoftwobiquadrantalversors,ofwhichtheaxes lieintheplaneperpendiculartotheaxisOfthegivenversor andincludebetweenthemanangleequaltoonehalfthe angleofthegivenversor. ForletI2bethegivenversor.Letaandbbeunitvectors perpendiculartotheaxis—.9xofthisversor.Furthermore lettheanglefromatobbeequaltoonehalftheangleof thisversor.Thenbytheforegoingtheorem .Q:(2bb—I). (14) Theresolutionofversorsintotheproductoftwobiquad rantalversorsaﬁordsanimmediateandsimplemethodfor compoundingtwoﬁniterotationsaboutaﬁxedpoint.Let TandTbetwogivenversors.Letbbeaunitvectorper 346 VECTORANALYSIS T5:4(a-b)2—1, T=4cobcb—2bb«_c +I, Tx=4cobc, TS=4(cob)2—1 ToT:4c-aca—c—2aa+I, (ToT)x=4c-ac, oe)3—1. axb c aXcHence Q18“HQ,b.c,Q3a”: [abc]b Q2XQ1aobbcc a-bb-c° But Hence c axb aXc Q2xg1b-c aob+ aobboc a-CQHence Q2XQ1=—Q1 8acc a-bboc QQa-bb oc aocbob 1 2 aobboc acbb-c aobb-c Ha’°enceaobhm1Q2Q1. Hence 08QIXQ2+Q2+Q1 ROTATIONSANDSTRAINS 347 Thisformulagivesthecompositionoftwoﬁniterotations. IftherotationsbeinﬁnitesimalQ1andQ2arebothinﬁnitesi mal.Neglectinginﬁnitesimalsofthesecondorderthefor mulareducesto 03QI'i"Q20 Theinﬁnitesimalrotationscombineaccordingtothelawof vectoraddition .Thisdemonstratestheparallelogramlawfor angularvelocities.Thesubjectwastreatedfromdifferent standpointsinArts.51and60. Cyclics,RightTensors,Tonics,andCyclotonics IfthedyadicTbeaversoritmaybewritteninthe form(4) +sing(kj—jk). Theaxisofrotationisiandtheangleofrotationaboutthat axisisg.LetTbeanotherversorwiththesameaxisand anangleofrotationequaltog’ T=ii+coSq’Cji+kk)+sing'(kj—jk). Multiplying: TT:ToT=iicos (jj+kk) sin +99as—jk). (16) Thisistheresultwhichwastobeexpectedtheproductof twoversorsofwhichtheaxesarecoincidentisaversorwith thesameaxisandwithanangleequaltothesumofthe anglesofthetwogivenversors. Ifaversorbemultipliedbyitself,geometricandanalytic considerationsalikemakeitevidentthat —jk), and sinng(kj—jk). 348 VECTORANALYSIS OntheotherhandletTlequaljjkk;andT2equal 1:j—jk.Then T”(iicosgTlsing TheproductofiiintoeitherTIorT2iszeroandintoitselfis ii.Hence T”=ii(cos9TIsrngT2)” T":ii cos”gT,”ncos”"1gsingTln-lT2 ThedyadicTIraisedtoanypowerreproducesitself.TI“TI. ThedyadicT2raisedtothesecondpowergivesthenegative ofTI;raisedtothethirdpower,thenegativeofT2;raised tothefourthpower,TI;raisedtotheﬁfthpower,T2andso on(Art. ThedyadicTImultipliedbyT2isequalto T2.Hence T”=ii+cos"gT1+ncos"‘1gsingT2 But EquatingcoefﬁcientsofTIandT2inthesetwoexpressions forT” cosnn(n_1) g—cosg2!cosgsn-j—H . —1—2 srnng=ncosklgsrngnm Thustheordinaryexpansionsforcosnqandsinnqare obtainedinamannerverysimilartothemannerinwhich theyaregenerallyobtained. Theexpressionforaversormaybegeneralizedasfollows. Leta,b,cbeanythreenon-OOplanarvectorsanda’ ,b' ,the reciprocalsystem .Considerthedyadic T=aa’+cosg(bb’+cc’)+sing(cb’(17) 350 VECTORANALYSIS thewholeellipseasgisto2'7r.lSuchadisplacementofthe radiusvectorrmaybecalledanellipticrotationthrougha sectorgfromitssimilaritytoanordinaryrotationofwhich itistheprojection . DeﬁnitionAdyadicToftheform T=aa’+cosg(bb’+cc’)sing(cb’—bc')(17) iscalledacyclicdyadic.Theversorisaspecialcaseofa cyclicdyadic. Itisevidentfromgeometricoranalyticconsiderationsthat thepowersofacyclicdyadicareformed,asthepowersofa versorwereformed,bymultiplyingthescalargbythepower towhichthedyadicistoberaised . T"aa’cosng(bb’cc’)sinng(cb’ Ifthescalargisanintegralsub-multipleof27r,thatis,if 271' 9m: itispossibletoraisethedyadicTtosuchanintegralpower, namely,thepowerm,thatitbecomestheidemfactor Tmaythenberegardedasthemthrootoftheidemfactor. Inlikemannerifgand271'arecommensurableitispossible toraiseTtosuchapowerthatitbecomesequaltotheidem factorandevenifgand271'areincommensurableapowerof Tmaybefoundwhichdiﬁersbyaslittleasonepleasesfrom theidemfactor.Henceanycyclicdyadicmayberegardedas arootoftheidemfactor. 1ItisevidentthatﬁxingtheresultoftheapplicationofTtoallradiivectors inanellipsepracticallyﬁxesitforallvectorsintheplaneofbandc.Forany vectorinthatplanemayberegardedasascalarmultipleofaradiusvectorof theellipse. ROTATIONSANDSTRAINS 351 Deﬁnition:Thetransformationrepresentedbythe T=aii+bjj+6kk (18) wherea,b,carepositivescalarsiscalledapurestrain .The dyadicitselfiscalledarighttensor. Arighttensormaybefactoredintothreefactors T:(aii+iibjjkk)+(ii+jjckk). Theorderinwhichthesefactorsoccurisimmaterial.The transformation or issuchthattheiandjcomponentsofavectorremainun alteredbutthek-componentisalteredintheratioofcto1. Thetransformationmaythereforebedescribedasastretchor elongationalongthedirectionk .Iftheconstantcisgreater thanunitytheelongationisatrueelongation:butifcisless thanunitytheelongationisreallyacompression,fortheratio ofelongationislessthanunity .Betweenthesetwocases comesthecaseinwhichtheconstantisunity .Thelengths ofthek-componentsarethennotaltered . ThetransformationduetothedyadicTmayberegarded asthesuccessiveorsimultaneouselongationofthecom ponentsofrparalleltoi,j,andkrespectivelyintheratios ato1,bto1,cto1 .Ifoneormoreoftheconstantsa,b,c islessthanunitytheelongationinthatorthosedirections becomesacompression .Ifoneormoreoftheconstantsis unity,componentsparalleltothatdirectionarenotaltered . Thedirectionsi,j,karecalledtheprincipalaxesofthestrain . Theirdirectionsarenotalteredbythestrainwhereas,ifthe constantsa,b,cbedifferent,everyotherdirectionisaltered. Thescalarsa,b,careknownastheprincipalratiosof elongation . InArt.115itwasseenthatanycompletedyadicwas reducibletothenormalform 352 VECTORANALYSIS wherea,b,carepositiveconstants.Thisexpressionmaybe factoredintotheproductoftwodyadics. T::t(ai’i'+bj’j'+ck’k') (19) or Thefactor i’i+j’jk'k whichisthesameineithermethodoffactoringisaversor. Itturnsthevectorsi,j,kintothevectorsi' ,j’ ,k' .This versormayberepresentedbyitsvectorsemi-tangentof Theotherfactor or aii+bjj+ckk isarighttensorandrepresentsapurestrain .Intheﬁrst casethestrainhasthelines j’ ,k’forprincipalaxes:in thesecond,i,j,k .Inbothcasestheratiosofelongationare thesame,—ato1,bto1,cto1.Ifthenegativesignoccurs beforetheproducttheversionandpurestrainmusthave associatedwiththemareversalofdirectionsofallvectorsin space—thatis,aperversion .Hence TheoremAnydyadicisreducibletotheproductofa versorandarighttensortakenineitherorderandapositive ornegativesign .Hencethemostgeneraltransformation representablebyadyadicconsistsoftheproductofarota tionorversionaboutadeﬁniteaxisthroughadeﬁniteangle accompaniedbyapurestraineitherwithorwithoutperver sion .Therotationandstrainmaybeperformedineither order.Inthetwocasestherotationandtheratiosofelonga tionofthestrainarethesame;buttheprincipalaxesofthe straindifferaccordingasitisperformedbeforeorafterthe 354 VECTORANALYSIS componentsarestretchedintheratiosatto1,bto1,cto1. Ifoneormoreoftheconstantsa,b,c'arenegativethecom ponentsparalleltothecorrespondingvector3,b,carere versedindirectionaswellaschangedinmagnitude.The tonicmaybefactoredintothreefactorsofwhicheach stretchesthecomponentsparalleltooneofthevectorsa,b,0 butleavesunchangedthecomponentsparalleltotheother two . ThevalueofatonicTisnotalteredifinplaceofa,b,c anythreevectorsrespectivelycollinearwiththembesub« stituted,providedofcoursethatthecorrespondingchanges whicharenecessarybemadeinthereciprocalsystema’ ,b’ , Butwiththeexceptionofthischange,adyadicwhichis expressibleintheformofatonicissoexpressibleinonly onewayiftheconstantsa,b,caredifferent.Iftwoofthe constantssayI)andcareequal,anytwovectorscoplanar withthecorrespondingvectorsband0maybesubstituted inplaceofbandc.Ifalltheconstantsareequalthetonic reducestoaconstantmultipleoftheidemfactor.Anythree non-COpIanarvectorsmaybetakenfora,b,c. Theproductoftwotonicsofwhichtheaxesa,b,carethe sameiscommutativeandisatonicwiththeseaxesand withscalarcoefﬁcientsequalrespectivelytotheproductsof thecorrespondingcoefﬁcientsofthetwodyadics. T:alaa’+hlbb’clec’ T=azaa’+bzbb’+czcc’ T(22) Thegeneralizationofthecyclicdyadic 32’cosg(bb’cc’)sing(cb’—bc’) is(23) ROTATIONSANDSTRAINS 355 wherea,b,carethreenon-coplanarvectorsofwhicha’ ,b’ ,c' isthereciprocalsystemandwherethequantitiesa,b,c,are positiveornegativescalars.Thisdyadicmaybechanged intoamoreconvenientformbydeterminingthepositive scalarpandthepositiveornegativescalarg(whichmay alwaysbechosenbetweenthelimitsi sothat b=pcosg and c=psing. (24) That18, p=+ c2 I_p—bIand tan2g 10+6(24) Then T=aaa’+19cosq(bb’+psing(cb’(25) Thismaybefactoredintotheproductofthreedyadics T:(aaa’bb’cc’)o(33’+pbb’+pcc’)o {22’cosg(bb’cc’)sin9(cb’ Theorderofthesefactorsisimmaterial.Theﬁrstisatonic whichleavesunchangedvectorsparalleltoband0but stretchesthoseparalleltoaintheratioofato1 .Ifais negativethestretchingmustbeaccompaniedbyreversal indirection .Thesecondfactorisalsoatonic.Itleaves unchangedvectorsparalleltoabutstretchesallvectorsin theplaneofbandcintheratiopto1.Thethirdisa cyclicfactor.Vectorsparalleltoaremainunchanged;but radiivectorsintheellipseofwhichbandcareconjugate semi-diametersarerotatedthroughavectorsuchthatthe areaofthevectoristotheareaofthewholeellipseasqto Othervectorsintheplaneofbandcmayberegarded asscalarmultiplesoftheradiivectorsoftheellipse. 