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Vector Analysis Gibbs Wilson

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Google Books scan of the Yale University Press textbook Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, by Edwin Bidwell Wilson from lectures of J. Willard Gibbs (copyright 1901, later printings to 1922). The preface describes chapters on vector addition and products, vector calculus with divergence and curl, and the linear vector function. This is a published book by others, kept in Phil's math book downloads.

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Willard(Edwin) lItl7'Jb,GoogletKl .:...-:-~__ "";;;''' .<'~ --..:-.:..."\"---.. .........-.... THEPENNSYLVANIA STATEUNIVERSITY LIBRARIES AICSLIBRARY DI17'Jb,Google JPale16icmtmmal 10Ublication~ VECTOR ANALYSIS fL"------------, THBPBNNSYLVANIA STATBUNIVERSITY LIBRARIES DI17'Jb,Google Jllale15icentennial ~bltcation~ WiththeapprO'UalofthePresident andFellows ofYaleUniversity, aseriesofvolumes hasbun prepared byanumberoftheProfessors andIn structors, tobeissuedinconnection withthe Bicentennial Anniversary, asapartialindica tionofthecharacterofthestudiesinwhichthe University teachersareengaged. erhisurusofvolumes isrespectfully dedicated tf) VECTOR ANALYSIS ATEXT-BOOK FORTHEUSEOFSTUDENTS OFMATHEMATICS ANDPHYSICS FOUNDED UPONTHELECTURES OF J.WILLARD GIBBS,PH.D.,LL.D. ProfmorofMatlltma'iical PhyJiai"YaleU"iflerJity BY EDWIN BIDW'ELL WILSON, PH.D. I"Jtructor i"MathematicJ i"YaleU"iflerJity NEWHAVEN YALEUNIVERSITY PRESS MDCCCCXXII Copyright, I90I, ByYALEUNIVERSITY P"bli.;',d, D"""btr, U)OJ &,o"dPri",i"l, Ja""ary, J9J] T;'irdPri"ti"I,J"Iy, J9J6 Four';'Pri"ti"l, April,J922 PREFACE BYPROFESSOR GIBBS SINCEtheprinting ofashortpamphlet ontheElemenu oj VectorAnalysisintheyears1881-84, -neverpublished, but somewhat widelycirculated amongthosewhowereknownto beinterested inthesubject,-thedesirehasbeenexpressed inmorethanonequarter, thatthesubstance ofthattrea tise,perhaps infullerform,shouldbemadeaccessible to thepublic. As,however, theyearspassedwithout myfindingthe leisuretomeetthiswant,whichseemedarealone,Iwas verygladtohaveoneofthehearersofmycourseonVector Analysis intheyear1899-1900 undertake thepreparation of atext-book onthesubject. IhavenotdesiredthatDr.Wilson shouldaimsimply atthereproduction ofmylectures, butratherthatheshould uschisownjudgment inallrespects fortheproduction ofa text-book inwhichthesubjectshouldbesoillu8trated byan adequate numberofexamples astomeetthewantsofstu dentsofgeometry andphysics. J.WILLARD GIBBS. YALE UNIVERSITY, September, 1901. GENERAL PREFACE WHENIundertook toadaptthelecturesofProfessor Gibbs ·onVECTOR ANALYSIS forpublication intheYaleBicenten nialSeriel:!,Professor Gibbshimself wasalreadysofully engaged uponhisworktoappearinthesameseries,Elementary Principles inStatistical Mechanics, thatitwasunderstood no material assistance inthecomposition ofthisbookcouldbe expected fromhim.Forthisreasonhewishedmetofeel entirelyfreetousemyowndiscretion alikeintheselection ofthetopicstobetreatedandinthemodeoftreatment. Ithasbeenmyendeavor tousethefreedom thusgranted onlyinsofaraswasnecessary forpresenting hismethodin text-book form. Byfarthegreaterpartofthe material usedinthefollow ingpageshasbeentakenfromthecourseoflectures on VectorAnalysis delivered annually a.ttheUniversity by Professor Gibbs.Someuse,however, hasbeenmadeofthe chapters onVectorAnalysis inMr.OliverHeaviside's Elec tromagnetic Theory(Electrician Series,1893)andinProfessor Foppl'slecturesonDieMaxwell'sche TheoNederElectricitcit (Teubner, 1894). Myprevious studyofQuaternions has alsobeenofgreatassistance. Thematerial thusobtained hasbeenarranged intheway whichseemsbestsuitedtoeasymastery ofthesubject. ThoseArts.whichitseemedbesttoincorporate inthe textbutwhichforvariousreasonsmaywellbeomittedat thefirstreadinghavebeenmarkedwithanasterisk(-).Nu merousillustrative examples havebeendrawnfromgeometry, mechanics, andphysics. Indeed,alargepartofthetexthas todowithapplications ofthemethod. Theseapplications havenotbeensetapartinchapters bythemselves, buthave x GENERAL PREFACE heendistributed throughout thebodyofthebookasfastas theanalysis hasbeendeveloped sufficiently fortheiradequate treatment. Itishopedthatbythismeansthereadermaybe betterenabledtomakepractical useofthebook.Greatcare hasbeentakeninavoiding theintroduction ofunnecessary ideas,andinsoillustrating eachideathatisintroduced as tomakeitsnecessity evidentanditsmeaning easytograsp. Thusthebookisnotintended asacomplete exposition of thetheoryofVectorAnalysis, butasatext-book fromwhich somuchofthesubjectasmayberequired forpractical appli cationsmaybelearned. Henceasummary, including alist ofthemoreimpOltant formulre, andanumberofexercises, havebeenplacedattheendofeachchapter, andmanyless essential pointBinthetexthavebeenindicated ratherthan fullyworkedout,inthehopethatthereaderwillsupplythe details. Thesummary maybefoundusefulinreviewsand forreference. ThesubjectofVectorAnalysis naturally dividesitselfinto threedistinctparts.First,thatwhichconcerns addition and thescalarandvectorproducts ofvectors. Second,thatwhich concerns thedifferential andintegral calculus initsrelations toscalarandvectorfunctions. Third,thatwhichcontains thetheoryofthelinearvectorfunction. Thefirstpartis anecessary introduction tobothotherparts.Thesecond andthirdaremutually independent. Eithermaybetaken upfirst.Forpractical purposes inmathematical physicsthe secondmustberegarded asmoreelementary thanthethird. Butastudentnotprimarily interested inphysicswouldnat urallypassfromthefirstparttothethird,whichhewould probably findmoreattractive andeasythanthesecond. Following thisdivision ofthesubje<;t, themainbodyof thebookisdivided intosixchapters ofwhichtwodealwith eachofthethreepartsintheordernamed. Chapters 1.and II.treatofaddition, subtraction, scalarmultiplication, and thescalarandvectorproducts ofvectors. Theexposition hasbeenmadequiteelementary. Itcanreadilybeunder stoodbyandiseRpecially suitedforsuchreadersashavea knowledge ofonlytheelements ofTrigonometry andAna- GENERAL PREFACE xi lyticGeometry. ThosewhoarewellversedinQuaternions oralliedsubjects mayperhapsneedtoreadonlythesum maries. Chapters HI.andIV.contain thetreatment of thosetopicsinVectorAnalysis which,thoughoflessvalue tothestudents ofpuremathematics, areoftheutmostimpor tancetostudents ofphysics. Chapters V.andVI.dealwith thelinearvectorfunction. Tostudents ofphysicsthelinear vectorfunction isofparticular importance inthemathemati caltreatment ofphenomena connected withnon-isotropic media;andtothestudentofpuremathematics thispartof thebookwillprobably bethemostinteresting ofall,owing tothefactthatitleadstoMultiple Algebra ortheTheory ofMatrices. Aconcluding chapter, VII.,whichcontains the development ofcertainhigherpartsofthetheory,anumber ofa.pplications, andashortsketchofimaginary orcomplex vectors,hasbeenadded. Inthetreatment oftheintegral calculus, Chapter IV., questions ofmathematical rigorarise.Although modern theorists aredevoting muchtimeandthought torigor,and although theywilldoubtless criticisethisportionofthebook adversely, ithasbeendeemedbesttogivebutlittleattention tothediscussion ofthissubject. Andthemoresoforthe reasonthatwhatever systemofnotation beemployed ques tionsofrigorareindissolubly associated withthecalculus andoccasion nonewdifficulty tothestudent ofVector Analytlis, whomustfirstlearnwhatthefacts!loreandmay postpone untillaterthedetailed consideration oftherestric tionsthatareputuponthosefacts. Notwithstanding theeffortBwhichhavebeenmadeduring morethanhalfacentury tointroduce Quaternions into physicsthefactrema.insthattheyhavenotfoundwidefavor. Ontheotherhandtherehasbeenagrowing tendency espe ciallyinthelastdecadetowardtheadoption ofsomeformof VectorAnalysis. TheworksofHeaviside andFopplre ferredtobeforemaybecitedinevidence. Asyethowever nosystemofVectorAnalysis whichmakesanyclaimto completeness hasbeenpublished. InfactHeaviside says: "IaminhopesthatthechapterwhichInowfinishmay xii GENERAL PREFACE serveasastopgaptillregularvectorial treatises cometobe writtensuitableforphysicists, baseduponthevectorial treat mentofvectors" (Electromagnetic Theory,Vol.1.,p.305). Elsewhere inthesamechapterHeaviside hassetforththe claimsofvectoranalysis asagainstQuaternions, andothers haveexpressed similarviews. Thekeynote, then,toanysystemofvectoranalysis must beitspractical utility. This,Ifeelco~fident, wasProfessor Gibbs'spointofviewinbuilding uphissystem. Heusesit entirelyinhiscoursesonElectricity andMagnetism andon Electromagnetic TheoryofLight.Inwriting thisbookI havetriedtopresentthesubjectfromthispractical stand point,andkeepclearlybeforethereader's mindtheques tions:Whatcombinations orfunctions ofvectorsoccurin physicsandgeometry? Andhowmaytheseberepresented symbolically inthewaybestsuitedtofacileanalytic manip ulation? Thetreatment ofthesequestions inmodernbooks onphysicshasbeentoomuchconfined totheaddition and subtraction ofvectors. Thisisscarcely enough. Ithas beentheaimheretogivealsoanexposition ofscalarand vectorproducts, oftheoperator \1,ofdivergence andcurl whichhavegainedsuchuniversal recognition sincetheap pearance ofMaxwell's Treatise onElectricity andMagnetism, ofslope,potential, linearvectorfunction, etc.,suchasshall beadequate fortheneedsofstudents ofphysicsatthe presentdayandadapted tothem. Ithasbeenasserted bysomethatQuaternions, Vector Analysis, andallsuchalgebras areoflittlevalueforinvesti gatingquestions inmathematical physics. Whether this assertion shallprovetrueornot,onemaystillmaintain that vectorsaretomathematical physicswhatinvariants areto geometry. Aseverygeometer mustbethoroughly conver~ santwiththeideasofinvariants, soeverystudentofphysics shouldbeabletothinkintermsofvectors. Andthereis nowayinwhichhe,especially atthebeginning ofhissci entificstudies, cancometosotrueanappreciation ofthe ~mportance ofvectorsandoftheideasconnected withthem asbyworking inVectorAnalysis anddealingdirectlywith GENERAL PREFACE xiii thevectorsthemselves. Tothosethatholdtheseviewsthe success ofProfessor Flippl's Vorlesungen werTechniscke Mechanilc (fourvolumes, Teubner, 1897-1900, alreadyina secondedition), inwhichthetheoryofmechanics isdevel opedbymeansofavectoranalysis, canbebutanencour agingsign. Itakepleasure inthanking mycolleagues, Dr.M.B.Porter andProf.H.A.Bumstead, forassisting mewiththemanu script.Thegoodservicesofthelatterhavebeenparticularly valuable inarranging Chapters III.andIV.intheirpresent formandinsuggesting manyoftheillustrations usedinthe work.Iamalsounderobligations tomyfather,Mr.Edwin H.Wilson,forhelpinconnection bothwiththeproofsand themanuscript. Finally,Iwishtoexpressmydeepindebt ednesstoProfessor Gibbs. Foralthough hehasbeenso preoccupied astobeunabletoreadeithermanuscript or proof,hehasalwaysbeenreadytotalkmattersoverwith me,anditishewhohasfurnished mewithinspiration suf ficienttocarrythrough thework. EDWIN BIDWELL WILSON. YALEUNIVERSITY, October, 1901. PREFACE TOTHESECOND EDITION THEonlychangeswhichhavebeenmadeinthiseditionare afewcorrections whichmyreadershavebeenkindenoughto pointouttome. E.B.W. TABLE OFCONTENTS PREFACE BYPROFESSOR GIBBS GENERAL PREFACE CHAPTER I ADDITIO~ ANDSCALAR MULTIPLICATIONPAGE vii ix ARTS. 1-3 4 5 6-7 8-10 11 12 13-16 17 18-19 20-22 23-24 25SCALARS ANDVECTORS EQUAL A:;'DNULLVECTORS THEPOINT OFVIKWOFTHISCHAPTER SCALAR MULTIPLICATION. THII:NEGATIVE SlON ADDITION. THE:PARALLELOGRAM LAW, SUBTRACTION , LAWSGOVERNING THEFOREGOI:;'G OPERATIONS COMPONENTS OFVECTORS. VECTOR .:QUATIONS THETHR.:E UNITVECTORS i,j,k . ApPLICATIONS TOSUNDRY pROBLKMS INGEOMETRY, VKCTOR RELATION" INDEPENDENT 0.'THEORlOlN . CKNTKRS OFGRA\'ITY. BARYCENTRIC COORDINATES THEUSEOFVECTORS TODENOTE ARKAS SUMMARY OFCHAPTER I • EXERCISES ONCHAPTKR I CHAPTER II DIRECT A~DSKEWPRODUCTS OFVECTORS1 4 6 7 8 21 27 39 46 51 52 27-28 29-30 31-33 34,-35 36THEDIRECT, SCALAR, ORDOTPRODUCT OFTWOVECTORS 5." THEDI8TIUBUTIVE LAWA:;'DAppLICATIO:;'S 58 Till,S!{I':W, VF.CTOR, ORCROSS PIlODUCT OFTWOVECTORS 60 THEDISTRIIIUTIVE LAWASIlAPPLICATIONS 63 THETRIPLE pRODlJCTA'B C 67 • XVI CONTENTS ARTS. 37-88 39-40 41-42 43-45 46-47 48-50 51 52 53 54THESCALAR TRIPLE PRODUCT A'B X C OR[ABC] THEVECTOR TRIPLE PRODUCT A X(BXC) PRODUCTS OJ'MORI<: THAN THREE VI<:CTORS WITH APPLI CATIONS TOTRIGONOMKTRY. RECIPROCAL SYSTEMS OF'THREE VECTORS. SoLUTioN OFSCALAR ANDVECTOR EQUATIONS LINEAR IN ANUNKNOWN VECTOR SYSTKMS OFFORCES ACTING ONARIGID BODY • KIXEMATICS 0."ARIGID BODY CONDITIONS FOREQUILIBRIUM OFARIGID BODY' RELATIONS BETWEEN TWORIGHT-HANDED SYSTEMS OF THREE PERPE:lrDICULAR U:lrITVECTORS • PROBLEMS INGEOMETRY. PLANAR COl)RDINAn:S SUMMARY OFCHAPTER II EXERCI!!ES O~CHAPTER IIPAGE 68 71 75 81 87 92 97 101 104 106 "109 113 CHAPTER III THEDIFFERENTIAL CALCULUS OFVECTORS i1 77 7872 73 74-7657 58-59 GO 61 62 63-67 68 69 70 159 166 170 172 177115 120 125 131 133 136 ]88 147 149 150 ]52 155 15755-56.DERIVATIVES AXDDIFFERENTIALS OFVECTOR FUNCTIO:"S WITHHr:SPECT TOA!!CALAH VARIABLE CURVATURE ANDTORSION OFGAUCHE CURVES. KINEMATICS 0."APARTICLE. THEHODOGRAPH THEINSTANTANEOUS AXISOFROTATION . INTEGIl ..TIONWITHAPPLICATIONS TOKINEMATICS SCALAR FUNCTIONS OFPOSITIO:Ir INSPACE THEVECTOR DIFFERENTIATING OPERATOR \1 THE!!CALAR OPEHATon A,\1 VECTOR FUNCTION!! OFPOSITION INSPACK THEDIVERG.:NCE \1.ANDTHECURL\1X INTERPHETATION OFTHEDIVERGENCE \1' INTERPRETATION OFTHECUnL\1X LAWSOFOPERATION OF\1,\1',\1X THEPARTIAL APPLICATION OF\1.EXPANSION OFAVEC TORFUNCTION ANALOGOUS TOTAYLOR'S, THEOREM. ApPLICATION TOHYDROMECHANICS THEDIFFERENTIATING OPERATORS OFTHESECOND ORDER GEOMETRIC INTERPRETATIO:ll OFLAPLACE'S OPERATOR \1.\1ASTHEDISPERSION SUMMARY OFCHAPTER III• EXERCISES ONCHAPTER III CONTENTS xvii CHAPTER IV THEINTEGRAL CALCULUS OFVECTORS 179 184187 198 197 200 205 211 215222 280228ARTS. PAG. 79-80 LINEINTEGRAI.8 OFVECTOR FUNCTIOK8 WITH APPLICA- TION8 ~GAU88'8 THEOREM I82,STOKEI:l'S THEOREM \83,CONVERI:l'K OFSTOKE8'8 THEOREM WITHAPPLICATION8 \84iTRAN8FORMATION8 OFLINK,8URFACE, ANDVOLUME IN· I / TF.GRALS. GREEN'S THEOREM. • • • . \85REMARKS ONMULTIPLE-VALUED FUNCTION8 • 86-81 POTENTIAL. THEINTEGRATING OPERATOR" POT" 88COMMUTATIVE PROPERTY OFPOTAND\l 89RKMARKS UPONTHEFOREGOING • 90THEINTEGRATING OPERATORS" NEW," "LAP," ..MAX" 91RELATIONS BETWEEN THEINTEGRATING ANDDIFFER· ENTIATING OPERATORS 92THEPOTENTIAL "POT" ISASOLUTION OFPOISSON'S EQI;ATION SOLENOIDAL ANDIRROTATIONAL PARTI:l OFAVECTOR FUNCTION. CERTAIN OPERATORS ANDTHEIR INVERSE •2lH MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND MAXWKLLlANS 96CERTAIN BOUNDARY VALUE THEOREMS SUMMARY OFCHAPTER IV EXERCISES ONCHAPTER IV.~- 95 CHAPTER V LINEAR VECTOR FUNCTIONS 97-98 99 100 ]01 102 108-104 105-107 108LINEAR VECTOR FUNCTIONS DEFINED 260 DyADICS DEFINED • 264 ANYLINEAR VECTOR FUNCTIO:IJ MAYBEREPRESENTED BYADYADJC. PROPERTIE8 OFDYADJCI:l 266 THKNONION FORM OFADYADIC. 269 THEDYAD ORINDETERMINATE PROIlUCT OFTWOVEC- TOR8ISTHEM08TGENERAL. FUNCTIONAL PROPERTY OFTHESCALAR ANDVECTOR PRODUCTS. 271 PRODUCTS OFDYADICS 276 DEGREEA OFNI1LLITY OFDYADICS 282 THEIDEXFACTOR 288 XVlll ARTS. 109-110 111 112-114 115-116 117 118-119 120 121 122CONTENTS PAGS RECIPROCAL DYADICS. POWER8 ANDROOT8 OFDYADIC8 290 CON.JUGATE DYADICS. SELF-CONJUGATE ANDANTI- 8ELF-CONJUGATE PARTS OFADYADIC 294 ANTI-SELF-CONJUGATE DYADIC8. THEVECTOR PROD- UCT. QUADRANTAL VER80R8 297 REDUCTION OFDYADICS TONORMAL FORM 302 DOUBLE MULTIPLICATION OFDYADICS . 806 THESECOND ANDTHIRD OFADYADIC. 810 CONDITIONS FORDIFFERENT DEGREES OJ'NULLITY 313 NONION .·OIlM. DETRRMINANTS • 315 INVARIANTS OFADYADIC. THEHAMILTON-CAYLEY EQUATION 319 SUMMARY OJ'CHAPTER V 321 EXERCISES ONCHAPTER V 829 CHAPTER VI ROTATIONS ANDSTRAINS 123-124 HOMOGENEOUS 8TRAIN REPRESENTED BYADYADIC 832 121)-126 RoTATIONll ABOUT AFIXED POINT. VER80R8 334 127THEVECTOR SEMI-TANGENT OFVER8ION 389 128BIQUADRANTAL VBR80RS ANDTHEIR PRODUCTS 343 129CYCLIC DYADIC8 . 347 130RIGHT TENSORS . 351 131TONICS ANDCYCLOTONIC8 353 132REDUCTION OFDYADICS TOCANONICAL FORMS, TONICS, CYCLOTONICS, 81MPLE ANDCOMPLEX SHEARERS 356 SUMMARY OFCHAPTER VI 868 CHAPTER VII MISCELLANEOUS APPLICATIONS 136-142 QUADRIC 8URFACES 143-146 THEPROPAGATION OFLIGHT INCRY8TALS 147-148 VARIABLE DYADICS 149-157 CURVATURE OF8URFACES 158-162 HARMONIC VIBRATION8 ANDBIVBCTORS372 392 403 411 426 VECTOR ANALYSIS | VECTOR ANALYSIS CHAPTER I ADDITION ANDSCALAR MULTIPLICATION 1.]INmathematics andespecially inphysicstwovery different kindsofquantity presentthemselves. Consider, for example, mass,time,density, temperature, force,displacement ofapoint,velocity, andacceleration. Ofthesequantities somecanberepresented adequately byasinglenumber temperature, bydegrees onathermometric scale;time,by years,days,orseconds; massanddensity, bynumerical val ueswhicharewhollydetermined whentheunitofthescale isfixed.Ontheotherhandtheremaining quantities arenot capableofsuchrepresentation. Forcetobe8ureissaidtobe ofsomanypoundsorgramsweight; velocity, ofsomany feetorcentimeters persecond. Butinadditiontothiseach ofthemmustbeconsidered ashavingdirection aswellas magnitude. AforcepointsNorth,South,East,West,up, down,orinsomeintermediate direction. Thesameistrue ofdisplacement, velocity, andacceleration. Noscaleofnum berscanrepresent themadequately. Itcanrepresent only theirmagnitude, nottheirdirection. 2.JDefinition: Avectorisaquantity whichisconsidered aspossessing direction aswellasmagnitude. Definition: Ascalarisaquantity whichisconsidered aspos sessingmagnitude butnodirection. 2 VECTOR ANALYSIS o FlO.1.pThepositiveandnegativenumbersojordinary algebraareeM typicalscalars. Forthisreasontheordinary algebraiscalled scalaralgebrawhennecessary todistinguish itfmmthevector algebraoranalysitl whichisthesubjectofthisbook. Thetypicalvectoristhedisplacement ojtranslatwn inspace. Consider firstapointP(Fig.1).LetPbedisplaced ina straight lineandtakeanewposition pl. Thischangeofpositionisrepresented bythe linePp/.Themagnitude ofthedisplace mentisthelengthofPpI;thedirection of itisthedirection ofthelinePpIJromPto P'.Nextconsider aditlplacement notofone, butofallthepointsinspace.Letallthe pointsmoveinstraightlinesinthesamedirection andforthe samedilltanceD.Thisisequivalent toshifting spaceasa rigidbodyinthatdirection through thedistance Dwithout rotation. Suchadisplacement iscalledatranslatwn. It possesses direction andmagnitude. Whenspaceundergoes atranslation T,eachpointofspaceundergoes adisplacement equaltoTinmagnitude anddirection; andconversely if thedisplacement PpIwhichanyoneparticular pointPsuf fersinthetranslation Tisknown,thenthatofanyother pointQisalsoknown: forQQ'mustbeequalandparallel toPr. Thetranslation Tisrepresented geometrically orgraphically byanarrowT(Fig.1)ofwhichthemagnitude anddirection areequaltothoseofthetranslation. Theabsolute position ofthisarrowinspaceisentirelyimmaterial. Technically the arrowiscalledastrOh"e.It,;tailorinitialpointisitsorigin; anditsheadorfiMlpoint,itsterminus. Inthefigurethe orig-inisdesig-nated by0andtheterminus byT.Thisgeo metricquantity, astroke,isusedasthemathematical symbol forallvectors,justastheordinary positiveandnegative num bersareusedasthesymbols forall8calars. ADDiTION ANDSCALAR MULTiPLiCATION 3 •3.]Asexamples ofscalarquantities mass,time,den sity,andtemperature havebeenmentioned. Othersaredis tance,volume, moment ofinertia,work,etc.Magnitude, however, isbynomeansthesoleproperty ofthesequantities. Eachimpliessomething besidesmagnitude. Eachhasits owndistinguishing characteristics, asanexample ofwhich itsdimensions inthesensewellknowntophysicists may becited.Adistance 3,atime3,awork3,etc.,arevery different. Themagnitude 3is,however, aproperty common tothemall-perhaps theonlyone.Ofallscalarquanti t4tiespurenumber isthesimplest.Itimpliesnothingbut magnitude. Itisthescalarparexcellence andconsequently itisusedasthemathematical symbolforallscalars. Asexamples ofvectorquantities force,displacement, velo city,andacceleration havebeengiven.Eachofthesehas othercharacteristics thanthosewhichbelongtoavectorpure andsimple. Theconcept ofvectorinvolves twoideasand twoalone-magnitude ofthe vector anddirection ofthe vector. Butforceismorecomplicated. Whenitisapplied toarigidbodythelineinwhichitactsmustbetakeninto consideration; magnitude anddirection alonedonotsuf fice.Andincaseitisappliedtoanon-rigid bodythepoint ofapplication oftheforceisasimportant asthemagnitude or direction. Suchisfrequently trueforvectorquantities other thanforce.Moreover thequestion ofdimensions ispresent asinthecaseofscalarquantities. Themathematical vector, thestroke,whichistheprimary objectofconsideration in thisbook,abstracts fromalldirected quantities theirmagni tudeanddirection and,nothingbutthese;justasthemathe matical scalar,purenumber, abstracts themagnitude and thatalone. Henceonemustbeonhisguardlestfrom analogy heattribute someproperties tothemathematical vectorwhichdonotbelongtoit;andhemustbeevenmore carefullestheobtainerroneous resultsbyconsidering the 4 VECTOR ANALYSIS vectorquantities ofphysicsaspossessing noproperties other thanthoseofthemathematical vector.Forexample itwould neverdotoconsider forceanditseffectsasunaltered by shifting itparallel toitself. Thiswarning maynotbe necessary, yetitmaypossibly savesomeconfusion. 4.]Inasmuch as,takeninitsentirety, avectororstroke isbutasingleconcept,itmayappropriately bedesignated by oneletter. Owinghowever tothefundamental difference between scalarsandvectors,itisnecessary todistinguish carefully theonefromtheother.Sometimes, asinmathe maticalphysics,thedistinction isfurnished bythephysical interpretation. Thusifnhetheindexofrefraction it mustbescalar; m,themass,andt,thetime,arealso scalars; but!,theforce,anda,theacceleration, are vectors. When,however, thelettersareregarded merely assymbols withnoparticular physical significance some typographical difference mustbereliedupontodistinguish vectorsfromscalars. HenceinthisbookClarendon typeis usedforsettingupvectorsandordinary typeforscalars. Thispermitstheuseofthesameletterdifferently printed torepresent thevectoranditsscalarmagnitude.lThusif Cbetheelectriccurrentinmagnitude anddirection, Cmay beusedtorepresent themagnitude ofthatcurrent; ifgbe thevectoracceleration duetogravity, gmaybethescalar valueofthatacceleration; ifvbethevelocityofamoving mass, 'Vmaybethemagnitude ofthatvelocity. Theuseof Clarendons todenotevectorsmakesitpossible topassfrom directed quantities totheirscalarmagnitudes byamere changeintheappearance ofaletterwithout anyconfusing changeintheletteritself. Definition: Twovectorsaresaidtobeequalwhentheyhave thesamemagnitude andthesamedirection. 1Thisconvention, however, isbynomeansin\"ariably followed. InBOrne instances itwouldprovejnst88undesirable 88itisconvenient illothers.Itis chieflyvaluable illtheapplication ofvectorstophysics. A=PP'= 0T=T.ADDITION ANDSCALAR MUI,TIPLICATION 5 Theequality oftwovectorsAandBisdenoted bythe usualsign=.Thul:l A=B. Evidently avectororstrokeisnotalteredbyshifting it aboutparalleltoitselfinspace.HenceanyvectorA=Ppi (Fig.1)maybedrawnfromanyassigned point0asorigin; forthesegment PP'maybemovedparalleltoitselfuntil thepointPfallsuponthepoint0andP'uponsomepointT. Then Inthiswayallvectorsinspacemaybereplaced bydirected segments radiating fromonefixedpointO.Equalvectors inspacewillofcoursecoincide, whenplacedwiththeirter--- miniatthesamepointO.Thus(Fig.1)A=PP',andB=QQ', bothfalluponT=0T. Forthenumerical determination ofavectorthreescalars arenecessary. Thesemaybechoseninavarietyofways. Hr,cp,8bepolarcoordinates inspaceanyvectorrdrawn withitsoriginattheoriginofc<X5rdinates mayberepresented bythethreescalarsr,cp,8whichdetermine theterminus of thevector. (,I,8)r......r,.,..,. Orifx,y,zbeCartesian coordinates inspaceavectorrmay beconsidered asgivenbythedifferences ofthec<X5rdinates x', y',z'ofitsterminus andthosex,y,zofitsorigin. r......(x'-x,y'-y,z'-z). Ifinparticular theoriginofthevectorcoincide withthe originofcoordinates, thevectorwillberepresented bythe threec<X5rdinates ofitsterminus r......(x',y',z'). Whentwovectorsareequalthethreescalarswhich repre sentthemmustbeequalrespectively eachtoeach.Hence onevectorequality impliesthreescalarequalities. 6 VECTOR ANALYSIS Definition: AvectorAissaidtobeequaltoUTOwhenits magnitude Aiszero. SuchavectorAiscalledanullorUTOvectorandiswritten equaltonaughtintheusualmanner. Thus A=0ifA=O. Allnullvectorsareregarded asequaltoeachotherwithout anyconsiderations ofdirection. Infactanullvectorfromageometrical standpoint would berepresented byalinearsegment oflengthzero-thatisto I:lay,byapoint.Itconsequently wouldhaveawhollyinde terminate direction or,whatamounts tothesamething,noneat all.If,however, itberegarded asthelimitapproached bya vectoroffinitelength,itmightbeconsidered tohavethat direction whichisthelimitapproached bythedirection ofthe finitevector,whenthclengthdecreases indefinitely andap proaches zeroasalimit.Thejustification fordisregarding thisdirection andlooking uponallnullvectorsasequalis thatwhentheyareadded(Art.8)toothervectorsnochange occursandwhenmultiplied (Arts.27,31)byothervectors theproductiszero. 5.]Inextending tovectorsthefundamental operations ofalgebraandarithmetic, namely,addition, subtraction, and mUltiplication, caremustbeexercised notonlytoavoidself contradictory definitions butalsotolaydownusefulones. Boththeseendsmaybeaccomplished mostnaturally and easilybylookingtophysics(forinthatsciencevectorscon tinually present themselves) andbyobserving howsuch quantities aretreatedthere.IfthenAbeagivendisplace ment,force,orvelocity, whatistwo,three,oringeneral x timesA?What,thenegative ofA?AndifBbeanother, whatisthesumofAandB?Thatistosay,whatisthe equivalent ofAandBtakentogether? Theobviousanswers tothesequestions suggestimmediately thedesireddefinitions. ..4DDITION ANDSCALAR MULTIPLICA TION 7 ScalarMultiplication 6.]Definition: Avectorissaidtobemultiplied bya positivescalarwhenitsmagnitude ismultiplied bythatscalar anditsdirection isleftunaltered. ThusifvbeavelocityofnineknotsEastbyNorth,21times visavelocity oftwenty-one knotswiththedirection still EastbyNorth. Oriffbetheforceexerted uponthescale panbyagramweight,1000timesfistheforceexertedbya kilogram. Thedirection inbothcasesisvertically down ward. IfAbethevectorandxthescalartheproductofxandAis denoted asusualby xAorAx. Itis,however, morecustomary toplacethescalarmultiplier beforethemultiplicand A.Thismultiplication byascalar iscalledscalarmultiplication, anditfollowstheatlSociative law x(yA)=(xy)A=Y(cA) asinordinary algebraandarithmetic. Thisstatement isim mediately obvious whenthefactistakenintoconsideration thatscalarmultiplication doesnotalterdirection butmerely multiplies thelength. Definition: Aunitvectorisonewhosemagnitude isunity. AnyvectorAmaybelookeduponastheproduct ofaunit "ectorainitsdirection bythepositive scalarA,itsmagni tude. A=Aa=aA. Theunitvectoramaysimilarly bewrittenastheproductof Abyl/Aorasthequotient ofAandA. FIG.2.VECTOR ANALYSIS 7.]Definition: Thenegative sign,-,prefixed toavectox reversesitsdirection butleavesitsmagnitude unchanged. Forexample ifAbeadisplacement fortwofeettotheright, - Aisadisplacement fortwofeettotheleft.Againifthe strokeA EbeA,thestrokeEA,whichisofthesamelength asA Ebutwhichisinthedirection fromEtoAinsteadof fromAtoE,willbe-A.Another illustration oftheuse ofthenegative signmaybetakenfromNewton's thirdlaw ofmotion.IfAdenotean"action," -Awilldenotethe "reaction." Thepositivesign,+,maybeprefixed toavec tortocallparticular attention tothefactthatthedirection hasnotbeenreversed. Thetwosigns+and-whenused inconnection withscalarmultiplication ofvectorsfollowthe samelawsofoperation asinordinary algebra. Theseare symbolically ++=+;+-=-; -+=-;--=+: -(mA)=m(-A). Theinterpretation isobvious. Addition andSubtraction 8.]Theaddition oftwovectorsorstrokesmaybeb"eated mostsimplybyregarding themasdefining translations in space(Art.2).LetSbeonevectorandTtheother.LetP beapointofspace(Fig.2).Thetrans lationScarriesPintopisuchthatthe linePpiisequaltoSinmagnitude and direction. Thetransformation Twillthen carrypiintoP"-thelinepip"being paralleltoTandequaltoitinmagnitude. Consequently theresultofSfollowed by TistocarrythepointPintothepoint P".IfnowQbeanyotherpointinspace,SwillcarryQ intoQ/suchthatQQ'=SandTwillthencarryQ'intoQ" A.DDITION ANDSCALAR MULTIPLICATION 9 suchthatQ'Q"=T.ThusSfollowed byTcarriesQintoQ". Moreover, thetriangleQQ'Q"isequaltoPP'P".For thetwosidesQQ'andQ'Q",beingequalandparallelto8 andTrespectively, mustbelikewise parallel toPP'and pIpI!respectively whicharealsoparallelto8andT.Hence thethirdsidesofthetriangles mustbeequalandparallel Thatis QQ"isequalandparalleltoPP". AsQisanypointinspacethisisequivalent tosayingthat bymeansofSfollowed byTallpointsofspacearedisplaced thesameamountandinthesamedirection. Thisdisplace mentistherefore atranslation. Consequently thetwo translations 8andTareequivalent toasingletranslation R. Moreover if S=PP'andT=P'P",thenR=PP". ThestrokeRiscalledtheresultant orsumofthetwo strokes8andTtowhichitisequivalent. Thissumisde notedintheusualmannerby R=8+T. Fromanalogywiththesumorresultant oftwotranslations thefollowing definition fortheaddition ofanytwovectorsis laiddown. . Definition: ThesumorreRultant oftwovectorsisfound byplacingtheoriginoftheseconduponthetermin'us ofthe firstanddrawing the vector fromtheoriginofthefirsttothe terminus ofthesecond. 9.]Theorem. TheorderinwhichtwovectorsSandTare addeddoesnotaffectthesum. Sfollowed byTgivesprecisely thesameresultasTfollowed by8.Forlet8carryPintoP'(Fig.3);andT,P'intoPII. 8+TthencarriesPintoPII.Suppose nowthatTcarriesP intoP'''.ThelineppIIIisequalandparalleltoP'P".Con- 10 VECTOR ANALYSIS sequentlythepoints FIG.3. eithercasethesame. bywritingP,pI,P",andpililieattheverticesof aparallelogram. Hence p'"p"isequalandpar alleltoPP'.Hence8 camesP'"intoPll.Tfol lowedbyStherefore car riesPintoP"throughpI, whereas 8followed byT camesPintopllthrough pili.Thefinalresultisin Thismaybedesignated symbolically R=8+T=T+ S. ItistobenoticedthatS=PpIandT=Ppiliarethetwosides oftheparallelogram PpIp"pilIwhichhavethepointPall common origin;andthatR=Ppllisthediagonal drawn through P.Thisleadstoanother verycommon wayof statingthedefinition ofthesumoftwovectors. Iftwovectorsbedrawnfromthesameoriginandaparallelo grambecOllstructed uponthemassides,theirsumwillbethat diagonal whichpassesthroughtheircommon origin. Thisisthewell-knowll "parallelogram law"according to whichthephysicalvectorquantities force,acceleration, veloc ity,andangularvelocityarecompounded. Itisimportant to notethatincasethevectorsliealongthesamelinevector additionbecomesequivalent toalgebl"'"d.ic Ill'alaraddition. The lengthsofthetwovectorstobeaddedareaddedifthevectol'll havethesamedirection; butsubtracted iftheyhaveoppo sitedirections. Ineithercasethesumhasthesamedirection asthatofthegreatervector. 10.]Afterthedefinition ofthesumoftwovectorshas beenlaiddown,thesumofseveralmaybefoundbyadding together thefirsttwo,tothissumthethird,tothisthefourth, andsoonuntilallthevectorshavebeencombined intoasin- ADDITION ANDSCALAR MULTIPLICATION 11 gleone.Thefinalresultisthesameasthatobtained byplacing theoriginofeachsucceeding vectorupontheterminus ofthe preeeding oneandthendrawing atoncethe vector from theoriginofthefirsttotheterminus ofthelast.Incase thesetwopointscoincide thevectorsformaclosedpolygon andtheirsumiszero.Interpreted geometrically thisstates thatifanumberofdisplacements ll.,8,T...aresuchthatthe strokesll.,8,T...formthesidesofaclosedpolygon takenin order,thentheeffectofcarrying outthedisplacements isnil. Eachpointofspaceisbroughtbacktoitsstartingpoint.In terpreted inmechanics itstatesthatifanynumber offorces actatapointandiftheyformthesidesofaclosedpolygon takeninorder,thentheresultant forceiszeroandthepoint isinequilibrium undertheactionoftheforces. Theorderofsequence ofthevectorsinasumisofnocon sequence. Thismaybeshownbyprovingthatanytwoadja centvectorsmaybeinterchanged withoutaffecting theresult. Toshow A+B+C+D+E=A+B+D+C+1 LetA=0A,B=AB,C=BC,D=CD,E=DE. Then 0 E=A+ B+ C+ D+ E. LetnowB Cf=D.ThenCfBCDisaparallelogram and consequently CfD=C.Hence OE=A+B+D+C+E, whichprovesthestatement. Sinceanytwoadjacent vectors maybeinterchanged, andsincethesummaybearranged in anyorderbysuccessiveinterchanges ofadjacent vectors,the orderinwhichthevectorsoccurinthesumisimmaterial. 11.]Definition: Avectorissaidtobesubtracted whenit isaddedafterreversalofdirection. Symbolically, A - B=A+(-B). Bythismeanssubtraction isreduced toaddition andneeds 12 VECTOR ANALYSIS nospecialconsideration. Thereishowever aninteresting and important wayofrepresenting thedifference oftwovectors geometrically. LetA=OA,B=OB(Fig.4).Complete theparallelogram ofwhichAandB <.t---::--:.carethesides.Thenthediagonal oC=CisthesumA+Bofthe twovectors. Nextcomplete the parallelogram ofwhicr,Aand- B =0B'arethesides.Thenthedi- E agonal0D=Dwillbethesumof Aandthenegative ofB.Butthe segment ODisparallelandequal toBA.HenceBAmaybetakenasthedifference tothetwo vectorsAandB.Thisleadstothefollowing rule:Thediffer enceoftwovectorswhicharedrawnfromthesameoriginis thevectordrawnfromtheterminus ofthevectortobeBUb tractedtotheterminus ofthevectorfromwhichitissub tracted. Thullthetwodiagonals oftheparallelogram, which isconstructed uponAandBassides,givethesumanddif ferenceofAandB. 12.]Intheforegoing paragraphs addition, subtraction, and scalarmultiplication ofvectorshavebeendefinedandinter preted. Tomakethedevelopment ofvectoralgebramathe matically exactandsystematic itwouldnowbecomenecessary todemonstrate thatthesethreefundamental operations follow thesameformallawsasintheordinary scalaralgebra.,al thoughfromthestandpoint ofthephysical andgeometrical interpretation ofvectorsthismayseemsuperfluous. These lawsare Ia: Ib: II: IlIa: IIIb: III.: .m(nA)=n(mA)=(mn)A, (A+B)+ C=A +(B+C), A+B=B+A, (m+n)A=mA +nA, m(A+B)=TIlA+mB, -(A+B)= -A-B. ADDTTION ANDSCALAR MULTIPLICATION 13 I..istheso-called lawofassociation andcommutation of thescalarfactorsinscalarmultiplication. Ibisthelawofassociation forvectorsinvectoraddition.It statesthatinaddingvectorsparentheses maybeinsertedat anypointswithout.altering theresult. IIisthecommutative lawofvectoraddition. III.isthedistributive lawforscalarsinscalarmUltipli cation. IIIbi8thedistributive lawforvectorsinscalarmultipli cation. III.isthedistributive lawforthenegative sign. Theproofsoftheselawsofoperation dependuponthose propositions inelementary geometry whichhavetodealwith thefirstproperties oftheparallelogram andsimilartriangles. Theywillnotbegivenhere;butitissuggested thatthe readerworkthemoutforthesakeoffixingthefundamental ideasofaddition, subtraction, andscalarmultiplication more clearlyinmind.Theresultofthelawsmaybesummed up inthestatement: Thelauswhichgovernaddition, subtraction, andscalar multiplication ofvectorsareidentical withthosegoverning these operations inO1'dinary scalaralgebra. Itisprecisely thisidentityofformallawswhichjustifies theextension oftheuseofthefamiliar signs=,+,and ofarithmetic tothealgebraofvectorsanditisalsothis whichensuresthecorrectness ofresultsobtained byoperat ingwiththosesignsintheusualmanner. Onecautiononly needbementioned. Scalarsandvectorsareentirelydifferent sortsofquantity. Forthisreasontheycanneverbeequated toeachother-exceptperhapsinthetrivialcasewhereeachis zero.Forthesamereasontheyarenottobeaddedtogether. Solongasthisisborneinmindnodifficulty needbeantici patedfromdealingwithvectorsmuchasiftheywerescalars. Thusfromequations inwhichthevectorsenterlinearlywith 14 VECTOR ANALYSIS scalarcoefficients unknown vectors maybeeliminated or foundby-solution inthesamewayandwiththesamelimita tionsasinordinary algebra; fortheeliminations andsolu tionsdependsolelyonthescalarcoefficients oftheequations andnotatallonwhatthevariables represent. Iffor instance aA +bB +cC+dD=0, thenA,B,C,orDmaybeexpressed intermsoftheother three as1D= -d(aA+bB+cC). Andtwovectorequations suchas and3A+4B=B 2A+3B=P andyieldbytheusualprocesses thesolutions A=3B-4P B=3P-2B. Components ofVectors 13.]Definition: Vectors aresaidtobecollinear when theyareparalleltothesameline;coplanar, whenparallel tothesameplane.Twoormorevectorstowhichnoline canbedrawnparallelaresaidtobenon-collinear. Threeor morevectorstowhichnoplanecanbedrawnparallelare saidtobenon-eoplanar. Obviously anytwovectorsare coplanar. Anyvectorbcollinear withamaybeexpressed asthe productofaandapositive ornegative scalarwhichisthe ratioofthemagnitude ofbtothatofa.Thesignispositive whenbandahavethesamedirection; negative, whenthey haveopposite directions. IfthenOA=a,thevectorrdrawn ADDITION ANDSCALAR MULTIPLICA TION 15 fromtheorigin0toanypointoftheline0Aproduced in eitherdirection is r=xa. (1) Ifxbeavariablescalarparameter thisequation maythere foreberegarded asthe(vector) equation ofallpointsinthe line0A.LetnowBbeanypointnot upontheline0Aorthatlineproduced -_!I'e. rR.--.-"ineitherdirection (Fig.5). Let0B=b.Thevectorbissurely notoftheformxa.DrawthroughB FIG.5. alineparallelto0AandletRbeany pointuponit.ThevectorBRiscollinear withaandis consequently expressible asxa.Hencethevectordrawn from0toRis orOR=OB+BR r=b+xa. (2) • FIG.6.Thisequation mayberegarded asthe(vector)equation of allthepointsinthelinewhichisparalleltoaandofwhich Bisonepoint. 14.]Anyvectorrcoplanar withtwonon-collinear vectors aandbmayberesolved intotwocomponents paralleltoa andbrespectively. Thisresolution may beaccomplished byconstructing thepar allelogram (Fig.6)ofwhichthesidesare paralleltoaandbandofwhichthedi agonalisr.Ofthesecomponents oneis :x;a;theother,yb.xandyarerespec tivelythescalarratios(takenwiththe propersign)ofthelengthsofthesecomponents tothelengths ofaandb.Hence r=xa+yb (2)' isatypicalformforanyvectorcoplanar withaandb.If severalvectorsrl,r2,ra.,.maybeexpressed inthisformas 16 VECTOR ANALYSU~ rl=Xla+Ylb, r2=x2a+Y2b, rs=x3a +Ysb. theirsumristhen r=rl+ r2+ rs+...=(xl+x2+Xs+ )a +(III+Y2+Ys+ )b. Thisisthewell-known theorem thatthecomponents ofa sumofvectorsarethesumsofthecomponents ofthose vectors.Ifthevectorriszeroeachofitscomponents must bezero.Consequently theonevectorequation r=0is equivalent tothetwoscalarequations Xl+x2+x3+=0>r=O. (3) Yl+Y2+Y3+=0 xa FIG.7.y'"o:13.•:1'--._- _.•_•••_. "':;a+'y~_ - :~-----_..--C,-------:_-/ ·ry15.]Anyvectorrinspacemayberesolved intothree components paralleltoanythreegivennon-coplanar vectors. Letthevectorsbea,b, andc.Theresolution maythenbeaccom plishedbyconstructing theparallelopiped (Fig. 7)ofwhichtheedges areparalleltoa,b,and candofwhichthedi agonalisr.Thispar allelopiped maybe drawneasilybypassing threeplanesparallelre spectively toaand.b,bandc,candathrough theorigin0 ofthevectorr;andasimilarsetofthreeplanesthrough its terminus R.Thesesixplaneswillthenbeparallel inpairs ADDITION ANDSCALAR MULTIPLICATION 17 andhenceformaparallelopiped. Thattheintersections of theplanesarelineswhichareparallel toa,orb,orcis obvious. Thethreecomponents ofrarexa,Yb,andZc; where :z;,y,andzarerespectively thescalarratios(takenwith thepropersign)ofthelengthsofthesecomponents tothe lengthofa,b,andc.Hence r=xa+yb+zc (4) (5)isatypicalformforanyvectorwhatsoever inspace.Several vectorsrl'r2'f8•••maybeexpressed inthisformas rl=Xla+Ylb+zlC, f2=x2a+Y2b+Z2C, ra=Xsa +Yab +zaC, Theirsumfisthen r=fl+f2+fa+...=(Xl+x2+xa+ )a +(Y1+Y2+Ya+ )b +(zl+Z2+za+..-)c. t Ifthevectorriszeroeachofitsthreecomponents iszero. Consequently theonevectorequation r=0isequivalent to thethreescalarequations Xl+x2+xa+=0> Y1+Y2+Ya+=0 r=O. ".~+Z2+Za+=0 Shouldthevectorsallbecoplanar withaandb,allthecom ponentsparalleltocvanish. Inthiscaaetherefore theabove equations reducetothosegivenbefore., 16.]Iftwoequalvectorsareexpressed intermsofthe samethreenon-coplanar vectors,thecorresponding scalarco efficients areequal. 18 LetVECTOR ANALYSIS r=r', r=xa+yb+zc, r'=x'a+y'b+z'C, Then x=x',!I=y',z=z'. Forr -r'=0=(x-x')a+(y-y')b+(z-z')c. Hence x-x'=0,y-y'=0,z-z'=O. Butthiswouldnotbetrueifa,b,andcwerecoplanar. In thatcaseoneofthethreevectorscouldbeexpressed interms oftheothertwoas c=ma+nb. TheJlr =xa+yb +zc=(x+mz)a +(y+nz)b, r'=x'a +y'b +z;c =(x'+mz')a +(y'+nz')b, r -r'=[(x+mz)-(x'+mz')]a, +[(y+nz)-(y'+nz')]b=O. Hencetheindividual components ofr -r'inthedirections aandb(supposed different) arezero. Hence x+m z=x'+mz' y+n z=y'+nz'. Butthisbynomeansnecessitates x,y,ztobeequalrespec tivelyto:rI,y',z'.Inasimilarmannerifaandbwerecol linearitisimpossible toinferthattheircoefficients vanish individually. Thetheoremmayperhapsbestatedasfollows: Incasetwoequal'Vectorsareexpressed intermsofone'Vector, ortwonon-collinear vectors,orthreenon-coplanar vectors,the corresponding scalarcoefficients areequal.Butthisisnotne cessarily trueiftMtwovectorsbecollinear; ortMthreevectors, coplanar. Thisprinciple willbeusedintheapplications (Arts.18etseq.). TMThreeUnitVectorsi,j,k. 17.]Intheforegoing paragraphs themethodofexpress ingvectorsintermsofthreegivennon-coplanar oneshasbeen explained. Thesimplestsetofthreesuchvectorsistherect- ADDITION ANDSCALAR MULTIPLICA.TION 19 angularsystemfamiliarinSolidCartesian Geometry. This rectangular systemmayhoweverbeeitheroftwoverydistinct types.Inonecase(Fig.8,firstpart)theZaxis 1liesupon thatsideoftheXY-planeonwhichrotation through aright anglefromtheX-axistotheY:axisappears counterclockwise orpositiveaccording totheconvention adopted inTrigonome try.Thisrelationmaybestatedinanotherform.IftheX axisbedirectedtotherightandthey:.axisvertically, the Z-axiswillbedirected towardtheobserver. OriftheX axispointtowardtheobserver andtheY-axistotheright, theZ-axiswillpointupward. Stillanother methodofstate- z z Left-handed.... ,.. yj--...;..-:;i Right-handed FIG.8. mentiscommoninmathematical physicsandengineering. If aright-handed screwbeturnedfromtheX-axistotheY axisitwilladvance alongthe(positive) Z-axis. Suchasys temofaxesiscalledright-handed, positive, orcounterclock wise.llItiseasytoseethattheY-axisliesuponthatsideof theZX-plane onwhichrotation fromtheZ-axistotheX axisiscounterclockwise; andtheX-axis,uponthatsideof 1BytheX-,yo,orZaxisthepoaitivehalfofthataxisismeant. TheXy. planemeanstheplanewhichcontains theX-andY-axis,i.e.,theplane %=O. ~Aconvenient right-handed systemandonewhichisalwaysavailable consists ofthethnmb,firstfinger,andsecondfingeroftherighthand.lfthethumband firstfingerbestretched outfromthepalmperpendicular toeachother,andifthe secondfingerbebentovertowardthepalmatrightanglestofirlltfinger,aright bandedsystemisformedbythefingerstakenintheorderthumb,firstfinger, IIeCOndfinger. 20 VECTOR A.NALYSIS theYZ-planeonwhichrotation fromtheY-axistotheZ. axisiscounterclockwise. Thusitappearsthattherelation between thethreeaxesisperfectly symmetrical 80longasthe samecyclicorderX YZX Y isobserved.Ifaright-handed screwisturnedfromoneaxistowardthenextitadvances along.thethird. Intheothercase(Fig.8,secondpart)theZ-axisliesupon thatsideoftheXY-planeonwhichrotationthrougharight anglefromtheX-axistotheY-axisappears clockwise orneg ative.TheY-axisthenliesuponthatsideoftheZX-plane onwhichrotation fromtheZ-axistotheX-axisappeam clockwise andasimilarstatement maybemadeconoerning theX-axisinitsrelationtotheYZ-plane. Inthiscase,too, therelation between thethreeaxesissymmetrical solong asthesamecyclicorderXYZXY ispreserved butitisjust theoppositeofthatintheformercase.Ifaleft-handed screw isturnedfromoneaxistowardthenextitadvances along thethird.HencethissJstemiscalledleft-handed, negative, orclockwise.) Thetwosystems arenotsuperposable. Theyaresym metric. Oneistheimageoftheotherasseenina mirror.IftheX-andY-axesofthetwodifferent systemsbe superimposed, theZ-axeswillpointinopposite directions. Thusonesystemmaybeobtained fromtheotherbyreversing thedirection ofoneoftheaxes.Alittlethought willshow thatiftwooftheaxesbereversed indirection thesystemwill notbealtered,butifallthreebesoreversed itwillbe. Whichofthetwosystems beused,matterslittle.Butin asmuchastheformulre ofgeometry andmechanics differ slightlyinthematterofsign,itisadvisable tosettleoncefor allwhichshallbeadopt.ed. Inthisbooktheright-handed or counterclockwise systemwillbeinvariably employed. 1Aleft-handed systemmaybeformedbythelefthandjustasaright-handed ouewasformedbytheright. ADDITION ANDSCALAR JIULT1PLlCATION 21 Definition,: Thethreelettersi,i.kwillbereserved tode notethreevectorsofunitlengthdrawnrespectively inthe directions oftheX-,Y-,andZ-axesofaright-handed rectan gularsystem. Intermsofthesevect.ors,anyvectormaybeexpressed as r=xi+Yi+zk. (6) Thecoefficients x.Y.zaretheordinary Cartesian coijrdinates oftheterminus ofrifitsoriginbesituatedatthe-originof coordinates. Thecomponents ofrparalleltotheX-,Y-,and Z-axesarerespectively xi,Yi.zk. Therotations aboutifromitok,aboutifromktoi,and aboutkfromitoiareallpositive. Bymeansofthesevectorsi,i,ksuchacorrespondence is established between vectoranalysis andtheanalysisinCar tesiancoOrdinates thatitbecomes pOSRible topassatwill fromeitheronetotheother.There il:lnothingcontradic torybetween them.Onthecontrary itisoftendesirable orevennecessary totranslate theformulre obtained by vectormethods intoCartesian coordinates forthesakeof comparing themwithresultsalready knownanditis stillmorefrequently convenient topassfromCartesian analysis tovectorsbothonaccount ofthebrevitythereby obtained andbecausethevectorexpressions showforththe intrinsic meaning oftheformulre. .Applieations -18.]Problems inplanegeometry mayfrequently besolved easilybyvectormethods. Anytwonon-collinear vectorsin theplanemaybetakenasthefundamental onesintemlsof whichallothersinthatplanemaybeexpressed. Theorigin mayalsobeselected atpleasure. Oftenitispossible to 22 VECTOR ANALYSIS FIG.9.makesuchanadvantageous choiceoftheoriginandfunda mentalvectorsthattheanalytic workofsolutionismaterially simplified. Theadaptability ofthevectormethodisa.bout thesameasthatofobliqueCartesian coordinates withdiffer entscalesuponthetwoaxes. Exa.mple1:Thelinewhichjoinsonevertexofaparallelo gramtothemiddlepointofanopposite sidetrisectsthediag onal(Fig.9). LetABeDbetheparallelogram, BEthelinejoiningthtl vertexBtothemiddlepointBoftheside AD,Rthepointinwhichthislinecutsthe diagonal AG.ToshowARisonethirdof AG.ChooseAasorigin,AiJandADasthe twofundamental vectors 8andT.Then ACisthesumof8andT.LetAR=R.Toshow 1R=8(8+T). R=AR=A.E+E R=!.T+x(8-!.T)2 2 ' wherexistheratioofERtoBB-an unknown scalar. And R=Y(8+T), whereyisthescalarratioofARtoACtobeshownequal 1toS' Hence or1 12T+x(8-2T)=Y(8+T) 1x8+2(1-x)T=Y8+YT. Hence,equating corresponding coefficients (Art.16), x=y, 12(1-x)=y. ADDITION ••NDSC.HAR MULTIPLICA TION 28 Fromwhich Inasmuch asxisalso}thelineEBmwitbetrisected 88 wellasthediagonal AC. Example ~:Ifthrough anypointwithinatriangle lines bedrawnparalleltothesidesthesumoftheratiosofthese linestotheircorresponding sidesis2. LetABCbethetriangle,Rthepointwithinit.Choose .Aasorigin,A BandACasthetwofundamental vectors8 andT.Let AR=R=m8+nT. (a) m8isthefractionofABwhichiscutoffbythelinethrough RparalleltoAC.Theremainder ofA Bmustbethefrac tion(1-m)8.Consequently bysimilartriangles theratioof thelineparallel toAatothelineAaitselfis(1-m). Similarly theratioofthelineparalleltoA BtothelineA B itselfis(1-n).NextexpressRintennsof8andT -8the thirdsideofthetriangle. Evidently from(a) R=(m+n)8+n(T-8). Hence(m+n)8isthefraction ofABwhichiscutoffbythe linethroughRparalleltoBC.Consequently bysimilartri anglestheratioofthislinetoBeitselfis(m+n).Adding thethreeratios (1-m)+(1-n)+(m+n)=2, andthetheorem isproved. Example3:Iffromanypointwithinaparallelogram lines bedrawnparalleltothesides,thediagonals oftheparallelo gramsthusfonnedintersect uponthediagonal ofthegiven parallelogram. LetABeD beaparallelogram, Rapointwithinit,K M andLNtwolinesthrough Rparallel respectively toABand VECTOR ANALYSIS AD,thepointsK,L,M,NlyinguponthesidesDA,AB. Ba,aDrespectively. Toshowthatthediagonals KNand LMofthetwoparallelograms KRND andLBMR meet onAa.ChooseAasorigin,ABandADasthetwofunda mentalvectors8andT.Let 11.=A R=m8+11,T, andletPbethepointofinte1'8ection ofK NwithLM. Then Hence andK N=K R+R N=m8+(1-11,)T, P=AP=AK+ xKN, LM=(1-m)8+11,T, P=AP=AL+yLM. P=11,T+x[m8+(1~11,)T], P=m8+Y[(1-m)8+11,TJ. Equatingcoefficien ta. xm=m+y(1-m) yn=n+x(1-n) Bysolution,11,x= ,m+n-l m y=m+n-t" Substituting eitherofthesesolutions intheexpression forp. theresultis mn P=;;-+-11,_1(8+T), whichshowsthatPiscollinear withAa. •19.]Problems inthreedimensional geometry maybe solvedinessentially thesamemannerasthoseintwo dimen sions.Inthiscasetherearethreefundamental vectorsin termsofwhichallotherscanbeexpressed. Themethodof solution isanalogous tothatinthesimpler case.Two andHenceADDITION ANDSCALAR MULTIPLICATION 25 expressions forthesamevectorareusuallyfound.Theco efficients ofthecorresponding termsareequated. Inthisway theequationsbetween threeunknown scalarsareobtained fromwhichthosescalarsmaybedetermined bysolution and thensubstituted ineitheroftheexpressions fortherequired vector. Thevectormethodhasthesamedegreeofadapta bilityastheCartesian methodinwhichobliqueaxeswith different scalesareemployed. Thefollowing examples like thoseintheforegoing sectionareworkedoutnotsomuchfor theirintrinsic valueasforgainingafamiliarity withvectors. Example1:LetABODbeatetrahedron andPany pointwithiuit.JointheverticestoPandproduce thelines untiltheyintersect theopposite facesinA',B',0',D'.To show PA'PB'PO'PD' AA'+BB'+00'+DD'=1. ChooseAasorigin,andtheedgesAB,A0,ADasthe threefundamental vectorsB,C,D.LetthevectorAPbe p=AP=1B+rnC+nD, A'=AA'=klP=kl(lB+rnC+nD). Also A'=AA'=AB+BA'. ThevectorBA'iscoplanar withBd=C-BandBD= D-B.Henceitmaybeexpressed intermsofthem. A'=B+Xl(C-B)+YI(D-B). Equating coefficients klm=Xl' kln=YI' kll=1 -Xl-Yl' k_1 l-l+m+n PA'kl-lAA'=-r;-=1-(l+m+n). 26 VECTOR ANALYSIS Inlikemanner AB'=x2C+'lI3D and AB'=AB+BB'= B+k2(P-B). Hence x3C+'lI3D= B+k2(lB+mC+nD-B) and 0=1+k3(l-1). x3=k3m. 'lI3=kzn. 1Hence k3=--l-l and PB'kz- 1l BB'=~= . Inthesamewayitmaybeshownthat pa' PD'aa,=mand DIJ,=n. Addingthefourratiostheresultis 1-(l+m+n)+l+m+n=1. Example 13:Tofindalinewhichpassesthroughagiven pointandcutstwogivenlinesinspace. Letthetwolinesbefixedrespectively bytwopointsA andB.aandDoneach.Let0bethegivenpoint.Choose itasoriginandlet A=0A.B=0B.C=0a.D=0D. AnypointPofABmaybeexpressed as P=op=0A+xA B=A+x(B-A). AnypointQofaDmaylikewise bewritten Q=0Q=(Fa+'lIaD=C+'lI(D-C). IfthepointsPandQlieinthesamelinethrough O.PandQ arecollinear Thatis P=zQ. ADDITION ANDSCALAR MULTIPLICA nON 27 Beforeitispossible toequatecoefficients oneofthefour vectorsmustbeexpressed intermsoftheotherthree. Let D=lA+mB+11.O. Then P=A+x(B-A) =z[0+Y(lA+mB+nO-0)]. Hence 1 -x=zYl, x=zym, o=z[1+y(11.-1)J. Hencemx=--,l+m 1y=--,I-n 1-nz=--'l+m A~O~8 FIG.10.Substituting inPandQ P =lA+mB, l+m nO-DQ= .11.-1 Eitherofthesemaybetakenasdefining alinedrawnfrom0 andcuttingABandCD. VectorRelations independent oftkeOrigin 20.]Example 1 :TodividealineABinagivenratio m:n(Fig.10). Chooseanyarbitrary point0as origin. Let0A=Aand0B=B. TofindthevectorP=0 Pofwhich theterminus PdividesABinthe ratiom:n. _ - m- mP=OP=OA+--AB=A+ --(B-A).m+n m+n Thatis, P=nA+mB. (7)m+n 28 VECTOR ANALYSIS Thecomponents ofPparalleltoAandDareininverseratio tothesegments A PandPBintowhichthelineA Bis dividedbythepointP.IfitshouldsohappenthatPdivided thelineABexternally, theratioA P/PBwouldbenega tive,andthesignsofmand 11,wouldbeopposite, butthe formula wouldholdwithoutchangeifthisdifference ofsign inmand11,betakenintoaccount. Example'2: Tofindthepointofintersection ofthemedians ofatriangle. ChoosetheOl;gin0atrandom. LetA Babethegiven triangle. Let0A=A,0 B=D,aud0()=O.LetA',B',a' berespectively themiddlepointsofthesidesopposite the verticesA,B,a.LetMbethepointofintersection ofthe medians andJiI=0JIthevectordrawntoit.Then JiI=0J[=0 A+xAA'=A+x[CD-A);(0-A)J and JiI=0ill=0 B+Y BB'=D+y[CO-D);(A-D)} Assuming that0hasbeenchosenoutsideoftheplaneofthe trianglesothatA,D,0arenon-coplanar, corresponding coeffi. cientsmaybeequated. 11-x=2"y, 1 "2x=1 -y, 1 1 2"x=2"y. Hence Hence2x=Y=s' ADDITION ANDSCALAR MULTIPLICA 7'ION 29 Thevectordrowntothemedianpointofatriangle isequal toonethirdofthesumofthevectorsdrawntothevertices. Intheproblems ofwhichthesolution hasjustbeengiven theorigincouldbechosenarbitrarily andtheresultisin dependent ofthatohoice. Henceitisevenpossible todisre gardtheoriginentirely andreplacethevectorsA,B,0,etc., bytheirterminiA,B,0,etc.Thusthepointsthemselves becomethesubjectsofanalysisandtheformulm read p=nA+mB m+n and1M=3(A+B+0). Thisistypicalofawholeclassofproblems solublebyvector methods. InfactanyJYUrelygeometric relation between the different partsofafiguremustnecessarily beindependent oftheoriginassumed fortheanalytic demonstration. In somecases,suchasthoseinArts.18, 19,thepositionofthe originmaybespecialized withregardtosomecrucialpoint ofthefiguresoastofacilitate thecomputation; butinmany othercasesthegenerality obtained byleavingtheoriginun specialized andundetermined leadstoasymmetry which renderstheresultsjustaseasytocompute andmoreeasy toremember. c-Theorem: Thenecessary andsufficient condition thata vectorequation represent arelationindependent oftheorigin isthatthesumofthescalarcoefficients ofthevectorsoc onesideofthesignofequality isequaltothesumofthe coefficients ofthevectorsupontheotherside.Orifallthe termsofavectorequation betransposed toonesideleaving zeroontheother,thesumofthescalarcoefficients must bezero. Lettheequation writteninthelatterformbe aA+bB+c0+dD+...=O. 80 VECTOR ANALYSIS Changetheoriginfrom0to0'byaddingaconstant vector R=00'toeachofthevectorsA,B,0,D·...Theequation thenbecomes a(A+R)+b(B+R)+c(0+R)+d(D+R)+=0 =aA+bB+cO+dD+.,.+R(a+b+c+d+). Ifthisistobeindependent oftheoriginthecoefficient ofR mustvanish. Hence . a+b+c+d+...=o. Thatthiscondition isfulfilled inthetwoexamples cited. isobvious. If Ifp=nA+mB, m+n n m1=--+-m+nm+n •=~(AtB+0), 1 1 11=3+3"+3' •21.]TheneoeS8ary andsufficient condition thattwo vectorssatisfyanequation, inwhichthesumofthescalar coefficients iszero,isthatthevectorsbeequalinmagnitude andindirection. Firstlet andaA+bB=O a+b=O. ItisofcourseRS8umed thatnotboththecoefficients aandb vanish.Iftheydidtheequation wouldmeannothing. Sub stitutethevalueofaobtained fromthesecondequation into thefirst. Hence-bA+bB=O. A=B. A-B=OADDITION A.NDSCALAR MULTIPLICATION 31 SecondlyifAandBareequalinmagnitude anddirection theequation subsistsbetween them.Thesumofthecoefficients iszero. 1.--Thenecessary andsufficient condition thatthreevectors satisfyanequation, inwhichthesumofthescalarcoeffiCients_~ iszero,isthatwhendrawnfromacommon origintheytermi- I/~ nateinthesamestraightline.l- Firstlet aA+bB+cC=0 and a+b+c=O. Nota.llthecoefficients a,b,c,vanishortheequations wouldbemeaningless. Letcbeanon-vanishing coefficient. Substitute thevalueofaobtained fromthesecondequation intothefirst. or-(b+c)A+bB+cC=0, c(0-A)=b(A-B). Hencethevectorwhichjoinstheextremities ofCandAis collinear withthatwhichjoinstheextremities ofAandB. HencethosethreepointsA,B,alieonaline.Secondly supposethreevectorsA=0A,B=0B,C=0adrawnfrom thesameorigin0terminate inastraight line.Thenthe vectors A B=B-AandAa=C-A arecollinear. Hencetheequation (B-A)=x(C-A) subsists. Thesumofthecoefficients onthetwosidesis thesame. £.Thenecessa.ry andsufficient condition thatanequation, inwhichthesumofthescalarcoefficients iszero,subsist 1Vectonwhichhaveacommon originandterminate inonelinearecalledby Hamilton" termino-collinear." 82 VECTOR ANALYSIS between fourvectors, istha.tifdrawnfromacommon origin theyterminate inoneplane.1 Firstlet andaA+bD+cC+dD=O a+b+c+d=O. Letdbeanon-vanishing coefficient. Substitute thevalue ofaobtained fromthelastequation intothefirst. -(b+c+d)A+bD+cC+dD=0, or d(D-A)=b(A-D)+c(A-C). ThelineADiscoplanar withABandAC.Henceallfour terminiA,B,C,DofA,D,C,Dlieinoneplane.Secondly supposethattheterminiofA,B,C,Ddolieinoneplane. ThenAD=D-A,A C=C-A,andA B=D-Aareco planarvectors. Oneofthemmaybeexpressed intermsof theothertwo.Thisleadstotheequation l(B-A)+m(C-A)+n(D-A)=0, wherel,m,andnarecertainscalars. Thesumofthecoeffi cientsinthisequation iszero. Between anynvevectorsthereexistsoneequation thesum ofwhosecoefficients iszero. LetA,D,C,D,Ebethefivegivenvectors. Formthe differences E-A,E-D,E-C,B-D. Oneofthesemaybeexpressed intermsoftheotherthree -orwhatamounts tothesamethingtheremustexistan equation betweenthem. k(E-A)+l(E-D)+m(E-C)+n(E-D)=O. Thesumofthecoefficients ofthisequation iszero. 1VectOrllwhichhaTeacommon originandterminate inoneplanearecalled byHamilton" Urmino<omplanar." c FIG.11.A.DDITION ANDSCA.LAR MULTIPLICATION 33 •22.]Theresultsoftheforegoing sectionaffordsimple solutions ofmanyproblems connected solelywiththegeo metricproperties offigures. Special theorems, thevector equations oflinesandplanes,andgeometric netsintwoand threedimensions aretakenupinorder. Example 1:Ifalinebedrawnparalleltothebaseofa triangle, thelinewhichjoinstheopposite vertextotheinter sectionofthediagonals ofthe trapezoid thusformedbisectsthe base(Fig.11). LetABCbethetriangle,ED thelineparalleltothebaseCB,A.. ;.:......'B Gthepointofintersection ofthe "'"f/::::.»?..... diagonals EBandDCof~hetra- '.::.;;...... \~jr.···pezoidCBDE,andFtheintersec- 0 tionofA GwithCB.Toshow thatFbisectsCB.Choosethe originatrandom. Letthevectorsdrawnfromittothe variouspointsofthefigurebedenoted bythecorresponding Clarendons asusual.ThensinceEDisbyhypothesis paral leltoCB,theequation E- D=n(0-B) holdstrue.Thesumofthecoefficients isevidently zeroas itshouldbe.Rearrange thetermssothattheequation takesonthefOnD E-nO=D-nB. ThevectorE-nOiscoplanar withEandO.Itmustcut thelineEC.TheequalvectorD-nBiscoplanar withD andB.ItmustcutthelineDB.Consequently thevector represented byeithersideofthisequation mustpassthrough the,aointA.Hence ' E-nO=D-nB=xA. 3 VECTOR ANALYSIS However thepointsE,(J,andAlieuponthesamestraight line.Hencetheequation whichconnects thevectors E,C, andAmustbesuchthatthesumofitscoefficients iszero. Thisdetermines xas1 -n. Hence E-nC=D-nB=(1-n)A. Byanotherrearrangement andsimilarreasoning E+nB=D+nC=(1+n)G. Subtract thefirstequation fromthesecond: n(B+C)=(1+n)G-(1-n)A. ThisvectorcutsB(JandAG.Itmusttherefore bea multiple ofFandsuchamultiple thatthesumofthecoeffi cientsoftheequations whichconnect B,C,andForG,A, andFshallbezero. Hence Hencen(B+C)=(1+n)G-(1-n)A=2nF. F=B+C, 2 andthetheorem hasbeenproved. Theproofhascovered considerable spacebecauseeachdetailofthereasoning has beengiven.Inreality,however, theactualanalysishascon sistedofjustfourequations obtained simplyfromthefirst. Example ~:Todetermine theequations ofthelineand plane. LetthelinebefixedbytwopointsAandBuponit.Let Pbeanypointoftheline.Chooseanarbitrary origin. ThevectorsA,B,andPterminate inthesameline.Hence and ThereforeaA+bB+pP=O a+b+P=o. p=aA+bB. a+b ADDITION ANDSCALAR MULTIPLICATION 85 and ThereforeFordifferent pointsPthescalarsaandbhavedifferent values. Theymaybereplaced byxandy,whichareused moregenerally torepresent variables. Then p=xA+yB. x+y Letaplanebedetermined bythreepoints.A.,B,andC. LetPbeanypointoftheplane.Chooseanarbitrary origin. ThevectorsA,B,C,andPterminate inoneplane.Hence aA+bB+cC+pP=O a+b+c+P=O. p=aA+bB+cC. a+b+c Asa,b,c,varyfordifferent pointsoftheplane,itismore customary towriteintheirsteadx,y.z. p=:rA+yB+zC. x+y+z ". FIG.12.Example3:Thelinewhichjoinsonevertexofacom pletequadrilateral totheintersection oftwodiagonals dividestheopposite sideshar monically (Fig.12). LetA,B,C,Dbefourvertices ofaquadrilateral. LetABmeet CDinafifthvertexE,and.A.D meetBCinthesixthvertexF. £·'::;'---~-L..::r--..::.:.l8 Letthetwodiagonals ACand BDintersect inG.Toshow thatFGintersects .A.BinapointE'andCDinapointE" suchthatthelines.A.BandCDaredivided internally at E'andE"inthesameratioastheyaredividedexternally byE.Thatisto'showthatthecrossratios (.A.B.EE')=(CD.EE")=-1. 