356 VECTORANALYSIS Deﬁnition:Adyadicwhichisreducibletotheform T aaa’+pcos9(bb’+19sin9(cb’(25) owingtothefactthatitcombinesthepropertiesofthe cyclicdyadicandthetoniciscalledacyclotonic. Theproductoftwocyclotonicswhichhavethesamethree vectors,a,b,casantecedentsandthereciprocalsystem a’ ,b’ ,c’forconsequentsisathirdcyclotonicandiscom mutative. T=alaa'+p1+plsin9,(cb’—bc’) T=azaa’+192cosy,(bb’+cc’)+p2sin92(cb’—bc’) ToT:ToT=a1a2aa’+plp2cos(q1(12)(bb’+ +101102sin(91+92)(oh'—bc')o (26) ReductionofDyadicstoCanonicalForms Theorem:IngeneralanydyadicTmaybereduced eithertoatonicortoacyclotonic.Thedyadicsforwhich thereductionisimpossiblemayberegardedaslimitingcases whichmayberepresentedtoanydesireddegreeofapproxi mationbytonicsorcyclotonics. Fromthistheoremtheimportanceofthetonicandcyclo tonicwhichhavebeentreatedasnaturalgeneralizationsof therighttensorandthecyclicdyadicmaybeseen .The proofofthetheorem,includingadiscussionofallthe specialcasesthatmayarise,islongandsomewhattedious. Themethodofprovingthetheoremingeneralhoweveris patent.Ifthreedirectionsa,b,0maybefoundwhichare leftunchangedbytheapplicationofTthenTmustbea tonic.Ifonlyonesuchdirectioncanbefound,thereexists aplaneinwhichthevectorssufferachangesuchasthatdue tothecyclotonicandthedyadicindeedprovestobesuch. 358 VECTORANALYSIS Anyvalueofa:whichsatisﬁesthisequationwillbesuch that (T (cl)30 . Thatistosay,thedyadicT :31isplanar.Avectorper pendiculartoitsconsequentsisreducedtozero.HenceT leavessuchadirectionunchanged .Thefurtherdiscussion ofthereductionofadyadictotheformofatonicoracyclo tonicdependsmerelyuponwhetherthecubicequationinx hasoneorthreerealroots. Theorem Ifthecubicequation 333x2¢8+:13$23$5: hasthreerealrootsthedyadicTmayingeneralbereduced toatonic. Forlet mza, bethethreerootsoftheequation .Thedyadics T—aI,T—bI,T—cI areingeneralplanar.Leta,b,cberespectivelythree vectorsdrawnperpendiculartotheplanesoftheconsequents ofthesedyadics. (T aI)oa0, (T—bI)abz0, (30) (T Then Toa=aa, Tob=bb, Toc=cc. Iftherootsa,b,caredistinctthevectorsa,b,carenon Coplanar.Forsuppose c=ma+nb (T ROTATIONSANDSTRAINS 359 mT-a—mca+nTob—ncb=0. But Toa=aa,Tob=bb . Hence m(a and m(a n(b Hence m :OOra=c,n=OOrb=c. Consequentlyifthevectorsa,b,carecoplanar,therootsare notdistinct;andthereforeiftherootsaredistinct,the vectorsa,b,carenecessarilynon-coplanar.Incasetheroots arenotdistinctitisstillalwayspossibletochoosethree non-OOplanarvectorsa,b,cinsuchamannerthattheequa tions(30)hold.Thisbeingso,thereexistsasystema’ ,b’ ,c’ reciprocaltoa,b,candthedyadicwhichcarriesa,b,cinto as,bh,ccisthetonic T=aaa’bbb’000. Theorem:Ifthecubicequation x3—22T3+xTzs—T3=O hasonerealrootthedyadicTmayingeneralbereducedto acyclotomic. Thecubicequationhasonerealroot.Thismustbeposi tiveornegativeaccordingasT3ispositiveornegative.Let therootbea .Determineaperpendiculartotheplaneof theconsequentsofT aI. (T—aI)oa=0. Determinea’alsosothat a’ o(T—aI)=0 andletthelengthsofaanda’besoadjustedthata’ Thiscannotbeaccomplishedinthespecialcaseinwhicha 360 VECTORANALYSIS anda'aremutuallyperpendicular.Letbbeanyvectorin theplaneperpendiculartoa’ . a’ Hence(T—aI)obisperpendiculartoa’ .HenceTobis perpendiculartoa’ .InasimilarmannerTzob,T3oh,and T‘Iob,TJ2ob,etc.,willallbeperpendiculartoa’andliein oneplane.ThevectorsTobandbcannotbeparallelorT wouldhavethedirectionbaswellasaunchangedand thusthecubicwouldhavemorethanonerealroot. ThedyadicTchangesa,Tob,bintoT-a,Tzob,Tobre spectively.Thevolumeoftheparallelopiped [ToaTZohTob]=T3[aTohb].(31) But Toazaa. Hence aa ThevectorsTzob,Tob,balllieinthesameplane.Their vectorproductsareparalleltoa’andtoeachother.Hence Tobxb . InasmuchasaandT8havethesamesign,let p2a‘1T3. (32) Letalso b8=p‘1Tob b2=p‘2T2ob,etc.(33) and boh b_2_p2T‘2-b,etc. bs lzblxbz, Thevectorsb2bandb1areparallel.Let b2+b=2nb1. Thenb3+b1:2nb2b1+b2=2nb3 etc., (35) b1+b_1=2nb b-1+b_2-2nb_1 etc. 362 VECTORANALYSIS Thereremaintwocases1inwhichthereduction isimpossible,ascanbeseenbylookingovertheproof.In theﬁrstplaceiftheconstantnusedinthereductiontocyclo tonicformbei1thereductionfallsthrough .Inthesecond placeiftheplaneoftheantecedentsof T—aI andtheplaneoftheconsequentsareperpendicularthe vectorsaanda’usedinthereductiontocyclotonicformare perpendicularanditisimpossibletodeterminea’suchthat aoa’shallbeunity .Thereductionfallsthrough. If n=i1, b_l+bl=i2bo b_lbl2ho Choose 0:b1—b=b—b_1. ConsiderthedyadicT=aaa’ p(bb’+cc’)pcb’ Toa:aa:Toa, T .c:pc:pb1—ph=T-c. Hence T:aaa’+p (37) Thetransformationduetothisdyadicmaybeseenbestby factoringitintothreefactorswhichareindependentofthe orderorarrangement T:(aaa’+bb’+cc’)009} (aa’bb’ 1Inthesecasesitwillbeseenthatthecubicequationhasthreerealroots. Inonecasetwoofthemareequalandintheothercasethreeofthem .Thus thesedyadicsmayberegardedaslimitingcaseslyingbetweenthecyclotonicin whichtwooftherootsareimaginaryandthetonicinwhichalltherootsarereal anddistinct.Thelimitmayberegardedastakingplaceeitherbythepure imaginarypartofthetwoimaginaryrootsofthecyclotonicbecomingzeroorby twooftherootsofthetonicapproachingeachother. ROTATIONSANDSTRAINS 363 Theﬁrstfactorrepresentsanelongationinthedirectionaina ratioato1 .Theplaneofbandcisundisturbed .The secondfactorrepresentsastretchingoftheplaneofbandcin theratiopto1.Thelastfactortakestheform I+cM (I+cb’) (I+cb’)xb::cb+we, (I eh’)oxc:xc. AdyadicoftheformI cb’leavesvectorsparalleltoaandc unaltered.Avectorxbparalleltobisincreasedbythevec torcmultipliedbytheratioofthevector .vbtob .Inother wordsthetransformationofpointsinSpaceissuchthatthe planeofaand0remainsﬁxedpointforpointbutthepoints inplanesparalleltothatplaneareshiftedinthedirectionc byanamountproportionaltothedistanceoftheplanein whichtheyliefromtheplaneofaand0. DeﬁnitionAdyadicreducibletotheform I+cb’ iscalledashearingdyadicorshearerandthegeometrical transformationwhichitcausesiscalledashear.Themore generaldyadic (37) willalsobecalledashearingdyadicorshearer.Thetrans formationtowhichitgivesriseis'ashearcombinedwith elongationsinthedirectionofaandisintheplaneofbandc. Ifn—1insteadofn+1,theresultismuchthesame. Thedyadicthenbecomes T:aaa’ —cb’ T:(aaa’bb’cc’)o{aa’-p(bb’(I 364 VECTORANALYSIS Thefactorsarethesameexceptthesecondwhichnowrepre sentsastretchingoftheplaneofband3combinedwitha reversalofallthevectorsinthatplane.Theshearingdyadic Tthenrepresentsanelongationinthedirectiona,anelonga tioncombinedwithareversalofdirectionintheplaneof bandc,andaShear. Supposethattheplaneoftheantecedentsandtheplaneof theconsequentsofthedyadicT—aIareperpendicular.Let theseplanesbetakenrespectivelyastheplaneofjandkand theplaneofiandjk .Thedyadicthentakestheform T ThecoefﬁcientBmustvanish .Forotherwisethedyadic T—aI—BI:(—Bi+Aj—Bk) isplanarandthescalaraBisarootofthecubicequation. Withthisrootthereductiontotheformofatonicmaybe carriedonasbefore.Nothingnewarises.ButifBvanishes anewcaseoccurs.Let Thismaybereducedasfollowstotheform ab’+bc’ where a-b’:a andbob’=l. SquareT Tzz—ADki=ac’ . Henceamustbechosenparalleltok;andc’ ,paralleltoi. ThedyadicTmaythenbetransformedinto 0i+DjT=ADk AD+A11 Then—ADk, b’=0i+Di AD b=Aj c’:i. 366 VECTORANALYSIS Amoresystematictreatmentofthevariouskinds ofdyadicswhichmayarisemaybegivenbymeansofthe Hamilton-Cayleyequation T3—T ,SrT2+T25T—T31=0 (39) andthecubicequationinx x3—Tgx2+Tzsx—T3=O . Ifa,b,caretherootsofthiscubictheHamilton-Cayley equationmaybewrittenas (T—aI)-(T—bI)o(T (40) If,however,thecubichasonlyoneroottheHamilton-Cayley equationtakestheform (T—aI)o(T2(41) IngeneraltheHamilton-Cayleyequationwhichisanequa tionofthethirddegreeinTistheequationoflowestdegree whichissatisﬁedbyT .Ingeneralthereforeoneoftheabove equationsandthecorrespondingreductionstothetonicor cyclotonicformhold .Inspecialcases,however,thedyadic Tmaysatisfyanequationoflowerdegree.Thatequation oflowestdegreewhichmaybesatisﬁedbyadyadiciscalled itscharacteristicequation.Thefollowingpossibilitiesoccur. I. (T—aI)-(T—bI)-(T II.(T—aI)o(T3 III. (T IV .(T—aI)o(T V .(T VI.(T (T ROTATIONSANDSTRAINS 367 Intheﬁrstcasethedyadicisatonicandmaybereduced totheform T:aaa’+bbb’+ccc' . Inthesecondcasethedyadicisacyclotonicandmaybe reducedtotheform InthethirdcasethedyadicisaSimpleShearerandmaybe reducedtotheform T:aaa’+b(bb’+cc’)+cb’ . Inthefourthcasethedyadicisagainatonic.Twoofthe ratiosofelongationarethesame.Thefollowingreduction maybeaccomplishedinaninﬁnitenumberofways. T:aaa’+b(bb’+ IntheﬁfthcasethedyadicisacomplexShearerandmaybe soexpressedthat T: InthesixthcasethedyadicisagainaSimpleshearerwhich maybereducedtotheform T:aI+cb’— .a(as’+bb’+cc’)+cb’ . Intheseventhcasethedyadicisagainatonicwhichmaybe reducedinadoublyinﬁnitenumberofwaystotheform Thesesevenaretheonlyessentiallydifferentformswhicha dyadicmaytake.Therearethenonlysevenreallydifferent kindsofdyadicsthreetonicsinwhichtheratiosofelonga tionarealldiﬁerent,twoalike,orallequal,andthecyclo tonictogetherwiththreelimitingcases,thetwosimpleand theonecomplexshearer. 