86 VECTOR ANALYSIS Choosetheoriginatrandom. ThefourvectorsA,B,C,D drawnfromittothepointsA,B,0,Dterminate inone plane.Hence andaA+bB+cC+dD=O a+b+c+d=O. orDivide:Separate theequations bytransposing twoterms: aA+cC=-(bB+dD), a+c= -(b+d).l G=aA+cC=bB+dD. a+c b+d Inlikemanner J!'=aA+dD=bB+cC. a+d b+c Form: (a+c)G-(a+d)J!'_cC-dD (a+c)-(a+d)(a+c)-(a+d) (a+.c)G-(a+d)J!' =cC-dD=E". c-d c-d(a) (b)Separate theequations againanddivide: aA+bB =cC+dD =E. a+b c+d HenceEdivideaABintheratioa:bandO.Dintheratio c:d.Butequation (a)showsthatE"dividesODinthe ratio-c:d.HenceEandE"divideODinternally and externally inthesameratio.Whichofthetwodivisions is internalandwhichexternal depends upontherelativesigns ofcandd.Iftheyhavethesamesigntheinternal point ofdivisionisE;ifopposite signs,itisE".Inasimilarway E'andEmaybeshowntodivideABharmonically. Example4-:Todiscussgeometric nets. Byageometric netinaplaneismeantafigurecomposed ofpointsandstraightlinesobtained inthefollowing manner. Startwithacertainnumberofpointsallofwhichlieiuone ADDITION ANDSCALAR MULTIPLICATION 37 plane.Drawallthelines joining thesepointsinpairs. Theselineswillintersect eachotherinanumberofpoints. Nextdrawallthelineswhichconnect thesepointsinpairs. Thissecondsetoflineswilldetermine astillgreaternumber ofpointswhichmayinturnbejoinedinpairsandsoon. Theconstruction maybekeptupindefinitely. Ateachstep thenumber ofpointsandlinesinthefigureincreases. Probably themostinteresting caseofaplanegeometric netis thatinwhichfourpointsaregiventocommence with. Joining thesetherearesixlineswhichintersect inthree pointsdifferent fromthegivenfour.Threenewlinesmay nowbedrawninthefigure. Thesecutoutsixnewpoints. Fromthesemorelinesmaybeobtained andsoon. Totreatthisnetanalytically writedowntheequations andaA+bB+cO+dD=0 a+b+c+d=O(c) . whichsubsistbetween thefourvectorsdrawnfromanunde- termined origintothefourgivenpoints. Fromtheseitis possible toobtain E=aA+bB =cO+dD, a+b c+d p=aA+cO =bB+dD, a+c b+d G=aA+~~ =bB:+-cO, a+d b+c bysplitting theequations intotwopartsanddividing. Next fourvectorssuchasA,D,E,Pmaybechosenandtheequa tionthesumofwhosecoefficients iszeromaybedetermined. Thiswouldbe -aA+dD+(a+b)E+(a+c)P=O. Bytreating thisequation as(c)wastreatednewpointsmay beobtained 38 VECTOR ANALYSIS H=-aA+dD=(a+b)E+(a+c)F, -a+d 2a+b+c 1=-aA+(a+b)E =dD+(a+c)F, b a+c+d X=-aA+(a+c)F =dD+(a+b)E. c a+b+d Equations between othersetsoffourvectorsselectedfrom A,B,C,D,E,F,Gmaybefound;andfromthesemorepoints obtained. Theprocessoffindingmorepointsgoesforward indefinitely. Afulleraccount ofgeometric netsmaybe foundinHamilton's" Elements ofQu,aternions," Book1. Asregards geometric netsinspacejustawordmaybe said.Fivepointsaregiven.Fromthesenewpointsmaybe obtained byfindingtheintersections ofplanespassedthrough setsofthreeofthegivenpointswithlinesconnecting the remaining pairs.ThecOll8truction maythenbecarriedfor wardwiththepointsthusobtained. Theanalytic treatment issimilartothatinthecaseofplanenets.Thereare fivevectorsdrawnfromanundetermined origintothegiven fivepoints. Between thesevectorsthereexists!Inequation thesumofwhosecoefficients iszero.Thisequation maybe separated intopartsasbeforeandthenewpointsmaythus beobtained. If aA+bB+ cC+dD+eE=O and thena+b+c+d+e=0, F =aA+bB=cC+dD+eE, a+b c+d+e aA+cC bB+dD+eEH= = ,a+b b+d+c aretwoofthepointsandothersmaybefoundinthesame way.Netsinspacearealsodiscussed byHamilton, lac.cit. .ADDITION ANDSCALAR MULTIPLICA TlON 39 OentersofGravity •23.]Thecenterofgravityofasystemofparticles may befoundveryeasilybyvectormethods. Thetwolawsof physicswhich will beassumed arethefollowing: 10 •Thecenterofgravityoftwomasses(considered as situatedatpoints)liesonthelineconnecting thetwomasses anddividesitintotwosegments whichareinversely pro portional tothemassesattheextremities. 20 •Infindingthecenterofgravityoftwosystems of masseseachsystemmaybereplac~d byasinglemassequal inmagnitude tothesumofthemassesinthesystemand situated atthecenterofgravityofthesystem. GiventwomassesaandbsituatedattwopointsAandB. TheircenterofgravityGisgivenby G_aA,+bB - a+b'(8) wberethevectorsarereferred toanyongmwhatsoever. Thisfollowsimmediately fromlaw1audtheformula (7) fordivision ofalineinagivenratio. Thecenterofgravityofthreemassesa,b,csituatedatthe threepointsA,B,0maybefoundbymeansoflaw2.The massesaandbmaybeconsidered asequivalent toasingle massa+bsituatedatthepoint Then HenceaA,+bB a+b• aA,+bBG=(a+b)----+cCa+b a+b+c G=aA,+bB+cC. a+b+c 40 VECTOR ANALYSIS Evidently thecenterofgravityofanynumberofmasses a,b,c,d,...situated atthepointsA,B,a,D,...may befoundinasimilarmanner. Theresultis aA+bB+cC+dD+···G= .(9)a+b+c+d+ ... Theorem1:Thelineswhichjointhecenterofgravityofa triangle tothevertices divideitintothreetriangles which areproportional tothemassesattheop positevertices(Fig.13).LetA,B,a betheverticesofatriangle weighted withmassesa,b,c.Let0bethecen terofgravity. JoinA,B,ato0and produce thelinesuntiltheyintersect theopposite sidesinA',B',a'respectively. Toshowthat theareas o Ba:0 a A:GAB:A B a=a:b:c:a+b+c. Thelastproportion between A Baanda+b+ccomes fromcompounding thefirstthree.Itis,however, usefulin thedemonstration. A B a A A'A 0 GA'_b+cLoBa=GA'=GA.'+GA'-a+ Hencea+b+c a InasimilarmannerBaAa+b+c GaA b andaABa+b+c=OAB c Hencetheproportion isproved. Theorem ~:Thelineswhichjointhecenterofgravityof atetrahedron totheverticesdividethetetrahedron intofour ADDITION ANDSCAL.4.R MUL7'lPLICATION 41 tetrahedra whichareproportional tothemallSesattheoppo sitevertices. LetA,B,a,Dbethevefticesofthetetrahedron weighted respectively withweightsa,b,c,d.Letabethecenterof gravity. JoinA,B,a,Dtoaandproduce thelinesuntil theymeettheopposite facesinA',B',a',D'.Toshowthat thevolumes BaDa:aDAa:DABa:ABaa:ABaD. =a:b:c:d:a+b+c+d. BaDA_AA'_Aa aA'_b+c+d1 BaDa-aA'-aA'+GA'-a+ a+b+c+d------a InlikemanneraDAaa+b+c+d----aDAB b andDABa a+b+c+d DABa-c andABaa a+b+c+d AB(!iJ-d whichprovestheproportion. •24.]Byasuitablechoiceofthethreemasses,a,b,clo catedattheverticesA,B,a,thecenterofgravityamay bemadetocoincide withanygivenpointPofthetriangle. Ifthisbenotobviousfromphysical considerations itcer tainlybecomes sointhelightoftheforegoing theorems. ForinorderthatthecenterofgravityfallatP,itisonly necessary tochoosethemassesa,b,cproportional tothe areasofthetriangles PBa,paA,andPABrespectively. Thusnotmerelyonesetofmassesa,b,cmaybefound,but aninfinitenumberofsetswhichdifferfromeachotheronly byacommon factorofproportionality. Thesequantities 42 VECTOR ANALYSIS a,b,cmaytherefore belookeduponascoordinates ofthe pointsPinsideofthetriangleABO.Toeachsetthere corresponds adefinite pointP,amdtoeachpointPthere corresponds aninfinitenumber ofsetsofquantities, which however donotdifferfromoneanother exceptforafactor ofproportionality. ToobtainthepointsPoftheplaneABOwhichlieoutside ofthetriangleABOonemayresorttotheconception of negative weights ormasses. Thecenterofgravityofthe .masses2and-1situatedatthepointsAandBreapectively wouldbeapoint0dividing thelineABexternally inthe ratio1 :2.Thatis OA:OB=1:2. AnypointofthelineA Bproduced mayberepresented by asuitablesetofmassesa,bwhichdifferinsign.Similarly anypointPoftheplaneABOmayberepresented bya suitable setofmassesa,b,cofwhichonewilldifferinsign fromtheothertwoifthepointPliesoutsideofthetriangle ABO.Inasmuch asonlytheratiosofa,b,andcareim portanttwoofthequantities mayalwaysbetakenpositive. Theideaofemploying themassessituatedatthevertices ascoordinates ofthecenterofgravityisduetoMobiusand wuspublished byhiminhisbookentitled" Derbarycentrische Ualcul,"in1827.Thismaybefairlyregarded asthestarting pointofmodernanalytic geometry. Theconception ofnegative masseswhichhavenoexistence innaturemaybeavoided byreplacing themassesatthe vertices bytheareasofthetriangles 0B0,0 0A,and GABtowhichtheyareproportional. Thecoordinates of apointPwouldthenbethreenumbers proportional tothe areasofthethreetriangles ofwhichPisthecommon vertex; andthesidesofagiventriangleABO,thebases.Thesign oftheseareasisdetermined bythefollowing definition. ADDITION ANDSCALAR MULTIPLICATION 43 Definition: Thearea.ABaofatriangle issaidtobe positive whenthevertices .A,B,afolloweachotherinthe positive orcounterclockwise direction uponthecirclede scribedthrough them.Theareaissaidtobenegative when thepointsfollowinthe negative orclockwise direction. Cyclicpermutation oftheletterstherefore doesnotalter thesignofthearea. .AB a=Ba.A=a.AB. Interchange oftwoletterswhichamounts toareversal of thecyclicorderchanges thesign. .Aa B=B.Aa=aB.A=-.ABa. IfPbeanypointwithinthetriangletheequation PAB+P B a+pa.A=.AB a musthold.ThesamewillalsoholdifPbeoutsideofthe triangle provided thesignsoftheareasbetakenintocon sideration. Theareasorthreequantities proportional to themmayberegarded ascoordinates ofthepointP. Theextension oftheideaof"barycentric" coordinates to spaceisimmediate. Thefourpoints.A,B,a,Dsituated at theverticesofatetrahedron areweighted withmassa,b,c,d respectively. Thecenterofgravity Gisrepresented by thesequantities orfourothersproportional tothem.To obtainpointsoutside ofthetetrahedron negative masses maybeemployed. Orinthelightoftheorem 2,page40, themassesmaybereplaced bythefourtetrahedra which areproportional tothem.Thentheideaofnegative vol umestakestheplaceofthatofnegative weights. Asthis ideaisofconsiderable importance later,abrieftreatment of itheremaynotbeoutofpla.ce. Definition: Thevolume.ABODofatetrahedron issaid tobepositive whenthetriangle .ABaappears positive to 44 VECTOR ANALYSIS theeyesituatedatthepointD.Thevolume isnegative iftheareaofthetriangle appearnegative. Toma.kethediscussion ofthesignsofthevarious tetrahedra perfectly clearitisalmostnecessary tohavea. solidmodeLAplanedrawing isscarcely sufficient. Itis difficulttoseefromitwhichtriangles appearpositiveand whichnegative. Thefollowing relations willbeseento holdifamodelbeexamined. Theinterchange oftwolettersinthetetrahedron ABOD changesthesign. ACBD=CBAD=BACD=DBCA =ADCB=A BDC=-ABCD. Thesignofthetetrahedron foranygivenoneofthepos sibletwenty-four arrangements ofthelettersmaybeobtained byreducing thatarrangement totheorderABC Dby meansofanumberofsuccessive interchanges oftwolettehl. Ifthenumberofinterchanges iseventhesignisthesame 88thatofABCD,.ifodd,opposite. Thus CADB=-CABD=+ACBD=-ABCft IfPisanypointinsideofthetetrahedron ABCDthe equation ABCP-BCDP+CDAP-DABP=ABCD holdsgood.ItstillistrueifPbewith0ut thetetrahedron provided thesignsofthevolumes betakenintoconsidera tion.Theequation maybeputintoafonnmoresymmetri calandmoreeasilyremembered bytransposing allthetenns toonenumber. Then ABCD+BCDP+CDPA+D P A B+PABC=O. Theproportion intheorem 2,page40,dresnotboldtrue ifthesignsoftbetetrabedra beregarded. Itshouldread BCDG: CDOA:DGAB: GABC:ABCD =a:b:c:d:a+b+c+d. ADD1T/ON ANDSCALAR MULTIPLICA 7'/ON 45 IfthepointGliesinsidethetetrahedron a,b,c,drepre sentquantities proportional tothemasseswhichmustbe locatedatthevertices.A,B,a,Drespectively ifGistobethe centerofgravity,IfGliesoutsideofthetetrahedron theymay stillberegarded asmassessomeofwhicharenegative -or perhapsbettermerelyasfournumbers whoseratiosdetermine thepositionofthepointG.Inthismannerasetof"bary centric" cotll'dinates isestablished forspace. ThevectorPdrawnfromanindeterminate origintoany pointoftheplaneA Bais(page35) p=xA+yB+zO. x+y+z Comparing thiswiththeexpression G=aA+bB+cO a+b+c itwillbeseenthatthequantities x,y,zareinrealitynothing morenorle88thanthebarycentric cotlrdinates ofthepointP withrespecttothetriangleABO,Inlikemannerfrom equation p=xA+yB+zO+wD x+y+z+w whichexpre88es anyvectorPdrawnfromanindeterminate originintermsoffourgivenvectorsA,B,0,Ddrawnfrom thesameorigin,itmaybeseenbycomparison with G=aA+bB+c0+dD a+b+c+d thatthefourquantities x,y,Z,wareprecisely thebary. centriccol5rdinates ofP,theterminus ofP,withrespectto thetetrahedron ABaD.Thusthevectormethods inwhich theoriginisundetermined andthemethods ofthe"Bard centricaalcul1U" arepractically co-extensive. Itwasmentioned beforeanditmaybewelltorepeathere 46 VECTOR ANALYSIS thattheoriginmaybeleftwhollyoutofconsideration and thevectorsreplaced bytheirtermini. Thevectorequations thenbecomepointequations p=xA+yB+zC x+y+z ~d p=xA+yB+zC+wD x+y+z+w. Thisstepbringsinthepointsthemselves lI.8theobjectsof analysisandleadsstillnearertothe"BarycentriscM CalcUl" ofMl5biusandthe"Ausdehnungslehre" ofGrassmann. TheUseofVectorstodenoteArea. 25.]Definition: AnarealyinginoneplaneMNand bounded byacontinuous curvePQRwhichnowhere cuts itselfissaidtoappearpositivefromthepoint0whenthe lettersPQRfolloweach A0 otherinthecounterclockwise Norpositive order;negative, ~+-\-------'7whentheyfollowinthe negative orclockwise order (Fig.14). It( Itisevidentthatanarea canhavenodetermined sign FIG.14. perse,butonlyinreference tothatdirection inwhichits boundary issupposed tobetracedandtosomepoint0out sideofitspl~e.FortheareaP RQisnegative relativeto PQRjandanareaviewedfrom0isnegative relativetothe sameareaviewedfromapoint0'uponthesideoftheplane opposite toO.AcirclelyingintheXY-planeanddescribed inthepositivetrigonometric orderappearspositivefromevery pointonthatsideoftheplaneonwhichthepositiveZ-axis lies,butnegative fromallpointsonthesideupon which ADDITION ANDSCALAR MULTIPLICATION 47 thenegative Z-axislies.Forthisreasonthepointofview andthedirection ofdescription oftheboundary mustbekept clearlyinmind. Another methodofstatingthedefinition isasfollows:If apersonwalking uponaplanetracesoutaclosedcurve,the areaenclosed issaidtobepositive ifitliesuponhisleft handside,negative ifuponhisright.Itisclearth!1tiftwo personsbeconsidered totraceouttogether thesamecurveby walking uponopposite sidesoftheplanetheareaenclosed willlieupontherighthandofoneandthelefthandofthe other.Tooneitwillconsequently appearpositive; tothe other,negative. Thatsideoftheplaneuponwhichthearea seemspositive iscalledthepositive side;thesideupon whichitappears negative, thenegative side.Thisideais familiar tostudents ofelectricity andmagnetism. Ifan electriccurrentflowaroundaclosedplanecurvethelinesof magnetic forcethrough thecircuitpassfromthenegative to thepositive sideoftheplane.Apositive magnetic pole placeduponthepositivesideoftheplanewillberepelled by thecircuit. Aplaneareamaybelookeduponaspossessing morethan positive ornegative magnitude. Itmaybeconsidered to possessdirection, namely,thedirection ofthenormaltothe positive sideoftheplaneinwhichitlies.Henceaplane areaisavectorquantity. Thefollowing theorems concerning areaswhenlookeduponasvectorsareimportant. Theorem 1 :Ifaplaneareabedenoted byavectorwhose magnitude isthenumerical valueofthatareaandwhose direction isthenormaluponthepositive sideoftheplane, thentheorthogonal projection ofthatareauponaplane willberepresented bythecomponent ofthatvectorinthe direction normaltotheplaneofprojection (Fig.15). LettheareaAlieintheplaneMN.Letitbeprojected orthogonally upontheplaneM'N'.LetMNandM'Nr'inter- 48 VECTOR ANALYSIS sectintheline1andletthediedmlanglebetween these twoplanesbex.Consider firstarectangle PQRSinMN whosesides,PQ,RSandQR,SParerespectively parallel 'andperpendicular totheline1.Thiswillprojectinto1\ rectangle P'Q'R'S'inJlf'N',ThesidesP'Q'andR'S' willbeequaltoPQandRSjbutthesidesQ'R'andS'P' willbeequaltoQRandSPmultiplied bythecosineofx, theanglebetween theplanes. Consequently therectangle P'Q'R'S' =PQRScosx. FIG.15. Hencerectangles, ofwhichthesidesarerespectively parallelandperpendicular to1,thelineofintersection ofthe twoplanes,projectintorectangles whosesidesarelikewise respectively parallelandperpendicular to1andwho.'ll!,areais equaltotheareaoftheoriginal rectangles multiplud bythe cosineoftheanglebetween theplanes. FromthisitfollowsthatanyareaAisprojected intoan areawhichisequaltothegivenareamultiplied bythecosine oftheanglebetween theplanes. ForanyareaAmaybedi videdupintoalargenumberofsmallrectangles bydrawing a seriesoflinesin!IfNparallelandperpendicular totheline1. ADDITION ANDSCALAR MULTIPLICA TION 49 Eachoftheserectangles whenprojected ismultiplied bythe cosineoftheanglebetween theplanesandhencethetotal areaisalsomultiplied bythecosineofthatangle.Onthe otherhandthecomponent A'ofthevectorA,whichrepre sentsthegivenarea,inthedirection normaltotheplane H'N'ofprojection isequaltothetotalvectorAmultiplied bythecosineoftheanglebetween itsdirection whichis thenormaltotheplaneM Nandthenormaltojl'N'.This angleisx;fortheanglebetween thenormalstotwoplanes isthesameastheanglebetween theplanes. Therelation between themagnitudes ofAandA'istherefore A'=Acosx, whichprovesthetheorem. 26.]Definition: Twoplaneareasregarded asvectorsare saidtobeaddedwhenthevectorswhichrepresent themare addeJ. Avectorareaisconsequently thesumofitsthreecom ponents obtainable byorthogonal projection uponthree mutually perpendicular planes. Moreover inaddingtwo areaseachmayberesolved intoitsthreecomponents, the corresponding components addedasscalarquantities, and thesesumscompounded asvectorsintotheresultant area. Ageneralization ofthisstatement tothecasewherethethree planesarenotmutually orthogonal andwheretheprojection i>obliqueexists. Asurfacemadeupofseveralplaneareasmayberepre sentedbythevectorwhichisthesumofallthevectors representing thoseareas.Incasethesurfacebelookedupon asforming theboundary oraportionoftheboundary ofa solid,thosesidesofthebounding planeswhichlieoutsideof thebodyareconventionally takentobepositive. Thevec torswhichrepresent thefacesofsolidsarealwa.ysdirected outfromthesolid,notintoit• 50 VECTOR ANALYSIS Theoremf:Thevectorwhichrepresents aclo&edpolyhedral surfaceiszero. Thismaybeprovedbymeansofcertainconsiderations of hydrostatics. Suppose thepolyhedron drawninabodyof fluidassumed tobefreefromallexternal forces,gravityin cluded.I Thefluidisinequilibrium underitsowninternal pressures. Theportionofthefluidbounded bytheclosed surfacemovesneitlieronewaynortheother.Uponeach face ofthesurfacethefluidexertsadefiniteforceproportiunAl totheareaofthefaceandnormaltoit.Theresultant ofall theseforcesmustbezero,asthefluidisinequilibrium. Hence thesumofallthevectorareasintheclosedsurfaceiszero. Theproofmaybegiveninapurelygeometric manner. Consider theorthogonal projection oftheclosedsurfaceupon anyplane.Thisconsistsofadoublearea.Thepartofthe surfacefarthestfromtheplaneprojects intopositive area; thepartnearesttheplane,intonegative area.Thusthe surfaceprojects intoacertainportionoftheplanewhichis coveredtwice,oncewithpositiveareaandoncewitbnegative. Thesecanceleachother.Hencethetotalprojection ofa closedsurfaceuponaplane(iftakenwithregardtosign)is zero.Butbytheorem 1theprojection ofanareaupona planeisequaltothecomponent ofthevectorrepresenting thatareainthedirection perpendicular tothatplane.Hence thevectorwhichrepresents aclosedsurfacehasnocomponent along the lineperpendicular totheplaneofprojection. This, however, wasanyplanewhatsoever. Hencethevectoris zero. Thetheorem hasbeenprovedforthecaseinwhichthe closedsurfaceconsists ofplanes. Incasethatsurfacebe 1Such&stAteofaffairsisrealized to&11practical PUrpOBeB inthecueofa polyhedron suspeuded intheatmosphere andconsequently subjected toatmos phericpretlBure. Theforceofgravityactsbutiscounterb&1anced bythetension inthesmpending string. ADDITION ANDSCALAR MULTIPLICATION 51 curveditmayberegarded asthelimitofapolyhedral surface whosenumber offacesincreases without limit.Hencethe vectorwhichrepresents anyclosedsurface polyhedral or curvediszero.Ifthesurfacebenotclosedbutbecurvedit mayberepresented byavectorjustasifitwerepolyhedral. Thatvectoristhelimit1approached bythevectorwhich represents thatpolyhedral surfaceofwhichthecurvedsurface isthelimitwhenthenumber offacesbecomes indefinitely great. SUMMARY OFCHAPTER I Avectorisaquantity considered aspossessing magnitude anddirection. Equal vec~rspossessthesamemagnitude andthesamedirection. Avectorisnotalteredbyshiftingit paralleltoitself.Anullorzerovectorisonewhosemag nitudeiszero.Tomultiply avectorbyapositive scalar multiply itslengthbythatscalarandleaveitsdirection unchanged. Tomultiply avectorbyanegative scalarmul tiplyitslengthbythatscalarandreverse itsdirection. Vectorsaddaccording totheparallelogram law.Tosubtract avectorreverseitsdirection andadd.Addition, subtrac tion,andmultiplication ofvectorsbyascalarfollowthesame lawsasaddition, subtraction, andmultiplication inordinary algebra. Avectormayberesolved intothreecomponents parallel toanythreenon-coplanar vectors. Thisresolution canbeaccomplished inonlyone way. r=xa+yb+zc. (4) Thecomponents ofequalvectors,parallel tothreegiven non-coplanar vectors, areequal,andconversely ifthecom ponents areequalthevectors areequal.Thethreeunit vectorsi,i,kformaright-handed rectangular system. In 1ThiHlimitexistllandu.unique. Itisindependent ofthemethodinwhich thepolyhedral surfaceapproachll8 thecurvedlurface. 52 VECTOR ANALYSIS termsofthemanyvectormaybeexpressed bymeansofthe Cartesian col:lrdinates x,y,z. r=xi+yj +zk. (6) Applications. Thepointwhichdividesalineinagiven ratiom:nisgivenbytheformula p=nA+mB. 1n+n(7) Thenecessary andsufficient condition thatavectorequation represent arelationindependent oftheoriginisthattheSUIll ofthescalarcoefficients intht:equation bezero.Between anyfourvectors thereexistsanequation withscalarcoeffi cients.Ifthesumofthecoefficients iszerothevectorsare termino-coplanar. Ifanequation thesumofwhosescala.r ~oefficients iszeroexistsbetween threevectors theyare termino-collinear. Thecenterofgravityofanumber of masses a,b,c...situated atthetermini ofthevectors A,B,C...supposed tobedrawnfromacommon origin 18 givenbytheformula G=aA+bB+cC+. (9) a+b+c+ ... Avectormaybeusedtodenoteanarea.Iftheareais planethemagnitude ofthevectorisequaltothemagnitude ofthearea,andthedirection ofthevectoristhedirection of thenormaluponthepositive sideoftheplane.Thevector representing aclosedsurfaceiszero. EXERCISES ONCHAPTER I 1.Demonstrate thelawsstatedinArt.12. 2.Atrianglemaybeconstructed whosesidesareparallel andequaltothemedians ofanygiventriangle. ADDITION .-1NDSCALAR MULTIPLICATION 53 3.Thesixpointsinwhichthethreediagonals ofacom pletequadmngle 1meetthepairsofopposite sidesliethree bythreeuponfourstraight lines. 4.Iftwotriangles aresosituated inspacethatthethree pointsofintersection ofcorresponding sideslieonaline,then thelinesjoiningthecorresponding vertices passthrough a common pointandconversely. 5.Givenaquadrilateml inspace.Findthemiddlepoint ofthelinewhichjoinsthemiddlepointsofthediagonals. Findthemiddlepointofthelinewhichjoinsthemiddle pointsoftwoopposite sides.Showthatthesetwopointsare thesameandcoincide withthecenterofgmvityof asystem ofequalIllassesplacedattheverticesofthequadrilateml. 6.Iftwoopposite sidesofaquadrilateml inspacebe dividedproportionally andiftwoquadrilaterals beformedby joiningthetwopointsofdivision, thenthecentersofgravity ofthesetwoquadrilatemls lieonalinewiththecenterof gmvityof theoriginalquadrilateral. Bythecenterofgmvity ismeantthecenterofgravityoffourequalmassesplacedat thevertices. Canthistheorem begeneralized tothecase wherethemassesarenotequal? 7.Thebisectors oftheanglesofatriangle meetina point. 8.Iftheedgesofahexahedron meetfourbvfourinthree points,thefourdiagonals ofthehexahedron meetIIIapoint. Inthespecialcaseinwhichthehexahedron 18aparallelopiped thethreepointsareataninfinitediStancb 9.Provethatthethreestmightlinestiuoug"ntnemiddle pointsofthesidesofanyfaceofatetrabearon. eacnparallel tothestraightlineconnecting afixedoointPwiththemid dlepointoftheopposite edgeofthetetrahedron. meetina 1Acomplete quadrangle consistaofthelOixstraight lineswhichmaybepas.sed through fonrpoiutanothreeofwhicharecollinear. Thedia~nals aretheHnlll whichjointhepointaofintersection ofpail"llofsides 54 VECTOR ANALYSIS pointEandthatthispointissuchthatPEpassesthrough andisbisectedbythecenterofgravityofthetetrahedron. 10.Showthatwitlwutexception thereexistsonevector equation withscalarcoefficients between anyfourgiven vectorsA,B,0,D. L11.Discusstheconditions imposed uponthree,four,or fivevectorsiftheysatisfytwoequations thesumoftheco efficientsineachofwhichiszero. CHAPTER II DIRECT ANDSKEW PRODUCTS OFVECTORS Products ojTwoVectors 27.]THEoperations ofaddition, subtraction, andscalar multiplication havebeendefined forvectors intheway suggested byphysicsandhavebeenemployed inafew applications. Itnowbecomes necessary tointroduce two newcombinations ofvectors. Thesewillbecalledpodu.cts becausetheyobeythefundamental lawofproducts; i.e.,the distributive lawwhichstatesthattheproduct ofAintothe sumofBandCisequaltothesumoftheproducts ofAinto BandAintoC. Definition: ThedirectproductoftwovectorsAandBis thescalarquantity obtained bymultiplying theproduct of themagnitudes ofthevectorsbythecosineoftheanglebe tweenthem. Thedirectproduct isdenoted bywritingthetwovectors withadotbetween themas A·B. ThisisreadAdotBandtherefore mayoftenbecalledthe dotproduct insteadofthedirectproduct. Itisalsocalled thescalarproduct owingtothefactthatitsvalueissca lar.If.Abethemagnitude ofAandBthatofB,thenby definition A•B=.ABcos(A,B). (1) Obviously thedirectproduct followsthecommutative law (2) 56 VECl'OR ANALYSIS (3)Ifeithervectorbemultiplied byascalartheproductis multiplied bythatscalar.Thatis (xA)•B=A·(xB)=x(A.B). IncasethetwovectorsAandBarecollinear theanglebe tweenthembecomes zerooronehundred andeightydegrees anditscosineistherefore equaltounitywiththepositiveor negative sign.,Hencethescalarproduct oftwoparallel vectorsisnumerically equaltotheproductoftheirlengths. Thesignoftheproductispositive whenthedirections of'the vectorsarethesame,negative whentheyareopposite. The productofavectorbyitselfistherefore equaltothesquare ofitslength Consequently iftheproductofavectorbyitselfvanishthe vectorisanullvector. IncasethetwovectorsAandBareperpendicular the anglebetween thembecomes plusorminusninetydegrees andthecosinevanishes. Hencetheproduct A•Bvanishes. Conversely ifthescalarproductA•Bvanishes. then ABcos(A.B)=O. HenceeitherAorBorcos(A.B)iszero,andeitherthe vectorsareperpendicular oroneofthemisnull.Thusthe condition fortheperpendicularity oftwovector.,neitMrof whichvanishes, 1'SA•B=O. 28.]Thescalarproducts ofthethreefundamental unit vectorsi,j,kareevidently ioi=j • j=k •k=t, i.j=j •k=k.i=O.(4.) Ifmoregenerally aandbareanytwounitvectorsthe product a·b=cos(a,b). , •DIRECT .4NDSKEWPRODUCTS OFVECTORS 57 Thusthescalarproduct determines thecosineoftheangle between twovectorsandisinacertainsenseequivalent to it.Forthisreasonitmightbebettertogiveapurely geometric definition oftheproduct ratherthanonewhich depends upontrigonometry. Thisiseasilyaccomplished as follows:Ifaandbaretwounitvectors,a.bisthelength oftheprojection ofeitherupontheother.Ifmoregenerally AandBareanytwovectorsA•Bistheproductofthelength ofeitherbythelengthofprojection oftheotheruponit. Fromthesedefinitions thefactsthattheproductofavector byitselfisthe square ofitslengthandtheproductoftwo perpendicular vectorsiszerofollowimmediately. Thetrigo nometric definition canalsoreadilybededuced. Thetlcalarproductoftwovectorswillappearwhenever the cosineoftheincluded angleisofimportance. Thefollowing examples maybecited.Theprojection ofavectorBupona vectorAis A.BABA.AA=AAAacos(A,B)=Bcos(A,B)a,(5) whereaisaunitvectorinthedirection of.A.IfAisitselfa unitvectortheformulareducesto CA.B)A=Bcos(A,B)A. IfAbeaconstant forceandBadisplacement theworkdone bytheforceAduringthedisplacement isA•B.IfArepre sentaplanearea(Art.25),andifBbea vectorinclined tothatplane,thescalarprod-~.l§1..!A.. uctA·Bwillbethevolumeofthecylinder .. ofwhichtheareaAisthebaseandof whichBisthedirected slantheight. For thevolume(Fig.16)isequaltothebase FIG.16. Amultiplied bythealtitude h.Thisis theprojection of]IuponAorBcos(A.B).Hence v=Ah=A Bcos(A,B)=A•B. 58 VECTOR ANALYSIS 29.]Thescalarordirectproduct followsthedistributive lawofmultiplication. Thatis (A+B) 0C=A0C+B0C. (6) and thenThismaybeprovedbymeansofprojections. LetCheequal toitsmagnitude Cmultiplied byaunitvector 0initsdirec tion.Toshow (A+B) 0(Co)=A0(Co)+Bo(Co) or (A+B) 00=A00+Boo. A00istheprojection ofAupon0;B0c,thatofBupon0; (A+B)0c,thatofA+Bupono.Buttheprojection ofthe sumA+Bisequaltothesumoftheprojections. Hence therelation (6)isproved. Byanimmediate generalization (A+B+...)0(P+Q+...)=A0P+A0Q+... +BoP+B0Q+..,(6)' + . Thescalarproductmaybeusedjustastheproduct inordi llaryalgebra.Ithasnopeculiardifficulties. IftwovectorsAandBareexpressed intermsofthe threeunitvectorsi,i,kas A=Ali+A2i+Ask, B=Bli+B2j+Bsk, A0B=(Ali+A2i+Ask) 0(Bli+B2i+Bsk) =AlBli0i+AlB2i0i+AlBsi0k +A2Bli0i+A2B2i0i+A2Bsi0k +AsBlk0i+AsB2koi+AsBsk0k. Bymeansof(4)thisreducesto A0B=AlBl+~B2+AsBs. (7) Ifinparticular AandBareunitvectors,theircomponents Al,A2,AsandBl,B2,Bsarethedirection cosinesofthe linesAandBreferredtoX,1';Z. DIRECT ANDSKEWPRODUCTS OFVECTORS 59 Al=cos(A,X),A2=COB(A,Y),Aa=cos(A,Z), B1=.cos(B,X),B2=COB(B,Y),Ba=cos(B,Z). Moreover A0Bisthecosineoftheincluded angle.Hence theequation becomes cos(A,B)=cos(A,X)cos(B,X)+cos(A,Y)COB(B,Y) +cos(A,Z)cos(B,Z). IncaseAandBareperpendicular thisreduces tothewell knownrelation 0=cos(A,X)cos(B,X)+COB(A,Y)cos(B,Y) +cos(A,Z)COB(B,Z) & FIG.17.~obetween thedirection cosinesofthe lineAandthelineB. 30.]IfAandBaretwosides0A and0Bofatriangle0AB,thethird sideABisC=B-A(Fig.17). CoC=(B-A) 0(B-A)=BoB+AoA-2AoB 02=A2+ B2_2A Bcos(AB). or Thatis,thesquareofonesideofatriangleisequaltothe sumofthesquaresoftheothertwosidesdiminished bytwice theirproduct timesthecOBineoftheanglebetween them. Or,thesquareofonesideofatriangleisequaltothesumof thesquaresoftheothertwosidesdiminished bytwicethe product ofeitherofthOBesidesbytheprojection oftheother uponit-thegeneralized PythagoNan theorem. IfAandBaretwosidesofaparallelogram, C=A+B andD=A-Barethediagonals. Then CoC=(A+B) 0(A+B)=A0A+ 2A0B +BoB, DoD=(A-B) 0(A-B)=A0A- 2A0B +BoB, CoC+DoD= 2(A 0A+BoB), or 02+D2=2(A2+B2). 60 VECTOR ANALYSIS Thatis,thesumofthesquaresofthediagonals ofaparallelo gramisequaltotwicetheBumofthesquaresoftwosides. Inlikemanneralso orC·C-D.D=4A·B C2-])2=4A Bcos(A,B). Thatis,thedifference ofthesquaresofthediagonals ofa parallelogram isequaltofourtimestheproductofoneofthe sidesbytheprojection oftheotheruponit. IfAisanyvectorexpressed intermsofi,i,kas thenA=Ati+A2j+Ask, A.A=A2=A12+A22+As2. (8) ButifAbeexpressed intermsofanythreenon-coplanar unit vectorsa,b,cas A=aa+bb+cc, A.A=A2=a2a.a+b2b •b+-c2c.c + 2aba.b +2bcb.c+2cac.a A2=a2+b2+",.2+ 2abcos(a,b)+2bccos(b,c) +2cacos(C,a). Thisformula isanalogous totheoneinCartesian geometry whichgivesthedistance between twopointsreferred to oblique axes.IfthepointsbeXt'Y1'%1'andx2,!h.'%2the distance squared is D2=(,1:2-X1)2+(Y2-Y1)2+(Z2-Zl)2 +2(x2-x~)(Y2-Y1)cos(X,Y) +2(Y2-Yt)(Z2-Zt)cos(Y,Z) +2(Z2-Zt)(x2-Xl)cos(Z,X). 31.] ])~finitwn: Theskewproduct ofthevectorAinto thevectorBisthet'ectorquantity Cwhosedirection isthe normaluponthatsideoftheplaneofAandBonwhich DIRECT ANDSKEWPRODUCTS OFVECTORS 61 rotation fromAtoBth~ough anangleoflessthanone hundred andeightydegrees appears positive orcounter clockwise; andwhosemagnitude isobtained bymultiplying theproductofthemagnitudes ofAandBbythesineofthe anglefromAtoB. Thedirection ofAx Bmayalsobedefinedasthatin whichanordinary right-handed screwadvances asitturnssoasC=t2=XB tocarryAtowardB(Fig.18). B ... Theskewproductisdenoted by ,."" IIcrossasthedirectproductwas .. byadot.Itiswritten FlO.18.A C=AxB andreadAcrossB.Forthisreasonitisoftencalledthecross product. Morefrequently, however, itiscalledthevectorprod uct,owingtothefactthatitisavectorquantity andincon trastwiththedirectorscalarproductwhosevalueisscalar. Thevectorproductisbydefinition C=Ax B=A Bsin(A,B)c, (9) whenAandBarethemagnitudes ofAandBrespectively and wherecisaunitvectorinthedirection ofC.IncaseAand Bareunitvectorstheskewproduct Ax Breducestothe unitvectorcmultiplied bythesineoftheanglefromAtoB. Obviously alsoifeithervectorAorBismUltiplied byascalar xtheirproductismultiplied bythatscalar. (xA)XB=Ax(xB)=xC. IfAandBareparalleltheanglebetweenthemiseitherzero oronehundred andeightydegrees. Ineithercasethesine vanishes andconsequently thevectorproductAXBisanull vector. Andconversely ifAXBiszero A Bsin(A,B)=O. 62 VECTOR ANALYSIS HenceAorBorsin(A,B)iszero.Thusthecondition for parallelism oftwovectorsneitherofwhichvanishes isAXB =O.Asacorollary thevectorproduct ofanyvectorinto itselfvanishes. 32.]Thevectorproductoftwovectorswillappearwher everthesineoftheincluded angleisofimportance, justas thescalarproductdidinthecaseofthecosine.Thetwoprod uctsareinacertainsensecomplementary. Theyhavebeen denoted bythetwocommon signsofmultiplication, thedot andthecross.Invectoranalysistheyoccupytheplaceheld bythetrigonometric functions ofscalaranalysis. Theyare atthesametimeamenable toalgebraic treatment, aswillbe seenlater.Atpresentafewusesofthevectorproductmay becited. IfAandB(Fig.18)arethetwoadjacent sidesofaparallel ogramthevectorproduct C=AXB=A Bsin(A,B)c represents theareaofthatparallelogram inmagnitude and direction (Art..25).Thisgeometric representation ofAxB isofsuchcommon occurrence andimportance thatitmight wellbetakenasthedefinition oftheproduct. Fromitthe trigonometric definition followsatonce.Thevectorproduct appearsinmechanics inconnection withcouples.IfAand -Aaretwoforcesforming acouple,themoment ofthe coupleisAXBprovided onlythatBisavectordrawnfrom anypointofAtoanypointof-A.Theproductmakesits appearance againinconsidering thevelocities oftheindivid ualparticles ofabodywhichisrotatingwithanangularve locitygiveninmagnitude anddirection byA.IfRbethe radiusvectordrawnfromanypointoftheaxisofrotationA theproductAXRwillgivethevelocityoftheextremity of R(Art.51).Thisvelocityisperpendicular aliketotheaxis ofrotationandtotheradiusvectorR. DIRECT ANDSKEW PRODUCTS OFVECTORS 63 33.]Thevectorproducts AXBandBXAarenotthe same.Theyareinfactthenegatives ofeachother.Forif rotation fromAtoBappearpositiveononesideoftheplane ofAandB,rotation fromBtoAwillappearpositiveonthe other.HenceAXBisthenormaltotheplaneofAandB uponthatsideopposite totheoneuponwhichBXAisthe normal. Themagnitudes ofAxBandBXAarethesame. Hence AxB=-BxA. (10) Thejactorsinavectorproductcanbeinterchangedifandonly ijthesignojtheproduct bereversed. Thisisthefirstinstance inwhichthelawsofoperation in vectoranalysisdifferessentially fromthoseofscalaranaly sis.Itmaybethatatfirstthischangeofsignwhichmust accompany theinterchange offactorsinavectorproductwill giverisetosomedifficulty andconfusion. Changes similarto thisare,however, veryfamiliar. Noonewouldthinkofinter changing theorderofxandyintheexpression sinex-y) withoutprefixing thenegative signtotheresult.Thus sin(y-x)= -sin(x-y), although thesignisnotrequired forthecaseofthecosine. cos(.1/-x)=cos(x-y). AgainifthecyclicorderofthelettersABCintheareaofa triangle bechanged, theareawillbechanged insign(Art. 25). ABC=-A CB. Inthesamemannerthisreversal ofsign,whichoccurs whentheorderofthefactorsinavectorproductisreversed, willappearafteralittlepractice andacquaintance justas naturalandconvenient asitisnecessary. 34.]Thedistributive lawofmultiplication holdsinthe. caseofvectorprodUCts justasinordinary algebra-except 64 VECTOR ANALYSIS ......--......-c..-..-."/' '-- / ..............,,,/thattkeorderoftkefactorsmustbecarefully maintained whenexpanding. (A+B)xC=AxC+BxC. (11) Averysimpleproofmaybegivenbymakinguseoftheideas developed inArt.26.Suppose thatC isnotcoplanar withAandB.LetA andBbetwosidesofatriangletaken inorder.Then-(A+B)willbethe &.::~~t--::~:a thirdside(Fig.19).Formtheprism ofwhichthistriangle isthebaseand ofwhichCistheslantheightoredge. Theareasofthelateralfacesofthis prismare AxC,BxC,-(A+B)xC. Theareasofthebasesare Butthesumofallthefacesoftheprismiszero;forthe prismisaclosedsurface. Hence . AxC+BxC-(A+B)xC+~(AxB)-~(AXB)=0, AxC+BxC-(A+B)xC=0, or AxC+BxC=(A+B)xC. (11) Therelation istherefore provedincaseCisnon-eoplanar withAandB.ShouldCbecoplanar withAandB,chooseD, anyvectoroutofthatplane.ThenC+Dalsowilllieputof thatplane.Henceby(11) Ax(C+D)+Bx(C+D)=(A+B)x(C+D). SincethethreevectorsineachsetA,C,D,andB.C,D,and A+B,C,Dwillbenon-eoplanar ifDisproperly chosen,the products maybeexpanded. DIRECT ANDSKEWPRODUCTS OFVECTORS 65 AXC+AxD+BxC+BxD =(A+B)x C+(A+B)xD. Butby(11)Ax D + B x D= (A+B)xD. Hence Ax C+ B x C= (A+B)xC. Thiscompletes thedemonstration. Thedistributive lawholds foravectorproduct. Thegeneralization isimmediate. (A+ B+...) X(P+ Q+..-)=Ax P +Ax Q+ ...(11)' +BxP+BxQ+ ..· + . 35.]Thevectorproducts ofthethreeunitvectorsi,i,kare easilyseenbymeansofArt.17tobe ixi= i x i = kxk=0, ixj=-jxi=k, (12) i xk=-kxj=i, kxi= -ixk=i. TheskewproductoftwoequalIvectorsofthesystemi,i,k iszero.Theproductoftwounequalvectorsisthethirdtaken withthepositivesignifthevectorsfollowinthecyclicorder iikbutwiththenegative signiftheydonot. IftwovectorsAandBareexpre88ed intermsofi,i,k, theirvectorproduct maybefoundbyexpanding according tothedistributive lawandsubstituting. A=Ali +Asi+ Aak, B =B1i+Bsi+Bak, Ax B =(AIi+Asi+ Aak)X(BIi+Bsi+ Bak) =AlB1ixi+AlBsixi+AlBai xk +AsBdxi+AsBsix i +AsBai xk, + AaB1kxi+ AaBskXi + AaBakxk. Hence Ax B=(AsBa-AaBs)i+(AaB1-AtBa)i +(AIBs-AsBJ)k. 1Thillfollowsa1Iofromthefactthatthe8~illchanged whentheorderof ladoraillrnened. HellCfliXj= -jXi=O. I) 66 VECTOR ANALYSIS Thismaybewrittenintheformofadeterminant 88 ijkI AxB=.At .A~.As BtB~BsI' Theformulre fortheBineandcOBineoftheBumordif ferenceoftwoanglesfollowimmediately fromthedotand croSBproducts. Letaandbbetwounitvectorslyinginthe ii-plane.IfxbetheanglethatamakeBwithi,and'!Ithe anglebmakeBwithi,then Hence If Hence Hence Hencea:::::tcosxi+sinxi, b =COBYi+sinYi, a·b=cos(a,b)=COB(y-x), a • b = COBxCOBY+BinxBiny. COB(y-x)=COBYcosx+sinysinx. b'=COBYi-BinYi, a •b'=COB(a,b') =COB(y+x). COB(y+x)=COBYCOBX-Binysinx. a x b = k sin(a,b)= kBin(y-x), a x b =k(sinyCOBx-sinxcosy). sin(y-x)=Binycosx-BinxCOBy. a xb'= ksin(a,b')=kBin(y+x), a xb'= k(sinyCO!'lx+sinxCOBy). sin(y+x)=Binycosx+Binxcosy. Ifl,m,nandl',m',n'arethedirection cOBinesoftwo unitvectorsaanda'referred toX,:Y,Z,then a=li+mi+nk, a'=l'i+m'i+n'k, a.a'=cos(a,a')=ll'+mm'+nn', ashasalreadybeenshowninArt.29.Thefamiliar formula. forthesquareofthesineoftheanglebetween aanda'may befound. DIRECT ANDSKEWPRODUCTS OFVECTORS 67 a xa'=sin(a,a')e=(m11,'-m'11,)i+(11,l'-11,'1)i +(1m'-l'm)k, whereeisaunitvecWrperpendicular waanda'. (axa')•(aXa')=sin2(a,a')e·e=sin2(a,a'). sin2(a,a')=(m11,'-m'11,)2+(11,l'-11,'l)2+(lm'-l'm)2. Thisleadstoaneasy.way ofestablishing theusefulidentity (mn'-m'n)2+(nl'-11,'l)2+(lm'-l'm)2 =(l2+m2+11,2)(l'2+m'2+11,'2)=(ll'+mm'+11,11,')2. ProductsofMorethanTwoVectors 36.]Upwthispointnothing hasbeensaidconcerning products inwhich'thenumber ofvectorsisgreaterthan two.IfthreevecWrsarecombined intoaproducttheresult iscalledatripleproduct. NextWthesimpleproducts A·BandAxBthetripleproducts arethemostimportant. Allhigherproducts maybereduced tothem. Thesimplest tripleproductisformedbymultiplying the scalarproductoftwovectorsAandBintoathirdCas (A.B)C. Thisinrealitydoesnotdifferessentially fromscalarmulti plication (Art.6).Thescalarinthiscasemerelyhappensto bethescalarproductofthetwovectorsAandB.Moreover inasmuch astwovectorscannotstandsidebysideinthe formofaproduct asBewithouteitheradotoracrossto unitethem,theparenthesis in(A.B)Cissuperfluous. The expression A.BC cannotbeinterpreted inanyotherway1thanastheproduct ofthevectorCbythescalarA·B. 1Later(Chap.V.)theproductBO,wherenosig'neitherdotorcrOllSoccurs, willbedefined. Butitwillbeseentherethat(A.B)0andA.(BO)areidentical andconsequently noambiguity canarisefromtheomill8ion oftheparenthesis. 68 VEC'10R ANALYSIS (14) A.(BxC) =vI. FIo.20.I.••37.]Thesecondtripleproduct isthescalarproductof twovectors,ofwhichoneisitselfavectorproduct, as A.(BxC) or(AxB).C. Thissortofproducthasascalarvalueandconsequently is oftencalledthescalartripleprod uct.Itsproperties areperhapsmost easily,deduced fromitscommonest geometrical interpretation. LetA,B, andCbeanythreevectorsdrawn fromthesameorigin(Fig.20). ThenBxCistheareaof thepar allelogram ofwhichBandCaretwoadjacent sides.The scalar (14) (AxB).C=A.(BxC)="'.willtherefore bethevolumeoftheparallelopiped ofwhich BxCisthebaseandAtheslantheightoredge.SeeArt.28. ThisvolumevispositiveifAandBxClieuponthesame sideoftheBC-plane; butnegative iftheylieonopposite sides.InotherwordsifA,B,Cformaright-handed or positivesystemofthreevectorsthescalarA.(BxC)isposi tive;butiftheyformaleft-handed ornegative system,it isnegative. IncaseA,B,andCarecoplanar thisvolumewillbe neitherpositive nornegative butzero.Andconversely if thevolumeiszerothethreeedgesA,B,Coftheparallelo pipedmustlieinoneplane.Hencethenecessary andtrUffi cientcondition forthecoplanarity ofthreevectorsA,B,Cnone ofwhichvanishes isA.(BxC)=O.Asacorollary thescalar tripleproductofthreevectorsofwhichtwoareequalor collinear mustvanish;foranytwovectorsarecoplanar. Thetwoproducts A.(BxC) and(AxB).C areequaltothe samevolumevoftheparallelopiped whoseconcurrent edges areA,B,C.Thesignofthevolumeisthesameinboth cases.Hence DIRECT ANDSKEWPRODUCTS OFVECTORS 69 Thisequality maybestatedasaruleofoperation. Thedot andthecrossinascalartripleproduct1naybeinterchanged withoutalteringthevalueoftheproduct. Itmayal,sobeseenthatthevectorsA,B,Cmaybeper mutedcycliclywithoutalteringtheproduct. A.(BxC) =B.(CxA) =C·(AxB). (15)..J.'-\ Foreachoftheexpressions givesthevolumeofthesame parallelopiped andthatvolumewillhaveineachcasethe samesign.becauseifAisuponthepositive sideoftheBC plane,Bwillbeonthepositive sideoftheCA-planeandC uponthepositivesideoftheAB-plane. Thetripleproduct maytherefore haveanyoneofsixequivalent forlU8 A'(BxC) =B·(CxA)=C.(AxB) (15)' =(AxB).C =(BxC).A =(CxA).B Ifhowever thecyclicorderofthelettersischanged the productwillchangesign. A.(BxC) = -A-(CxB). (16) Thismaybeseenfromthefigureorfromthefactthat BxC=-CxB. Hence: Asc,.,lllrtripleproductisnotalteredbyinterchanging thed{)torthecrossorbypermuting cycliclytheorderofthe vectors,butitisreversedinsignifthecyclicorder b~changed. 38.]Awordisnecessary uponthesubjectofparentheses inthistripleproduct. Cantheybeomitted without am biguity? TheycanTheexpression A·BxC canhaveonlytheoneinterpretation A.(BxC). Fortheexpression (A.B)xC ismeaningless. Itisimpos sibletoformtheskewproduct ofascalarA.Bandavector 70 VECTOR ANALYSIS [ABCJC.HenceasthereisonlyonewayinwhichA·BxCmay beinterpreted, noconfusion canarisefromomitting the parentheses. Furthermore owingtothefactthatthereare sixscalartripleproducts ofA,B,andCwhichhavethesame valueandareconsequently generally notworthdistinguish ingtheonefromanother,itisoftenconvenient tousethe symbol thentodenoteanyoneofthesixequalproducts. [ABCJ=A·BxC=B·CxA=C·AxB =AxB·C=BxC·A=CxA·B [ABC]=-[ACB].(15)' (16)' Thescalartripleproducts ofthethreeunitvectorsi,j,k allvanishexceptthetwowhichcontainthethreedifferent vectors. [ijk]=-[ikj]=1. (17) (18)'HenceifthreevectorsA,B,Cbeexpressed intermsofi,j,k as A=Aii+A2j+Aak, B=B1i+B2j+Bak, C=01i+02j+0aIt, then[ABC]=A}B20a+B}02Aa+01A2Ba(18) -AiBa02-B}0aA2-O}AaB2. Thismaybeobtained byactually performing themultiplica tionswhich ar~indicated inthetripleproduct. Theresult maybewrittenintheformofadeterminant.} A}A2Aal [ABC]=B}B2Bal o}020a 1Thisistheformula giveninsolidanalytic geometry forthevolumeofa tetrahedron oneofwhoseverticesisattheorigin. lo'oramoregeneral formula Beeexercises. DIRECT ANDSKEWPRODUCTS OFVECTORS 71 Ifmoregenerally A,B,Careexpressed intermsofanythree non-eoplanar vectors a,b,0whicharenotnecessarily unit vectors, (19)' orA=ala+a2b+as0 B=bla+b2b+bs0 C=cia+c2b+Cs0 whereai'a2,as,.bl,b2,bs,.andcI'c2'Csarecertaincon stants,then [ABCJ=(alb2Cs+blc2as+cia2bs(19)- aIbsC2 -bICSa2-CIaab2)[aboJ. ata2as [ABCJ=blb2ba[aboJ C1c2Cs 39.]Thethirdtypeoftripleproductisthevectorproduct oftwovectorsofwhichoneisitselfavectorproduct. Such are AX(BxC) and(AxB)xC. ThevectorAx(BxC) isperpendicular toAandto(BxC). But(BxC)isperpendicular totheplaneofBandC.Hence Ax(BxC). beingperpendicular to(BxC)mustlieinthe planeof]IandCandthustaketheform Ax(BxC) =xB+YC, wherexandyaretwoscalars. Inlikemanner alsothe vector(AxB)xC, bemgperpendicular to(AxB)mustlie intheplaneofAandB.Henceitwillbeoftheform (AxB)xC =mA+nB wheremandnaretwoscalars. Fromthisitisevidentthat ingeneral (AxB)xC isnotequaltoAx(BxC). Theparentheses therefore cannotberemoved orinter changed. Itisessential toknowwhichcrossproduct is 72 VECTOR ANALYSIS B if FIG.21.formedfirstandwhichsecond. Thisproduct istermedthe vectortripleproductincontrast tothesca.lartripleproduct. Thevectortripleproductmaybeusedtoexpressthatcom pOnentofavectorBwhichisperpendicular toagivenvector A.Thisgeometric useoftheproductisvaluable notonlyin itaelfbutforthelightitsheds upontheproperties oftheproduct. LetA(Fig.21)beagivenvector andBanother vectorwhosecom ponenta parallelandperpendicular AtoAaretobefound. Letthe components ofBparalleland,per pendicular toAbeB'andB"re spectively. DrawAandBfroma common origin.TheproductAxB isperpendicular totheplaneofAandB.Theproduct Ax(AxB) liesintheplaneofAandB.Itisfurthermore perpehdicular toA.Henceitiscollinear withB".An examination ofthefigurewillshowthatthedirection of Ax(AxB) isopposite tothatofB".Hence AX(AxB) = -cB", wherecissomescalarconstant. Now Ax(AxB) = -A2Bsin(A,B)b" hut -aB"=-aBsin(.6.,B)b", ifb"beaunitvectorinthedirection ofB". Hence Hencec=A2=A.A. B"= _Ax(AxB) . A.A(20) Thecomponent ofBperpendicular toAhasbeenexpressed intermsofthevectortripleproduct ofA,A,andB.The component B'paralleltoAwasfoundinArt.28tobe DIRECT ANDSKEW PRODUG"1'S OFVECTORS 'IS B'=A·BA (21) A·A B=B'+BI!=A·BA_Ax(AxB). (22) A·A A·A 40.]ThevectortripleproductAx(BxC)maybeexpressed asthesumoftwotermsas Ax(BxC) =A·OB -A·BC Inthefirstplaceconsider theproduct whentwoofthe vectorsarethesame.Byequation (22) orA.AB=A.BA-AX(AxB) Ax(AxB)=A.BA-A.AB(22) (2S) Thisprovestheformulaincasetwovectorsarethesame. ToproveitingeneralexpressAintermsofthethree non-coplanar vectorsB,C,andBxC. A=bB+eC+a(BxC), wherea,b,carescalarconstants. Then(I) Ax(BxC)=bBx(BxC)+cCx(BxC) (II) +a(BxC)x(BxC). Thevectorproductofanyvectorbyitselfiszero.Hence (BxC)x(BxC)=0 Ax(BxC) =bBx(BxC)+cCX(BxC). (II)' By(23) BX(BxC) =B.CB-B.BC Cx(BxC) = -CX(CxB) = -C·BC+C.CB. HenceAX(BxC)=[(bB.C+cC·C)B- (bB.B+cC.B)C]. (II)n Butfrom(I)A.B=bB·B+cC·B+a(BxC).B and A·C=bB·C+cC.C+a(BxC).C. ByArt.37(BxC).B=0and(BxC).C=o. Hence A.B=bB.B+cC.B, A.C=bB.C+eC.C.'l "'t\=-I~)\-1'-~.A '7'6~1\·Y)~t;d-~ T4 VECTOR ANALYSIS and HenceSubstituting thesevaluesin(II)", Ax(BxC)=A·CB-A·BC. (24) Therelationistherefore provedforanythreevectorsA,B,C. Another methodofgivingthedemonstration isasfollows. ItwasshownthatthevectortripleproductAx(BxC) was oftheform AX(BxC) =xB+yC. SinceAx(t><C)isperpendicular toA,thedirectproductof itbyAiszero.Hence A.[Ax(BxC)] =xA·B+yA.C=0 x:y=A.C:-A.B. AX(BxC) =n(A.CB-A·BC), wherenisascalarconstant.Itremains toshown=1. Multiply byB. Ax(BxC).B=n(A.CB·B-A·BC.B). Thescalartripleproductallowsaninterchange ofdotand cross.Hence Ax(BxC).B =A.(BxC)xB =-A.[Bx(BxC)], iftheorderofthefactors(BxC)andBbeinverted. -A.[Bx(BxC)] = -A•[B.CB-B·BC] = -B.CA·B+B.BA.C. Hencen=1and Ax(BxC) =A·CB -A.BC..(24) FromthethrcelettersA,B,Cbydifferent arrangements, fouralliedproducts ineachofwhichBandCareincluded in parentheses maybeformed. Theseare Ax(BxC), Ax(CxB), (CxB)xA, (BxC)xA. Asavectorproductchanges itssignwhenever theorderof twofactorsisinterchanged, theaboveproducts evidently satisfytheequations Ax(BxC) = -Ax(CxB) =(CxB)xA = -(BxC)xA. DIRECT ANDSKEWPRODUCTS OFVECTORS 75 Theexpansion foravectortripleproductinwhichthe parenthesis comesfirstmaytherefore beobtained directly fromthatalreadyfoundwhentheparenthesis comeslast. (AxB)xC=-Cx(AxB) =-C·BA+C·AB. Theformulre thenbecome andAx(BxC) =A·CB-A·BC (AxB)xC =A·CB-C·BA.(24) (24)' Thesereduction formulreareofsuchconstant occurrence and greatimportance thattheyshouldbecommitted tomemory. Theircontentmaybestatedinthefollowing rule.Toexpand avectortripleproductfirstmultiply theexteriorfactorintothe remoter termintkeparentkesis toformascalarcoefficient for tkenearerone,tkenmultiply tkeexte1'iorfactorintotkenearer termintheparentkesis toformascalarcoefficient fortke remoterone,andmbtracttkisresultfromtkefirst. 41.]Asfarasthepractical applications ofvectoranalysis areconcerned, onecangenel"cllly getalongwithout any fOl'mulre morecomplicated thanthatforthevectortriple product. Butitisfrequently moreconvenient tohaveat handotherreduction formulre ofwhichaUmaybederived simplybymakinguseoftheexpansion forthetripleproduct Ax(BxC) andoftherulesofoperation withthetriplepro ductA·BxC. Toreduceascalarproductoftwovectorseachofwhich isitselfavectorproductoftwovectors,as (AxB).(CxD). Letthisberegarded asascalartripleproductofthethree vectorsA,B,andCxD-thus AxB.(CxD). Interchange thedotandthecross. 76 HenceVECTOR ANALYSIS AxB.(OxD) =A.Bx(OxD) BxCOxD) =B·D0 -B·OD. (AxB).(CxD) =A.OB·D-A·DB·C.(25) Thismaybewrittenindeterminantal form. A·OA·DI(AxB).(CxD) =IB.CB.D (25)' IfAandDbecalledtheextremes: Band0themeans;A andCtheantecedents: BandDtheconsequents inthis productaccording tothefamiliar usageinproportions, then theexpansion maybestatedinwords. Thescalarproduct oftwovectorproduct'i isequaltothe(scalar)productofthe antecedents tjmesthe(scalar) product oftheconsequents diminished bythe(scalar)product ofthemean~timesthe (scalar)product oftheextremes. Toreduceavectorproduct oftwovectorseachofwhich isitselfavectorproductoftwovectors,as (AxB)x(CxD). LetCxD=B.Theproductbecomes (AxB)xE =A.BB-B.BA. Substituting thevalueofEbackintotheequation: (AxB)x(CxD) =(A.OxD)B -(B.CxD) A.(26) LetF=AxB.Theproductthenbecomes Fx(OxD)=F.DC-F·CD (AxB)x(CxD) =(AxB.D)C -(AxB.C) D.(26)' Byequating thesetwoequivalent result'iandtransposi!1g allthetermstoonesideoftheequation, [BCD]A-[CDA]B+[DAB]0 -[AB0]D=O.(27) Thisisanequation withscalarcoefficients between thefour vectorsA,B,C,D.Thereisingeneralonlyonesuchequa.- DIRECT ANDSKEWPRODUCTS OFVECTORS 77 tion,becauseanyoneofthevectorscanbeexpressed inonly vnewayintermsoftheotherthree:thusthescalarcoeffi-· dentsofthatequation whichexistsbetween fourvectorsare foundtobenothingbutthefourscalartripleproducts of thosevectorstakenthreeatatime.Theequation mayalso bewrittenintheform [ABCD]=[BCD]A+[CAD]B+[ABD]C.(27)' Moreexamples ofreduction formulre, ofwhichsomeare important, aregivenamongtheexercises attheendofthe chapter. Inviewoftheseitbecomes fairlyobvious that thecombination ofanynumber ofvectorsconnected in anylegitimate waypydotsandcrossesortheproductofany number ofsuchcombinations canbeultimately reduced to asumoftermseachofwhichcontains onlyonecrossatmost. Theproofofthistheorem depends solelyuponanalyzing the possible combinations ofvectorsandshowing thattheyall fallunderthereduction formulre insuchawaythatthe crossesmayberemoved twoatatimeuntilnotmorethan oneremains. •42.JTheformulre developed intheforegoing articlehave interesting geometric interpretations. Theyalso·afforda simplemeansofdeducing theformulre ofSpherical Trigo nometry. Thesedonotoccurinthevectoranalysis proper. Theirplaceistakenbythetwoquadruple products, (AxB).(CxD) =A·CB·D-B·CA·D (25) and (AxB)x(CxD) =[ACD]B-[BCD]A =[ABDJC-[ABC]D,(26) whicharenowtobeinterpreted. Letaunitsphere(Fig.22)begiven.Letthevectors A.B,C,Dbeunitvectorsdrawnfromacommon origin,the centreofthesphere.andterminating inthesurfaceofthe sphereatthepointsA,B,C,D.Thegreatcirculararcs 78 VECTOR ANALYSIS FIG.23.AB,AC,etc.,givetheanglesbetween thevectorsAandB, AandC,etc.ThepointsA,B,C,Ddetermine aquadrilateral uponthesphere. A CandBDareone pairofopposite sides;A DandBC,the other.ABandCDarethediagonals. (AxB).(CxD) =A·CB·D-A·DB·C /AxBI=sin(A,B),ICx~=sin(C.D). Theanglebetween AxBandCxDisthe Fanglebetween thenormalstotheAB-IG.22. andCD-planes. Thisisthesameas theanglebetween theplanesthemselves. Letitbedenoted byx.Then (AxB)·(CxD) =sin(A,B)sin(C,D)cosx. Theangles(A,B),(C,D)maybereplaced bythegreat circulararcsAB,CDwhichmeasure them.Then (AxB).(CxD) =sinA BsinCDcosX, A·CB·D-A·DB·C=cosA CcosBD-cosADcosBC. Hence sinA BsinCDcosx=cosA CcosBD-cosA DcosBC. Inwords:Theproductofthecosinesoftwoopposite sides ofaspherical quadrilateral lesstheproductofthecosinesof theothertwoopposite sidesisequaltotheproduct ofthe sinesofthediagonals multiplied bythe cosineoftheanglebetween them.This theorem iscreditedtoGauss. LetA,B,C(Fig.23)beaspherical tri angle,t.hesidesofwhicharearcsofgreat circles. Letthesidesbedenoted bya,b,c respectivel~r. LetA,B,Cbetheunitvectors drawnfromthecenterofthespheretothepointsA,B,C. Furthermore letP,.,Pb'Pcbethegreatcircular arcsdropped DIRECT ANDSKEWPRODUCTS OFVECTORS 79 perpendicularly fromtheverticesA,B,Ctothesidesa,b,a. Interpret theformula (AxB).(CxA) =A·CB·A-B·CA.A. (AxB)=sin(A,B)=sinc,(CxA)=sin(C,A)=sinb. Then (AxB)·(CxA) =sincsinbcosx, wherexistheanglebetween AxBandCxA.This angleisequaltotheanglebetween theplaneofA,Bandthe planeofC,A.Itis,however, nottheinteriorangleAwhich isoneoftheanglesofthetriangle: butitistheexterior angle1800-A,asanexamination ofthefigurewillshow. Hence (AxB).(CxA) =sincsinbcos(1800 -A) = -sincsinbCOBA A.CB·A-B·CA·A=cosbcosc-COBa1. Byequating theresultsandtransposing, COBa=cosbcosa-sinbsinccosA COBb=COBCCOBa-sincsinacosB cosC=cosacosb-sinasinbcosC. Thelasttwomaybeobtained bycyclicpermutation ofthe lettersorfromtheidentities (BxC).{AxB) =B.AC·B-C·A, (CxA).(BxC) =C·BA·C-B.C. Nextinterpret theidentity(AxB)x(CxD) inthespecial casesinwhichoneofthevectorsisrepeated. (AxB)x(AxC) =[ABC]A. Letthethreevectorsa,b,cbeunitvectorsinthedirection of BxC,CxA,AxBrespectively. Then AxB=0sinc,AxC= -bsinb (AxB)x(AxC) =-cxbsinasinb=AsincsinbsinA [ABC]=(AxB).C=O.Csinc=cos(900 -Pc)sinc lAB(;]A=sincsinPcA. 80 VECTOR ANA.LYSIS Hence a+b+c=OByequating theresultsandcancelling thecommon factor, sinP.=sinbsin.A smp,.=sincsinB sinPb=sinasinC. Thelasttwomaybeobtained bycyclicpermutation ofthe letters. Theformulre givethesinesofthealtitudes ofthe triangleintermsofthesinesoftheangleandsides.Again write (AxB)x(AxC) =[ABC]A (BxC)x(BxA) =[BCA]B (CxA)x(CxB) =[CAB]C. sincsinbsin.A=[ABC] sinasincsinB=[BCA] sinbsinasinC=[CAB]. Theexpressions [ABC],[BCA], [CAB]areequal.Equate theresultsinpairsandtheformulre sinbsinA=sinasinB sincsinB=sinbsinC sinasinC=sincsinA areobtained. Thesemaybewritteninasingleline. sinAsinBsinC sina=sinb=sinc • Theformulre ofPlaneTrigonometry areevenmoreeasyto obtain. 1f.AB Cbeatriangle, thesumofthesidestaken asvectorsiszero-forthetriangle isa.closedpolygon. Fromthisequation almostalltheelementary formulre followimmediately. It istobenoticedthattheanglesfromatob,frombtoc,from DIRECT A.NDSKEWP/WDUCTS OFVECTORS 81 atoaarenottheinterioranglesA,B,C,buttheexterior angles1800-A,11:100-B,1800-C. -a=b+o a·a=(b+e).(b+0)=b·b+0·0+ 2b·e. Ua,b,cbethelengthofthesidesa,b,0,thisbecomes a2=b2+c2-2bccosA b2=c2+a2-2cacosB c2=a2+b2-2abcosC. Thelasttwoareobtained ina.mannersimilartothefirst oneorbycyclicpermutation oftheletters. Theareaofthetriangleis 1 1 1~axb=2bxe=2exa= 1b'CIb•AI. B2asm=2csm.n.=2casm. Ifeachofthelastthreeequalities bedividedbytheproduct Iabc,thefundamental relation sinAsinBsinC--=--=---a bc (bxe).(bxc) =(cxa).(axb) .2Area(bcsinA)=(casinB)(absinC) 2Aa2sinBsinCrea= .sinAisobtained. Another formulafortheareamaybefoundfrom theproduct Reciprocal SystemsofThreeVectors. SolutionofEquations 43.]Theproblem ofexpressing anyvectorrintermsof threenon-eoplanar vectorsa,b,cmaybesolvedasfollows. Letr=aa+bb+cc 6 82 VECTOR. ANALYSIS (28) Hencewherea,b,carethreescalarconstants tobedetermined. Multiply by 0bxe. robxe=aaobxe+bbobxe+ceobxo or [rbe]=a[ab0]. Inlikemannerbymultiplying theequation by00xaand oa xbthecoefficients bandcmaybefound. [rea]=b[bea] [rab]=c[cab] r =[rbe]a +[rea]b+[fab]e. [abe] [boa] [eab] Thedenominators areallequal. Hencethisgivesthe equation [abe]r -[ber]a +[era]b -[rab]0= 0 orwhichmustexistbetweenthefourvectorsr,a,b,o. Theequation mayalsobewritten robxe rooxab+roaxbr=- ~a+ e[abo] [abe] [abe] bxe exa axbr=ro[abe]a+ro[abe]b+ro[abe]o. Thethreevectorswhichappearheremultiplied byro,namely bxeexaaxb --,-_.--[abe] [abe] [abe] areveryimportant. Theyareperpendicular respectively to theplanesofbande.eanda,aandb.Theyoccuroverand overagaininalargenumberofimportant relations. For thisreasontheymeritadistinctive nameandnotation. Definition: Thesystemofthreevectors bxe-_. [abe]c x a ~--,[abe]axb [abo] DIRECT ANDSKEWPRODUCTS OPVECTORS 88 whicharefoundbydividing thethreevectorproducts bxc, cXa,aXbofthreeno1HXJplanar vectorsa,b,cbythescalar product[abc]iscalledthereciprocal systemtoa,b,c. Thewordnon-coplanar isimportant. Ifa,b,cwereco planarthescalartripleproduct [abc]wouldvanishand consequently thefractions bxccxaaxb--'--,[abc] [abc] [abo] wouldallbecomemeaningless. Threecoplanar vectorshave noreciprocal s),stem. Thismustbecarefully remembered. Hereafter whenthetermreciprocal system.isused,itwillbe understood thatthethreevectorsa,b,0arenotcoplanar. Thesystemofthreevectorsreciprocal tosystema,b,c willbedenoted byprimesasa',b',c'. a'=~, b'=0xa,0'= a x b . (29) [abo] [abc] [abc] Theexpression forrreducesthentotheverysimpleform r =roa'a +rob'b+roc'c. (30) Thevectorrmaybeexpressed intermsofthereciprocal systema',b',c'insteadofintermsofa,b,o.Inthefirst placeitisnecessary tonotethatifa,b,carenon-eoplanar, a',b',c'whicharethenormals totheplanesofbandc, canda,aandbmustalsobenon-coplanar. Hencermay beexpressed intermsofthembymeansofproperscalar coefficients x,y,z. r=xa'+yb'+zc' or [abc]r =xb x c + yc x a +za xb. Multiply successively byoa,ob,oc.Thisgives [abc]roa=x[bca],x=roa [abc]rob=y[cab], y=rob [abo]roc=z[abc], z=roo Hence r =roaa'+robb'+roc0'. (31) 84 VECTOR ANALYS1S 44.]Ifa',b',0'bethesystemreciprocal toa,b,0the scalarproductofanyvectorofthereciprocal systemintothe corresponding vectorofthegivensystemisunity;but theproductoftwonon-corresponding vectorsiszero.Thatis a'.a=b'·b=0"0=1 a'.b=a'.c=b'·a=b'.o=c'.a=c'.b=O.