368 VECTORANALYSIS SummaryofChapterVI Thetransformationduetoadyadicisalinearhomogeneou strain.Thedyadicitselfgivesthetransformationofthe pointsinspace.Thesecondofthedyadicgivesthetrans formationofplaneareas .Thethirdofthedyadicgivesthe ratioinwhichvolumesarechanged . Thenecessaryandsufﬁcientconditionthatadyadicrepre sentarotationaboutadeﬁniteaxisisthatitbereducibleto theform (1) orthat ToTc:IT31 (2) orthat T0T0=IT3>0 Thenecessaryandsufﬁcientconditionthatadyadicrepre sentarotationcombinedwithatransformationofreﬂection bywhicheachﬁgureisreplacedbyonesymmetricaltoitis that orthat ToTc: T3:—1 orthat ToTc T80. (3) Adyadicoftheform(1)iscalledaversoroneoftheform aperversor. Iftheaxisofrotationofaversorbechosenorthei-axis theversorreducesto —jk)(4) or T=ii+cosg(I (5) Ifanyunitvector8.isdirectedalongtheaxisofrotation T:aa+cosg(I—aa)+singlxa (6) TheaxisoftheversorcoincidesindirectionwithTX. 370 VECTORANALYSIS ductoftwocyclicdyadicswhichhavethesameantecedents a,b,candconsequentsa’b’c’isobtainedbyaddingtheir angles9.Acyclicdyadicmayberegardedasarootofthe idemfactor.Adyadicreducibletotheform T:aii+bjji +ckk (18) wherea,b,carepositivescalarsiscalledarighttensor.It representsastretchingalongtheprincipalaxisi,j,kinthe ratioato1,bto1,cto1whicharecalledtheprincipalratios ofelongation .Thistransformationisapurestrain. Anydyadicmaybeexpressedastheproductofaversor, arighttensor,andapositiveor.negativeSign . T:i(ai’i’bj’j’ck’k’)(i’i+j’jk’k) or (19) Consequentlyanylinearhomogeneousstrainmayberegarded asacombinationofarotationandapurestrainaccompanied orunaccompaniedbyaperversion . Theimmediategeneralizationsoftherighttensorandthe cyclicdyadicistothetonic (21)andcyclotonic —bc) (23) orT:aaa’+pcos9—bc’)(25) —bwhere 2 2dtalJ’ 24'p +0an n zq Anydyadicingeneralmaybereducedeithertotheform andisthereforeatonic,ortotheform andis thereforeacyclotonic.Theconditionthatadyadicbea tonicisthatthecubicequation x3T,o ROTATIONSANDSTRAINS 371 Shallhavethreerealroots.Specialcasesinwhichthe reductionmaybeaccomplishedinmorewaysthanonearise whentheequationhasequalroots.Theconditionthata dyadicbeacyclotonicisthatthiscubicequationshallhave onlyonerealroot.Thereoccurtwolimitingcasesinwhich thedyadiccannotbereducedtocyclotonicform.Inthese casesitmaybewrittenas (37) andisasimpleshearer,orittakestheform (38) andisacomplexShearer.Dyadicsmaybeclassiﬁedaccord ingtotheircharacteristicequations (T—aI)-(T—bI)o(T—cl)=0 tonic (T—aI)3(T2—2pcoq +p21):O cyclotonic (T aI)o(T bI)20 simpleshearer (T—al)o(T—bI):0 specialtonic (T aI)30 complexshearer (T aI)20 specialsimpleshearer (T a1)0 Specialtonic. CHAPTERVII MISCELLANEOUSAPPLICATIONS QuadricSurfaces IfTbeanyconstantdyadictheequation 1'oTorconst. (1) isquadraticinr.Theconstant,incaseitbenotzero,may bedividedintothedyadicTandhencetheequationtakes theform roTor1, or r-Tor=0. (2) ThedyadicTmaybeassumedtobeself-conjugate .Forif Tisananti-self-conjugatedyadic,theproductroToris identicallyzeroforallvaluesofr.Theproofofthisstate mentisleftasanexercise.ByArt.116anyself—conjugate dyadicisreducibletotheform :1kk T—zt azzt bzi c. (3) r oTor::l:—:l:—:l: (4) Hencetheequation reTor1 representsaquadricsurfacerealorimaginary. Thedifferentcaseswhicharisearefourinnumber.Ifthe signsareallpositive,thequadricisarealellipsoid.Ifone Signisnegativeitisanhyperboloidofonesheet;iftwoare 374 VECTORANALYSIS Hencethemostgeneralquadraticexpressionmaybereduced to r-T whereTisaconstantdyadic,Aaconstantvector,and0’ aconstantscalar.Thedyadicmayberegardedasself conjugateifdesired . ToberidofthelineartermroA,makeachangeoforigin byreplacingrbyr’t. (r’—t)To(r’—t)+(r’—t)oA+ r’oTor’—toTor’—r’-T-t+toTot +r’ oA—toA+C-=O. SinceTisself-conjugatethesecondandthirdtermsare equal.Hence r'. -A IfnowTiscompletethevectortmaybechosensothat 1 1 §A=Totort— éT1oA . Hencethequadricisreducibletothecentralform r’oTor’const. IncaseTisincompleteitisuniplanarorunilinearbecause Tisself-conjugate.IfAliesintheplaneofTorintheline ofTasthecasemaybetheequation 1 issolublefortandthereductiontocentralformisstillpos sible.ButunlessAissosituatedthereductionisimpossible . Thequadricsurfaceisnotacentralsurface. Thediscussionandclassiﬁcationofthevariousnon-central quadricsisaninterestingexercise.Itwillnotbetakenup here.Thepresentobjectistodevelopsomuchofthetheory QUADRICSURFACES 375 ofquadricsurfacesaswillbeusefulinapplicationstomathe maticalphysicswithespecialreferencetonon-isotropicmedia. Hereafterthereforethecentralquadricsandinparticularthe ellipsoidwillbediscussed . Thetangentplanemaybefoundbydifferentiation . roT-r=l. droTor+roTodr=0 . SinceTisself-conjugatethesetwotermsareequaland droTor0. (5) TheincrementdrisperpendiculartoTor.HenceToris normaltothesurfaceattheextremityofthevectorr.Let thisnormalbedenotedbyNandlettheunitnormalbe11. N=Tor (6) Letpbethevectordrawnfromtheoriginperpendicularto thetangentplane.pisparalleltonTheperpendicular distancefromtheorigintothetangentplaneisthesquare rootofpop.Itisalsoequaltothesquarerootofrop. rop:rcos Hence rop=pop. Orpip —rp1. P‘P P‘P But r-T-r:r-N:1 . HenceinasmuchaspandNareparallel,theyareequal. PT-r:N_ 7 PP 376 VECTORANALYSIS Onpage108itwasseenthatthevectorwhichhasthedirec tionofthenormaltoaplaneandwhichisinmagnitudeequal tothereciprocalofthedistancefromtheorigintotheplane maybetakenasthevectorcoo‘rdinateofthatplane.Hence theaboveequationshowsthatTorisnotmerelynormalto thetangentplane,butisalsotheco' Ordinateoftheplane. Thatis,thelengthofToristhereciprocalofthedistance fromtheorigintotheplanetangenttotheellipsoidat theextremityofthevectorr. TheequationoftheellipsoidinplanecoOrdinatesmaybe foundbyeliminatingrfromthetwoequations. roT-r=l, Tor=N . r=T‘1oN:N-T‘I Hence r oTor:N-T“I-ToT‘1-N= .T1. Hencethedesiredequationis N-T(8) iijjkkIf T‘l L617 and whereu,v,warethereciprocalsoftheinterceptsofthe planeNupontheaxesi,j,k .Thentheellipsoidmaybe writtenineitherofthetwoformsfamiliarinCartesian geometry. 378 VECTORANALYSIS Thevectorsa,b,carechangedintoToa,Tob,Tocby thedyadicT .Let a’:Toa, arr-Too. Thevectorsa’ ,b’ ,0’formthesystemreciprocaltoa,b,c. For aoa’:a-Toa=1, O and coa’zcowa—“O. ThedyadicTmaybethereforeexpressedintheforms T a’s’b’b’c’c’ , (12) and T—l Ifforconveniencethethreedirectionsa,b,c,becalleda Systemofthreeconjugateradiivectors,andifinaSimilar mannerthethreetangentplanesattheirextremitiesbecalled asystemofthreeconjugatetangentplanes,anumberof geometrictheoremsmaybeobtainedfrominterpretingthe invariantsofT .Asystemofthreeconjugateradiivectors maybeobtainedinadoublyinﬁnitenumberOfways. Thevolumeofaparallelopipedofwhichthreeconcurrent edgesconstituteasystemofthreeconjugateradiivectorsis constantandequalinmagnitudetotherectangularparallelo pipedconstructeduponthethreesemi-axesoftheellipsoid. Forleta,b,cbeanysystemofthreeconjugateaxes. T“I ThedeterminantorthirdofT”Iisaninvariantandinde pendentoftheforminwhichTisexpressed. T3‘12[abc]2 QUADRICSURFACES 379 Butif T‘l T3“1a262c” . Hence [abc]abc. Thisdemonstratesthetheorem .Inlikemannerbyinter pretingT3,T;1 ,andTSitispossibletoShowthat: Thesumofthesquaresoftheradiivectorsdrawntoan ellipsoidinaSystemofthreeconjugatedirectionsisconstant andequaltothesumofthesquaresofthesemi-axes. Thevolumeoftheparallelopiped,whosethreeconcurrent edgesareinthedirectionsoftheperpendicularsuponasystem ofthreeconjugatetangentplanesandinmagnitudeequalto thereciprocalsofthedistancesofthoseplanesfromthe centeroftheellipsoid,isconstantandequaltothereciprocal oftheparallelopipedconstructeduponthesemi-axesofthe ellipsoid . Thesumofthesquaresofthereciprocalsofthethreeper pendicularsdroppedfromtheoriginuponasystemofthree conjugatetangentplanesisconstantandequaltothesumof thesquaresofthereciprocalsofthesemi-axes. Ifi,j,kbethreemutuallyperpendicularunitvectors T,-I—i.T‘loi+joT“1-j+k-T'1ok Leta,b,cbethreeradiivectorsintheellipsoiddrawn respectivelyparalleltoi,j,k . aoToa=b ioToijoT-jkoT-k 0 Hencea Butthethreetermsinthisexpressionarethesquaresofthe reciprocalsoftheradiivectorsdrawnrespectivelyinthei,j, kdirections.Hence: 380 VECTORANALYSIS Thesumofthesquaresofthereciprocalsofthreemutually perpendicularradiivectorsinanellipsoidisconstant.And inaSimilarmanner:thesumoftheSquaresoftheperpen dicularsdroppedfromtheoriginuponthreemutuallyperpen diculartangentplanesisconstant. Theequationofthepolarplaneofthepointdeter minedbythevectorais1 SOT (13) Forletsbethevectorofapointinthepolarplane.The vectorofanypointuponthelinewhichjoinstheterminusof sandtheterminusofais ys :ca a:y Ifthispointliesuponthesurface ys+xays+xa x+yx+y o (My):(Me): IftheterminusofSliesinthepolarplaneofathetwovalues oftheratiomydeterminedbythisequationmustbeequal inmagnitudeandoppositeinSign .Hencetheterminmy vanishes. Hence soToa:1 isthedesiredequationofthepolarplaneoftheterminus ofa. Letabereplacedbyza.Thepolarplanebecomes s-Toza=1, or soToazl z 1ItisevidentlyimmaterialwhetherthecentralquadricdeterminedbyTbe realorimaginary,ellipsoidorhyperboloid. 382 VECTORANALYSIS IfAisanypointoutsideofthequadricandifallthetangent planeswhichpassthroughAaredrawn,theseplanesenvelop acone.Thisconetouchesthequadricalongaplanecurve theplaneofthecurvebeingthepolarplaneofthepointA . ForletabethevectordrawntothepointA .Theequation ofanytangentplanetothequadricis 80 0r1 . IfthisplanecontainsA,itsequationissatisﬁedbya.Hence theconditionswhichmustbesatisﬁedbyrifitstangent planepassesthroughAare a.oTo1‘1, roTor1 . Thepointsrthereforelieinaplane1'o(Toa)1which oncomparisonwith(13)isseentobethepolarplaneofA . Thequadricwhichpassesthroughthecurveofintersection ofthispolarplanewiththegivenquadricandwhichtouches thequadricalongthatcurveis (reTor Tor IfthispassesthroughthepointA, (a.Toa T-a Hence(reTor—1)(a-T-a—1)—(aoTor BytransformingtheorigintothepointAthisiseasilyseen tobeaconewhosevertexisatthatpoint. LetTbeanyself-conjugatedyadic.Itisexpres sibleintheform T=Aii+Bjj+Ckk whereA,B,0’arepositiveornegativescalars.Further I moreet A<B<0 T—BI=(C—B)kk—(B—A)ii. QUADRIC'SURFACES 383 Let ThenTBI=cc—aa=§+a)(o Let c+a=pand c—a=q. Then T=EIé—(pq+qp). (14) ThedyadicThasbeenexpressedasthesumofaconstant multipleoftheidemfactorandonehalfthesum pq+qp~ ThereductionhasassumedtacitlythattheconstantsA,B,C aredifferentfromeachotherandfromzero. ThisexpressionforTiscloselyrelatedtothecircular sectionsofthequadricsurface roT-r=1. SubstitutingthevalueofT,roTor1becomes Dr-r-l-ropq-r:1. Let ropn beanyplaneperpendiculartop.Bysubstitution B r or+nqor—1:O . Thisisaspherebecausethetermsofthesecondorderall havethesamecoefﬁcientB .IftheequationofthisSphere besubtractedfromthatofthegivenquadric,theresulting equationisthatofaquadricwhichpassesthroughtheinter sectionoftheSphereandthegivenquadric.Thedifference 18 qor(rop HencetheSphereandthequadricintersectintwoplane curveslyingintheplanes q.t=oand r op:n . 384 VECTORANALYSIS Inasmuchasthesecurveslieuponaspheretheyarecircles. Henceplanesperpendiculartopoutthequadricincircles. InlikemanneritmaybeShownthatplanesperpendicularto qcutthequadricincircles.Theproofmaybeconductedas follows: Br Ifrisaradiusvectorintheplanepassedthroughthecenter ofthequadricperpendiculartoporq,thetermropqorvan ishes.Hencethevectorrinthisplanesatisﬁestheequation Dr-r=1 andisofconstantlength .Thesectionisthereforeacircular section .Theradiusofthesectionisequalinlengthtothe meansemi—axisofthequadric. Forconvenienceletthequadricbeanellipsoid.Thecon stantsA,B,Carethenpositive.ThereciprocaldyadicT"l maybereducedinasimilarmanner. _1_ii kk (D A+ B+ C 1_1.0T B11. 1 1 1 1L d: et 1‘ 0kand A Br ThenT‘IéI:ff—dd—d) (f+d) Let f+d=uandf—d:v. 11Then T1: -I+ (15) 386 VECTORANALYSIS therotationthesectionisacircle.Inlikemannerconsider theprojectionorshadowoftheellipsoidcastuponaplane paralleltothemeanaxisbyapointataninﬁnitedistance fromthatplaneandinadirectionperpendiculartoit.ASthe ellipsoidisrotatedaboutitsmeanaxis,fromthepositionin whichthemajoraxisisperpendiculartotheplaneofprojec tiontothepositioninwhichtheminoraxisisperpendicular tothatplane,theshadowandtheprojectingcylinderhavethe meanaxisoftheellipsoidasoneaxis.Theotheraxischanges fromtheminoraxisoftheellipsoidtothemajorandhenceat somestageoftherotationitpassesthroughavalueequalto themeanaxis.Atthisstagetheshadowandprojecting cylinderarecircular. Thenecessaryandsufﬁcientconditionthatrbethemajor orminorsemi-axisofthesectionoftheellipsoidroTor1 byaplanepassingthroughthecenterandperpendiculartoa isthata,r,andTorbecoplanar. Let roT.r:1 and r-a:0. Diﬁerentiate: drTor0, droa:0. Furthermore dror0, ifristobeamajororminoraxisofthesection;forrisa maximumoramininumandhenceisperpendiculartodr. ThesethreeequationsShowthata,r,andTorareallortho gonaltothesamevectordr.Hencetheyarecoplanar. [arTor]:O . (16) Converselyif [arTor]0, drmaybechosenperpendiculartotheircommonplane. Then dror0. QUADRIC'SURFACES 387 Hencerisamaximumoramininumandisoneoftheprin eipelsemi-axesofthesectionperpendiculartoa. Itisfrequentlyanadvantagetowritetheequation ofanellipsoidintheform roTzor:l, (17) insteadof roT-r:l. Thismaybedone;becauseif iijju 11jjkk T a+ b c(18) isadyadicsuchthatT3isequaltoT .Tmayberegardedas asquarerootofTandwrittenasTl.Butitmustbere memberedthatthereareothersquarerootsofT for ii kk +55 a b c and11jjkk a b c Forthisreasonitisnecessarytobearinmindthatthesquare rootwhichismeantbyTiisthatparticularonewhichhas beendenotedbyT Theequationoftheellipsoidmaybewrittenintheform roWoWor1, or (T-r)-(T Letr’betheradiusvectorofaunitsphere.Theequationof thesphereis r’or’1. 388 VECTORANALYSIS Ifr’:Toritbecomesevidentthatanellipsoidmaybe transformedintoaunitspherebyapplyingtheoperatorT toeachradiusvectorr,andviceversa,theunitspheremay betransformedintoanellipsoidbyapplyingtheinverseOper. atorT"Itoeachradiusvectorr’ .Furthermoreifa,b,care asystemofthreeconjugateradiivectorsinanellipsoid a.wzoazbo WOCZI, aTzobzboTzoc:cT2.a=0. Ifforthemomenta’ ,b’ ,0’denoterespectivelyToa,Tob, Toe, doU=Uod=d-T:Q Hencethethreeradiivectorsa' ,b’ ,c’oftheunitsphereinto whichthreeconjugateradiivectorsintheellipsoidaretrans formedbytheoperatorT‘1aremutuallyorthogonal.They formaright-handedorleft-handedsystemofthreemutually perpendicularunitvectors. Theorem:Anyellipsoidmaybetransformedintoanyother ellipsoidbymeansofahomogeneousstrain. Lettheequationsof .theellipsoidsbe r and r BymeansofthestrainTitheradiivectorsroftheﬁrst ellipsoidarechangedintotheradiivectorsr’ofaunitsphere BymeansofthestrainT”!theradiivectorsr’ofthisunit Spherearetransformedinlikemannerintotheradiivectorsr ofthesecondellipsoid .Hencebytheproductrischanged rT“ 0TI0r. (19) 390 VECTORANALYSIS Iftwoconfocalquadricsintersect,theydosoatrightangles. Letthequadricsbe roTor1, and roTor1 . Let r=T“1osandr:T‘los’ . ThenthequadricsmaybewrittenintermsofsandS’as soT‘los:1, and s’oT"Ios’1, wherebytheconfocalproperty, T“)T1xI. Ifthequadricsintersectatrtheconditionforperpendicularity isthatthenormalsTorandTorbeperpendicular.Thatis, s But—1+xl)os T‘los+xs, T‘loS’—soT'los’=1—soT‘los’ . Inlikemanner ros’ —as’ . as’-s’=s-T"1os’os=s-T‘1os’—1 . Add: 2338 Hence andthetheoremisproved. Iftheparameternbeallowedtovaryfrom coto cothe resultingconfocalquadricswillconsistofthreefamiliesof whichoneisellipsoids;another,hyperboloidsofonesheet; andthethird,hyperboloidsoftwoSheets.Bytheforegoingx1) soT‘l so(T‘lT‘1)o sos:0, QUADRICSURFACES 391 theoremeachsurfaceofanyonefamilycutseverysurface oftheothertwoorthogonally .Thesurfacesformatriply orthogonalsystem .Thelinesofintersectionoftwofamilies (saythefamilyofone-sheetedandthefamilyoftwo-sheeted hyperboloids)cutorthogonallytheotherfamily—thefamily ofellipsoids.Thepointsinwhichtwoellipsoidsarecutby theselinesarecalledcorrespondingpointsuponthetwoellip soids.ItmaybeShownthattheratiosofthecomponentsof theradiusvectorofapointtotheaxesoftheellipsoid throughthatpointarethesameforanytwocorresponding points. Forletanyellipsoidbegivenbythedyadic ii kk T a,J b—J ,02. Theneighboringellipsoidinthefamilyisrepresentedbythe dyadic o O o g}. r1 JJ kk az—dnbz—dn 02—377} T‘IT“IIdn . InasmuchasTandTarehomologous(seeEx .8,p .330) dyadicstheymaybetreatedasordinaryscalarsinalgebra. Thereforeiftermsoforderhigherthantheﬁrstindnbe “ﬂuted:T TIdn . Thetwoneighboringellipsoidsarethen rT-r=l, and By(19)Tier, i=(I+Tdn)‘ior, d r—(I—éTdn)or:r 2” T.r. 392 VECTORANALYSIS ThevectorsfandrdifferbyamultipleofTorwhichis perpendiculartotheellipsoidT .Hencetheterminioffand rarecorrespondingpoints,fortheylieupononeofthelines whichcutthefamilyofellipsoidsorthogonally.Thecom ponentsofrandfinthedirection1areroi aand Tn.wdnx r.1:x:r01— 2—1° 'r—x 2a2. ih ts. a: 1dn Theratio0tesecomponen is x 2a2 TheaxesoftheellipsoidsinthedirectioniareVa2dnand a.Theirratiois Vaz—d’n a ad”:x a 2a2x a 2_Inlikemannerv622andV0 d”z b 3/ c 2 Hencetheratiosofthecomponentsofthevectors1“andr drawntocorrespondingpointsupontwoneighboringellip soidsonlydifferatmostbytermsofthesecondorderindn fromtheratiosoftheaxesofthoseellipsoids.Itfollows immediatelythattheratiosofthecomponentsofthevectors drawntocorrespondingpointsuponanytwoellipsoids,sepa ratedbyaﬁnitevariationintheparametern,onlydifferat mostbytermsoftheﬁrstorderindnfromtheratiosofthe axesoftheellipsoidsandhencemustbeidenticalwiththem . Thiscompletesthedemonstration . ThePropagationofLightinCrystalsl Theelectromagneticequationsoftheetherorofany inﬁniteisotropicmediumwhichistransparenttoelectromag neticwavesmaybewrittenintheform 1Thefollowingdiscussionmustberegardedasmathematicalnotphysical. Totreatthesubjectfromthestandpointofphysicswouldbeoutofplacehere. 