(32) Hence Fromthesecond FromthethirdThismaybeseenmosteasilybyexpressing a',b',c'in termsofthemselves according totheformula(31) r =r'aa'+r·bb'+r·oc'. a'=a'.aa'+a'.bb'+a'·oc' b'=b'.aa'+b'.bb'+b'.cc' 0'=o'.aa'+o'.bb'+0"00'. Sincea',b',0'arenon-coplanar thecorresponding coeffi. cientsonthetwosidesofeachofthesethreeequations must; beequal.Hencefromthefirst 1 =a'.a0 =a'.b0 =a'c. 0=b'.a1 =b'.b0 =b'.o. 0=o'.a0 =o'.b1 =0'.0. Thisprovestherelations. Theymayalsobeproved directlyfromthedefinitions ofa',b',c'. a'.a=bxo.a=bxo.a=[boa]=l [abo] [abc] [abo] a,.b=~.b= bxo.b=_O_=O [abo] [abo] [abc] andsoforth. Conversely iftwosetsofthreevectorseach,sayA,B.C, anda,b,c,satisfytherelations A..a=B·b =C.c=1 A..a=A.·c=B·a=B·o=C·a=C·b=0 DIRECT ANDSKEWPRODUCTS OFVECTORS 85 thenthesetA,B,Cisthesystemreciprocal toa,b,c. Byreasoning similartothatbefore A=A·aa'+A.bb'+A·cc' B=B.aa'+B.bb'+B.cc' C=C·aa'+C·bb'+C·cc'. Substituting intheseequations thegivenrelations there sultis A=a',B=b',C=c', Hence Theorem: Thenecessary andsufficient conditions thatthe setofvectorsa',b',c'bethel'eciprocals ofa,b,cisthat theysatisfytheequations a'.a=b'·b=o'·c=1 (32) a'.b=a'.c=b'.a=b'.c=o'.a=o'.b=O. Astheseequations areperfectly symmetrical withrespect toa',b',c'anda,b.citisevidentthatthesystema,b,0may belookeduponasthereciprocal ofthesystema',b',0'just asthesystema',b',c'mayberegarded asthereciprocal of a,b,c.Thatistosay, Theorem:Ifa',b!,0'bethereciprocal systemofa,b.0, thena,b.cwillbethereciprocal systemofa',b',0'. ·b'xc'a=--··-'[a'b'0']c'xa' a'xb'b=-_.--,c=--~-'[a'b'c'] [a'b'c'](29)' Theserelations maybedemonstrated directly fromthe definition~ ofa',b'.c'.Thedemonstration isstraightfor ward,butratherlongandtediousasitdepends oncompli catedreduction formulre. Theproofgivenaboveisasshort ascouldbedesired. Therelationll between a',b',0'and a,b,caresymmetrical andhenceifa',b'.c'illthereciprocal systemofa,b,0,thena,b,cmustbethereciprocal systemof a',b',c'. 86 VECTOR A/I,"ALYSIS 45.]Theorem:Ifa',b',0'anda,b,ebereciprocal systems thescalartripleproducts[a'b'c']and[abe]arenumerical reciprocals. Thatis [a'b'e'][abe]= 1 (33) ['"[bXCcxaaXbJabc]=[abc][abe][abe] 1 -[abc]3[bxooxaaXb]. [bxeexaaxb]=(bxc)x(exa).(axb). But (bxe)x(exa) =[abc]c. Hence [bxccxaaxb]=[abe]c.axb=[abcpo Hence [a'b'c'] =[a111 «:]3[abep= [a~c]' (33)' Bymeansofthisrelation between [a'b'e']and[abe]it ispossibletoproveanimportant reduction formula, P·AP·Bp·c (P.QxB)(A.BxC) =Q.AQ.BQ·e, (34) R·AR.BR.C whichreplaces thetwoscalartripleproducts byasumof ninetermseachofwhichistheproductofthreedirectpro ducts.Thusthetwocrosseswhichoccurinthetwoscalar products areremoved. Togiv.etheproofletP,Q,Rbe expressed as P=poAA'+P·BB'+p·ce' Q=QoAA'+Q.BB'+QoCC' R=RoAA'+R.BB'+R·eC'. P.AP.BpoC Then [PQB]=Q.AQ.BQoe[A'B'C'l. RoAR·BR.C But [A'B'C']=_1_.[ABC] DIRECT ANDSKEWPRODUCTS OFVECTORS 87 HenceP.AP·Bp.C [PCUl][ABC]=Q.AQ.BQ.C R·AR·BR·C Thesystemofthreeunitvectorsi.j,kisitsownreciprocal system. i'=jXk_!_..I _kxi_.k'_ iXj _(35)[ijk]-t-1,J-[ijkJ- J,-[ijk]-k. Forthisreasontheprimesi',jI,k'arenotneededtodenote asystemofvectorsrecipr6Cal toi,j,k.Theprimeswill therefore belliledinthefuturetodenoteanothersetofrect angular axesi,j,k,justasX',Y',Z'areusedtodenotea setofaxesdifferent fromX,Y,Z. Theonlysystemsofthreevectorswhicharethtirownreciprocals aretheright-handed andlejt-handed systemsofthreellnit vectors. Thatisthesystemi,j,kandthesystemi,i,-k. LetA,B,Cbeasetofvectorswhichisitsownreciprocal. Thenby(32) A·A=B.B=C.C=1. Hencethevectorsareallunitvectors. A.B=A.C=O.• HenceA.isperpendicular toBandC. B·A=B·C=O. HenceBisperpendicular toAandC. C·A=C·B=O. HenceCisperpendicular toAandB. HenceA,B,Cmustbeasystemlikei,j,korlikei,j,-k. *46.]Ascalarequation ofthefirstdegreeinavectorris anequation ineachtermofwhichioccursnotmorethan once.Thevalueofeachtermmustbescalar.Asanexam pleofsuchanequation thefollowing maybegiven. aa.bxr+b(cxd).(exr) +cf.r+d=0, 88 VECTOR ANA.LYSIS wherea,b,0,d,e,fareknownvectors; anda,b,c,d,known scalars. Obviously anyscalarequation ofthefirstdegreein anunknown vectorrmaybereducedtotheform r.A=a whereAisaknownvector;anda,aknownscalar.Toac complish thisresultinthecas~ofthegivenequation proceed asfollows. aaxb.r+b(cxd)xe.r+cf.r+d=0 {aaxb+b(cxd)xe+cf}.r= -do Inmorecomplicated formsitmaybenecessary tomakeuse ofvariousreduction formulre beforetheequation canbemade totakethedesiredform, r.A=a. (36) r.A=a isinteresting. Letrbea.variable vector (Fig.24)drawnfromafixedorigin.Let Abeafixedvectordrawnfromthesame Theequation thenbecomesh0--eJ' FIG.24. origin.Asavectorhasthreedegrees offreedom itisclearthatone scalarequation isinsufficient todetermine avector. Three scalarequations arenecestmry. Thegeometric interpretation oftheequa tion rAcos(r,A)=a, ora rcos(r,A)=A' ifrbethemagnitude ofr;andAthatofA..Theexpression rcos(r,A) istheprojection ofruponA.Theequation therefore states thattheprojection ofruponacertainfixedvectorAmust DIRECT ANDSKEWPRODUCTS OFVECTORS 89 alwaysbeconstant andequaltoa/A.Consequently theter minusofrmusttraceoutaplaneperpendicular tothevector A.atadistance equaltoa/Afromtheorigin. Theprojec tionuponAofanyradiusvectordrawnfromtheorigintoa pointofthisplaneisconstant andequaltoa/A.Thisgives thefollowing theorem. Theortm: Ascalarequation inanunknown vectormaybe regarded astheequation ofaplane,whichisthelocusofthe terminus oftheunknown vectorifitsoriginbefixed. Itiseasytoseewhythreescalarequations inanunknown vectordetermine thevectorcompletely. Eachequation de termines aplaneinwhichtheterminus ofrmustlie.The threeplanesintersect illonecommon point.Henceonevec torrisdetermined. Theanalytic solution ofthreescalar equations isextremely easy.Iftheequations are r.A=a r·B=b r.C=c, itisonlynecessary tocalltomindtheformula r=r.AA'+r·BB'+r·CC'.(37) Hence r=aA'+bB'+cC'. (38) Thesolution i8therefore accomplished. Itisexpressed in termsA',B',C'whichisthereciprocal systemtoA.,B,C.One cautionmusthowever beobserved. Thevectors A.,B,Cwill havenoreciprocal systemiftheyarecoplanar. Hencethe solution willfail.Inthiscase,however, thethreeplanesde termined bythethreeequations willbeparallel toaline. Theywilltherefore eithernotintersect (asinthecaseofthe lateralfacesofatriangular prism)ortheywillintersect ina common line.Hencethere.willbeeithernosolution forror therewillQeaninfinitenumber. 90 VECTOR ANALYSIS Fromfourscalar equations r.A=a r.B=b r.C=c r.D=d(39) thevectorrmaybeentirelyeliminated. Toaccomplish this solvethreeoftheequations andsubstitute thevalueinthe fourth.. r=aA'+bB'+cC' aA'·D+bB'·D+cC'.D=d ora[BCD]+b[CAD]+c[ABD]=d[ABC]. (40) •47.]Avectorequation ofthefirstdegreeinanunknown vectorisanequation eachtermofwhichisavectorquantity containing theunknown vectornotmorethanonce.Such anequation is (AxB)x(Cxr) +DE.r+nr+F=0, whereA,B,C,D,E,Fareknownvectors,na.knownscalar, andrtheunknown vector. Onesuchequation mayingen eralbesolvedforr.Thatistosay,onevectorequation isin generalsufficient todetermine theunknown vectorwhichis contained inittothefirstdegree. Themethodofsolvingavectorequation istomultiplyit withadotsuccessively bythreearbitrary known non~oplanar vectors. Thusthreescalar equations areobtained. These maybesolvedbythemethods oftheforegoing article. Inthe firstplacelettheequation be Aa.r+Bb.r+Co·r=D, whereA,B,C,D,a,b,0areknownvectors. Noscalarcoeffi cientsarewrittenintheterms,fortheymaybeincorporated in thevectors. Multiply theequation successively byA',B',C', Itisunderstood ofcoursethatA,B,Carenon~opla.nar, DIRECT ANDSKEW PRODUCTS OFVECTORS 91 But HenceIl.or=DoA' bor=DoB' cor=DoC'. r=a'aor+b'bor+c'cor; r=DoA'a'+DoB'b'+DoC'c'. Thesolution hItherefore accomplished incaseA,B,Carenon coplanar anda,b,calsonon-coplanar. Thespecialcasesin whicheitherofthesesetsofthreevectorsiscoplanar willnot bediscussed here. Themostgeneralvectorequation ofthefirstdegreeinan unknown vectorrcontains termsofthetypes Aaor, 11.r,Exr,D. Thatisitwillcontain termswhichconsistofaknown vectormultiplied bythescalarproductofanotherknownvec torandtheunknown yector:termswhicharescalarmulti plesoftheunknown vector; termswhicharethevector productofaknownandtheunknown vector;andconstant terms.ThetermsofthetypeAaormayalwaysbereduc'ed tothreeinnumber. Forthevectorsa,b,C,'"whichare multiplied intormayallbeexpressed intermsofthreenon coplanar vectors. Hencealltheproducts aor,bor,cor,... maybeexpressed int~rmsofthree.Thesumofalltermsof thetypeAaortherefore reducestoanexpression ofthree terms,as Aaor+ Bbor+ Ccor. Thetermsofthetypes 11.randExrmayalsobeexpressed inthisform. 11.r=11.a'aor+11.b'bor+11.c'cor Exr=Exa'aor+Exb'bor+Exc' cor. Addingallthesetermstogether thewholeequation reduces totheform Laor+J(bor+Iic.r=K. 92 VECTOR ANALYSIS Thishasalreadybeensolvedas r=K.L'a.'+K.JiI'b'+K.N'c'. Thesolution isintermsofthreenon-coplanar vectorsa',b',c'. Theseformthesystemreciprocal toa,b,cintermsofwhich theproducts containing theunknown vectorrwereexpressed. •SUNDRY ApPLICATIONS OFPRODUCTS Applications toIt/echanics 48.]Inthemechanics ofarigidbodyaforceisnota. vectorinthesenseunderstood inthisbook.SeeArt.3. Aforcehasmagnitude anddirection; butithasalsoaline ofapplication. Twoforceswhiaharealikeinmagnitude anddirection, butwhichlieupondifferent linesinthebody donotproduce thesameeffect.Nevertheless vectorsare sufficiently likeforcestobeusefulintreating them. .Ifl\number offorcesfl'f2•fs,···actonabodyatthe samepoint0,thesumoftheforcesaddedasvectorsiscalled theresultant R. II=f1+f2+fs+... Inthesamewayiff1,f2•fs'"donotactatthesamepoint thetermresultant isstillappliedtothesumoftheseforces addedjustasiftheywerevectors. (41) Theideaoftheresultant therefore doesnotintroduce the lineofactionofaforce.Asfarastheresultant isconcerned aforcedoesnotdifferfromavector. D,jinition: Themomentofaforcefaboutthepoint0is equaltotheproduct oftheforcebytheperpendicular dis tancefrom0tothelineofactionoftheforce.Themoment however isbestlookeduponasavectorquantity. Itsmag nitudeisasdefinedabove.Itsdirection isusuallytakento DIRECT ANDSKEWPRODUCTS OFVECTORS 93 bethenormalonthatsideoftheplanepl\S8edthrough the point0andthelinefuponwhichtheforceappears topro duceatendency torotationaboutthepoint0inthepositive trigonometric direction. Another method ofdefining the moment ofaforcef=PQaboutthepoint0isasfollows: Themoment oftheforcef=PQaboutthepoint0isequal totwicetheareaofthetriangle0 PQ.Thisincludesatonce boththemagnitude anddirection ofthemoment (Art.25). ThepointPissupposed tobetheorigin;andthepointQ, theterminus ofthearrowwhichrepresents theforcef.The letter.willbeusedtodenotethemoment. Asubscript will beattached todesignate thepointaboutwhichthemomentis taken..0{f}=2OPQ. Themoment ofanumberofforcesfl'f2,•••isthe(vector) sumofthemoments oftheindividual forces. If f1=P1Ql'f2=P2Q2'" .0{f1,f2•...}=2(OP1Q1+OP2Q2+...). Thisisknownasthetotalorresultant moment oftheforces f1,fS'.... 49.]Iffbeaforceactingonabodyandifdbethevector drawnfromthepoint0toanypointinthelineofactionof theforce,themoment oftheforceaboutthepoint0isthe vectorproductofdintof. For.0if}=dxf dxf=dIsin(d,f)e,(42) ifebeaunitvectorinthedirection ofdxf. dxf=dsin(d,f)Ie. Nowdsin(d,f)istheperpendicular distance from0tof. Themagnitude ofdxfisaccordingly equaltothisperpen diculardistance multiplied byI,themagnitude oftheforce. 94 VECTOR ANALYSIS Thisisthemagnitude ofthemoment 110{t}.Thedirection ofdxfisthesameasthedirection ofthemoment. Hence therelationisproved. I110{f}=dxf. Thesumofthemoments about0ofanumberofforces fl'f2,."actingatthesamepointPisequaltothemoment oftheresultant IIoftheforcesactingatthatpoint.Forlet dbethevectorfrom0toP.Then 110{fl}=dxfl 110{f2~=dxf2 110{fl} +110{f2} +...=dxfl+dxf2+... (43) =dx(fl+ f2+..-)=dxR Thetotalmomentabout0'ofanynumberofforcesfl'f2•·.. actingonarigidbodyisequaltothetotalmoment ofthose forcesabout0increased bythemoment about0'ofthe resultant lloconsidered asactingato. lilY{fl.f2•"'}=1I0 {fl'f2•...} +lilY{Ro~.(44) Letdl,d2••••bevectorsdrawnfrom0toanypointin fl'f2••.•respectively. Letdl'.d2'••.•bethevectorsdrawn from0'tothesamepointsinfl'f2••..respectively. Let0 bethevectorfrom0to0'.Then dl=dl' +C,d2=d2' +Co••• 110{fl, f2•...}=dlxfl+ d2xf2+ . 110'{fl'f2•... ~=dl'xfl+ d2'xf2+ . =(dl-c)xfl+(d2-c)xf2+... =dlXfl+~xf2+'" -cx(fl+f2+"'J But- cisthevectordrawnfrom0'too.Hence-cxf, isthemoment about0'ofaforceequalinmagnitude and parallelindirection tof1butsituatedato.Hence DIRECT ANDSKEWPRODUCTS OFVECTORS 95 -ex(fl+'2+...)= -eXRo=XO'{Boo}. HenceXO'{fl,t,p.,.}= Xo{fl'f2,••.}+XO'{RoI.(44) Thetheoremistherefore proved. Theresultant RisofcourSethesameatallpointe. The subscript 0isattached merelytoshowatwhatpointitis supposed toactwhenthemoment about0'istaken.For thepointofapplication ofRaffectsthevalueofthatmoment. Thescalarproductofthetotalmoment andtheresultant isthesamenomatteraboutwhatpointthemoment betaken. Inotherwordstheproductofthetotalmoment, theresult ant,andthecosineoftheanglebetween themisinvariant forallpointsofspace. R·Xo'{fl,f2,••.}=R·Xo {fi'f2,...} where0'and0areanytwopointsinspace.Thisimportant relationfollowsimmediately fromtheequation . XO'{f1'f2,···}= Xo{fl'f2,···}+XO'{Boo}. ForR.XO'Ifl'f2,.•.}=R.Xo{fl'f2,···}+R.XO'{Ro}. Butthemoment ofRisperpendicular toRnomatterwhat thepoint0ofapplication be.Hence R.XO'IRo}= 0 andtherelationisproved. Thevariation inthetotal moment duetoavariation ofthepointaboutwhichthe moment istakenisalwaysperpendicular totheresultant. 50.]Apoint0'maybefoundsuchthatthetotalmoment aboutitisparallel totheresultant. Thecondition for parallelism is RxXO'{fl,f2,•..}= 0 RxXO'Jfl,f2,···1=RxXo{fl,f2,••.} +RxXO'IRoI=0 VECTOR ANALYSIS where0isanypointchosenatrandom. Replace XO'{Ro} byitsvalueandforbrevityomittowritethefl,f3,.••inthe braces{}.Then B.xXO'=B.xXo-B.x(exB.)=O. Theproblemistosolvethisequation forc. B.xXo-RoB.e+B..eB.=O. NowB.isaknownquantity. Xoisalsosupposed tobe known. Letcbechosenintheplanethrough0perpen diculartoB..ThenB.·c=0andtheequation reducesto B.xXo=RoB.c B.xXoe= .B..B. Ifcbechosenequaltothisvectorthetotalmoment about thepoint0',whichisatavectordistancefrom0equaltoC, willbeparalleltoB..Moreover, sincethescalarproductof thetotalmomentandtheresultant isconstant andsincethe , resultant itselfisconstant itisclearthatinthecasewhere theyareparallelthenumerical valueofthetotalmoment willbeaminimum. Thetotalmoment isunchanged bydisplacing thepoint aboutwhichitistakeninthedirection oftheresultant. For XO'If1,f3•••.}=Xo{fl,f3,·•.} -exB.. Ife=V(jisparalleltoB.,cxB.vanishes andthemoment about0'isequaltothataboutO.Henceitispossibleto findnotmerelyonepoint0'aboutwhichthetotalmoment isparalleltotheresultant; butthetotalmoment aboutany pointinthelinedrawnthrough0'paralleltoB.isparallel toB..Furthermore thesolution foundinequation forcis theonlyonewhichexistsintheplaneperpendicular toB. unlesstheresultant B.vanishes. Theresultsthathavebeen obtained maybesummed upasfollows: DIRECT ANDSKEWPRODUCTS OFVECTORS 9. Ifanysystemofforcesfl'flI'..•whoseresultant isnot zeroactuponarigidbody,thenthereexistsinspaceone andonlyonelinesuchthatthetotalmoment aboutany pointofitisparalleltotheresultant. Thislineisitself paralleltotheresultant. Thetotalmoment aboutallpoints ofitisthesameandisnumerically lessthanthataboutany otherpointinspace. Thistheorem isequivalent totheonewhichstatesthat anysystemofforcesactinguponarigidbodyisequivalent toasingleforce(theresultant) actinginadefinitelineand acoupleofwhichtheplaneisperpendicuhr totheresultant andofwhichthemomentisaminimum. Asystemofforces maybereducedtoasingleforce(theresultant) actingatany desiredpoint0ofspaceandacouplethemoment ofwhich (regarded asavectorquantity) isequaltothetotalmoment about0oftheforcesactingonthebody.Butingeneralthe planeofthiscouplewillnotbeperpendicular totheresult ant,norwillitsmomentbeaminimum. Thosewhowouldpursuethestudyofsystems offorces actingonarigidbodyfurtherandmorethoroughly may consulttheTraiUdeMecanique Rationnelle 1byP.ApPELL. Thefirstchapterofthefirstvolumeisentirelydevoted to thediscussion ofsystemsofforces.Appelldefinesavector 88aquantity possessing magnitude, direction, andpointof application. Hisvectorsareconsequently notthesameas thoseusedinthisbook.Thetreatment ofhisvectorsis carriedthrough intheCartesian coordinates. Eachstep however maybeeasilyconverted intothenotation ofvector analysis. Anumber ofexercises isgivenatthecloseof thechapter. 51.]Suppose abodyberotatingaboutanaxiswithacon stantangularvelocity a.Thepointsinthebodydescribe circlesconcentric withtheaxisinplanesperpendicular to 1Paris, Gauthie~ VillarsetFila.1893. 7 98 VECTOR ANALYSIS theaxis.Thevelocity ofanypointinitscircleisequal totheproductoftheangularvelocityandtheradiusofthe circle.Itistherefore equaltotheproduct oftheangular velocity andtheperpendicular dis tancefromthepointtotheaxis. Thedirection ofthevelocity is perpendicular totheaxisandto theradiusofthecircledescribed bythepoint. Leta(Fig.25)beavectordrawn alongtheaxisofrotation inthat direction inwhicharight-handed screwwouldadvance ifturnedin FIG.25. thedirection inwhichthebodyis rotating. Letthemagnitude of& bea,theangular velocity. Thevectoramaybetakento represent therotation ofthebody.Letrbearadiusvector drawnfromanypointoftheaxisofrotation toapointinthe body.Thevectorproduct axr=a rsin(a,r) isequalinmagnitude anddirection tothevelocity vofthe terminus ofr.Foritsdirection isperpendicular toaandr anditsmagnitude istheproduct ofaandtheperpendicular distance rsin(a,r)fromthepointtothelinea.Thatis v=axr. (45) v1=&lxr1 v2=&2Xf2 va=&axrsIfthebodyberotating simultaneously aboutseveralaxes &1'&2'&3...whichpassthrough thesamepointasinthe caseofthegyroscope, thevelocities duetothevarious rotations are DlilEC1' ANDSKEWPRODUCTS OFVECTORS 99 wheref1,f2,fa'...aretheradiivectores drawnfrompoints ontheaxisaI'a2,as'tothesamepointofthebody.Let thevectorsf1,r2,rs'bedrawnfromthecommon pointof intersection oftheaxes.Then and v=v1+v2+Vs+...=a1xr+a2xf+asxf+... =(a1+a2+aa+ "·)Xf. Thisshowsthatthebodymovesasifrotating withthe angular velocity whichisthevectorsum'oftheangular velocities a!'a2,as'...Thistheorem issometimes known astheparallelogram lawofangularvelocities. Itwillbeshownlater(Art.)60thatthemotionofany rigidbodyonepointofwhichisfixedisateachinstantof t.imearotation aboutsomeaxisdrawnthrough thatpoint. Thisaxisiscalledtheinstantaneous axisofrotation. The axisisnotthesameforalltime,butconstantly changes its position. Themotionofarigidbodyonepointofwhichis fixedistherefore represented by v=axr (45) whereaistheinstantaneous angular velocity; andf,the radiusvectordrawnfromthefixedpointtoanypointofthe body. Themostgeneralmotionofarigidbodynopoin~ofwhich isfixedmaybetreated asfollows. Chooseanarbitrary pointO.Atanyinstantthispointwillhaveavelocity YO' Relative tothepoint0thebodywillhaveamotionofrotation aboutsomeaxisdrawnthrough O.Hencethevelocityvof anypointofthebodymayberepresented bythesumof Vothevelocity of0andaXfthevelocity ofthatpoint relative toO. v=Vo+aXf. (46) 100 VECTOR ANALYSIS IncaseVoisparalleltoa,thebodymovesaroundaand alongasimultaneously. Thisisprecisely themotionofa screwadvancing alonga.IncaseVoispel'pendicular toa,it ispossibletofindapoint,givenbythevectorr,suchthat itsvelocityiszero.Thatis axt= -Vo' Thismaybedoneasfollows. Multiply byX&. (axr)xa= -voxa or Letrbechosenperpendicular toa.Thena·riszeroand a·ar= -Vox a - VoXar= .a·a Thepointr,thusdetermined, hastheproperty thatitsveloc ityiszero.Ifalinebedrawnthroughthispointparallelto a,themotionofthebodyisoneofinstantaneous rotation aboutthisnewaxis. IncaseVoisneitherparallelnorperpendicular toaitmay beresolved intotwocomponents Vo=vo'+vo" whicharerespectively parallelandperpendicular toa. v=vo'+vo"+axr Apointmaynowbefoundsuchthat vo"= -axr. Letthedifferent pointsofthebodyreferredtothispointbe denotedbyr'.Thentheequation becomes v=vo'+axr'. (46)' Themotionhereexpressed consistsofrotationaboutanaxis aandtranslation alongthataxis.Itistherefore seenthat themostgeneralmotionofarigidbodyisatanyinstant D.R=0DIRECT ANDSKEWPRODUCTS OFVECTORS 101 themotionofascrewadvancing atacertainratealonga definiteaxisainspace.Theaxisofthescrewanditsrate ofadvancing perunitofrotation (i.e.itspitch)changefrom instanttoinstant. 52.]Theconditions forequilibrium asobtained bythe principle ofvirtualvelocities maybetreatedbyvector methods. Suppose anysystemofforcesfl'f2,.••actona rigidbody.Ifthebodybedisplaced tlu."oughavectordis tanceDwhether thisdistance befiniteorinfinitesimal the workdonebytheforcesis D.fl,D.~,... Thetotalworkdoneistherefore W=D·f1+D·f2+... Ifthebodybeinequilibrium undertheactionoftheforces theworkdonemustbezero. W=D.fl+D.f2,+...=D·Cfl+f2+...)=D.R=O. Theworkdonebytheforcesisequaltotheworkdoneby theirresultant. Thismustbezeroforeverydisplacement D.Theequation holdsforallvectorsD.Hence R=O. Thetotalresultant mustbezeroifthebodybeinequilibrium. Theworkdonebya.forcefwhentherigidbodyisdis placedbyarotationofangull\rvelocityaforaninfinitesimal timetisapproximately a·dxft, wheredisavectordrawnfromanypointoftheaxisofrota tionatoanypointoff.Toprovethisbreakupfintotwo components f',f"parallelandperpendicular respectively toa- a·dxf=a.dxf'+a.dxf". 102 VECTOR ANALYSIS Asf'isparalleltoathescalarproduct[adfl]vanishes. a·dxf=a.dxf". Ontheotherhandtheworkdonebyf"isequaltothework donebyfduringthedisplacement. Forf'beingparallelto aisperpendicular toitslineofaction.Ifhbethecommon vectorperpendicular fromthelineatotheforcef",thework donebyf"duringarotation ofangularvelocityafortime tisapproximately W=hI"at=a.hxf"t. Thevectorddrawnfromanypointofatoanypointoffmay bebrokenupintothreecomponents ofwhichoneish,another isparalleltoa,andthethirdisparallel tof".Inthescalar tripleproduct [adf"]onlythatcomponent ofdwhichis perpendicular aliketoaandf"hasanyeffect.Hence W=a·hxf"t=a.dxf't'=a·dxft. IfarigidbodyuponwhichtheforcesfI,f2,..•actbedis placedbyanangular velocity aforaninfinitesimal timet andifdI,d2,••.bethevectorsdrawnfromanypoint0of atoanypointsoffl'f2'...respectively, thentheworkdone bytheforcesfI,f2,...willbeapproximately W=(a.dlxfl+a.d2xf2+ )t =a·CdlxfI+d2xf2+ )t =a·.o{fpf2,.•}t. Ifthebodybeinequilibrium thisworkmustbezero. Hence Thescalarproduct oftheangular velocity aandthetotal moment oftheforcesfl'f2,•••aboutanypoint0mustbe zero.Asamaybeanyvectorwhatsoever themoment itself mustvanish. HenceDIRECT ANDSKEWPRODUCTS OFVECTORS 103 Thenecessary conditions thatarigidbodybeinequilib riumundertheactionofasystemofforcesisthattheresult antofthoseforcesandthetotalmoment aboutanypointin spaceshallvanish. Conversely iftheresultant ofasystemofforcesandthe moment ofthoseIorcesaboutanyonepalticular pointinspace vanishsimultaneously, thebodywillbeinequilibrium. IfR=0,thenforanydisplacement oftranslation D D.R=O. W=D.fl+D·f2+...=0 andthetotalworkdoneiszero,whenthebodysuffersany displacement oftranslation. Let.0{fl'f2•.··1bezeroforagivenpointO.Thenfor anyotherpoint0' .0'{fl,f3•···1=.0Ifl,f2•...}+.0'{Ro}. Butbyhypothesis Risalsozero.Hence .0'{fl,f2•..·1=O. a·.o'{fl'f2•••·1t=0 whereaisanyvectorwhatsoever. Butthisexpression is equaltotheworkdonebytheforceswhenthebodyisrotated foratimetwithanangular velocity aaboutthelinea passingthroughthepoint0'.Thisworkiszero. Anydisplacement ofarigidbodymayberegarded asa translation through adistance Dcombined witharotation foratimetwithangularvelocity aaboutasuitable lineain space.Ithasbeenprovedthatthetotalworkdonebythe forcesduringthisdisplacement iszero.Hencetheforces mustbeinequilibrium. Thetheorem isproved. 104 VECTOR ANALYSIS Applications toGeometry 53,]Relations between tworight-handed systemsofthree mutually perpendicular unitvectors.-Leti,i,kandi',i',It' betwosuchsystems, Theyformtheirownreciprocal systems. Hence and Fromthisr =r·ii+r·jj+r·kk r =r.i'i'+r.j'j'+r.k'k',(47) {i'=i'.ii+i'.jj+i'.kk= ali+asj+aak j'=j'.ii+j'.jj+j'.kk=hIi+hsj+hak k'=k'·ii+k'.jj+k'.kk= c1i+Csj+cak.(47") ThescalarsaI'a2,as;hI'hs•ha;C]'02'Caarerespectively the direction cosinesofi';j';k'withrespecttoi,j,k. Thatis ("') al=cos1,1 hI=cos(j',i) c1=cos(k',i)("') as=cos 1,J h2=cos(j',j) c2=cos(k',j)as=cos(i',k) ha=cos(j',k) ca=cos(k',k).(48) andandrnthesamemanner ~i =i.i'i'+i.j'j'+i.k'k'=a]i'+h]j'+c]k' j =j.i'i'+j.j'j'+j.k'k'=a2i'+hsj'+c2k' k=k.i'i'+k.j'j'+k.k'k'=aai'+haj'+Csk' ~i'.i'=1=al2+a22+ass j'.j'=1=b]2+bs2+bas k'.k'=1=C12+c.l+ca2 ~i.i=1=a12+b]2+C]2 joj=1=a22+b22+c22 k·k=1=as2+ba2+ca2 !i'.j'=0=a]b]+a2b2+aaba j'.k'=0=b]c1+b2c2+baca k'.i'=0=c1at+c2as+caaa(47')" (49) (49)' (50) DIRECT ANDSKEWPRODUCTS OFVECTORS 105 and and But Hence1~.j=0=ala2+blb';&+CIC';& J.t=0=a';&as+b';&bs+c';&Cs k·i=0=asal+bsbl+CsCI ala';&as [ijkJ=[i'j't']=1=blb';&bs CIC';&Cs t'=i'xj'=(a';&bs-asb';&)i+(asbl-albs)J +(alb';&-a';&bl)t. t'=CIi+c';&j+Csk. 1CI=(a';&bs-asb';&), c';&=(asbl- UIbs)' Cs=(alb';&-a';&bl)'(50)' (51) (52) OrCI=I~:::I,c';&=I~:~:I,c3=I~:~I' andsimilarrelations maybefoundfortheothersixquantities ai'a,;&,as;bl,b';&,ba-Allthesescalarrelations between the coefficients ofatransformation whichexpresses onesetof orthogonal axesX',Y',Z'intermsofanothersetX,Y,Zare important andwellknowntostudents ofCartesian methods. Theeasewithwhichtheyareobtained heremaybenote worthy. Anumberof'Vectorrelations, whichareperhapsnotsowell known,butnevertheless important, maybefoundbymulti plyingtheequations i'=aIi+a2j+ast ali'+blj'+cik'=i invectormultiplication. blk'-cd'=asj-a2k. (58) Thequantity oneithersideofthisequalityisavector.From itBformupontherightitisseentopossessnocomponent in 106 VECTOR ANALYSIS theidirection buttoliewhollyintheik-plane; andfrom itaformupontheleftitisseentolieinthei'k'-plane. Henceitmustbethelineofintersection ofth08etwoplanes. Itamagnitude isv'a33+as3orv'b13+C13•Thisgivesthe scalarrelations a33+(1/'=b/'+c13=1 -a13, Themagnitude 1 -a12isthesquareofthesine oftheangle between thevectorsiandif.Hencethe vector (58) isthelineofintersection ofthei'k'-andik-planes, and itsmagnitude is'thesineoftheanglebetween theplanes. Eightothersimilarvectorsmaybefound,eachofwhichgives oneoftheninelinesofintersection ofthetwosetaofmu tuallyorthogonal planes. Themagnitude ofthevectorisin eachcasethesineoftheanglebetween theplanes. 54.]Various examples inPlaneandSolidGeometry may besolvedbymeansofproducts. Examp]e 1:Theperpendiculars fromtheverticesofatrian gletotheopposite sidesmeetinapoint.LetABObethe triangle. Letthe perpendiculars fromAtoB0andfromB to0AmeetinthepointO.Toshow00isperpendicular toAB.Choose0asoriginandlet0A=A,0B=B,and 00-=C.Then BO=C-B, OA=A-C, AB=B-A. Byhypothesis and Subtract;A.(C-B)=0 B.(A-C)=o. C.(B-A)=o~ whichprovesthetheorem. E:cample 1]:Tofindthevectorequation ofalinedrawn through thepointBparalleltoagivenvectorA. HenceDIRECT ANDSKEWPRODUCTS OFVECTORS 107 Let0betheoriginandBthevectorOB.LetRbethera diusvectorfrom0toanypointoftherequired line.Then ]I.-BisparalleltoA.Hencethevectorproductvanishes. AX(R-B)=O. Thisisthedesiredequation.Itisavectorequation inthe unknown vectorR.Theequation ofaplanewasseen(page 88)tobeascalarequation suchas R·C=c intheunknown vectorR. Thepointofintersection ofalineandaplanemaybe foundatonce.Theequations are ~AX(R-B)=0 ~R.C=c AxR=AxB (AxR)xC =(AxB)xC A·CR-C·RA=(AxB)xC A·CR-cA=(AxB)xC (AxB)xC+cA R= A.C Thesolutionevidently failswhenA·C=O.Inthiscasebow everthelineisparalleltotheplaneandthereisnosolution; or,ifitliesintheplane,thereareaninfinitenumberofsolu tions. Example:3:Theintroduction ofvectorstorepresent planes. Heretofore vectorshavebeenusedtodenoteplaneareasof definiteextent. Thedirection ofthevectorwasnormalto theplaneandthemagnitude wasequaltotheareatobere presented. Butitispossibletousevectorstodenotenota planeareabuttheentireplaneitself,justasavectorrepresents apoint.Theresultisanalogous totheplanecoordinates of analytic geometry. Let0beanassumed origin.LetMNbe aplaneinspace.TheplaneMNistobedenotedbyavector 108 VECTOR ANALYS1S Hencewhosedirection isthedirection oftheperpendicular dropped upontheplanefromtheorigin0andwhosemagnitude isthe reciprocal ofthelengthofthatperpendicular. Thusthenearer aplaneistotheoriginthelongerwillbethevectorwhich represents it. Ifrbeanyradiusvectordrawnfromtheo.rigintoapoint intheplaneandifpbethevectorwhichdenotestheplane, then rop=1 istheequation oftheplane.For rop=rcos(r,p)p. Nowp,thelengthofpisthereciprocal oftheperpendicular distance from0totheplane.Ontheotherhandrcos(r,p) isthatperpendicular distance. Henceropmustbeunity. Ifrandpbeexpressed intermsofi,j,k r=xi+yj+zk p=ui+vj+wk rop=oXu+yv+zw=1. Thequantities u,v,warethereciprocals oftheintercepts of theplanepupontheaxes. Therelationbetweenrandpissymmetrical. Itisarela tionofduality.Ifintheequation rop=1 rberegarded asvariable, theequation represents aplanep whichisthelocusofallpointsgivenbyr.Ifhowever pbe regarded asvariable andrasconstant, theequation repre sentsapointrthrough whichalltheplanesppass.The development oftheideaofdualitywillnotbecarriedout. Itisfamiliartoallstudents ofgeometry. Theuseofvec torstodenoteplaneswillscarcely bealluded toagainuntil ChapterVII. DIRECT ANDSKEWPRODUCTS OFVECTORS 109 SUMMARY OFCHAPTERiI Thescalarproduct oftwovectorsisequaltotheproduct oftheirlengthsmultiplied bythecosineoftheanglebetween them. A·B=A Bcos(A,B) A·B=B·A A.A=A2.