394 VECTORANALYSIS Then D=Acos(mor—nt) whereAandmareconstantvectorsandnaconstantscalar representsatrainofwaves.Thevibrationstakeplacein thedirectionA .Thatis,thewaveisplanepolarized.The waveadvancesinthedirectionm .Thevelocityvofthatad vanceisthequotientofnbym,themagnitudeofthevector m .Ifthiswaveisanelectromagneticwaveinthemedium considereditmustsatisfythetwoequationsofthatmedium . SubstitutethevalueofDinthoseequations. ThevalueofVoD,VoV-D,andVV¢cDmaybe obtainedmosteasilybyassumingthedirectionitobecoinci dentwithm .morthenreducestomiorwhichisequalto mx.Thevariablesyand2nolongeroccurinD .Hence D=Acos(mx—nt) 9D Von zlo ax—i-Amsin(mx—nt) VoVQ-D=—m2(D oAcos(mm—nt) VV¢oD=—m2iiQ-Acos(mac—n0. Hence VoD z—moAsiu(m-r—ui) V-V¢-D=—mom Moreover n2D . HenceiftheharmonicvibrationDistosatisfytheequa tions(4)ofthemedium n2D=mom(0-D—mmo¢-D (5) andnoA=0. (6) THEPROPAGATIONOFLIGHTINCRYSTALS395 Thelatterequationstatesatoncethatthevibrationsmust betransversetothedirectionmofpropagationofthewaves. Theformerequationmaybeputintheform mom mm 1) 2.4043. ’IL mIntroduce Thevector8isinthedirectionofadvancem .Themagnitude ofsisthequotientofmbyn .Thisisthereciprocalofthe velocityofthewave.Thevector3maythereforebecalled thewave-slowness. D=sos¢cD—sso¢oD . Thismayalsobewrittenas D= -D)Xs. Dividingbythescalarfactorcos(mx nt), A=sx(¢-A)xs=sosQoA—sso¢oA .(7) Itisevidentthatthewaveslowness3dependsnotatall uponthephraseofthevibrationbutonlyuponitsdirection . Themotionofawavenotplanepolarizedmaybediscussedby decomposingthewaveintowaveswhichareplanepolarized . LetabeavectordrawninthedirectionAofthe displacementandletthemagnitudeofabesodetermined that ao¢oa=l. (8) Theequation(7)thenbecomesreducedtotheform o(0 oa(9) ao¢o3=1 . (8) Thesearetheequationsbywhichthediscussionofthevelocity orrathertheslownessofpropagationofawaveindifferent directionsinanon-isotropicmediummaybecarriedon . a-azsosa-(Poazsos. (10) 396 VECTORANALYSIS Hencethewaveslowness8duetoadisplacementinthe directionaisequalinmagnitude(butnotindirection)tothe radiusvectordrawnintheellipsoidao(Poa1inthat direction . axa=0=sosax¢oa—axsso¢oa O=s-s(aX(D -a so¢oa. Buttheﬁrsttermcontains atwiceandvanishes.Hence axso¢oa=[a s (11) Thewave-slowness3thereforeliesinaplanewiththe directionaofdisplacementandthenormaloadrawntothe ellipsoidao(0oa1attheterminusofa.Since8isperpen diculartoaandequalinmagnitudetoaitisevidentlycom pletelydeterminedexceptasregardssignwhenthedirection aisknown .Giventhedirectionofdisplacementthelineof advanceofthewavecompatiblewiththedisplacementiscom pletelydetermined,thevelocityoftheadvanceislikewise known.Thewavehowevermayadvanceineitherdirection alongthatline.Byreferencetopage386,equation(11)isseen tobetheconditionthatashallbeoneoftheprincipalaxesof theellipsoidformedbypassingaplanethroughtheellipsoid perpendicularto8.Henceforanygivendirectionofadvance therearetwopossiblelinesofdisplacement.Thesearethe principalaxesoftheellipsecutfromtheellipsoidaoa1 byaplanepassedthroughthecenterperpendiculartothe lineofadvance.Tothesestatementsconcerningthedeter minatenessofswhenaisgivenandofawhen8isgivenjust suchexceptionsoccurasareobviousgeometrically .Ifaand oa.areparallel3mayhaveanydirectionperpendiculartoa. Thishappenswhenaisdirectedalongoneoftheprincipal axesoftheellipsoid .Ifsisperpendiculartooneofthe circularsectionsoftheellipsoidamayhaveanydirectioninthe planeofthesection. 398 VECTORANALYSIS Iii JJ+ Let s=xi+yj+zkand ThentheequationofthesurfaceinCartesiancoordinatesis x2y222 132 132 1320' a.2b202 TheequationinCartesiancoordinatesmaybeobtained directlyfrom —sosI+ss)8_-O. Thedeterminantofthisdyadicis a2—32+x2myxz myI)2yz :62yz 02 Bymeansoftherelation32x2y222thisassumesthe forms sz—az-i- sz —e21 czzz or 82—a2+ 32—b2+ 32—020’ 30222 01‘ 182+ 320. a z?1' 7 Thisequationappearstobeofthesixthdegree.Itishow everofonlythefourth.Thetermsofthesixthordercancel out. Thevectornrepresentsthewave-slowness.Supposethata planewavepolarizedinthedirectionapassestheoriginata THEPROPAGATIONOFLIGHTINCRYSTALS399 certaininstantoftimewiththisslowness.Attheendofa unitoftimeitwillhavetravelledinthedirection3,adistance equaltothereciprocalofthemagnitudeofs.Theplanewill beinthispositionrepresentedbythevector5(page If s=ui+vj+wk theplaneattheexpirationoftheunittimecutsoffintercepts upontheaxesequaltothereciprocalsofu,v,w .These quantitiesarethereforetheplanecoordinatesoftheplane. Theyareconnectedwiththecoordinatesofthepointsinthe planebytherelation ux=vy+wz=L Ifdifferentplanewavespolarizedinallpossibledifferent directionsabesupposedtopassthroughtheoriginatthe sameinstanttheywillenvelopasurfaceattheendofaunit oftime.Thissurfaceisknownasthewave-surface.The perpendicularuponatangentplaneofthewave-surfaceisthe reciprocaloftheslownessandgivesthevelocitywithwhich thewavetravelsinthatdirection .Theequationofthewave surfaceinplanecoordinatesu,v,wisidenticalwiththeequa tionforthelocusoftheterminusoftheslownessvector3. Theequationis u2v2w2 182+ 32+ a21 b202(15) wheres2u2v2w” .Thismaybewritteninanyofthe formsgivenpreviously .ThesurfaceisknownasFresnel’ s WaveSurfaee.Theequationsinvectorformaregivenon page397ifthevariablevector8beregardedasdetermininga planeinsteadofapoint. Inanisotropicmediumthedirectionofarayof lightisperpendiculartothewave-front.Itisthesameas thedirectionofthewave’sadvance.Thevelocityoftheray 400 VECTORANALYSIS isequaltothevelocityofthewave.Inanon-isotropic mediumthisisnolongertrue.Theraydoesnottravelper pendiculartothewave-front—thatis,inthedirectionofthe wave’sadvance.Andthevelocitywithwhichtheraytravels isgreaterthanthevelocityofthewave.Infact,whereasthe wave-fronttravelsoffalwaystangenttothewave-surface,the raytravelsalongtheradiusvectordrawntothepointoftan gencyofthewave-plane.Thewave-planesenvelopthe wave-surface;theterminioftheraysaresituateduponit. Thusinthewave-surfacetheradiusvectorrepresentsinmag nitudeanddirectionthevelocityofarayandtheperpen dicularuponthetangentplanerepresentsinmagnitudeand directionthevelocityofthewave.Ifinsteadofthewave surfacethesurfacewhichisthelocusoftheextremityofthe waveslownessbeconsidereditisseenthattheradiusvector representstheslownessofthewave;andtheperpendicular uponthetangentplane,theslownessoftheray . Letv’bethevelocityoftheray .Then8ov'1because theextremityofv’liesintheplanedenotedby8.Moreover theconditionthatv’bethepointoftangencygivesdv’per pendiculartos.Inlikemannerifs’betheslownessofthe rayandvthevelocityofthewave,8' ov1andthecondition oftangencygivesds’perpendiculartov.Hence s (16) andsodv'=0,v v’ ods=0,s' odv=0, v’maybeexpressedintermsofa,s,and(Dasfollows. da=25ods¢oa—soOoads+sos¢oda — ~sdso¢-a=ss-(D-da. Multiplybyaand‘takeaccountoftherelationsa osOand aoToda=0andaoa=soaThen 402 VECTORANALYSIS andthenormaltotheellipsoid,thewave-slowness,andthe ellipsoidO—1ontheother. Itwasseenthatifswasnormaltooneofthecir onlarsectionsofOthedisplacementacouldtakeplaceinany directionintheplaneofthatsection .Foralldirectionsin thisplanethewave-slownesshadthesamedirectionandthe samemagnitude.Hencethewave-surfacehasasingular planeperpendiculartos.Thisplaneistangenttothesurface alongacurveinsteadofatasinglepoint.Henceifawave travelsinthedirection8theraytravelsalongtheelementsof theconedrawnfromthecenterofthewave-surfacetothis curveinwhichthesingularplanetouchesthesurface.The twodirections3whicharenormaltothecircularsectionsofO arecalledtheprimaryOpticaxes.Thesearetheaxesofequal wavevelocitiesbutunequalrayvelocities. Inlikemannerv’being00planarwithaandOoa [Ooav' Thelastequationstatesthatifaplanebepassedthrough thecenteroftheellipsoidO“Iperpendiculartov’ ,thena’ whichisequaltoOoawillbedirectedalongoneoftheprin cipalaxesofthesection .Henceifarayistotakeadeﬁnite directiona’mayhaveoneoftwodirections.Itismorecon venienthowevertoregardv’asavectordeterminingaplane. Theﬁrstequation [Ooav'a]0 statesthataistheradiusvectordrawnintheellipsoidOto thepointoftangencyofoneoftheprincipalelementsofthe cylindercircumscribedaboutOparalleltov'ifbyaprincipal elementismeantanelementpassingthroughtheextremities ofthemajororminoraxesoforthogonalplanesections ofthatcylinder.Hencegiventhedirectionv’oftheray,the twopossibledirectionsofdisplacementarethoseradiivectors VARIABLEDYADICS 403 oftheellipsoidwhichlieintheprincipalplanesofthecylin dercircumscribedabouttheellipsoidparalleltov' Ifthecylinderisoneofthetwocircularcylinderswhich maybecircumscribedaboutOthedirectionofdisplacement maybeanydirectionintheplanepassedthroughthecenter oftheellipsoidandcontainingthecommoncurveoftangency ofthecylinderwiththeellipsoid .Theray-velocityforall thesedirectionsofdisplacementhasthesamedirectionand thesamemagnitude.Itisthereforealinedrawntoone ofthesingularpointsofthewave-surface.Atthissingular pointthereareaninﬁnitenumberoftangentplanesenvelop ingacone.Thewave-velocitymaybeequalinmagnitude anddirectiontotheperpendiculardrawnfromtheoriginto anyoftheseplanes.Thedirectionsoftheaxesofthetwo circularcylinderscircumscriptibleabouttheellipsoidOare thedirectionsofequalray-velocitybutunequalwave-velocity . Theyaretheradiidrawntothesingularpointsofthewave surfaceandarecalledthesecondaryopticaxes.Ifaray travelsalongoneofthesecondaryopticaxesthewaveplanes travelalongtheelementsofacone. VariableDyadics.TheserentialandIntegralCalculus Hithertothedyadicsconsideredhavebeenconstant. Thevectorswhichenteredintotheirmakeupandthescalar coefﬁcientswhichoccurredintheexpansioninnonionform havebeenconstants.Fortheelementsofthetheoryandfor elementaryapplicationstheseconstantdyadicssufﬁce.The introductionofvariabledyadics,however,leadstoasimpliﬁcar tionanduniﬁcationofthedifferentialandintegralcalculusof vectors,andfurthermorevariabledyadicsbecomeanecessity inthemoreadvancedapplicationsforinstance,inthetheory ofthecurvatureofsurfacesandinthedynamicsofarigid bodyonepointofwhichisﬁxed . 404 VECTORANALYSIS LetWbeavectorfunctionofpositioninspace.Letrbe thevectordrawnfromaﬁxedorigintoanypointinspace. 9W 9W 9W dW=dcc 9x+dyay+dz az. Hence dW=droiaW +jaW +kaW 9x 9y 8z Theexpressionenclosedinthebracesisadyadic.Itthus appearsthatthedifferentialWisalinearfunctionofdr,the differentialchangeofposition .Theantecedentsarei,j,k, andtheconsequentstheﬁrstpartialderivativesofWwithre spectofx,y,z.Theexpressionisfoundinamannerprecisely analogoustodelandwillinfactbedenotedbyVW . ,9W,9W 9WVW.—1 9x+J9y+k az. (1) Then dW2drVW . (2) Thisequationisliketheoneforthediﬁerentialofascalar functionV . dV:drVV ItmayberegardedasdeﬁningVW .Ifexpandedinto nonionformVWbecomes X Y vw=iia +ij‘9 +jk Ox are tax 91/ o o t o 0k 3 +Jiay+uay+i ,OX .917 406 VECTORANALYSIS IfanattemptweremadetoapplytheoperatorVsymboli callytoascalarfunctionVthreetimes,theresultwouldbea sumoftwenty-seventermslike ,93V 93V 111 9933,rjk axayazmtc. Thisisatriadic.Threevectorsareplacedinjuxtaposition withoutanysignofmultiplication .Suchexpressionswill notbediscussedhere.InasimilarmanneriftheoperatorV beappliedtwicetoavectorfunction,oroncetoadyadicfunc tionofpositioninspace,theresultwillbeatriadicandhence outsidethelimitssettothediscussionhere.Theoperators VXandVomayhoweverbeappliedtoadyadicOtoyield respectivelyadyadicandavector. 9O 9O 9O (7) .9O .907 9O (8) If whereu,v,warevectorfunctionsofpositioninspace, and VoO=Voui+Vovj+Vowk. Orif O=iu+jv+kW, 9w9v 9u9w «9v c911VX(D:I] 93;92HOz996+k 9x9y(7) 9n0v9Wad VoO n 9w+9y+az. (8) Inasimilarmannerthescalaroperators(aV)and(VV) maybeappliedtoO .Theresultisineachcaseadyadic, VARIABLEDYADICS 407 aa) aO 9(I) (9) QZOOZOO2O (10) Theoperators3.aVandVoVasappliedtovectorfunc tionsarenolongernecessarilytoberegardedassingleOper ators.Theindividualstepsmaybecarriedoutbymeansof thedyadicVW . (a (V -VW . Butwhenappliedtoadyadictheoperatorscannotbeinter pretedasmadeupoftwosuccessivestepswithoutmakinguse ofthetriadicVO .Theparentheseshowevermayberemoved withoutdangerofconfusionjustastheywereremovedin caseofavectorfunctionbeforetheintroductionofthedyadic. Formulaesimilartothoseuponpage176maybegivenfor differentiatingproductsinthecasethatthedifferentiation leadtodyadics. Vx(vxw)=w-Vv—Vo;w—voVw+V-wv, V(vow)=Vv-w+Vwov, V -vw+v-Vw . VX(vw)=v w—v w, V VxVxO:VV-O—V-VO,etc . Theprincipleintheseandallsimilarcasesisthatenun ciatedbefore,namely:TheoperatorVmaybetreatedsym 408 VECTORANALYSIS bolicallyasavector.Thedifferentiationswhichitimplies mustbecarriedoutinturnuponeachfactorofaproduct towhichitisapplied.Thus Vx(vw)[Vx [Vx [VX w , [Vx Hence VX(vw)=v w Again [V [V—w v. Hence Itwasseen(Art.79)thatif0denoteacurveof whichtheinitialpointisandtheﬁnalpointisrthelinein tegralofthederivativeofascalarfunctiontakenalongthe curveisequaltothediﬁerencebetweenthevaluesofthat functionatrandr0. fdrVV=V(r) 0 InlikemannerfdrVW W W(to): C andfdr-VW=O . 0 Itmaybewelltonotethattheintegrals fanvw andfk zr arebynomeansthesamething .VWisadyadic.The vectordrcannotbeplacedarbitrarilyuponeithersideofit. 410 VECTORANALYSIS Thesurfaceintegralsaretakenoverthecompletebounding surfaceoftheregionthroughoutwhichthevolumeintegrals aretaken . Numerousformulaeofintegrationbypartslikethoseupon page250mightbeadded .Thereaderwillﬁndnodifﬁcultyin obtainingthemforhimself.Theintegratingoperatorsmay alsobeextendedtoothercases.Tothepotentialsofscalar andvectorfunctionsthepotential,PotO,ofadyadicmaybe added .TheNewtonianofavectorfunctionandtheLapla cianandMaxwellianofdyadicsmaybedeﬁned . Newwsfr12w(x 32$y2’z2)dv2,r12 O Lap(0: X 9222)d 12 MaxO=fffrm°(Dy2’22)dv2. 12 Theanalytictheoryoftheseintegralsmaybedevelopedas before.Themostnaturalwayinwhichthedemonstrations maybegivenisbyconsideringthevectorfunctionWasthe sumofitscomponents, W=Xi+Yj+Zk andthedyadicOasexpressedwiththeconstantconsequents i,j,kandvariableantecedentsu,v,w,or'viceversa, Thesematterswillbeleftatthispoint.Theobjectofeu teringuponthematallwastoindicatethenaturalextensions whichoccurwhenvariabledyadicsareconsidered.Theseex tensionsdiffersoslightlyfromthesimplecaseswhichhave THECURVATUREOFSURFACES 411 gonebeforethatitisfarbettertoleavethedetailstobeworked outorassumedfromanalogywhenevertheymaybeneeded ratherthantoattempttodeveloptheminadvance.Itissufﬁ cientmerelytomentionwhattheextensionsareandhowthey maybetreated . TheCurvatureofSurfaces1 Therearetwodiﬁerentmethodsoftreatingthecur vatureofsurfaces.Inonethesurfaceisexpressedinpara meticformbythreeequations “3=f1(“av),y=f2 z=f3(u, or rf(u,v). Thisisanalogoustothemethodfollowed(Art.57)indealing withcurvatureandtorsionofcurvesanditisthemethod employedbyFehrinthebooktowhichreferencewasmade. Inthesecondmethodthesurfaceisexpressedbyasingle equationconnectingthevariablesx,y,z—thus F(x,y,z)0. Thelattermethodoftreatmentsaffordsasimpleapplicationof thedifferentialcalculusofvariabledyadics.Moreover,the dyadicsleadnaturallytothemostimportantresultsconnected withtheelementarytheoryofsurfaces. Letrbearadiusvectordrawnfromanarbitraryﬁxed origintoavariablepointofthesurface.Theincrementdr liesinthesurfaceorinthetangentplanedrawntothesurface attheterminusofr. s dI'OVFZ-‘O. HencethederivativeVFiscollinearwiththenormaltothe surface.Moreover,inasmuchasFandthenegativeofFwhen 1Muchofwhatfollowsispracticallyfreefromtheuseofdyadics.Thisis especiallytrueofthetreatmentofgeodetics,Arts.155-157. 412 VECTORANALYSIS equatedtozerogivethesamegeometricsurface,VFmaybe consideredasthenormaluponeithersideofthesurface.In casethesurfacebelongstothefamilydeﬁnedby F(a),y,z)const. thenormalVFliesuponthatsideuponwhichtheconstant increases.LetVFberepresentedbyNthemagnitudeof whichmaybedenotedbyN,andletnbeaunitnormaldrawn inthedirectionofN .Then (1) 1 n N Ifsisthevectordrawntoanypointinthetangentplaneat theterminusofr,s-randnareperpendicular.Consequently theequationofthetangentplaneis —ﬁ-VF=& andinlikemannertheequationofthenormallineis (sr)XVFO, m s=r+hVF wherekisavariableparameter.Theseequationsmaybe translatedintoCartesianformandgivethefamiliarresults. Thevariationdnoftheunitnormaltoasurface playsanimportantpartinthetheoryofcurvature.dnis perpendicularto11because11isaunitvector. 1 N2N 414 VECTORANALYSIS thesurface.Itispossible(Art.116)toreduceOtothe form (5) wherei’andj’aretwoperpendicularunitvectorslyinginthe tangentplaneandaandbarepositiveornegativescalars. dn=dro(ai’i'+b Thevectorsi’ ,j’andthescalarsa,bvaryfrompointtopoint ofthesurface.ThedyadicOisvariable. TheconicroOor1iscalledtheindicatrixofthe surfaceatthepointinquestion.Ifthisconicisanellipse, thatis,ifaandbhavethesamesign,thesurfaceisconvexat thepoint;butiftheconicisanhyperbola,thatis,ifaandb haveoppositesignsthesurfaceisconcavo-convex .Thecurve 1'oOor1mayberegardedasapproximatelyequaltothe intersectionofthesurfacewithaplanedrawnparalleltothe tangentplaneandneartoit.IfroOorbesetequaltozero theresultisapairofstraightlines.Thesearetheasymp totesoftheconic.Iftheyarerealtheconicisanhyperbola; ifimaginary,anellipse.Twodirectionsonthesurfacewhich areparalleltoconjugatediametersoftheconicarecalledcon jugatedirections.Thedirectionsonthesurfacewhichcoin cidewiththedirectionsoftheprincipalaxesi’ ,j’ofthe indicatrixareknownastheprincipaldirections.Theyarea specialcaseofconjugatedirections.Thedirectionsuponthe surfacewhichcoincidewiththedirectionsoftheasymptotes oftheindicatrixareknownasasymptoticdirections.Incase thesurfaceisconvex,theindicatrixisanellipseandthe asymptoticdirectionsareimaginary. InspecialcasesthedyadicOmaybesuchthatthecoefﬁ cientsaandbareequal.Omaythenbereducedtothe form O a(i’i’j'j') THECURVATUREOFSURFACES 415 inaninﬁnitenumberofways.Thedirectionsi’andj’maybe anytwoperpendiculardirections.Theindicatrixbecomesa circle.Anypairofperpendiculardiametersofthiscircle giveprincipaldirectionsuponthesurface.Suchapointis calledanumbilic.Thesurfaceintheneighborhoodofan umbilicisconvex .Theasymptoticdirectionsareimaginary . InanotherspecialcasethedyadicObecomeslinearandredu tothe£01mw aifif . Theindicatrixconsistsofa ofparallellinesperpendicular toi’ .Suchapointiscalledaparabolicpointofthesurface. Thefurtherdiscussionoftheseandotherspecialcaseswillbe omitted . Thequadricsurfacesaffordexamplesofthevariouskinds ofpoints.Theellipsoidandthehyperboloidoftwosheets areconvex .Theindicatrixofpointsuponthemisanellipse. Thehyperboloidofonesheetisconcavo-convex .Thein dicatrixofpointsuponitisanhyperbola.Theindicatrix ofanypointuponasphereisacircle.Thepointsareall umbilics.Theindicatrixofanypointuponaconeorcylinder isapairofparallellines.Thepointsareparabolic.Asur faceingeneralmayhaveuponitpointsofalltypes—elliptic, hyperbolic,parabolic,andumbilical. Alineofprincipalcurvatureuponasurfaceisa curvewhichhasateachpointthedirectionofoneoftheprin cipalaxesoftheindicatrix .Thedirectionofthecurveata pointisalwaysoneoftheprincipaldirectionsonthesurfaceat