(1) (2) (H) Thenecessary andsufficient condition fortheperpendicularity oftwovectorsneitherofwhichvanishes isthattheirscalar product vanishes. Thescalarproducts ofthevectorsi,j.t are i·i=joj=t.t=1 i.j=j.t=k.i=0 A·B=AlB1+A2B2+AsBs A.A=A2=AI2+A22+A32. Iftheprojection ofavectorBuponavectorAisB', B'=A·BA A·A.(4) (7) (8) ,(5) Thevectorproductoftwovectorsisequalinmagnitude to theproductoftheirlengthsmultiplied bythesineofttwan glebetween them.Thedirection ofthevectorproduct ISthe normaltotheplaneofthetwovectorsonthatsideonwhich arotation oflessthan1800fromthefirstvectortothesecond appearspositive. AxB=A Bsin(A,B)c. (9) Thevectorproductisequalinmagnitude anddirection tothe veotorwhichrepresents theparallelogram ofwhichAandB arethetwoadjacent sides.Thenecessary andsufficient con ditionfortheparallelism oftwovectorsneitherofwhich 110 VECTOR ANALYSIS vanishes isthattheirvectorproduct vanishes. Thecom· mutative lawsdonothold. AxB= -BxA (10) ixi=jxj=kxk=0 ixj=-jxi=k (12) jxk=-kxi=i kxi= -ixk=j AxB=(AllDs-AsBlI)i+(AsB1-A1Bs)j +(A1BII-AllB1)k(13) ijk AxB=A1AllAs (13)' B1BIIBs Thescalartripleproduct ofthreevectors[ABC]isequal tothevolumeoftheparallelopiped ofwhichA,B,Carethree edgeswhichmeetinapoint. [ABC]=A·BxC=B·CxA=C·AxB =AxB·C=BxC·A=CxA·B [ABC]= -[ACB].(15)' (16)' Thedotandthecrossinascalartripleproductmaybeinter- chan.d lindtheorderofthelettersmaybepermuted cyclicly without altering thevalueoftheproduct; butachangeof cyclicorderchangesthesign. A1AllAs [ABC]=B1BIIBs C1CllCs a1allas [ABC]=b1b'},bs[abc] c1cllc8(18)' (19)' DIRECT ANDSKEWPRODUCTS OFVECTORS 111 Ifthecomponent ofBperpendicular toAbeB", B"=-Ax(AxB) (20) A·A AX(BxC) =A·CB -A·BC (24) (AxB)xC =A·CB -C·BA (24)' (AxB),(CxD) =A·CB·D-A.DB.C (25) (AxB)x(CxD)=[ACD]B -fBCD]A =[ABD]C-[ABC]D. (26) Theequation whichsubsistsbetween fourvectors A.,B,C,D is [BCD]A-[CDA]B+[DAB]C-[ABC]D=O.(27) Application offormulm ofvectoranalysistoobtainthefor· mulmofPlaneandSpherical Trigonometry. Thesystemofvectorsa',b',~issaidtobereciprocal tothe systemofthreenon.coplanar vectors a,b,0 when a'=~' b'=0xa,o,=axb.(29) [abo] [abo] [abo] Thenecessary andsufficient conditions thatthetwosystemsof non.coplanar vectorsa,b,0anda',b',0'bereciprocals isthat--(30) (31)Avectorrmaybeexpressed inter~ofasetofvectorsand itsreciprocal intwosimilarways r =r.a'a+r.b'b+r.o'0 r =r.aa'+r.bb'+r.oo'.or a'.a=b'.b=0'.0= 1 a'.b=a'.o=b'.o=b'.a=o'.a=o'.b=o.(32) Ifa',b',0'formasystemreciprocal toa,b,o;thena,b,0will formasystemreciprocal toa',b',0'. 1[a'b'O']=-- (33)'[ab0] 112 VECTOR Al\TALYSIS P·AP·Bp·C [PQR][.6.BCJ=Q·AQ.BQ.C (34) R.AR.BR.C Thesystemi,j,kisitaownreciprocal andifconversely a. systembeitsownreciprocal itmustbearightorlefthanded systemofthreemutually perpendicular unitvectors. Appli cationofthetheoryofreciprocal systems tothesolution of scalarandvectorequations ofthefirstdegreeinanunknown vector. Thevectorequation ofaplaneis r.A=a. (36) Applications ofthemethods developed inChapterII.,tothe treatment ofasystemofforcesactingonarigidbodyandin particular tothereduction ofanysystemofforcestoasingle forceandacoupleofwhichtheplaneisperpendicular tothat force.Application ofthemethods tothetreatment of instantaneous motionofarigidbodyobtaining v=Vo+aXr (46) wherevisthevelocityofanypoint, Voatranslational veloc ityinthedirection a,andathevectorangularvelocityofro tation. Further application ofthemethods toobtainthe conditions forequilibrium bymakingnseoftheprinciple of virtualvelocities. Applications ofthemethod toobtain therelations whichexistbetween theninedirection cosines oftheanglesbetween twosystems ofmutually orthogonal axes.Application tospecialproblems ingeometry including theformunderwhichplanecoordinates maketheirappear anceinvectoranalysisandthemethodbywhichplanes(as distinguished fromfiniteplaneareas)mayberepresented byvectors. DIRECT ANDSKEWPRODUCTS OFVECTORS 113 EXERCISES ONCHAPTERII Provethefollowing reduction formulre 1.Ax{Bx(CxD)} =[ACD]B -A·BCxD =B·DAxO-B·CAxD. 2.[AxBCxDExF]=[ABD][OEF]-[ABC][DEF] =[ABE][FCD]-[ABF][EOD] =[0DA][BEF]-[0DB][AEF]. 3.[AxBBxOCxA]=[ABC]2. P·AP·BP v4.[PQR](AxB)=Q.AQ·BQ. R·AR·BR 5.AX(BxC)+BX(CxA)+Cx(AxB) =O. 6.[AxPBxQCXRJ+[AxQBxRCXP] +[AxRBxPOxQ]=O. 7.Obtainformula (34)inthetextbyexpanding [(AxB)xP} [Cx(QxR)] intwodifferent waysandequating theresults. 8.Demonstrate directly bytheaboveformulre thatif a',b', c'formareciprocal systemtoa,b,c;thena,b,cform asystemreciprocal toa',b',c'. 9.Showtheconnection between reciprocal systemsofvec~ torsandpolartriangles uponasphere. Obtainsomeofthe geometrical forrnulre connected withpolartriangles byinter pretingvectorformulre suchas(3)intheabovelist. 10.Theperpendicular bisectors ofthesidesofatriangle meetinapoint. 11.Findanexpression forthecommon perpendicular to twolinesnotlyinginthesameplane. s~{l'!) 114 VECTOR ANALYSIS 12.Showbyvectormethodsthattheformulre forthevol umeofatetrahedron whosefourverticesare (Xl'Yl'Zl)(X2,Y2'Z2)(Xa'Ya'Za)(x"y"zJ is Xl'!IIZl1 1x2'!I2z21 6x8'!Iaza1 x,y,z,1 13.Makinguseofformula(34)ofthetextshowthat ~1,nm [ab0]=abcnIl . mIl whereu,b,carethelengthsofa,b,orespectively andwhere I=cos(b,o),m=cos(0,a),n=cos(a,b). 14.Determine theperpendicular (asavectorquantity) whichisdroppedfromtheoriginuponaplanedetermined by theterminiofthevectorsa,b,o.Usethemethodofsolution giveninArt.46. 15.Showthatthevolumeofatetrahedron isequaltoone sixthoftheproductoftwooppositeedgesbytheperpendicu lardistancebetweenthemandthesineoftheincluded angle. 16.Ifalineisdrawnineachfaceplaneofanytriedralangle throughthevertexandperpendicular tothethirdedge,the threelinesthusobtained lieinaplane. CHAPTER III THEDIFFERENTIAL CALCULUS OFVECTORS lJifferentiation OfFunctions ofOneScalarVariable 55.]IFavectorvariesandchangesfromrtor'theincre mentofrwillbethedifference betweenr'andrandwillbe denotedasusualby~r.. ~r=r'-r, (1) FIG.26.owhere ~rmustbeavectorquantity.Ifthevariable rbe unrestricted theincrement ~risofcoursealsounrestricted: itmayhaveanymagnitude andanydirection. If,however, thevectorrberegarded asafunction (avectorfunction) of asinglescalarvariabletthevalueof~rwillbecompletely determined whenthetwovaluestandt'oft,whichgivethe twovaluesrandr',areknown. Toobtainaclearerconception ofthequantities involved itwillbeadvantageous tothinkofthevectorrasdrawn fromafixedorigin0(Fig.26).When theindependent variabletchanges its valuethevectorrwillchange,andast possesses onedegreeoffreedom rwill varyinsuchawaythatitsterminus describes acurveinspace.rwillbe theradiusvectorofonepointPof thecurve;r',ofaneighboring pointP'.~rwillbethe chordPP'ofthecurve.Theratio 116 VECTOR ANALYSIS willbeavectorcollinear withthechordPpIbutmagnified intheratio1:at.Whenatapproaches zeroP'willap proachP,thechordPpIwillapproach thetangentatP,and thevectorar. dratwillapproach dt whichisavectortangent tothecurveatPdirected inthat senseinwhichthevariabletincreases alongthecurve. Ifrbeexpressed intermsofi,j,kas r=r1i+1'2j+r3k thecomponen~ r1,r2,rswillbefUllctio.ns ofthescalart. r'=(r1+ar1)i+(r2+ar2)j+ers+ars)k ar=r'- r=ar1i+ar2j+arsk ar _ar1iar2•arskat-at +atJ+at and (2) Hencethecomponents ofthefirstderivative ofrwithre specttotarethefirstderivatives withrespecttotofthe components ofr.Thesameistrueforthesecondandhigher derivatives. (2)' Inasimilarmannerifrbeexpressed intermsofanythree non-eoplanar vectorsa,b,Cas r=aa+bb+cc d"rd"ad"b dnc dtn=dtna+dt"b+dtnc. THEDIFFERENTIAL CALCULUS OFVECTORS 117 Example: Let r=acost+bsint. Thevectorrwillthendescribe anellipseofwhichaandb aretwoconjugate diameters. Thismaybeseenbyassum ingasetofobliqueCartesian axesX,Ycoincident witha andb.Thenx=acost.Y=bsint, XlIy.a a2+b'J.=1, whichistheequation ofanellipsereferred toapairofcon jugatediameters oflengthsaandbrespectively. dr . b-dt= -aS111t+cost. , dr . Hence dt=acos(t+90°)+bSill(t+90°). Thetangent tothecurveisparallel totheradiusvector for(t+90°).d2r dt2= -(acost+bsint). Thesecondderivative isthe negative ofr.Hence d2r-=-rdt2 isevidently adifferential equation satisfied bytheellipse. EXa1nple: Let r=acosht+bsinht. Thevectorrwillthendescribe anhyperbola ofwhichaand baretwoconjugate diameters. dr- =asinht+bcosht.dt d2rand -2=acosht+bsinht.dt d2r Hence dt2=r isadifferential equation satisfied bythehyperbola. 118 VECTOR .AN.ALYSIS 56.]Acombination ofvectorsallofwhichdependonthe samescalarvariabletmaybedifferentiated verymuchasin ordinary calculus. d (db)(da) -(a°b)=a° - + -°b.dt dtdt For (a+~b)°(b+~b)=a°b +a°~b +~a°b +~a°~b ~(a°b)=(a+~a)°(b+~b)-a°b =ao~b+~aob+~ao~b ~(aob)_ °~b· ~aob ~ao~b ~t-a·~t+~t+~t Henceinthelimitwhen~t=0, d dbdadt(aob)=a°it+dtob (3) :t(aXb)=.aX(~:)+(~:)Xb (4) ddt(aobXC)=aobX (~~)+ao (~:)xc +(~:)°bXo. (5) :t(aX[~Xc])=aX[bX(~~)] +aX[(~:)xc] r +(~:) X[bX0]. (6) Thelastthreeoftheseformulre maybedemonstrated exactly asthefirstwas. Theformalprocessofdifferentiation invectoranalysis differsinnowayfromthatinscalaranalysis exceptinthis onepointinwhichvectoranalysis alwaysdiffersfromscalar analysis, namely: Theorderofthefactorsinavectorproduct THEDIFFERENTIAL CALr;ULUS OFVECTORS 119 cannotbechanged withoutchanging thesignoftheproduct. Henceofthetwoformullll d(da)(db)dt(axb)= dtxb+dtxa and:t(axb)=(~;)x b + a x (~:) thefirstisevidently incorrect, butthesecondcorrect. In otherwords,scalardifferentiation musttakeplacewithout altering theorderofthefactorsofavectorproduct. The factorsmustbedifferentiated insitu.Thisofcoursewasto beexpected. Incasethevectorsdependuponmorethanonevariable theresultsarepractically thesame.Inplaceoftotalderiva tiveswithrespecttothescalarvariables, partialderivatives occur.Suppose a.andbaretwovectorswhichdependon threescalarvariables x,y,z.Thescalarproduct a·bwill dependuponthesethreevariables, anditwillhavethree partialderivatives ofthefirstorder. CJ (CJa) (CJb)~x(a.b)= CJx.b+a. CJx ~(CJa) (CJb) -(a.b)= - • b+a·- ~y CJy CJy ~ (CJa) (CJb) -(a.b)= -.b+a. - . ~z CJz CJz(7) Thesecondpartialderivatives areformedinthesameway. ~2(CJ2a)(CJa)(Clb) --(a.b)=--•b+-. - ~x~y CJxCJy Clx CJy +(Cla).(CJb)+a.(~). CJy CJx CJxCJy 120 VECTOR ANALYSIS Oftenitismoreconvenient tousenotthederivatives but thedifferentials. Thisisparticularly truewhendealingwith firstdifferentials. Theformulre (3),(4)become d(a.b)=da.b+a.db, (3)' d(axb)=daxb +axdb, (4)' and80forth.Asanillustration consider thefollowing example.Ifrbeaunitvector r·r=1. Thelocusoftheterminus ofrisaspherical surfaceofunit radiusdescribed abouttheorigin. rdepends upontwovari ables.Differentiate theequation. Hence(dr).r+r.(dr)=2r.(dr)=O. r.dr=O. Hencetheincrement drofaunitvectorisperpendicular to thevector. Thiscanbeseengeometrically. Ifrtracesa spherethevariation drmustbeateachpointinthetangent planeandhenceperpendicular tor. -57.JVectormethods maybeemployed advantageously inthediscussion ofcurvature andtorsionofcurves. Letr denotetheradiusvectorofacurve r=f(t), wherefissomevectorfunction ofthescalart.Inmostappli cationsinphysicsandmechanics trepresents thetime.Let sbethelengthofarcmeasured fromsomedefinitepointof thecurveasorigin. Theincrementaristhechordofthe curve.Hencear /asisapproximately equalin.magnitude tounityandapproaches unityasitslimitwhenasbecomes infinitesimal. Hencedr /dswillbeaunitvectortangent to thecurveandwillbedirected towardthatportionofthe THEDIFFERENTIAL CALCULUS OFVECTORS 121 curvealongwhichsisincre8.lling (Fig.27).Lettbethe unittangent tuc c...7i FIG.27.(8)dr-=t.ds Thecurvature ofthecurveisthe limitoftheratiooftheanglethrough whichthetangentturnstothelength ofthearc.Thetangentchanges bytheincrementat.Ast isofunitlength,thelengthofatisapproximately theangle through whichthetangent hasturnedmeasured incircular measure. Hencethedirected curvature Cis (9) I(=txC.ThevectorCiscollinear withatandhenceperpendicular to t;forinasmuch astisaunitvectoratisperpendicular tot. Thetortuosity ofacurveisthelimitoftheratioofthe anglethrough whichtheosculating planeturnstothelength ofthearc.Theosculating planeistheplaneofthetangent vectortandthecurvature vectorC.Thenormaltot~is !>laneis Ifcbeaunitvectorcollinear withC D=txc 'OK' •FIG.28.willbeaunitnormal(Fig.28)totheosculating planeand thethreevectorst,C,Dformani,j,ksystem, thatis,aright-handed rectangular system. Thentheanglethrough whichtheosculating planeturnswillbegivenapproximately by ADandhencethetortuosity isbydefinition dDjds. Fromthefactthatt,c,Dformani,j,ksystemofunit vectors 122 andVECTOR ANALYSIS tot=Coc=DoD=l toe=CoD=Dot=O. Differentiating thefirstset t0dt =codc =DodD=0, andthesecond t0de+dtoo=codD+dCoD=Dodt+dDot=O. Butdtisparallelto0andconsequently perpendicular toD. Dodt=o. Hence dDot=O. Theincrement ofDisperpendicular tot.Buttheincrement ofDisalsoperpendicular toD.Itistherefore paralleltoc. Asthetortuosity isT=dDids,itisparalleltodDandhence too. Thetortuosity Tis d d(drdlr1)T=ds(txc)=dsdsXds2v'CoC (11) d2rd2rId rd8r 1T=-x ---+-x---dS2dS2v'CoCds ds3v'c--;C drd2rd1+--x-- .d s d 82d8v'C0C Thefirsttermofthisexpres8ion vanishes. Tmoreover has beenseentobeparalleltoC=d2ridS2.Consequently the magnitude ofTisthescalarproductofTbytheunitvec torcinthedirection ofC.Itisdesirable however tohave thetortuosity positivewhenthenormalDappearstoturnin thepositive orcounterclockwise direction ifviewedfrom thatsideoftheDo-plane uponwhichtor thepositivepart ofthecurvelies.Withthisconvention dDappearstomove inthedirection - cwhenthetortuosity ispositive, thatis,D turnsawayfromc.Thescalarvalueofthetortuosity will therefore begivenby-0 0T. (12) HenceTHEDIFFERENTIAL CALCULUS OFVECTORS 123 drd8r1 drd2rd1-CoT=-oo-x -----Oo-x ---_.d8ds8~ d8ds2dsv'CoC But0isparalleltothevectord2r/d82•Hence drd2r o0d8 Xd82=o. And0isaunitvectorinthedirection C.Hence C d2r1 0=v'CoC= d82v'CoC' d2rdrd8r1T=-coT=-- o-x--_·d82d8ds8C0C (13) T= Or[drd2rd8r] d8dS2d 83 d2rd2r if8'J 0dS2 Thetortuosity maybeobtained byanothermethodwhich issomewhat shorterifnotquitesostraightforward. toc= 0 0n =not=O. Hence dt00= -de0t doon= -d~oc dnot=-dton. Nowdtisparallelto0;henceperpendicular ton.Hence dton=O.Hencedn 0t =O.Butdnisperpendicular ton. Hencednmustbeparalleltoc.Thetortuosity isthemag nitudeofdn!dstakenhowever withthenegative sign becausednappearsclockwise fromthepositive direction of thecurve. Hence thescalartortuosity Tmaybegivenby dn doT=- - 00=n0-, (14)ds ds doT=txco- , (14)'ds 124 VECTOR ANALYSIS (13) (18)'Cc= , v'C~ _dC d dev'C•C(~- CJ;v'C-:-C -ds C.C de dC_ dtxc·-=txc·-v'c--:C-txc·C-v'c-:l'.ds ds ds But txc.C=O. dCtxc.-VC:-O T- ds - C·C ' dCtxC·ds T=C.C ' Cdrdr2dar] T=dSJ:;2dsa d2rd2r ds~•ITS'i InCartesian coordinates thisbecomes d.vdydz dsdsds d2x d2yd2z d82dS2dS2 dSXdSydSz (fS8([;;sdsST=------- (d2 _or)2(d2Y)2(d2Z)2 d82+d82+d82• Thosewhowouldpursuethestudyoftwistedcurvesand surfaces inspacefurtherfromthestandpoint ofvectorswill findthebook"Application delaMethode Vertorielle deGrass mannIilaGeomCtrie Infinithimah" 1byFEHRextremely 1Paris,CarreetNaud,1899. THEDIFFERENTIAL CA.LCULUS OFVECTORS 125 helpful. Heworkswithvectorsconstantly. Thetreatment iselegant. Thenotation usedishowevet slightlydifferent fromthatusedbythepresent writer. Thefundamental pointsofdifference areexhibited inthistable '-toa2-[a1a2] a1xa2 -[a1Ia2] a1•a2Xaa=[a1a2aa]-[a1a2aa]' Oneusedtoeithermethodneedhavenodifficulty withthe other.Alltheimportant elementary properties ofcurves andsurfaces aretheretreated. Theywillnotbetaken uphere. •Kinematics 58.]Letrbearadiusvectordrawnfromafixedoriginto amovingpointorparticle. Lettbethetime.Theequation ofthepathisthen r=f(t). Thevelocityoftheparticleisitsrateofchangeofposition. Thisisthelimitoftheincrement ~rtotheincrement ~t. (15) Thisvelocity isavectorquantity. Itsdirection isthe direction ofthetangent ofthecurvedescribed bythepar ticle.Thetermspeedisusedfrequently todenotemerely thescalarvalueofthevelocity. .Thisconvention willbe followed here.Then (16) ifsbethelengthofthearcmeasured fromsomefixedpoint ofthecurve.Itisfoundconvenient inmechanics todenote differentiationR withrespecttothetimebydotsplacedover thequantity differentiated. Thisistheoldfluxional notation 126 VECTOR ANALYSIS introduced byNewton. Itwillalsobeconvenient todenote theunittangenttothecurvebytoTheequations become dr(15) v=r=-dt d8(16) t7=8=-dl v=vt. (17) TheacceleTation istherateofchangeofvelocity. It isavectorquantity. Letitbedenoted byA.Thenby definition andLIM.:1vdv. A=.:1t::!:::O.:1t=di=v A';v=~~=ddt(~~)=~2t:=r. (18) Differentiate theexpression v=vt. dvd(vt) dvdtA=- =--=-t+'17-,dtdtdtdl dvd28••dt=dt2=8, dtdtd8C dt=dsd:i="', whereCisthe(vector) curvature ofthecurveandvisthe speedinthecurve.Substituting thesevaluesintheequation theresultis A=8t+'172C. Theacceleration ofaparticlemovinginacurvehasthere forebeenbrokenupintotwocomponents ofwhichoneisparal leltothetangenttandofwhichtheotherisparallel tothe curvature C,thatis,perpendicular tothetangent. Thatthis resolution hasbeenaccomplished wouldbeunimportant were (15)' (16)' (18)'THEDIFFERENTIAL CALCULUS OFVECTORS 127 itnotfortheremarkable factwhichitbringstolight.The component of theacceleration paralleltothetangentisequal inmagnitude totherateofchangeofspeed.Itisentirely independent ofwhatsortofcurvetheparticleisdescribing. Itwouldbethesameiftheparticledescribed arightline withthesamespeedasitdescribes thecurve.Ontheother handthecomponent oftheacceleration normaltothetangent isequalinmagnitude totheproductof thesquareofthe speedoftheparticleandthecurvature ofthecurve.The sharperthecurve,thegreaterthiscomponent. Thegreater thespeedoftheparticle,thegreaterthecomponent. Butthe rateofchangeofspeedinpathhasnoeffectatallonthis normalcomponent oftheacceleration. Ifrbeexpressed intermsofi,i,kas r=xi+yi+zk, v=r=xi+Yi+zk, v=V.i;1l+fill+z2, A=v=r=xi+yi +lilt, A...xx+yy+zz-v-s- .- - -VxII+fill+z2 Fromtheseformulre thedifference between S,therateof changeofspeed,andA=r,therateofchangeofvelocity, isapparent. Justwhenthisdifference firstbecameclearly recognized wouldbehardtosay.Butcertainitisthat Newtonmusthavehaditinmindwhenhestatedhissecond lawofmotion. Therateofchangeofvelocityisproportional totheimpreslled force;butrateofchangeofspeedisnot. ~9.]Thelwdograph wasintroduced byHamilton asan aidtothestudyofthecurvilinear motionofaparticle. With'anyassumed originthevectorvelocityrislaidoff. Thelocusofitsterminus isthehodograph. Inotherwords, theradiusvectorinthehodograph givesthevelocityofthe 128 VECTOR ANALYSIS 'D=ar. Letrbetheradiusvector drawntotheparticle. The perpendicular torandtoa.ItisFlo.29. velocity visparticle inmagnitude anddirection atanyinstant.Itis possible toproceedonestepfurtherandconstruct thehodo graphofthehodograph. Thisisdonebylayingoffthe vectoracceleration A.=rfromanassumed origin. The radiusvectorinthehodograph ofthehodograph therefore givestheacceleration ateachinstant. Example1:Letaparticle revolveinacircle(Fig.29) ofradiusrwithauniform angular velocity a.The speedoftheparticlewillthen beequalto r=v=a xr. Thevectorvisalwaysperpendicular andofconstant magni tude.Thehodograph istherefore acircleofradius 'D=ar. Theradiusvectorrinthiscircleisjustninetydegreesin advance oftheradiusvectorrinitscircle,anditconse quently describes thecirclewiththesameangular velocity a.Theacceleration A.whichistherateofchangeofvis alwaysperpendicular tovandequalinmagnitude to A=a v=a2r. r=A=-a.ar=-a2r.Theacceleration Amaybegivenbytheformula r=A=axv'=ax(axr) =a.ra -a.ar. Butasaisperpendicular totheplaneinwhichrlies,a.r=O. Hence Theacceleration duetotheuniform motionofaparticlein acircleisdirected towardthecentreandisequalinmagni tudetothesquareoftheangular velocity multiplied bythe radiusofthecircle. THEDIFFERENTIAL CALCULUS OFVECTORS 129 Example ~:Consider themotionofaprojectile. The acceleration inthiscaseistheacceleration gduetogravity. r=A=g. Thehodograph ofthehodograph reduces toaconstant vector. Thecurveismerelyapoint.Itiseasytofind thehodograph. LetVobethevelocity oftheprojectile inpathatanygiveninstant. Atalaterinstantthevelocity willbe v=Vo+tg. Thusthehodograph isastraight lineparalleltogandpass ingthrough theextremity ofVO'Thehodograph ofa particle moving undertheinfluence ofgravityishencea straight line.Thepathiswellknowntobea parabola. Example3:Incaseaparticle moveunderany central acceleration r=A=f(r). Thetangents tothehodograph ofraretheaccelerations r. Butthesetangents areapproximately collinear withthe chordsbetween twosuccessive valuesrandrooftheradius vectorinthehodogl"cl.ph. Thatisapproximately Multiply byrx...r - ror=--·A.t ..(r-ro)rxr=rx •A.t Sincerandrareparallel r x(r-ro)=o. Hence r xr=r xroo Buttr xristherateofdescription ofarea.Hencethe equation statesthatwhenaparticle movesunderanac celeration directed towards thecentre,equalareasareswept overinequaltimesbytheradiusvector. 9 130 VECTOR ANALYSIS Perhapsitwouldbewelltogoalittlemorecarefully into thisquestion.Ifrbetheradiusvectoroftheparticlein itspathatoneinstant,theradiusvectoratthenextinstant isr +Ar.Theareaofthevectorofwhichrandr +Arare thebounding radiiisapproximately equaltotheareaofthe triangle enclosed byr,r+Ar,andthechordAr.This areais 1 1 1 12rx(r+Ar)=~rXr +~rXAr=:lrxAr. Therateofdescription ofareabytheradiusvectoris consequently LIM1rX(r+Ar)LIM1Ar1 At==02At=At..:.-02r x.;it=~r xr. Letrand10betwovaluesofthevelocityattwopoints PandPowhichareneartogether. Theacceleration roatPo isthelimitof r-roAr ~=At' A'• . Breakupthevector_r_=r - rointotwocomponents oneatAt parallelandtheotherperpendicular totheacceleration fo• Ai..At=xro+YD, ifDbeanormaltothevectorrooThequantity :lJap proaches unitywhenAtapproaches zero.Thequantity 11 approaches zerowhenAtapproaches zero. Ar=r-ro=:lJAtro+yAtDo Hence r x(r-ro)=xAtr xro+yAtr xDo (.')' . ( ArA)·r x r - r o=r x r - r o+Attxr•• THEDlFFERENTIAL CALCULUS OFVECTORS 131 Hence . . &r.A A ., Arxr - rOxrO= -XrO~t+x~trxrO+Y~trxD.&t Buteachofthethreetermsupontheright-hand sideisan infinitesimal ofthesecondorder.Hencetheratesofdescrip tionofareaatPandPodifferbyaninfinitesimal ofthe secondorderwithrespecttothetime.Thisistrueforany pointofthecurve.Hencetheratesmustbeexactlyequal atallpoints. Thisprovesthetheorem. 60.]Themotionofarigidbodyonepointofwhichis fixedisatanyinstantarotationaboutaninstantaneous axis passingthroughthefixedpoint. Leti,j,kbethreeaxesfixedinthebodybutmovingin space.Lettheradiusvectorrbedrawnfromthefixedpoint toanypointofthebody.Then r=xi+yj+zk, dr=xdi+ydj+zdk. But dr=(droi)i+(dr.j)j+(dr·k)k. Substituting thevaluesofdr·i,dr·j,dr.kobtained from thesecondequation dr=(xi •di+yi •dj+zi •dk)i +(xj •di+yj •dj+zj •dk)j +(xk•di+Yk•dj+zk•dk)k. But i •j=j •k=k•i=O. Hence i •dj+j •di=0orj.di= -i .dj j •dk+k..dj=0ork.dj=-j •dk k•di+i .dk=0ori.dk=-k•di. Moreover Hencei • i=j •j=k•k=1. i.di=j •dj=k.dk=0:' 132 VECTOR ANALYSIS LetSubstituting thesevaluesintheexpression fordr. dr=(d•dk-Yj •di)i+(xj •di -zk•dj)j +(yk•dj -xi.dk)k. Thislsavectorproduct. dr=(k.dji+i.dkj+j.dik)x(xi+Yj+zk). kdj..dk..dia='dtl+l'JtJ+J'dtk. Then . drr =-=axr.dt Thisshowsthattheinstantaneous motionofthebodyisone ofrotation withtheangularvelocity aaboutthelinea. Thisangularvelocitychangesfrominstanttoinstant. The proofofthistheoremfillsthelacunaintbeworkinArt.51. Twoinfinitesimal rotations maybeaddedlikevectors. Leta1andatbetwoangularvelocities. Tbedisplacements duetothemare d1r=a1x rdt, dtr=atx rdt. Ifrbedisplaced bya,itbecomes r+d1r=r+a1x rdt. Ifitthenbedisplaced byat,itbecomes r+dr=r+d1r+~x[r+(a1xr)dt]dt. Hence dr=a1x rdt+0.2Xrdt+0.2X(0.1Xr)(dt)2. Iftheinfinitesimals (dt)toforderhigherthanthefirstbe neglected, dr=0.1Xrdt+atx rdt=(a1+at)Xrdt, whichprovesthetheorem.Ifbothsidesbedividedbyd, . dr r=fit=(0.1+at)Xr. THEDIFFEREN7'IAL CALCULUS OFVECTORS 138 Thisistheparallelogram lawforangular velocities. It wasobtained before(Art.51)inadifferent way. Incasethedirection ofa,theinstantaneous axis,iscon stant,themotionreducestooneofsteadyrotationabouta. dr=aXrdt, r=a xr. Theaccelerationr=ax r+axr=a.x r+ax(axr). Asadoesnotchangeitsdirectionamustbecollinear with aandhenceax rispamlleltoaxr.Thatis,itisperpen diculartor.Ontheotherhanda x(axr)isparalleltor. Inasmuch asallpointsoftherotating bodymoveincon centriccirclesaboutainplanesperpendicular toa,itis unnecessary toconsider morethanonesuchplane. Thepartoftheacceleration ofaparticletowardthecentre ofthecircleinwhichitmovesis ax(axr). Thisisequalinmagnitude tothesquareoftheangular velocitymultiplied bytheradiusofthecircle.Itdoesnot dependupontheangularaccelemtionaatall.Itcorresponds towhatisknownascentrifugal force.Ontheotherhand theacceleration normaltothe·radiusofthecircleis axr. Thisisequalinmagnitude to therateofchangeofangular velocity multiplied bythemdiusofthecircle.Itdoesnot dependinanywayupon the angularvelocityitselfbutonly uponitsrateofchange. 61.]Thesubjectofintegration ofvectorequations inwhich thedifferentials dependuponscalarvariables needsbuta word.Itisprecisely likeintegration inordinary calculus. Ifthen dr=dI, r=8+C, 134 VECTOR ANALYSIS whereCissomeconstant vector. Toaccomplish theintegra. tioninanyparticular casemaybeamatterofsomedifficulty justasitisinthecaseofordinary integration ofscalars. Example 1:Integrate theequation ofmotionofa. projectile. Theequation ofmotionissimply r=g, whichexpresses thefactthattheacceleration isalwaysver ticallydownward andduetogravity. r=gt+b, wherebisaconstant ofintegration. Itisevidently the velocityatthetimet=O. 1r=2gtS+bt+c. cisanotherconstant ofintegration. Itisthepositionvector ofthepointattimet=O.Thepathwhichisgivenbythis lastequation isaparabola. Thatthisissomaybeseenby expressing itintermsofxand'!Iandeliminating t. Example 13:Therateofdescription ofareaswhenapar ticlemovesunderacentralacceleration isconstant. r=f(r). Sincetheacceleration isparalleltotheradius, rXr=O. B-d .ut rXr=de(rXr). F d(")" - or dtrXr=rXr+rXr. Hd.ence dt(rXr)=0 and rXr=C, whichprovesthestatement. THEDIFFERENTIAL CALCULUS OFVECTORS 185 Exampk3:Integrate theequation ofmotionforaparticle movingwithanacceleration towardthecentreandequalto aconstant multiple oftheinversesquareofthedistance fromthecentre. Given Then Hence..c2 r=-li r.r r xr=O. r xr=o. Multiply theequations together withx. rx0-1(.)-1{ . '}--2-=-8 r x r x r =-8r·rr-r·rr. err r • r=r2• Differentiate. Then. .r.r=rr. Hence rx0r ~-r Eachsideofthisequalityisaperfectdifferential. d(rc~O)=d(~} Integrate. Thenrx0r--=-+el,c2r whereeIisthevectorconstant ofintegration. eisitsmagni tudeandIaunitvectorinitsdirection. Multiply theequ&o tionbyr•. r.i-xOr.r-----::-2-=- +er.I.c r r·rx0Butrxi-.O 0·0 --c-=-2- -c2=7' 136 VECTOR ANALYSIS Let c·op=-2-andcosu=cos(r,I).c p=r+ercosu. Then Or p.r=-:---=-----1+ecosu Thisistheequation oftheellipseofwhicheistheeccentri· city.ThevectorIisdrawninthedirection ofthemajor axis.Thelengthofthisaxisis a=-p-.1-e2 Itispossibletocan'ytheintegration furtherandobtain thetime.Sofarmerelythepathhasbeenfound. ScalarFunctions ofPositioninSpace. TheOperator\J 62.jAfunctionV(x,y,z)whichtakesonadefinitesoa.lar valueforeachsetofcoordinates x,y,zinspaceiscalleda scalarfunction ofpositioninspace.Suchafunction, forex ample,is v(x,y,z)=x2+y2+z2=rl. Thisfunction givesthesquareofthedistance ofthepoint (x,y,z)fromtheorigin. ThefunctionVwillbesupposed to beingeneralcontinuous andsingle-valued. Inphysicsscalar functions ofposition areofconstant occurrence. Inthe theoryofheatthetemperature Tatanypointofabodyisa scalarfunction oftheposition ofthatpoint.Inmechanics andtheories ofattraction thepotential istheall-important function. This,too,isascalarfunction ofposition. Ifascalarfunction Vbesetequaltoaconstant, theequa tion V(x,y,z)=c. (20) definesasurfaceinspacesuchthatateverypointofitthe function Vhasthesamevaluec.IncaseVbethetempera- THEDIFFERENTIAL CALCULUS OFVECTORS 137 ture,thisisasurfaceofconstant temperature. Itiscalledan isothermal surface. IncaseVbethepotential, thissurfaceof constant potential isknownasanequipotential surface. As thepotential isatypicalscalarfunction ofpositioninspace, andasitisperhaps themostimportant ofallsuchfunctions owingtoitsmanifold applications, thesurface V(x,y,z)=c obtained bysettingVequaltoaconstant isfrequently spoken ofasanequipotential surfaceeveninthecasewhereVhas noconnection withthepotential, butisanyscalarfunction ofpositions inspace. Therateatwhichthefunction Vincreases intheXdirec tion-thatis,whenxchanges tox+~xandyandzremain constant -is LUI[V(x+~x,y,z)-V(x,y,z)] ~x=O ~x . Thisisthepartialderivative ofVwithrespecttox.Hence theratesatwhichVincreases inthedirections ofthethree axesX,Y,Zarerespectively C)VC)VC)V-,-,C)xC)yC)Z Inasmuch astheseareratesinacertaindirection, theymay bewrittenappropriately asvectors. Leti,i,kbeasystem ofunitvectors coincident withtherectangular sJ'stemof axesX,1';Z.Theratesofincrease ofVare .C)V.C)VkC)V lC).c'JC)y' ~ Thesumofthesethreevectorswouldtherefore appeartobe avectorwhichrepresents bothinmagnitude anddirection theresultant ormostrapidrateofincrease ofV.Thatthis isactually thecasewillbeshownlater(Art.64). 138 VECTOR .ANALYSIS (21) (22)(21)'63.]Thevectorsumwhichistheresultant rateofincrease ofVisdenotedby\JV. \lV_av.~V'-~V -l~X+J ~y+a.