thatpoint.Throughanygivenpointuponasurfacetwoper pendicularlinesofprincipalcurvaturepass.Thusthelines ofcurvaturedividethesurfaceintoasystemofinﬁnitesi malrectangles.Anasymptoticlineuponasurfaceisacurve whichhasateachpointthedirectionoftheasymptotesofthe indicatrix .Thedirectionofthecurveatapointisalways oneoftheasymptoticdirectionsuponthesurface.Through 416 VECTORANALYSIS anygivenpointofasurfacetwoasymptoticlinespass.These linesareimaginaryifthesurfaceisconvex.Evenwhenreal theydonotingeneralintersectatrightangles.Theangle betweenthetwoasymptoticlinesatanypointisbisectedby thelinesofcurvaturewhichpassthroughthatpoint. Thenecessaryandsufﬁcientconditionthatacurveupona surfacebealineofprincipalcurvatureisthatasoneadvances alongthatcurve,theincrementofdn,theunitnormaltothe surfaceisparalleltothelineofadvance.For dn:O-dr=(a dr=ce Thenevidentlydnanddrareparallelwhenandonlywhen drisparalleltoi’orj’ .Thestatementisthereforeproved . Itisfrequentlytakenasthedeﬁnitionoflinesofcurvature. Thedifferentialequationofalineofcurvatureis dnxdr=0. (6) Anothermethodofstatementisthatthenormaltothesurface, theincrementdnofthenormal,andtheelementdrofthe surfacelieinoneplanewhenandonlywhentheelementdr isanelementofalineofprincipalcurvature.Thedifferential equationthenbecomes [11dndr]=0. (7) Thenecessaryandsufﬁcientconditionthatacurveupona surfacebeanasymptoticline,isthatasoneadvancesalong thatcurvetheincrementoftheunitnormaltothesurfaceis perpendiculartothelineofadvance.For dn=droO dnodr=droOodr. IfthendnodriszerodroOdriszero.Hencedrisan asymptoticdirection .Thestatementisthereforeproved .It 418 VECTORANALYSIS (i' odrr (j' odrvo—a dr-dr+b drodr Hence 0 a0082(i’ ,dr)6cos2(j’ ,dr), or C acos2(i’ ,dr)6sin2(i’ ,dr).(10) Theinterpretationofthisformulaforthecurvatureofa normalsectionisasfollows:Whentheplanepturnsabout thenormaltothesurfacefromi’toj’ ,thecurvatureCofthe planesectionvariesfromthevalueawhentheplanepasses throughtheprincipaldirectioni’ ,tothevaluebwhenit passesthroughtheotherprincipaldirectionj’ .Thevalues ofthecurvaturehavealgebraicallyamaximumandminimum inthedirectionsoftheprincipallinesofcurvature.Ifaandbhaveunlikesigns,thatis,ifthesurfaceisconcavo-convex atP,thereexisttwodirectionsforwhichthecurvatureofa normalsectionvanishes.Thesearetheasymptoticdirections. Thesumofthecurvaturesintwonormalsections atrightanglestooneanotherisconstantandindependentof theactualpositionofthosesections.Forthecurvaturein onesectionis a0082(i’ ,dr)6sin2(i’ ,dr), andinthesectionatrightanglestothis asin2(i’ ,dr)1)cos2(i’ ,dr). Hence 01+02=a+b=OS (11) whichprovesthestatement. ItiseasytoshowthattheinvariantOzsisequaltothepro ductofthecurvaturesaandbofthelinesofprincipalcurv ature. $25a6 Hencetheequation x2OSa:O25 (12) THECURVATUREOFSURFACES 419 isthequadraticequationwhichdeterminestheprincipalcurv aturesaandbatanypointofthesurface.Bymeansofthis equationthescalarquantitiesaandbmaybefoundinterms ofF(x,y,z). (I—nu)oVVF(I—nu)w N VVF—2nnVVF+nnoVVFonn (nu-VVF nn)5=(nu-nnoVVF)s=(nnoVVF)5 (VVF)S(nuVVF)S N (VVF)S=V.VF,Hence O8 -VVFou. VovFVFVF3VVFHence O5N3 (13) VoVFVF-VVF-VE or O ,gNa TheseexpressionsmaybewrittenoutinCartesiancoordinates, buttheyareextremelylong.TheCartesianexpressionsfor O25areevenlonger.Thevectorexpressionmaybeobtained asfollows: w(I—nn)e—nn)2 N2 (I nn)2nu. Hence a)VF.(VVF)2VFVFVF:(VVF)2 ZSN4N4 (14) Givenanycurveuponasurface.Lettbeaunit tangenttothecurve,11aunitnormaltothesurfaceandma 420 VECTORANALYSIS vectordeﬁnedasnXt.Thethreevectorsn,t,mconstitute ani,j,ksystem .Thevectortisparalleltotheelementd Hencetheconditionforalineofcurvaturebecomes t n=0. (15) Hence modn=0 odu+n-dm . Hence ndm 0. Moreover modm 0. Hence tXdm 0, (16) or dmXdn=o. Theincrementsofmandofnandofrareallparallelincaseof alineofprincipalcurvature. Ageodeticlineuponasurfaceisacurvewhoseosculating planeateachpointisperpendiculartothesurface.Thatthe geodeticlineistheshortestlinewhichcanbedrawnbetween twopointsuponasurfacemaybeseenfromthefollowing considerationsofmechanics.Letthesurfacebesmoothand letasmoothelasticstringwhichisconstrainedtolieinthe surfacebestretchedbetweenanytwopointsofit.Thestring actingunderitsowntensionswilltakeapositionofequili briumalongtheshortestcurvewhichcanbedrawnuponthe surfacebetweenthetwogivenpoints.Inasmuchasthe stringrsatrestuponthesurfacethenormalreactionsofthe surfacemustlieintheosculatingplaneofthecurve.Hence thatplaneisnormaltothesurfaceateverypointofthecurve andthecurveitselfisageodeticline. Thevectorstanddtlieintheosculatingplaneanddeter minethatplane.Incasethecurveisageodetic,thenormal totheosculatingplaneliesinthesurfaceandconsequentlyis perpendiculartothenormaln.Hence 422 VECTORANALYSIS givensurfaceislaidoff.TheterminusP’ofthisnormallies uponthesurfaceofasphere.Ifthenormalstoasurfaceatall pointsPofacurvearethusconstructedfromthesameorigin, thepointsP’willtraceacurveuponthesurfaceofaunit sphere.Thiscurveiscalledthesphericalimageofthegiven curve.InlikemannerawholeregionTofthesurfacemay bemappeduponaregionI"thesphere.TheregionT’upon thespherehasbeencalledthehodogramoftheregionTupon thesurface.Ifdrbeanelementofareuponthesurfacethe correspondingelementupontheunitsphereis dnOodr. Ifdabeanelementofareauponthesurface,thecorre spondingelementuponthesphereisda’where(Art. da’:O20da . O=arr+byyO O2=abi’Xj’i’Xj’=ab1111. Hence da’abnnda. (19) TheratioofanelementofsurfaceatapointPtotheareaof itshodogramisequaltotheproductoftheprincipalradiiof curvatureatPortothereciprocaloftheproductoftheprin cipalcurvaturesatP . Itwasseenthatthemeasureofturningtotherightorleft ismdt.IfthenCisanycurvedrawnuponasurfacethe totalamountofturninginadvancingalongthecurveisthe integral. fmdt. (20) C Foranyclosedcurvethisintegralmaybeevaluatedina manneranalogoustothatemployed(page190)intheproof ofStokes’stheorem .Considertwocurves0and0”near THECURVATUREOFSURFACES 423 together.Thevariationwhichtheintegralundergoeswhen thecurveofintegrationischangedfromCtoO”is 8fm-dt=f8modt—fdmo6t+fd(m-8t). Theintegraloftheperfectdiﬁerentiald(m3t)vanishes whentakenaroundaclosedcurve.Hence Bfm-dtsmodt—fdmodt Theidemfactoris Smodt=8moIodt=8monnodt fort cltand8momvanish.Asimilartransformationmay beeffecteduponthetermatmo8t.Then Sfmodt=f(8monn-dt—dmounoBt). Bydiﬁerentiatingtherelationsmonz:0andnot0itis seenthat 8mou—mo3n noSt—3not dm-nz—m-dn n-dtz—dn-t. Hence8fm-dt=f(mo8nt-dn—moduta) —foSna . 424 VECTORANALYSIS Thedifferential8nXdnrepresentstheelementofareain thehodogramupontheunitsphere.Theintegral fnaazfnoda’ representsthetotalareaofthehodogramofthestripof surfacewhichliesbetweenthecurves0and Letthe curve0startatapointuponthesurfaceandspreadoutto anydesiredsize.Thetotalamountofturningwhichisre quiredinmakinganinﬁnitesimalcircuitaboutthepointis 2 Thetotalvariationintheintegralis f8fmodt=fmodt—27r. (21) ButifHdenotethetotalareaofthehodogram . Hencefm-dtz27r—H, or H=27r—fmodt, (22) 01‘H+fmodt=2rn Theareaofthehodogramoftheregionenclosed closedcurveplusthetotalamountofturningalongthatcurve isequalto27r.Ifthesurfaceinquestionisconvexthearea uponthespherewillappearpositivewhenthecurveuponthe surfaceissodescribedthattheenclosedareaappearspositive. If,however,thesurfaceisconcavo-convextheareauponthe spherewillappearnegative.Thismatterofthesignofthe hodogrammustbetakenintoaccountinthestatementmade above. 426 VECTORANALYSIS HarmonicVibrationsandBivectors Thedifferentialequationofrectilinear motionis d2so 2nx. dt2 Theintegralofthisequationmaybereducedbyasuitable choiceoftheconstantstotheform mzAsinnt. ThisrepresentsavibrationbackandforthalongtheX-axis aboutthepointa;0 .Letthedisplacementbedenotedby Dinplaceof:e.Theequationmaybewritten D=iAsinnt. Consider D iAsinntcosmx. Thisisadisplacementnotmerelynearthepointa: 0 2hbutalongtheentireaxisof:23.Atpointsx m” ,where hisapositiveornegativeinteger,thedisplacementisatall timesequaltozero .Theequationrepresentsastationary wavewithnodesatthesepoints.Atpointsmidwaybetween thesethewavehaspointsofmaximumvibration .Ifthe equationberegardedasinthreevariablesx,y,2itrepre sentsaplanewavetheplaneofwhichisperpendicularto theaxisofthevariablex . Thedisplacementgivenbytheequation D1=iAlcos(mac—nt) (1) islikewiseaplanewaveperpendiculartotheaxisofa:but notstationary .Thevibrationisharmonicandadvances alongthedirectioniwithavelocityequaltothequotientof HARMONICVIBRATIONSANDBIVECTORS 427 nbym .Ifvbethevelocity;ptheperiod;andlthewave length, n 271'272' 1 l fly— r727p n’m’ v2; Thedisplacement D2=jA2cos(ma—nt)(2) differsfromD1intheparticularthatthedisplacementtakes placeinthedirectionj,notinthedirectioni.Thewaveas beforeproceedsinthedirectionofa:withthesamevelocity. Thisvibrationistransverseinsteadoflongitudinal.Bya simpleextensionitisseenthat D=Acos(mac—nt) isadisplacementinthedirectionA .Thewaveadvances alongthedirectionof .13.Hencethevibrationisobliqueto thewave-front.Astillmoregeneralformmaybeobtained bysubstitutingmorform:13.Then D=Acos(mor—nt). (3) ThisisadisplacementinthedirectionA .Themaximum amountofthatdisplacementisthemagnitudeofA .The waveadvancesinthedirectionmobliquetothedisplace ment;thevelocity,period,andwave-lengthareasbefore. Somuchforrectilinearharmonicmotion .Elliptichar monicmotionmaybedeﬁnedbytheequation(p. d2r 2 dtzur. Thegeneralintegralisobtainedas r=Acosnt+Bsinnt. Thediscussionofwavesmaybecarriedthroughaspre viously.Thegeneralwaveofellipticharmonicmotion advancinginthedirection111isseentobe 428 VECTORANALYSIS D=Acos(mor—nt)—Bsin(mor—nt).