~z· vVrepresents adirected rateofchangeofV-adirected orvectorderivative ofV;sotospeak.Forthisreason\lV willbecalledthederivative ofV;and~theprimitive of VV.Thetermsgradient andslopeofVarealsousedfor \lV.Itiscustomary toregardVasanoperatorwhichobtains avector"VVfromascalarfunctionVofpositioninspace. "VV=(i~+j~+k~)V ~xay ~z V=i~+j~+k~-.axayaz Thissymbolic operator "Vwasintroduced bySirW.R. Hamilton andisnowinuniversal employment. There seems,however, tobenouniversally recognized name1forit, although owingtothefrequent occurrence ofthesymbol somenameisapractical necessity. Ithasbeenfoundby experience thatthemonosyllable delissoshortandeasyto pronounce thatevenincomplicated formulminwhich"Voccurs anumberoftimesnoinconvenience tothespeakerorhearer arisesfromtherepetition. "VVisreadsimplyas"delV." Although thisoperator "Vhasbeendefinedas t"'7 •~ •a ~ v=1-+J-+k-,axay ~z 1SomeIl88thetermNablaowingtoitafanciedl'8Iemblance toanAII8Jrian harp.Othel'llhaTenoteditslikenl!ll8 toaninTerted4andhaveeOll8llqnently coinedthenonetooenphoniona nameAIledbyinverting theorderofthelettel'llin thewordDella.FopplinhisEinfi1lrung indieMazwelf,cM TAswdtll'Eke In"citiitavoidsanyspecialdesignation andrefel'lltothesymbolu "dieOperalion V."Howthisistobereadisnotdivulged. Indeed,forprinting noparticular Dameisnecessary, butforlecturing andpurpoees ofinstruction something isre quired-something toothatdoesnotconfusethespeakerorhearereTenwhen oftenrepeated. THEDIFFERENTIAL CALCULUS OFVECTORS 139 ---(22)',"a,,(J ,~ 'V=1a31+Jay'+kaz" andsothatitappears todependuponthechoiceoftheaxes,it isinrealityindependent ofthem.Thiswouldbesurmised fromtheinterpretation of'Vasthemagnitude anddirection ofthemostrapidincrease ofV.Todemonstrate theinde pendence takeanothersetofaxes,i',j',k'andanewsetof variables 31,y',z'referred tothem.Then'Vreferred tothis systemis Bymakinguseoftheformulre (47)'and(47)",Art.53,page 104,fortransformation ofaxesfromi,j,ktvi',J',It'ana,rby actually carrying outthedifferentiations andfinallyby takingintoaccount theidentities (49)and(50),'V'may actuallybetransformed into'V. 'V'='V, Thedetailsoftheproofareomitted here,beca.use a.nother shortermethodofdemonstration istobegiven. 64.] Conside~ twosurfaces (Fig.30) V(x,y,z)=c V(x,y,z)=c+de, FIG.30.DuponwhichVisconstant andwhicharemoreover infinitely neartogether. Letx,y,zbeagivenpointuponthesurface V=c.Letrdenotethera diusvectordrawntothis pointfromanyfixedorigin. Thenanypointnearhyin theneighboring surfaceV =c+dcmayberepresented bytheradiusvectorr+dr. TheactualincreaseofVfrom thefirstsurfacetothesecond isafixedquantity de.Therateofincrease isavariable 140 VECTOR ANALYSIS de (23)quantity anddepends uponthedirection drwhichisfol lowedwhenpassingfromonesurfacetotheother.Therate ofincrease willbethequotient oftheactualincrease deand thedistanceVdr.drbetween thesurfaces atthepoint x,y,zinthedirection dr.LetDbeaunitnormaltothe surfaces anddnthesegment ofthatnormalintercepted between thesurfaces, Ddnwillthenbetheleastvaluefor dr.Thequotient Vdr·dr willt.~are beamaximum whendrisparalleltoDand eqrlatin magnitude ofdn.Theexpression de ~Ddn istherefore avectorofwhichthedirection isthedirection of mostrapidincrease ofVandofwhichthemagnitude isthe rateofthatincrease. Thisvectorisentirelyindependent of theaxesX,Y,Z.Letdebereplaced byitsequald Vwhich istheincrement ofVinpassingfromthefirstsurfacetothe second. Thenlet'VVbedefinedagainas dV'VV= - D. (24)dn Fromthisdefinition, 'VViscertainly thevectorwhich givesthedirection ofmostrapidincreaseofVandtherate inthatdirection. Moreover 'VVisindependent oftheaxes. Itremainstoshowthatthisdefinition isequivalent totheone firstgiven.Todothismultiply by•dr. dV'VV.dr=-D.dr. (25)dn Disaunitnormal. HenceD•dristheprojection ofdron Dandmustbeequaltotheperpendicular distance dnbetween thesurfaces. ButTHEDIFFERENTIAL CALCULUS OFVECTORS 141 ~v ~v ~vdV="dx+-;s-d'!J+"dz, ~x ~'!J ~z where Ifdrtakesonsuccessively thevaluesidx,jdy,kdzthe equation (25)'takesonthevalues Y'V.idx=:V dx ~x Y'V.jdy=~;dy ~J' Y'V.kdz=~zdz.(26) Ifthefactorsdx,dy,dzbecancelled theseequations state thatthecomponents V'V•i,V'V·j,V'V.kofV'Vinthe i,j,kdirections rel:\pectively areequalto Y'V=(Y'V.i)i+(Y'V.j)j+(Y'V.k)k. .~V ,~V ~V Henceby(26)Y'V=1~;£+J~Y+k~z' (21) Theseconddefinition (24)hasbeenreduced tothefirst andconsequently isequivalent toit. -65.]Theequation (25)'foundaboveisoftentakenasa definition ofY'V,According toordinary calculus thederiv- .dy'fith .atlvedxsat18eseequatIOn dydxd~=dy. 142 VECTOR ANALYSIS Moreover thisequation definesdy/dx.Inasimilarmanner itispossibletolaydownthefollowing definition. Definition: Thederivative \lVofascalarfunction of position inspaceshallsatisfytheequation forallvaluesofdr. Thisdefinition iscertainly themostnaturalandimportant fromtheoretical considerations. Butforpractical purposes eitherofthedefinitions beforegivenseemstobebetter. Theyaremoretangible. Therealsignificance ofthislast definition cannotbeappreciated untilthesubjectoflinear vectorfunctions hasbeentreated. SeeChapterVII. Thecomputation ofthederivative\lofafunction ismost frequently carried onbymeansoftheordinary partial differentiation. Example1:LetV(x,y,z)=r=yxll+y2+z2. n' .~r .~r k~rvr=l-+J-+ -. ~x ~y ~z \lr=i x+i yyx3+y3+z3Yx2+y2+z3 +k z#+yll+z2 Hence and1\lr=y 22(ix+jy+kz) xli+y+z r r\lr=--=-'.yr:r r Thederivative ofrisaunitvectorinthedirection ofr. Thisisevidently thedirection ofmostrapidincrease of,. andtherateofthatincrease. THEDIFFERENTIAL CALCULUS OFVECTORS 148 Example 13:Let 1 1V(x,y,z)=-=.r.vX2+y2+Zl Hence-k Z (X2+y2+z2)1 \7!= 1(-ix-jy-kz)r(x2+y2+z2)i and1-r-r\7-=--=_. r(ror)i r3/,.j','-_.!' I and HenceThederivative of11risavectorwhosedirection isthat of-r,andwhosemagnitude isequaltothereciprocal ofthe squareofthelengthr. " I ~-I r / Example 3:\7r"=n rIr=nr"- • /ror Theproofislefttothereader.rvt""1,!. ~ Example4-:Let Vex,y,z)=log.vx2+y'l.. x y\7logvx:!+y2=i2 2+j2 2+0Itx+y x+y 1 - 2 2 (ix+jy).x+Y Ifrdenotethevectordrawnfrom th~origintothepoint (x,y,z)ofspace,thefunctionVmaybewrittenas V(x,y,z) =log.vror -(kor)2 ix+jy=r - kkor. r -kkor\7logVx2+y2=----:::---"r0r-(kor)2 l'-kkor---------(r-kkor) 0(r-kkor) 144 VECTOR ANA.LYSIS Thereisanothermethodofcomputing \1whichisbased upontheidentity , dr.\1V=dV. Example1:Let v=vr:r=r. Hencedror r dV=--=dro---:-==dr o\1JT..vr:r "t/r·r r r\1V---- .-v'r.r-r E:.cample 1!: HenceLetV= r01'whereaisaconstant vector. dV=droa=dr,l\1 y~ \1V=a. Example3:LetV=(rxa)•(rxb),whereaandbare constant vectors. V=r.raob-roarob. dV= 2droraob-dr.arob-drobroa=dr 0\1JT. Hence \1V= 2raob - arob - broa \1V=(raob-arob)+(nob-broa) =bx(rxa)+a x(rxb). Whichofthesetwomethods forcomputing \lshallbe appliedinaparticular casedepends entirely upontheir relativeeaseofexecution inthatcase.Thelattermethodis independent ofthecoordinate axesandmaytherefore be prefelTed. Itisalsoshorterincasethefunction Vcanbe expressed easilyintermsofr.ButwhenVcannotbeso expressed theformermethodhastoberesorted to. -66.]Thegreatimportance oftheoperator \1inmathe maticalphysicsmaybeseenfromafewillustrations. ::;up poseT(x,y,z)bethetemperature atthepointx,y,zofa THEDIFFERENTIAL CALCULUS OFVECTORS 145 heatedbody.Thatdirection inwhichthetemperature de creasesmostrapidlygivesthedirection oftheflowofheat. 'VT,ashasbeenseen,givesthedirection ofmostrapid increaseoftemperature. Hencetheflowofheatfis f=-k'VT, wherekisaconstant depending uponthematerial ofthe body.Suppose againthatVbethegravitational potential duetoafixedbody.Theforceactinguponaunitmassat thepoint(x,'!I,z)isinthedirection ofmostrapidincreaseof potential andisinmagnitude equaltotherateofincrease perunitlengthinthatdirection. LetPbetheforceperunit mass.Then P='VV. Asdifferent writersusedifferent conventions asregardsthe si!lnofthegravitational potential, itmightbewelltostate thatthepotential Vreferredtoherehastheoppositesignto thepotential energy.IfWdenoted thepotential energyof amassmsituatedatx,'!I.z,theforceactinguponthatmass wouldbe P=-'VW. IncaseVrepresent theelectricormagnetic potential due toadefiniteelectricchargeortoadefinitemagnetic polere spectively theforcePactinguponaunitchargeorunitpole asthecasemightbeis P=-'VV. Theforceisinthedirection ofmostrapiddecrease of potential. Indealingwithelectricity andmagnetism poten tialandpotential energyhavethesamesign;whereas in attraction problems theyaregenerally considered tohave oppositesigns.Thedirection oftheforceineithercaseisin thedirection ofmostrapiddecreaseofpotential energy. The difference between potential andpotential energyisthis. 10 146 VECTOR ANALYSIS Potential inelectricity ormagnetism it!thepotential energy perunitchargeorpole;andpotential inattraction problems ispotential energyperunitmasstaken,however, withthe negative sign. -67.JItisoftenconvenient totreatanoperator asa quantity provided itobeysthesameformallawsasthat quantity. Consider forexample thepartialdifferentiators ~ ~ ~-,-,_. ~x ~y ~z Asfarascombinations oftheseareconcerned, theformallaws areprecisely whattheywouldbeifinsteadofdifferentiato1'8 threetruescalars a,b,c weregiven.Forinstance thecommutative law ~~ ~~ ~x~y=~y~x theassociative law-ab=bCl, !.-(~!.-)=(!.-!.-)!.-_a(bc)=(ab)c, ~x ~'!J~z ~x~y ~z andthedistributive law ~(~ ~) ~~ ~~ .- - +- =- -+- ---a(b+C)=ab+ac ~x~y ~z ~x~y ~x~z holdforthedifferentiators justasforscalars. Ofcoursesuch formulre as ~ .~ u~x=~xu, where 'II.isafunction ofxcannotholdonaccountofthe properties ofdifferentiators. Ascalarfunction 'II.cannotbe placedundertheinfluence ofthesignofdifferentiators. Suchapatenterrormaybeavoidedbyremembering thatan operand mustbeunderstood uponwhich ~/~xistooperate. THEDIFFERENTIA.L CALCULUS OF'VECTORS 147 Inthesamewayagreatadvantage maybeobtained by looking upon 'r7.~ .~ '-~v=l-+J-+.- ~x ~y ~z asavector.Itisnotatruevector,forthecoefficients ~ ~ ~-,-, - ~x~y ~z arenottruescalars.Itisavectordiflerentiator andof courseanoperand isalwaysimpliedwithit.Asfarasformal operations areconcerned itbehaves likeavector. For instance 'l(u+v)='lu+'lv, 'l(uv)=('lu)v+u('lv), c'lu=V(cu), ifuandvareanytwoscalarfunctions ofthescalarvariables x,y,zandifcbeascalarindependent ofthevariables with regardtowhichthedifferentiations areperformed. 68.]IfArepresent anyvectortheformalcombination A·Vis ~ ~ ~A•V=At-+A2-+As-, (27) ~x ~y ~z provided A=Ati+A2j+Ask. Thisoperator A·Visascalardifferentiator. Whenapplied toa8calarfunction V(x,y,z)itgivesascalar. Suppose forconvenience thatAisaunitvector L 148 VECTOR ANALYSIS whereal,a2,aaarethedirection cosinesofthelineareferred totheaxesX,Y,Z.Consequently (a0\7)Vappears asthe well-known dilectional derivative ofVinthedirection a. Thisisoftenwritten ~Vav ~Vav-=a-+a-+a-. (29)' ~8lax2ay a~z Itexpresses themagnitude oftherateofincrease ofVin thedirection a.Intheparticular casewherethisdirection is thenormalntoasurfaceofconstant value of V,thisrelation becomes thenormalderivative. (~VaVavavaV(29)"nov)=-=n l-+n 2-+na,,\z'-anax ay g ifnt,n2,nabethedirection cosinesofthenormal. Theoperator a0\lappliedtoascalarfunction ofposition Vyieldsthesameresultasthedirectproduct ofaandthe vector\7V. (a"\7)V=a0(\7V). (30) Forthisreasoneitheroperation maybedenoted simplyby a"\7V without parentheses andnoambiguity canresultfromthe omission. Thetwodifferent forms(a 0\7)Vanda0(\7V) mayhowever beinterpreted inanimportant theorem. (80\7)Visthedirectional derivative ofVinthedirection a.Ontheotherhanda"(\7V)isthecomponent of\7Vin thedirection a.Hence:Thedirectional derivative ofVin anydirection isequaltothecomponent ofthederivative \7Vinthatdirection. IfVdenotegravitational po~ntial the theorem becomes: Thedirectional derivative ofthepotential inanydirection givesthecomponent oftheforceperunit massinthatdirection. IncaseVbeelectric ormagnetic potential adifference ofsignmustbeobserved. THEDIFFERENTIA.L CALCULUS OFVECTORS 149 VectorFunctions ojPosition inSpa~ 69.]Avectorfunction ofposition inspaceisafunction v(x,y,z) whichassociates with.eachpointx,y,zin,spaceadefinite vector. Thefunction maybebrokenupintoitsthreecom ponents v(x,y,z)=VI(x,y,z)i+V\l(x,y,z)j+Va(x,y,z)k. Examples ofvectorfunctions areverynumerous inphysics. Already thefunction \7Vhasoccurred. Ateachpointof space\7Vhasingeneraladefinitevectorvalue.Inmechan icsofrigidbodiesthevelocity ofeachpointofthebodyisa vectorfunction ofthepositionofthepoint.Fluxesofheat, electricity, magnetic foree,fluids,etc.,areallvectorfunctions ofpositioninspace. Thescalaroperatora·\7maybeappliedtoavectorfunc tionVtoyieldanothervectorfunction. LetV=V1(,r,y,z)i+V\l(x,y,z)j+Va(x,y,z)k and Thena=ali+l!\Ij+a3k. (a•\l)V=(a•\7)J"1i+(a.\7)V2j+(a•\7)~k (~V ~V ~V):loud(a.\7)V=a1__1+all_1+as-~i ~x ~y ~z (31) 150 VECTOR ANALYSIS Thismaybewritteninthefol"'".11 dVI.dV2•.9J~ (31I(a•\7)V=as1+a8J+~k. ) Hence(a.\7)Visthedirectional derivative ofthevector function Vinthedirection LItispossible towrite (a•\7)V=a •\7V without parentheses. Forthemeaning ofthevectorsymbol \7whenapplied toavectorfunction Vhasnotyetbeen defined. Hencefromthepresentstandpoint theexpression o..VVcanhavebuttheoneinterpretation giventoitby (a.\7)V. 70.]Although theoperation 'v'Vhasnotbeendefinedand cannotbeatpresent,l twoformalcombinations ofthevector operator \7andavectorfunction Vmaybetreated. These arethe(formal) scalarproductandthe(formal) vectorprod uctof\7intoV.Theyare and\7•V=(i~+j~+k~)• Vaxayaz \7x V =(i:x+j:y+k:z)xV.(32) (33) (32)'V• VisreaddelMtV;and\7xV,delcrOBBV. Thedifferentiators ~,:':,beingscalaroperators, passdXClyClZ bythedotandthecross.Thatis .dV.dVkdVV.V=l'-+J--+ .-.9xdY ~z .~V . ~V ~V33''7x V=1 X~X+J xdy+kx~z'() Thesemaybeexpressed intermsofthecomponents VI'~'V. ofV. 1Adf'ftnition ofVVwillbegiTenInCbapterVIL ./;.'-t)'1 ," J9AOa.r°NSAriAlA II/(SS) ·I~liexe-I-.A.x.6,eee - ][r, lUllU!Ul.I~ap 'IIJOUl.IOJalfJU!u~!-1.A\aq..f'l!ms!1{.r. (lieXe) •lAe-~e][+ (xeZr.l)(zelie)Ilsg) sAe-lA"er+zAe-sAe,=.A.X.6.90uaH zezeze 'riAe,-lAer=.A.ex][ lielielie 'rAe][-sAe'=.A.exr ,xexexe sAer-riAe][=.A.eX, 90U9H uaq.r./I(gs) (ts)zeliexe ·sAe+riAe+lAe=.A.0.6, zeze ·sAe=.A.C'•][ lielie'-=- orr;Ae.A.e. xexe'-=- 0t lAe.A.e. zezezeze·][-+r-+t-=-sAe '~e '~e.A.e lielielielie'][-+r-+l-=~e.r;Ae.lAe.A.e xexexexe '][sAe+r:te+,lAe=.A.e J&.ON 1:91:SUO.L:XiJA dOSfl7fl:J7Y:J 7YIJ.N:FlU:fIddla :FlHJ. 152 VECTOR ANALYSIS Itistobeunderstood thattheoperators aretobeappliedto thefunctions VI'V~,Vawhenexpanding thedeterminant. Fromsomestandpointtl objections maybebrought forward againsttreatingVasasymbolic vectorandintroducing V•V andVxVrespectively asthesymbolic scalarandvector products ofVintoV.Theseobjections maybeavoidedby simplylayingdownthedefinitiou thatthesymbols V•and Vx,whichmaybelookeduponasentirely newoperators quitedistinctfrolDV,shallbe and~.~V.~Vk~Vy·V=l'-+J'-+ .- ~x ~y ~z .dV.~Vk~VVXV=lX-+JX-+ X-'dX ~y ~z(32)' (33)' Butforpractical purposes- andforremembering formulreit seemsbyallmeansadvisable toregard 't"'7 •~ •~kdv=l-+J-+ -dX ~!J ~z asasymbolic vectordifferentiator. Thissymbolobeysthe samelawsasavectorjustinsofarasthedifferentiators !-,:':obeythesamelawsasordinary scalarquantities.oXgygZ . 71.]Thatthe'twofunctions V•VandVXVhavevery important physical meanings inconnection withthe vector function Vmaybeeasilyrecognized. Bythestraight forward proofindicated inArt.63itwasseenthatthe operator Visindependent ofthechoiceofaxes.Fromthis facttheinference isimmediate thatV•Vand"VxVrepresent intrinsic properties ofVinvariant ofchoiceofaxes.Inorder toperceive theseproperties itisconvenient toattribute tothe. function Vsomedefinitephysical meaning suchasfluxor Bowofafluidsubstance. Lettherefore thevectorVdenote THEDIFFERENTIAL CALCULUS OFVECTORS 153 ateachpointofspacethedirection andthemagnitude ofthe flowofsomefluid.Thismaybeamaterial fluidaswater orgas,orafictitious oneasheatorelectricity. Toobtainas greatclearness aspossible letthefluidbematerial butnot necessarily restricted toincompressibility likewater. Then iscalledthedivergence ofVandit!oftenwrittenv::. V.V=divV. Thereasonforthistermisthat\1.Vgivesateachpointthe rateperunitvolumeperunittimeatwhichfluidisleaving thatpoint-therateofdiminution ofdensity. Toprove thisconsider asmallcubeofmatter(Fig.31).Lettheedges ofthecubebedx,dy,anddzret!pectively. Let V(x,y,z)=VI(x,y,z)i+V2(.t,y,z)j+Va(x,y,z)k. -jdo)'dz-----R--- id)'d.~xJ'.Z x+dx,)'/1. x FlO.31.y -i.V(.r,y,z)dydz. Thenormaltotheoppo-Z siteface,thefacewhose xcoordinate isgreaterbytheamountdx,is+iandtheflux through itisthereforeConsider theamount offluidwhichpassesthrough those facesofthecubewhicharepa.ralleltotheYZ-plane, i.c. perpendicular totheX axis.Thenormaltothe facewhosexcoordinate is thelesser,thatis,thenor maltotheleft-hand face ofthecubeis-i.Theflux ofsubstance through this faceis 1£4 VECTOR ANALYSIS ()V dV i0hdxdydz=()Xldxdydz.i0V(x+dx,y,z)dydz=i0[V(x,y,z)+::dxJdydz =i0V(x,y,z)dydz+i0~Vdxdydz. ~x Thetotalfluxoutward fromthecubethrough thesetwo facesistherefore thealgebraic sumofthesequantities. This issimply Inlikemannerthefluxesthrough theotherpairsoffacesof thecubeare .~V ~VJ0"§J;dxdydzandk0fY~dxdydz. Thetotalfluxoutfromthecubeistherefore (.dV .dV k ()V)d d d10-+Jo-+ 0- xyz.(}x fYy ~z Thisisthenetquantity offluidwhichleavesthecubeper unittime.Thequotient ofthisbythevolumedxdydzof thecubegivestherateofdiminution ofdensity. Thisis .(}V.dV dV ~Vl ~V2dVS'V0V=10 -+ J 0 -+k0 -= -+ - +-.dXdy (}z ~x(}y(}z Because 'V0Vthusrepresents thediminution ofdensity ortherateatwhichmatterisleavingapointperunitvolume perunittime,itiscalledthedivergence. Maxwell employed thetermconvergence todenotetherateatwhichfluidap proaches apointperunitvolumeperunittime.Thisisthe negative ofthedivergence. Incasethefluidisincompressible, asmuchmattermustleavethecubeasentersit.Thetotal changeofcontents mm,ttherefore bezero.Forthisreason thecharacteristic differential equation whichanyincompres siblefluidmmltsatisfyis ,'VoV=O THEDIFFERENTIAL CALCULUS OFVECTORS 155 whereVisthefluxofthefluid.Thisequation isoften knownasthehydrodynarni£ equation.Itissatisfied byauy flowofwater,sincewaterispractically incompressible. The greatimportance oftheequation forworkinelectricity isdue tothefactthataccording toMaxwell's hypothesis electricdis placement obeysthesamelawsasanincompressible fluid.If thenDbetheelectricdisplacement, divD=\1.D=O. 12.]Totheoperator\lxMaxwell gavethenamecurl. Thisnomenclature hasbecome widelyaccepted. "VXV=curlV. Thecurlofavectorfunction Visitselfavectorfunction ofposition inspace.Asthenameindicates, itisclosely connected withtheangular velocity orspinofthefluxat eachpoint.Buttheinterpretation ofthecurlisneitherso easilyobtained norsosimpleasthatofthedivergence. Consider asbeforethatVrepresents thefluxofafluid. Takeatadefiniteinstantaninfinitesimal sphereaboutany point(x,y,z).Atthenextinstantwhathasbecomeofthe sphere? Inthefirstplaceitmayhavemovedoffasawhole inacertaindirection byanamountdr.Inotherwordsit mayhaveatranslational velocity ofdr/dt.Inaddition to thisitmayhaveundergone suchadeformation thatitisno longerasphere.Itmayhavebeensubjected toastrainby virtueofwhichitbecomes slightly ellipsoidal inshape. Finallyitmayhavebeenrotatedasawholeaboutsome axisthrough anangledw.Thatistosay,itmayhavean angular velocity themagnitude ofwhichisdw/dt. An infinitesimal spheretherefore mayhaveanyone ofthree distinct typesofmotionorallofthemcombined. First,a translation withdefinitevelocity. Second,astrainwiththree definiteratesofelongation alongtheaxesofanellipsoid. 156 VECTOR ANALYSIS Third,anangula.r velocity aboutadefiniteaxis.Itisthis thirdtypeofmotionwhichisgivenbythecurl.Infact, thecurlofthefluxVisavectorwhichhasateachpointof spacethedirection oftheinstantaneous axisofrotationat thatpointandamagnitude equaltotwicetheinstantaneous angularvelocityaboutthataxis. Theanalytic discussion ofthemotiohofafluidpresents moredifficulties thanitisnecessary tointroduce intreating thecurl.Themotionofarigidbodyissufficiently complex togiveanadequate ideaoftheoperation. Itwasseen(Art. 51)thatthevelocityoftheparticles ofarigidbodyatany instantisgivenbytheformula v=Vo+aXr. curlv=\1Xv=\1X Vo+\1X(aXr). Let a=ali+a2j+aak r=r1i+r2j+rak=xi+Yj+zk expand\1X(aXr)formally asifitwerethevectortriple productof\1,a,andr.Then \1Xv=\1X Vo+(\1•r)a -(\1•a)r. Voisa.constant vector. HencethetermV'X Vovanillhes. C)xC)y ~z \1•r=;;-+"+"=3.CI;];ctyctZ Asaisa:constant vectoritmaybeplacedupontheotherside ofthedifferential operator, V'•a=a •\1. (~ C) C)) • • kC".V'r=a1;;-+a2.,-+a8 - r=all+a2J+a3=a. • ClX Cly C)z I Hence \1Xv=3a-a=2a. Therefore inthecaseofthemotionofarigidbodythecurl ofthelinearvelocity atanypointisequaltotwicethe angular velocity inmagnitude andindirection. THEDIFFERENTIAL CALCULUS OFVECTORS 157 '1x v=curlv=2a, 1 1a=2'1x v=~curlv. 1 1V=v~+2('1xv)x r = v o+~(curlv)xr.(34) Theexpansion of\1x(axr)formally maybeavoidedby multiplying a.x routandthenapplying theoperntor\1xto theresult. 73.]Itfrequently happens, asinthecaseoftheapplica tionjustcited,thattheoperators \1,\1.,\1x,haveto~ appliedtocombinations ofscalarfunctions, vectorfunctions, orboth.Thefollowing rulesofoperntion willbefound useful. Let1£,'"bescalarfunctions andu,vvectorfunc tionsofposition inspace.Then '1(1£+'V)='11£+'1v (35) '1.(u+v)='1.u +'1.v (36) '1x(u+v)='1x u +'1x v (37) '1(UV)=V'11t+1t'1'V (38) '1.(uv)='1u•v +u'1• v (39) '1x(uv)='1uxv+u'1xv (40)7, '1.(u.v)=v.'1u+u.'1v (41) •1/,/+Vx('1xu)+u x("ZxV)l '1.(uxv)=v/'1xu)-u.'1 x v (42) /'I \'Vx(uxv)={V'?-V'1"u-~·'1f+U'1.V.l (43) Awordisnecessary uponthematteroftheinterpretation ofsuchexpressions as '11£'V,'1u.v,'1uxv. Therulefollowed inthisbookisthattheoperator\1applies tothenearesttermonly.Thatis, 1ByArt.69theexpl'8llllionl ••\1Uandu·\1.aretobeinterpreted u (.'\1)uand (u'\1)•. 158 VECTOR ANALYSIS 'Vuv=('Vu)v 'Vu•V=('Vu)•V 'VuX V=('Vu)Xv. If\1istobeappliedtomorethantheonetermwhichfollows it,theterIll8towhichitisappliedareenclosed inaparen thesisasupontheleft-hand sideoftheaboveequatioDs. Theproofsoftheformulre maybegivenmostnaturally byexpanding theexpressions intermsofthreeassumed unit vectorsi,j,k.Thesign~ofsummation willbefoundcon .venient. Bymeansofittheoperators \1,\1.,b,.Xtakethe form 'V=~i~~x' 'V.=~i~, ~x Thesummation extendsoverx,y,z. Todemonstrate 'Vx(uv)='VuX v +u'VXv. 'VX(uv)=~ix ~(uV)=~iX (~UV+1£~V), ~x ~x ~x 'VX(uv)=~i X(~:v ) +~i X (u~:) ~(.~U) ~. ~v=~ l~xXV+~U1X~x' Hence 'VX(uv)='VuX v +U'VXv. Todemonstrate 'V(u•v)=v •'Vu+u·'Vv + v X ('VXu)+uX('VXv). THEDIFFERENTIAL CALCULUS OFVECTORS 159 .~ ~. ( u ~V)V(Uov)=~1-(UoV)=~1 - 0V+ U0-~~X ~X ~X ~•~U ~. ~VV(U0V)=1 - 0V+~1U 0 - ~X ~X orNow ~. ~U~ ~ll.~ .~uvX(Vxu)=vX~1X.-=~V0 -1-~Vol-. ~X ~X ~X ~~U. ('r"7) ~ .~UVo-l=VX vXU+~VOl- ~X ~X ~V0~Ui=VX(\1XU)+V0VU. ~X Inlikemanner~U0~Vi =U X(\1Xv)+u0VV.~~x Hence V(uov) ~VoVu+ UoVv +VX(VXu)+UX(Vxv). (44) V(u0v)..Theotherformulm aredemonstrated inasimilarmanner. 74.]Thenotation 1 willbeusedtodenotethatinapplying theoperator\7 tothe product (u0v),thequantity uistoberegarded asconstant. Thatis,theoperation \7iscarriedoutonlypartially upon theproducteu 0v).Ingeneralif\7istobecarriedout partially uponanynumber offunctions whichoccurafter itinaparenthesis, thosefunctions whichareconstant forthe differentiations arewrittenaftertheparenthesis assubscripts. Let u="'1i+u2i+Uak, v=VIi+v2i+Vak. 1Thisideaandnotation ofapartialV80tospeakmaybeavoidedbymeans oftheformula 41.But"certaiu"mouut ofcompactuess andsimplicity is 10M;thereby. TheideaofV(11•Y).issurelynomorecomplicated than11·VYor .,X(VX11). 160 thenVECTOR ANALYSIS Hence "Y(n.v)=VI"Yu1+v~"Yu2+va"Yu. + u1"YvI+u~"Yv~+ua"Yva' But "Y(u,v),,=ul"Yvl+u2"Yv~+ua"Yva (44)' and "Y(n.v)..=vI"Yu1+v2"Yu2+va"Yua' Hence "Y(n.v)="Y(n.v)..+"Y(n.v)..,(45) Thisformula corresponds tothefollowing oneintheno~ tionofdifferentials ord(n.v)=d(n•v)..+d(n.v).. d(n.v)=u.dv+ dn·v, Theformulre (35)-(43) givenabove(Art.73)maybe writteninthefollowing manner,asisobviousfromanalogy withthecorresponding formulre indifferentials : "Y(u+v)="Y(u+v)"+"Y(u+v).(35)' "Y.(n+v)="Y.(n+V)II+"Y.(n+v)..(36)' "Yx(n+v)="Yx(n+v)lI+"YX(u+v)..C37}' THEDIFFERENTIAL CALCULUS OFVECTORS 161 '1(uv)='1(uv)..+'1(uv). (38)' '10(uv)='10(uv)..+'10(uv)..(39)' '1x(uv)='1x(uv).+'1x(uv)..(:to)' '1(uov)='1(uov)lI+'1(uov).. (41)' '1o(uxv)='1o(u Xv)lI+'1o(uxv)..(42)' '1x(uxv)='1x(uXv)1l+'1x(uxv)...(43)' Thisnotation isparticularly usefulinthecaseofthe scalarproductu0vandforthisreasonitwasintroduced. Inalmostallothercasesitcanbedoneawaywithoutlossof simplicity. Takeforinstance (43)'.Expand'1x(uXv)lI formally. '1x(uXv)lI=('10v)u -(\lou)v, whereitmustbeunderstood thatuisconstant forthediffer entiations whichoccurin\1..Theninthelasttermthe factorumaybeplacedbeforethesign\1.Hence '1x(uXv)lI= u'10v = u 0'1v. Inlikemanner '1x(uxv)..= v 0'1u -v'1oU. Hence '1x(uxv)=v0'1u -v'1 0u - u 0'1v+u'1 0v. 75.]Thereareanumberofimportant relations inwhich thepartialoperation \1(u0v)lIfigures. or oru x('1xv)='1(u0v)lI- U 0'1v, '1(u0v)1l= U0'1v+u x('1xv), u0'1v='1(u0v)lI+('1xv)xu.(46) (46)' (46)" Theproofofthisrelation maybegivenbyexpanding in termsof1,j,k.Amethodofremembering theresulteasily isasfollows. Expandtheproduct u x('1xv) 11 162 VECTOR ANALYSIS formally asifV",U,vwereallrealvectors. Then ux(V"xv)=u.vV"-u.V"v. Thesecondtermiscapableofinterpretation asitstands. Thefirstterm,however, isnot.Theoperator\lhasnothing uponwhichtooperate.Ittherefore mustbetransposed so thatitshallhaveu.vasanoperand. Butubeingoutside oftheparenthesis inu x(V"xv)isconstant forthedifferen tiations. Hence andu·vV"=V"(u.v)a Ux(V"xv)=V"(u.v)a-u·V"v.(46) Ifubeaunitvector,saya,theformula a·V"v=V"(a.v).+(V"xv)Xa(47) expresses thefactthatthedirectional derivative a.'V'vofa vectorfunction vinthedirection aisequaltothederivative oftheprojection ofthevectorvinthatdirection plusthe vectorproductofthecurlofvintothedirection a. Consider thevaluesofvattwoneighboring points. v(x,y,z) and v(x+dx,y+dy,z+dz) dv=v(x+dx,y+dy,z+dz)-v(x,y,z). Let v=vii+112i+Vak dv=dv1i+ dv2i+dvak. Butby(25)' dv1=dr,V"vl dv2=dr,V"vt dva=dr.V"v a· Hence dV=dr.('V'v1i+ 'V'v2i+V"vak). Hence dv=dr.V"v, By(46)" dv=V"(dr,v)dr+ (V"xv)xdr. (48) THEDIFFERENTIAL CALCULUS OFVECTORS 163 OrifVodenotethevalueofvatthepoint(x,y,z)andvthe valueataneighboring point v=vo+'V(drov)dr+('VX v)xdr.(49) Thisexpression ofvintermsofitsvalueVoatagivenpoint, thedels,andthedisplacement drisanalogous totheexpan sionofascalarfun~ofonevariable byTaylor's theorem, /(x)=/(xo)+/'(xo)dx. Thederivative of(r0v)whenvisconstantisequaltovo Thatis For Hence'1(rov)y=v. '1(r.v)v=v o'Vr-('Vxr)Xv, v=VIi+V2j+vak, ~ ~ ~vo'1=v 1-+v2-+va-' ~x ~y ~z r=xi+Yi+zk, v0'1r =VIi +v2i+vak=v, '1X r=O. '1(r•v)y=v. Inlikemannerifinsteadofthefinitevectorr,aninfinitesimal vectordrbesubstituted, theresultstillis '1(dr.v)y=v. By(47) v = Vo+'1(dr0V)dr+('1Xv)Xdr '1(drov)='1(dr,v)<l; +'1(drov)yo Hence '1(dr0v)<lr='1(dr0v)-v. Substituting: 1 1 1V=2Vo+2'1(dr0v)+2('1Xv)Xdr.(50) Thisgivesanotherformof(49)whichissometimes more convenient. Itisalsoslightly moresymmetrical. 164 VEC1'OR ANALYSIS •76.]Consider amovingfluid.Letv(x,1/,z,t)bethe velocityofthefluidatthepoint(x,1/,z)atthetimet.Sur roundapoint(xo'1/111%0)withasma.11sphere. dr.dr=c3• Ateachpointofthisspherethevelocityis v=Vo+dr.'\lv. Intheincrement oftime8tthepointsofthisspherewillhave movedthedistance (vo+dr.'\lv)8t. Thepointatthecenterwillhavemovedthedistance Thedistance between thecenterandthepointsthatwere uponthesphereofradiusdratthecommencement ofthe interval8thasbecomeattheendofthatinterval8t dr'=dr+dr.'\lv8t. Tofindthelocusoftheextremity ofdr'itisnecessary to eliminate drfromtheequations dr'=dr+dr.'\lv8t, c2=dr.dr. Thefirstequation maybesolvedfordrbythemethodof Art.47,page90,andthesolution substituted intothesecond. Theresultwillshowthattheinfinitesimal sphere dr·dr=c3 hasbeentransformed intoanellipsoid bythemotionofthe fluidduringthetime8t. Amoredefiniteaccountofthechangethathastakenplace maybeobtained bymakinguseofequation (50) THEDIFFERENTIAL CALCULUS OFVECTORS 166 1 1 1 V=2v0+~"\7(dr0v)+i("\7xv)xdr, oroftheequation (49) V=vo+"V(drov)dr+ ("\7xv)xdr, v=Vo+["\7(dr0V)dr+}("\7xv)xdrJ+~("\7xv)xdr. ThefirsttermVointheseequationR expresses thefactthat theinfinitesimal sphereismovingasawholewithaninstan taneousvelocityequaltoYO'Thisisthetranslational element ofthemotion. Thelastterm ~("\7xv)xdr=~curlvxdr showsthatthesphereisundergoing arotation aboutan instnntaneous axisinthedirection ofcurlvandwithanangu larvelocity equalinmagnitude toonehalfthemagnitude of curlv.Themiddleterm 1ii"\7(drov)-v'" or expresses thefactthatthesphereisuntlergoing adefor mationknown ashomogeneous strainbyvirtueofwhichit becomes ellipsoidal. Forthistermisequalto ifVI'V2, V3berespectively thecomponents ofvinthedirec tionsi,j.k.Itisfairlyobvious thatatanygivenpoint (xo•Yo.zo)asetofthreemutually perpendicular axesi,j.k maybechosensuchthatatthatpoint"\7vl'"Vv2,"Vvaarere- 166 VECTOR ANALYSIS spectively paralleltothem.ThentheexpreBBion above becomes simply ~vl' ~'V2' ~'V8dx-1+dY-J+dz-k. ~x ~y ~z Thepointwhosecoordinates referred tothecenterofthe infinitesimal sphereare dx,dy,dz istherefore endowed withthisvelocity. Inthetime ~tit willhavemovedtoanewposition dX(l+~:l~t). dY(l+~~'j~t). d~(1+~~Vz8~t} Thetotalityofthepointsuponthesphere dr.dr=dx2+dy2+dz2=c2 goesoverintothetotalityofpointsupontheellipsoid of whichtheequation is x2 y2 z2 (1+~~'Vx1~t)'},+(1+~'Vy'J~tY+(1+~~~~8~t)2 Thestatements madebefore(Art.72)concerning thethree typesofmotionwhichaninfinitesimal sphereoffluidmay possesshavetherefore DOWbeendemonstrated. 