(4) (ID dt—nAsin(mor-nt)i(5) isthevelocityofthedisplacedpointatanymomentinthe ellipseinwhichitvibrates.Thisisofcourseentirelydiffer entfromthevelocityofthewave. Aninterestingresultisobtainedbysubtractingthedis placementfromthevelocitymultipliedbytheimaginary unitv 1anddividedbyn . D n dt-Acos(m-r—nt)—Bsin(mor—nt) (6) +x/—1§Asin(m-r—nt)+Bsin(m-r V: .r-nD n dt—IB)6wn 0 Theexpressionhereobtained,asfarasitsformisconcerned, isanimaginaryvector.Itisthesumoftworealvectorsof whichonehasbeenmultipliedbytheimaginaryscalar1/1. Suchavectoriscalledabivectororimaginaryvector.The ordinaryimaginaryscalarsmaybecalledbiscalars.Theuse ofbivectorsisfoundveryconvenientinthediscussionof ellipticharmonicmotion .Indeedanyundampedelliptichar monroplanewavemayberepresentedasabovebythepro ductofabivectorandanexponentialfactor.Therealpart oftheproductgivesthedisplacementofanypointandthe pureimaginarypartgivesthevelocityofdisplacement reversedinsignanddividedbyn . Theanalytictheoryofbivectorsdiffersfromthatof realvectorsverymuchastheanalytictheoryofbiscalars diﬁersfromthatofrealscalars.Itisunnecessarytohave anydistinguishingcharacterforbivectorsjustasitisneed 430 VECTORANALYSIS r rXs=(r1X81 rs=(r181+r2 Twobivectorsorbiscalarsaresaidtobeconjugatewhen theirrealpartsareequalandtheirpureimaginaryparts diﬁeronlyinsign .Theconjugateofarealscalarorvector isequaltothescalarorvectoritself.Theconjugateofany sortofproductofbivectorsandbiscalarsisequaltothepro ductoftheconjugatestakeninthesameorder.Asimilar statementmaybemadeconcerningsumsanddifferences. (rl+ir2)X(rl 01+732)_irz)(r1r11'r2'i02l' l"l' lta)° Ifthebivectorr r1ir2bemultipliedbyarootofunity orcyclicfactorasitisfrequentlycalled,thatis,byanimagi naryscalaroftheform (7) where a2b21, theconjugateismultipliedbyaib,andhencethefour products areunalteredbymultiplyingthebivectorrbysuchafactor. Thusif (114 I I I I_r10r1 HARAIONICVIBRATIONSANDBIVECTORS 431 Acloserexaminationoftheeffectofmultiplyinga bivectorbyacyclicfactoryieldsinterestingandimportant geometricresults.Let r1’+’5 (COS97:Sin9)(11’5ta)°(8) Then r1’r1cosyr2sin9, o r2_rzcosq+rlsmg. ByreferencetoArt.129itwillbeseenthatthechangepro ducedintherealandimaginaryvectorpartsofabivectorby multiplicationwithacyclicfactor,ispreciselythesameas wouldbeproduceduponthosevectorsbyacyclicdyadic O=aa’+cosg—sing(cb’—bc’) usedasaprefactor.bandcaresupposedtobetwovectors collinearrespectivelywithrlandr2.8isanyvectornotin theirplane.Considertheellipseofwhichr1andr2area pairofconjugatesemi-diameters.Itthenappearsthatrl’ andr2’arealsoapairofconjugatesemi-diametersofthat ellipse .Theyarerotatedintheellipsefromrltowardr2,by asectorofwhichtheareaistotheareaofthewholeellipse asqisto2 Suchachangeofpositionhasbeencalledan ellipticrotationthroughthesectorq. Theellipseofwhichrlandr2areapairofconjugatesemi diametersiscalledthedirectionalellipseofthebivectorr. Whenthebivectorhasarealdirectionthedirectionalellipse reducestoarightlineinthatdirection .Whenthebivector hasacomplexdirectiontheellipseisatrueellipse.The angulardirectionfromtherealpartl‘ ltothecomplexpartr2 isconsideredasthepositivedirectioninthedirectional ellipse,andmustalwaysbeknown .Iftherealandimagi narypartsofabivectorturninthepositivedirectioninthe ellipsetheyaresaidtobeadvancedifinthenegativedirec tiontheyaresaidtoberetarded .Hencemultiplicationofa 432 VECTORANALYSIS bivectorbyacyclicfactorretardsitinitsdirectionalellipseby asectorequaltotheangleofthecyclicfactor. Itisalwayspossibletomultiplyabivectorbysuchacyclic factorthattherealandimaginarypartsbecomecoincident withtheaxesoftheellipseandareperpendicular. r=(cosg+isinq)(a+ib)wherea-bzo. ToaccomplishthereductionproceedasfollowsForm (a+ib) Ifaob=0, r—bob). Let ror=a+ib, band tan2g= a Withthisvalueofgtheaxesofthedirectionalellipseare givenbytheequation a+ib=(cosg—ising)r. Incasetherealandimaginarypartsaandbofabivector areequalinmagnitudeandperpendicularindirectionbotha andbintheexpressionforrorvanish .Hencetheangle gisindeterminate.Thedirectionalellipseisacircle.A bivectorwhosedirectionalellipseisacircleiscalledacircu larbivector.Thenecessaryandsufﬁcientconditionthata non-vanishingbivectorrbecircularis ror0, rcircular. If TheconditionrorO,whichforrealvectorsimpliesr0, isnotsufﬁcienttoensurethevanishingofabivector.The 434 VECTORANALYSIS Theﬁrstequationshowsthatr2and32areperpendicularand hences1and82areperpendicular.Moreover,thesecond showsthattheangulardirectionsfromr1toandfrom31to 32arethesame,andthattheaxesofthedirectionalellipses ofrandsareproportional. Hencetheconditionsforperpendicularityoftwobivectors whoseplanescoincidearethattheirdirectionalellipsesare similar,theangulardirectioninbothisthesame,andthe majoraxesoftheellipsesareperpendicular.1Ifbothvectors haverealdirectionstheconditionsdegenerateintotheper pendicularityofthosedirections.Theconditionstherefore holdforrealaswellasforimaginaryvectors. Letrandsbetwoperpendicularbivectorstheplanesof whichdonotcoincide.Resolver1andr2eachintotwocom ponentsrespectivelyparallelandperpendiculartotheplane ofs.Thecomponentsperpendiculartothatplanecontribute nothingtothevalueofro5.Hencethecomponentsofr1 andr2paralleltotheplaneof8formabivectorr’whichis perpendicularto8.Tothisbivectorandstheconditions statedaboveapply .Thedirectionalellipseofthebivectorr’ isevidentlytheprojectionofthedirectionalellipseofrupon theplaneof3. Hence,iftwobivectorsareperpendicularthedirectional ellipseofeitherbivectorandthedirectionalellipseofthe otherprojectedupontheplaneofthatonearesimilar,have thesameangulardirection,andhavetheirmajoraxesper pendicular. Considerabivectorofthetype nA (9) whereAandmarebivectorsandnisabiscalar.risthe positionvectorofapointinspace.Itisthereforetobecon 1Itshouldbenotedthattheconditionofperpendicularityofmajoraxesisnot thesameastheconditionofperpendicularityofrealpartsandimaginaryparts. HARMONICVIBRATIONSANDBIVECTORS 435 sideredasreal.tisthescalarvariabletimeandisalsoto beconsideredasreal.Let AAIiA2, m=m1imz, n=n1inz,D(AI—n1t—tngt) D(AIi e‘m’"e""(10) Ashasbeenseenbefore,thefactor(AI+iA2) representsatrainofplanewavesofellipticharmonicvibra tions.ThevibrationstakeplaceintheplaneofAIandA2, inanellipseofwhichAIandA2areconjugatesemi-diam eters.Thedisplacementofthevibratingpointfromthe centeroftheellipseisgivenbytherealpartofthefactor. Thevelocityofthepointreversedindirectionanddivided byu1isgivenbythepureimaginarypart.Thewavead vancesinthedirectionml.Theotherfactorsintheexpres sionaredampers.Thefactore‘m"risadamperinthe directionm2.Asthewaveproceedsinthedirection1112it diesaway .Thefactore""isadamperintime.Ifn2is negativethewavediesawayastimegoeson .Ifn2isposi tivethewaveincreasesinenergyastimeincreases.The presence(forunlimitedtime)ofanysuchfactorinanex pressionwhichrepresentsanactualvibrationisclearlyinad missible.Itcontradictsthelawofconservationofenergy . Inanyphysicalvibrationofaconservativesystemn2isne cessarilynegativeorzero. Thegeneralexpression(9)thereforerepresentsatrainof planewavesofellipticharmonicvibrationsdampedina deﬁnitedirectionandintime.Twosuchwavesmaybecom poundedbyaddingthebivectorswhichrepresentthem .If theexponentmor atisthesameforboththeresulting trainofwavesadvancesinthesamedirectionandhasthe 436 VECTORANALYSIS sameperiodandwave-lengthastheindividualwaves.The vibrations,however,takeplaceinadifferentellipse.Ifthe wavesare Aei(mor—nt)andBeumor—ar) theresultantis (A B)gum .r-no. Bycombiningtwotrainsofwaveswhichadvanceinopposite directionsbutwhichareinotherrespectsequalasystemof stationarywavesisobtained . Ae—mgcr .r—nt)Ae—mz'rei(—m1-r—nt) 6_tnt(6im1or6—tm1.r)2ACOS(m1106—mad: e—znt Thetheoryofbivectorsandtheirapplicationswillnotbe carriedfurther.Theobjectinenteringatalluponthisvery shortandcondenseddiscussionofbivectorswasﬁrsttoshow thereaderhowthesimpleideaofadirectionhastogiveway tothemorecomplicatedbutnolessusefulideaofadirectional ellipsewhenthegeneralizationfromrealtoimaginaryvectors ismade,andsecondtosetforththemannerinwhichasingle bivectorDmaybeemployedtorepresentatrainofplane wavesofellipticharmonicvibrations.Thisapplicationofhi vectorsmaybeusedtogivetheTheoryofLightawonderfully simpleandeleganttreatment .1 1SuchuseofbivectorsismadebyProfessorGibbsinhiscourseoflectureson TheElectromagneticTheoryofLight,”deliveredbiannuallyatYaleUniversity . Bivectorswerenotusedinthesecondpartofthischapter,becauseintheOpinion ofthepresentauthortheypossessnoessentialadvantageoverrealvectorsuntil themoreadvancedpartsofthetheory,rotationoftheplaneofpolarizationby magnetsandcrystals,totalandmetallicreﬂection,etc.,arereached.