77.]Thesymbolic operator 'Vmaybeappliedseveraltimes insuccession. Thiswillcorrespond inageneral wayto forming derivatives ofanorderhigherthanthefirst.The expreBBions foundbythusrepeating 'Vwillallbeindepend entoftheaxesbecause'Vitselfis.Therearesixofthese delsofthesecondorder. LetV(x,y,z)beascalarfunction ofposition inspace. Thederivative 'VVisavectorfunction andhencehasacurl andadivergence. Therefore 'V.'VV, V'X'VV THEDiFPERENTIAL CALCULUS OFVECTORS 167 arethetwoderivatives ofthesecondorderwhichmaybe obtained fromV. \7.\7V=div'VV \7X\7V=curl'VV.(51) (52) Symbolically,Thesecondexpression \7x'VVvanishes identically. Thatis, tluderivative ofanyscalarfunction Vpossesses nocurl.This maybeseenbyexpanding \7x'VVintermsofi,j,k.All thetermscancelout.Later(Art.83)itwillbeshowncon verselythatifavectorfunction Wpossesses nocurl,i.e.if \7xW=curlW=0,thenW='VV, Wisthederivative ofsomescalarfunction V. Thefirstexpression \7.\7Vwhenexpanded intermsof i,j,kbecomes ~2V ~2V ~2V'V.\7V=-+-+_· (51)' ~X2 ~y2 ~Z2 ~2 ~2 ~2\7.\7=-+-+-, ~x2~y2;;Z2 The_operator \7.\7istherefore thewell-known operator of Laplace. Laplace's Equation ~2VCJ2V ~2V.6.V=-+-+-=0 (53) ~x2 ~y2 ~z2 becomes inthenotation hereemployed (53)' Whenappliedtoascalarfunction Vtheoperator \7•\7yields ascalarfunction whichis,moreover, thedivergence ofthe derivative. LetTbethetemperature inabody.Letcbethecon ductivity, pthedensity, andkthespecific heat.The flowfis f=-c\7T. 168 VECTOR ANALYSIS Therateatwhichheatisleavingapointperunitvolumeper unittimeis"V•f.Theincrement oftemperature is 1d T=- -"Vofdt.. pie dT=~"V."VT.dtpie ThisisFourier's equation fortherateofchangeoftempera ture. LetVbeavectorfunction, andVl'J~,Jl;sits three com ponents. Theoperator "V•"VofLaplacemaybeappliedtov. Ifavectorfunction Vsatisfies Laplace's Equation, eachof its three scalarcomponents does.Otherdelsofthesecond ordermaybeobtained byconsidering thedivergence andcurl ofV.Thedivergence "V.Vhasaderivative "V"V.V="VdivV. Thecurl"VxVhasinturnadivergence andacurl,(55) and and"V."VxV,"VX"VXV. "V."VXV=divcurlV "VX"Vx V=curl curl V.(56) (57) Oftheseexpressions "V."Vx Vvanishes identically. Thatis, thedivergence ofthecurlofanyvectoriszero.Thismaybe seenbyexpanding "V•'Vx Vintermsofi,j,k.Later(Art. 83)itwillbeshownconversely thatifthedivergence ofa vectorfunction Wvanishes identically, i.e.if "V•W=div W=0,thenW='Vx V=curlV, Wisthecurlofsomevectorfunction V. TlIEDIFFERENTIAL CALCULUS OFVECTORS 169 Iftheexpression \7X(\7XV)wereexpanded formally according tothelawofthetriplevectorproduct, \7X(\7XV)=\7•V\l-\7•\7V. Theterms\7.V\7ismeaningless until\7betransposed to thebeginning sothatitoperates uponV. or\7X\7XV=\7\7•V -\7•\7V, curlcurlV=\7divV -\7.\lV.(58) (58)' Thisformula isveryimportant. Itexpresses thecurlofthe curlofavectorintermsofthederivative ofthedivergence andtheoperator ofLaplace. Shouldthevectorfunction V satisfyLaplace's Equation, \7•\7V=0and curlcurlV=\7divV. Shouldthedivergence ofVbezero, curlcurlV=-\7.\7V. ShouldthecurlofthecurlofVvanish, \7divV=\7•\7V. Tosumup.Therearesixofthedclsofthesecondorder. \7.\7V, \7x\7V, \7•\7V,\7\7•V,\7•\7xV,\7x\7XV. Ofthese,twovanishidentically. \7x\7V=0,\7.\lx V=O. Athirdmaybeexpressed intermsoftwoothers. \7X\7XV=\7\7•V-\7•\7V. (58) Theoperator\7.\7isequivalent totheoperator ofLaplace. 170 VECTOR ANALYSIS •78.]Thegeometric interpretation of'V.'VUisinteresting. Itdepends uponageometric interpretation ofthesecond derivative ofascalarfunctionFoftheonescalarvariable x. LetUibethevalueoffatthepointXi-Letitberequired tofindtheseconddenvative ofUwithrespecttozatthe point XO'LetXlandX,betwopointsequidistant fromXO' Thatis,let Then 92u92u--,-, 9x29y2 192uLm 2"~X2=a=:Oistheratioofthedifference between theaverageofuatthe points XlandX2andthevalueofuatxOtothesquareofthe distanceofthepoints Xl'X2fromXO'That iseasilyprovedbyTaylor's theorem. Letubeascalarfunction ofposition inspace.Choose threemutually orthogonal linesi,j,tandevaluate the expressions Letx2andX1betwopointsonthelineiatadistance afrom Xo;x~andxs'twopointsoniatthesamedistance afrom "0iX.andx6'twopointsoutatthesamedistance afromXo' Ul+u22 - Uo I THEDIFFERENTIAL CALCULUS OFVECTORS 171 A.dd: As\1and\1.a.reindependent oftheparticular axeschosen, thisexpression ma.ybeevaluated foradifferent setofaxes, thenforstilladifferent one,etc.Byaddingtogether all theseresults U1+U2+···611,terms . .. -Uo!\1•\1U=LIM 6_11,-;;-- _ 6 a=O a2 Let11,becomeinfiniteandatthesametimeletthedifferent setsofaxespointineverydirection issuingfromXo-The fraction U1+u2+...611,terms 611, thenapproaches theaverage valueofuuponthesurfaceofa sphereofradiusasurroun4ing thepoint XO'Denotethis byu". .!..\1.\1u=LIMU"-Uo• 6 a=Oa2 \1•\1Uisequa.ltosixtimesthelimitapproached bytheratio oftheexcessofuonthesurfaceofasphereabovethevalue atthecentertothesquareoftheradiusofthesphere. The samereasoning holdsincaseuisavectorfunction. Ifubethetemperature ofabody\1.\1u(exceptfora constant factorwhichdepends uponthematerial ofthe 172 VECTOR ANALYSIS body)isequaltotherateofincrease oftemperature (Art. 77).IfV'.V'uispositive theaveragetemperature upona smallsphereisgrea.terthanthetemperature atthecenter. Thecenterofthesphereisgrowing warmer. Inthecase ofasteadyflowthetemperature atthecentermustremain constant. Evidently therefore thecondition forasteady flowis V'.V'u=O. Thatis,thetemperature isasolutionofLapla.ce's Equation. Maxwell gavethenamecuncentratwn to-V'.V'uwhether ubeascalarorvectorfunction. ConsequeJjly V'.'lumay becalledthedispersion of thefunction uwhetheritbescalar orvector. Thedispersion isprot>ortional totheexcesso-f theaveragevalueofthefunction onaninfinitesimal surface <t.bovethevalueatthecenter. Incaseuisavectorfunction theaverage isavectoraverage. Theadditions initare vectoradditions. SU)IMARY OFCHAPTER III Ifavectorrisafunction ofascalartthederivative of rwithrespecttotisavectorquantity whosedirection is thatofthetangent tothecurvedescribed bytheterminus ofrandwhosemagnitude isequaltotherateofadvance of thatterminus alongthecurveperunitchangeoft.The derivatives ofthecomponents ofavectorarethecomponents ofthederivatives. Acombination ofvectorsorofvectorsandscalarsmaybe differentiated justasinordinary scalaranalysis exceptthat thedifferentiations mustbeperformed insitu. THEDIFFERENTIAL CALCULUS OFVECTORS 173 d da db dt(a.b)=dt.b+a•de' (3) d da db d t(aXb)=dtXb+aXdi' (4) or d(a•b)=da• b+a •db, d(aXb)=daXb+a xdb,.(3)' (4)' (8) (9)andsoforth.Thedifferential ofaunitvectorisperpendicu lartothatvector. Thederivativeofavectorrwithrespecttothearc8of thecurvewhichtheterminus ofthevectordescribes is theunittangenttothecurvesdirected towardthatpartofthe curvealong which 8issupposed toincrease. dr-=t.d8 Thederivative oftwithrespecttothearc,isavectorwhose direction isnormaltothecurveontheconcave sideand whosemagnitude isequaltothecurvature ofthecurve. C=dt=dlr. d8d82 Thetortuosity ofacurveinspaceisthederivative ofthe unitnormalntotheosculating planewithrespecttothe arc8.Tdnd(drdlr1) =ds=d8d8 Xd82•v'c.c . Themagnitude ofthetortuosity is [drdlrd8rJ d8d82d88T=-----d2rdlr (fBi•d81(11) (13) 174 VECTOR ANALYSIS Ifrdenotetheposition ofamoving particle, tthetime, vthevelocity, Atheacceleration, v=~=r (15)dt ds.tI=-=Bdt(16) . dVd2r.. A=v=Cit=dt2=r. (18) (19) A=8t+v2C.Theacceleration maybebrokenupintotwocomponents of whichoneisparallel tothetangent anddepends uponthe rateofchangeofthescalarvelocity voftheparticle inita path,andofwhichtheotherisperpendicular tothetangent anddepends uponthevelocity oftheparticleandthecurva tureofthepath. Applications tothehodograph, inparticular motioninl\ circle,parabola., orunderacentralacceleration. Application totheproofofthetheorem thatthemotionofarigidbody ~:mepointofwhichisfixedisaninstantaneous rotation about anaxisthrough thefixedpoint. Integration withrespecttoascalarismerelytheinverse ofdifferentiation. Application tofindingthepathsdueto givenaccelerations. Theoperator'lappliedtoascalarfunction ofpositionin spacegivesavectorwhosedirection isthatofmostrapid increase ofthatfunction andwhosemagnitude isequalto therateofthatincrease perunitchangeofposition inthat direction (21) (221 (24)dV"VV=-n,dnTHEDIFFERENTIAL CALCULUS OFVECTORS 175 Itmaybe Theoperator "Visinvariant oftheaxesi,j,k. definedbytheequation or "VV.dr=dV. (25)' Computation ofthederivative "VVbytwomethods depend inguponequations (21)and(25)'.Illustration oftheoc currence of"Vinmathematical physics. "Vmaybelookeduponasafictitious vector,avector differentiator. Itobeystheformallawsofvectorsjustin 80farasthescalardifferentiators of~/ax,~/~y,d/~zobey theformallawsofscalarquantities ~V ~V ~VA.."VV=A1-+.A2-+As-. (28)ax Ciy az Ifabeaunitvectora."VVisthedirectional derivative ofV inthedirection a. a•"VV=(a•"V)V=a•("VV). (30) IfVisavectorfunctiona·"VVisthedirectional derivative ofthatvectorfunction inthedirection a. (32)' (33)' (32)" (33)" 176 VECTOR ANALYSIS (85) (36) (37) (38) (39) (40)Proofthat"i1•Visthedivergence ofVand"i1XV,thecurl ofV. "i1.V=divV, "i1XV=curlV. "i1(u+v)="i1u+"i1'D, V'.(u+v)='V.u+"i1.v, V'X(u+v)="i1Xu+'VXv, "i1(uv)=v"i1u+u"i1'V, "i1•(uv)="i1u•v+u"i1•v, "i1X(uv)="i1uXv+u"i1Xv, V'(u•v)=V •"i1u+u•"i1v+vX("i1xu) +u)(("i1Xv),(41) "i1.(uXv)=v."i1xu-u•"i1Xv,(42) "i1X(uXv)=v •"i1u-v"i1•u-u•"i1v+u"i1•v.(43) Ifabeaunitvectorthedirectional derivative a·"i1v="i1(8•v).+("i1Xv)Xa.(47) Theexpansion ofanyvectorfunctionvintheneighborhood ofapoint(x",y~zo)atwhichittakesonthevalueofVois v=Vo+"i1(dr·V)dr+("i1Xv)Xdr,(49) or v=~Vo+"i1(dr •v)+~("i1Xv)Xdr.(50) Application tohydrodynamics. Thedelsofthesecondorderaresixinnumber. THEDIFFERENTIAL CALCULUS OFVECTORS 177 "Vx"VV=curl"VV=0, (52) ~2V~2V ~2V"V."V.Vdiv"VV= -+ - +-,(51) ~x2~y2 ~z2 V."VisLaplace's operator.If"V."VV=0,Vsatisfies La place'sEquation. Theoperator maybeappliedtoavector. ~2V ~2V ~2V "V•"VV=~x2+~y2+~z2' "V"V.V="VdivV, (55) "V."VxV=divcurlV=0, (56) "Vx"VXV=curl curl V="V"V•V-"V."VV.(58) Thegeometric interpretation of"V.'Vasgivingthedisper sWnofafunction. EXERCISES ONCHAPTER III 1.Givenaparticle moving inaplanecurve.Letthe planebetheii-plane. Obtaintheformulm forthecompo nentsofthevemcity parallelandpt\rpendicular totheradius vectorr.Theseare .rAr-,t1kxr,r where (Jistheangletheradiusvectorrmakeswithi,andk isthenormaltotheplane. 2.Obtaintheacceleratiqns oftheparticle parallel and perpendicular totheradiusvector. Theseare . r - r(r-r(2)-,(1'8+210iJ)kX-.r l' Express theseformulm intheusualmannerintermsofx andy. 1~ 178 VECTOR ANALYSIS 3.Obtaintheaccelerations ofamovingparticleparallel andperpendicular tothetangent tothepathandreducethe resultBtotheusualform. 4.IfT,4>,ebeasystemofpolarcoordinates inspace, where Tisthedistance ofapointfromtheorigin, 4>the meridianal angle,andethepolarangle;obtaintheexpressions forthecomponentB ofthevelocityandacceleration alongthe radiusvector,ameridian, andaparalleloflatitude. Reduce theseexpressions totheordinary formintermsofx,y,z. 5.Showbythedirectmethodsuggested inArt.68that theoperator \1isindependent oftheaxes. 6.Bythesecondmethod givenforcomputing \1find thederivative \1ofatripleproduct[ab0]eachtermofwhich isafunction ofx,y,zincase a=(r•r)r,b=(r•a)e,0=r x( whered,e,fareconstant vectors. 1 17.Compute \1.\1VwhenVisT~,r,;'or;:i. 8.Compute \1.\1V,\1\1•V,and\1x\1xVwhenVis equaltorandwhenVisequaltors'andshowthatintheser casestheformula (58)holds. 9.Expand \1x\1Vand\1•\1XVintermsofi,j,tand showthattheyvanish(Art.77)- 10.Showbyexpanding intermsofi,j,tthat \1x\1xV=\1\1•V-\1.\1V. 11.Prove A.•\1(V•W)=VA..\1W+WA.•\1V, and (\1xV)xW=\1x(VxW)....+W\1•V -\1(V•W).... CHAPTER IV THEINTF..GRAL CALCULUS OFVECTORS 79.]LetW(x,y,z)beavectorfunction ofposition in space.LetCbeanycurveinspace,andrtheradiusvector drawnfromsomefixedorigintothepointsofthecurve. Dividethecurveintoinfinitesimal elements dr.Fromthe sumofthescalarproductoftheseelements drandthevalue ofthefunction Watsomepointoftheelement- thus ~W.dr. Thelimitofthissumwhentheelementsdrbecomeinfinite innumber,eachapproaching zero,iscalledthelineintegralof WalongthecurveCandiswritten feW.dr. If andW=WIi+W2j+Wsk, dr=idx+jdy+kdz, Thedefinition ofthelineintegral therefore coincides with thedefinition usuallygiven.Itishowever necessary to specifyinwhichdirection theradiusvectorrissupposed to describethecurveduringtheintegration. Fortheelements drhaveoppositesignswhenthecurveisdescribed inoppo- 180 VECTOR ANALYSIS sitedirections. Ifonemethodofdescription bedenoted by aandtheotherby-c, fw·dr=-fw·dr.-0 Jc IncasethecurveCisaclosedcurve boundin~ aportionof surfacethecurvewillalwaysberegarded asdescribed in suchadirection thattheenclosed areaappears positive (Art.25). Iffdenotetheforcewhichmaybesupposed tovaryfrom pointtopointalongthecurveC,theworkdonebytheforce whenitspointofapplication ismovedfromtheinitialpoint roofthecurveCtoitsfinalpointristhelineintegral ff •dr=frf •dr.J0Jro Theorem: Thelineintegral ofthederivative VVofa, scalarfunction V(x,y,z)alonganycurvefromthepoint rotothepointrisequaltothedifference between thevalues ofthefunctionV(x,y,z)atthepointrandatthepointroo Thatis. f:VV.dr= V(r)-V(ro)=V(x,y,z) -V(xooy",zo). o Bydefinition dr.VV=dV idV=V(r)-V(ro)=V(x,y,z) -V(xo,yo.zo). (2) o Theorem: Thelineintegral ofthederivative \1Vofa singlevaluedscalarfunction ofposition Vtakenarounda closedcurvevanishes. Thefactthattheintegralistakenaroundaclosedcurve isdenotedbywritingacircleatthefootoftheintegralsign. ToshowJVV.dr=O. (3) o THEINTEGRAL CALCULUS OFVECTORS 181 Theinitialpointroandthefinalpointrcoincide. Hence Henceby(2)1:"VVodr=O. Theorem: Conversely ifthelineintegralofWaboutevery closedcurvevanishes, Wisthederivative ofsomescalar function V(x,y,z)ofpositioninspace. Given ToshowJoWodr=O. W="VF: Letrobeanyfixedpointinspacea.ndravariable point. Thelineintegral rJ:w 0dr o isindependent ofthepathofintegration O.Forletanytwo paths0and0'bedrawnbetween roandr.Thecurvewhich consistsofthepath0fromrotorandthepath-0'fromr toroisaclosedcurve.Hencebyhypothesis fWodr+ !Wodr=O,o -0' fWodr= -fwodr. -0' c' HencefWodr=fWodr.o 0' Hencethevalueoftheintegral isindependent of thepath ofintegration anddependsonlyuponthefinalpointr. 182 VECTOR .ANALYSIS Thevalueoftheintegralistherefore ascalarfunction of theposition ofthepointrwhosecoOrdinates areZ,y,z. r J:W·dr=V(x,y,z). o W="Vv.W.dr=dV(x,y,z). "VV.dr=dV, Butbydefinition HenceLettheintegralbetakenbetween twopointsinfinitely near together. Thetheoremistherefore demonstrated. 80.]Letfbetheforcewhichactsuponaunitmassnear thesurfaceoftheearthundertheinfluence ofgravity. Let asystemofaxesi,j,kbechosensothatkisvertical. Then f= -gk. Theworkdonebytheforcewhenitspointofapplication movesfromthepositionrotothepositionris r r r w=J:f.dr=1:-gk·dr=- !gd%. o 0 0 Hence w=-9(z-%0)=9(zo-z)..... Theforcefissaidtobederivable fromaforce-function V whenthereexistsascalarfunction ofposition Vsuchthat theforceisequalateachpointofthederivative "VV. Evidently ifVisoneforce-function, anothermaybeobtained byaddingtoVanyarbitrary constant. Intheaboveex· ampletheforce-function is Ormoresimply Theforceisv=w=9(zo-%). V=-gz. f="VV=-gk. f=VV=Vw.THEINTEGRAL CALCULUS OFVECTORS 183 Thenecessary andsufficient condition thataforce-function V(x,y,z)exist,istha.ttheworkdonebytheforcewhenits pointofapplication movesaroundaclosedcircuitbezero. Theworkdonebytheforceis w=ff·dr. Ifthisintegral vanishes whentakenaroundeveryclosed contour Andconversely iff=VV theintegralvanishes. Theforce-function andtheworkdone differonlybyaconstant. V=w+const. Incasethereisfrictionnoforce-function canexist.Forthe workdonebyfrictionwhenaparticleismovedaroundina closedcircuitisneverzero. Theforceofattraction exerted byafixedmassMupon aunitmassisdirected towardthefixedmassandispropor tionaltotheinversesquareofthedistance between the masses. Mf=-c3r.r Thisisthelawofuniversal gravitation asstatedbyNewton. Itiseasytoseethatthisforceisderivable fromaforce function V.Choosetheoriginofcoordinates atthecenter oftheattracting massM.Thenthework done is rM w=-fc3r·dr. rro Bui r.dr=rdr, w=-eMfrdr= _eM!! _.!.}. r'1'2 ~r"0o 184 VECTOR ANALYSIS Byaproperchoiceofunitstheconstant cmaybemade equaltounity. Theforce-function Vmaytherefore be chosenas Mv=--·r Iftherehadbeenseveralattracting bodiesMI,M2,Ms'··· theforce-function wouldhavebeen whererI,r2,TS'•.•arethedistances oftheattracted unit massfromtheattracting massesMI'M2•Ms... Thelawoftheconservation ofmechanical energyrequires thattheworkdonebytheforceswhenapointismoved aroundaclosedcurveshallbezero.Thisisontheassump tionthatnone ofthemechanical energyhasbeenconverted intootherformsofenergyduringthemotion. Thelawof conservation ofenergytherefore requires theforcestobe derivable fromaforce-function. Conversely ifaforee function existstheworkdonebytheforceswhenapointis carriedaroundaclosedcurveiszeroandconsequently there isnolossofenergy. Amechanical systemforwhichaforce function existsiscalledacon8ervative system. Fromthe example justcitedaboveitisclearthatbodiesmovingunder thelawofuniversal gravitation formaconservative system atleastsolongastheydonotcollide. 81.]LetW(x,y,z)beanyvectorfunction ofpositionin space.Let8beanysurface. Dividethissurfaceintoin finitesimal elements. Theseelements mayberegarded as planeandmayberepresented byinfinitesimal vectorsof whichthedirection isateachpointthedirection ofthe normaltothesurfaceatthatpointandofwhichthemagni tudeisequaltothemagnitude oftheareaoftheinfinitesimal THEINTEGR.4.L CALCULUS OFVECTORS 185 element. Letthisinfinitesimal vectorwhichrepresents the elementofsurfaceinmagnitude anddirection bedenotedby da.Formthesum l:W oda, whichisthesumofthescalarproducts ofthevalueofW ateachelement ofsurfaceandthe(vector) element of surface. Thelimitofthissumwhentheelements ofsur faceapproach zeroiscalledthesurfaceintegralofWover thesurfaceS,andiswritten (4) Thevalueoftheintegral isscalar.IfWanddabeex pressedintennsoftheirthreecomponents paralleltoi,j,t W=Wii+W2j+Wst, orda=(daoi)i+(daoj)j+(daot)t, da=dydzi+dzdxj+dxdyt, Thesurfaceintegral therefore hasbeendefinedasiscus tomaryinordinary analysis. Itishowever necessary to determine withthegreatestcarewhichnormaltothesurface dais.Thatis,whichsideofthesurface(sotospeak)the integral istakenover.Forthenormalsuponthetwosides arethenegatives ofeachother.Hencethesurfaceintegrals takenoverthetwosid~willdifferinsign.Incasethe surfacebelookeduponasbounding aportionofspaceda isalwaysconsidered tobetheexterior normal. Iffdenotethefluxofanysubstance thesurfaceintegral 186 VECTOR ANALYSIS givestheamountofthatsubstance whichispll8Bingthrough thesurfaceperunittime.Itwasseenbefore(Art.71)that therateatwhichmatterwasleaving apointperunit volumeperunittimewas"i1•f.Thetotalamountofmat terwhichleavesaclosedspacebounded byasurfaceSper unittimeistheordinary tripleintegral J J J"i1ofdv. (6) Hencetheveryimportant relation connecting asurfacein tegralofafluxtakenoveraclosedsurfaceandthevolume integral ofthedivergence ofthefluxtakenoverthespace enclosed bythesurface- Written outinthenotation oftheordinary calculus this becomes JJ[Xdydz +Ydzdx+Zdxdy] J.1.1(ClXClyClZ)= - +- + - dxdydzClxClyClz(8) whereX;Y,Zarethethreecomponents ofthefluxf.The theorem isperhapsstillmorefamiliarwheneachofthethree components istreatedseparately. JJXdxdy=JJJ~~ dxdydz. (8)' ThisisknownasGauss'sThe01'em.Itstatesthatthesurface integral (takenoveraclosedsurface) oftheproduct ofa. function Xandthecosineoftheanglewhichtheexterior normaltothatsm::facemakeswiththeX-axisisequalto thevolumeintegralofthepartialderivative ofthatfunction ff/oda=\1ofdVTHEINTEGRAL CALCULUS OFVECTORS 187 withrespecttoxtakenthroughout thevolumeenclosed by thatsurface. IfthesurfaceSbethesurfacebounding aninfinitesima.l sphereorcube wheredvisthevolumeofthatsphereorcube.Hence \1of=~J.'rf0da. (9)dvJ8 Thisequation maybetakenasadefinition ofthedivergence \10f.Thedivergence ofavectorfunction fisequaltothe limitapproached bythesurfaceintegralofftakenoverasur· facebounding aninfinitesimal body dividedbythatvolume whenthevolumeapproaches zeroasitslimit.Thatis LIM1J.f \1.f=d .0 - foda.v=dvs(10) Fromthisdefinition whichisevidently independent ofthe axesalltheproperties ofthedivergence maybededuced. In ordertomakeuseofthisdefinition itisnecessary todevelop atleasttheelements oftheintegralcalculus ofvectorsbefore thedifferentiating operators canbetreated. Thisdefinition of\10fconsequently isinteresting morefromatheoretical thanfromapractical standpoint. 82.]Theorem: Thesurfaceintegralofthecurlofavector function isequaltothelineintegral ofthatvectorfUllction takenaroundtheclosedcurvebounding tha.tsurface. Thisisthecelebrated theorem ofStokes. Onaccountofits greatimportance inallbranches ofmathematical physicsa numberofdifferent proofswillbegiven. 188 VECTOR ANALYSIS FirstProof:Consider asmalltriangle123uponthesurface S(Fig.32).LetthevalueofWatthevertex1beWoo Thenby(50),Chap.III.,thevalueatanyneighboring pointis W=~1Wo+V'(W.~r)+(V'XW)X0r}, wherethesymbol ~rhasbeenintroduced forthesakeofdis tinguishing itfromdrwhichistobeusedastheelementof integration. Theintegral ofWtakenaroundthetriangle 1i3is FIG.32. +~i(V'XW)X0r •tlr. Thefirstterm vanishes becausetheintegralofdraroundaclosedfigure,in thiscaseasmalltriangle, iszero.Thesecondterm vanishes byvirtueof(3)page180.Hence THEINTEGRAL CALCULUS OFVECTORS 189 iw.dr =~J.1.'1x W x ~r •dr. Interchange thedotandthecrossinthistripleproduct. J~.dr =~i'1XW·~rXdr. Whendrisequaltotheside12ofthetriangle, ~risalso equaltothisside.Hencetheproduct ~rxdr vanishes because ~randdrarecollinear. Inlikemanner whendristheside31,~risthesameside13,buttaken intheopposite direction. Hencethevectorproductvanishes. Whendristheside23,~risalinedrawnfromthevertex 1atwhichW=Wotothisside23.Hencetheproduct ~rXdr istwicetheareaofthetriangle. Thisarea,moreover, isthe positivearea1123.Hence 12~rxdr=da, wheredadenotesthepositiveareaofthetriangular element ofsurface. Fortheinfinitesimal triangle therefore the relationf ~W•dr='1XW •da holds. LetthesurfaceSbedivided intoelementary triangles. Forconvenience letthecurvewhichbounds thesurface bemadeupofthesidesofthesetriangles. Perfonn the integration aroundeachofthesetriangles andaddtheresultstogether. ~fW.dr=~'1XW.dL..J~ .. 190 VECTOR ANA.LYSIS Thesecondmember ~V'xW·da 8 isthesurfaceintegralofthecurlofW. ~V'xW·da-:JJaV'xW·dL Inaddingtogether thelineintegrals whichoccurinthefirst memberitisnecessary tonoticethatalltheside8oftheele mentary triangles exceptth08ewhichliealongthebounding curveofthesurfacearetracedtwiceinopposiudirections. Henceallthetermsinthesum whicharisefromthosesidesofthetriangles lyingwithinthe surfaceScancelout,leavinginthesumonlytheterIllB whicharisefromthosesideswhichmakeupthebounding curveofthesurface. Hencethesumreducestothelinein tegralofWalongthecurvewhichboundsthesurfaceS. ~fW.dr=fw.dr. aA 0 Hence (11) FIG.33.SecondProof:Let0beanyclosed contour drawnuponthesurfaceS (Fig.33).Itwillbeassumed thatC iscontinuous anddoesnotcutitself. Let0'beanothersuchcontournear toO.Consider thevariation Swhich takesplaceinthelineintegral ofW inpassingfromthecontour0tothe contour0'. THEINTEGRAL CALCULUS OFVECTORS 191 ~JWodr =JJfodr- lWodr, ~JWodr= J~(Wodr)= JWoSdr+ JSWodr. But d(W 0~r)=dW0~r+W0d~r and ~dr=d~r. Theexpression d(W 0~r)isbyitsformaperfectdifferential. Thevalueoftheintegralofthatexpression willtherefore be thedifference between thevaluesofW0drattheendandat thebeginning ofthepathofintegration. Inthiscasethe integral istakenaroundtheclosedcontourC.Hence Hence and But or andJW0~dr= -JdW0~r, SJW0dr-'J~W0dr -J dWoOr, ~JWodr=JloWodr-dWoSr}. ~W ~W ~W.dW=- dx+--dy+-d;" ~x ~y ~z ~W ~W. dWdW=-iodr+-Jodr+-kodr, ~x ~y dZ dW. dW. ~W ~W=,,-100r+,,-J •0r+--;:;-k0~roaX uy gZ 192 VECTOR ANALYSIS Substituting thesevalues JIiC)W .C)WSW.dr= -.dt1.Sr--.SrC)X dx +similartermsinyandz.~• B~tby(25)page111 (.CJW) C)W. C)W .1X';\-•(C)r xdr)= - • drl'C)r --'•C)rl'dr. ~x ~x C)x Hence orsJw.dr=Jtix~:.SrXdr +similartermsinyandzI. sJW•dr =J'Vxw·Sr xdr. InFig.33itwillbeseenthatdristheelement ofarc alongthecurveaandSristhedistance fromthecurveato thecurvea'.HenceSrXdrisequaltotheareaofanele mentary parallelogram included betweenaandA'uponthe surfaceS.Thatis Srxdr=da, SJw.dr-:J'V XW·dL Letthecurveastartingatapoint0inSexpanduntilit coincides withthecontourbounding S.Thelineintegral JW.dr willvaryfromthevalue0atthepoint0tothevalue .£W·dr THEINTEGRAL CALCULUS OFVECTORS 1l:}3 takenaroundthe\contourwhichboundsthesurfaceS.This totalvariation oftheintegralwillbeequaltothesumofthe variations ~ ~~IW •dr=~I'VxW •da. Or .faW •dr=II 8'VXw·da. (11) 83.]Stokes's theoremthatthesurfaceintegralofthecurl ofavectorfunction isequaltothelineintegral ofthefunc tiontakenalongtheclosedcurvewhichboundsthesurface hasbeenproved. Theconverse isalsotrue.Ifthesur/ace integralofa~·ect01'function 11isequaltothelineintegral0/the function Wtakenaroundthecurvebounding thesur/aceandif thisrelaticmholds/01'allsur/acesinspace,then11isthecurl0/ W.Thatis ifI1811.da=faW•dr,then11='Vxw.(12) Formthesurfaceintegralofthedifference between 11and 'Vxw. II.(11-'VxW)•da-J:W•dr -10w·dr=O. orI18(11-'VxW)•da=O. LetthesurfaceSoverwhichtheintegration isperformed be infinitesimal. Theintegralreducestomerelyasingleterm (11-'VxW)•da=O. Asthisequation holdsforanyelementofsurfaceda,the firstfactorvanishes. Hence 11-'VxW=O. Hence 11='VxW. Theconverse istherefore demonstrated. 18 194 VECTOR ANALYSIS Adefinition of'lx Wwhichisindependent oftheaxes i,j,tmaybeobtained byapplying Stokes's theorem toanin finitesimal planearea.Consider apointP.Passaplane throughPanddrawinit,concentric withP,asmallcircleof areada. \lxW•da-'10W·dr. (13) (13)'Whendahasthesamedirection as'lx Wthevalueofthe lineintegralwillbeamaximum, forthecosineoftheangle between'lxWanddawillbeequaltounity.Forthis valueofda, LIM[daIe] 'lxW-d'0 W·dr.a=da.da 0 Hencethecurl'lx Wofavectorfunction Whasateach pointofspacethedirection ofthenormaltothatplanein whichthelineintegral ofWtakenaboutasmallcirclecon centricwiththepointinquestion isamaximum. Themag nitudeofthecurlatthepointisequaltothemagnitude of thatlineintegralofmaximum valuedivided bytheareaof thecircleaboutwhichitistaken.Thisdefinition likethe onegiveninArt.81forthedivergence isinteresting more fromtheoretical thanfrompractical consideratioDs. Stokes's theorem orratheritsconverse maybeusedtode duceMaxwell's equations oftheelectro-magnetic fieldina. simplemanner. LetEbetheelectricforce,Bthemagnetic induction, Hthemagnetic force,andCthefluxofelectricity perunitareaperunittime(i.e.thecurrentdeDsity). Itisafactlearnedfromexperiment thatthetotalelectro motiveforcearoundaclosedcircuitisequaltothenegative oftherateofchangeoftotalmagnetic induction through thecircuit. Thetotalelectromotive forceisthelineintegral oftheelectricforcetakenaroundthecircuit. Thatis 1:B.dr. THEINTEGRA.L CA.LCULUS OFVECTORS 195 Thetotalmagnetic induction throughthecircuitisthesur faceintegralofthemagnetic induction Btakenoverasurface bounded bythecircuit. Thatis Experiment therefore showsthat or Hencebytheconverse ofStokes's theorem 'VxB= -:B,curlB= -i. Itisalsoafactofexperiment thattheworkdoneincarry ingaunitpositive magnetic polearoundaclosedcircuitis equalto47Ttimesthetotalelectricfluxthroughthecircuit. Theworkdoneincarrying aunitpolearoundacircuitis thelineintegral ofHaroundthecircuit. Thatis Thetotalfluxofelectricity through thecircuitisthe surfaceintegralofCtakenoverasurfacebounded bythe circuit. Thatis Experiment therefore teachesthat 196 VECTOR ANALYSIS Bytheconverse ofStokes'stheorem \/XH=4'1ro. Withaproperinterpretation ofthecurrent0,asthedis placement currentinaddition totheconduction curren~ aninterpretation depending upononeofMaxwell's primary hypotheses, thisrelationandthepreceding onearethefunda mentalequations ofMaxwell's theory,intheformusedby Heaviside andHertz. Thetheorems ofStokesandGa1lS8maybeusedtodemon stmtetheidentities. \/.\/xW=0,divcurlW=O. \/x\/r=0,curl\/V=O. According toGauss'stheorem According toStokes'stheorem HenceJJJ\/.\/xWdv=1:W•dr. Applythistoaninfinitesimal sphere. Thesurfacebounding thesphereisclosed. Henceitsbounding curvereducestoSo point;andtheintegralaroundit,tozero. \/•'VxWdv=faW•dr=0, \/.\/xW=o. THEINTEGRAL CALCULUS OFVECTORS 191 Againaccording toStokes's theorem ff8\1x\1V.da=.fa\1V.dr. Applythistoanyinfinitesimal portionofsurface. Thecurve bounding thissurfaceisclosed. Hencethelineintegral of thederivative VVvanishes. vxVV.da= O. Asthisequation holdsforanyda,itfollowsthat VxVV=O. Inasimilarmanuel' theconverse theorems maybe demonstrated. Ifthedivergence \1•Uofavectorfunction Uiseverywhere zero,thenUisthecurlofsomevector function W. U=VxW. Ifthecurl\1XUofavectorfunctionUiseverywhere zero, thenUisthederivative ofsomescalarfunction V, 84.]Bymakinguseofthethreefundamental relations between theline,surface, andvolumeintegrals, andthe lkis,viz.: ffVV.dr=VCr)-V(ro) , (2) fo itispossible toobtainalargenumberofformulre forthe transformation ofintegrals. Theseformulre correspond to 198 VECTOR ANALYSIS thoseconnected withhintegration byparts" lDordinary calculus. Theyareobtained byintegrating bothsidesofthe formullll, page161,fordifferentiating. First \!(uv)=u\!v+v\!u. Ja\!(uv)0dr=Jau\!v0dr=J0'"\!u0dr. HenceJu\!vodr=[uv]:o -Jv\!uodr. (14) Theexpressionr[uv]ro represents thedifference between thevalueof(uv)atr,the endofthepath,andthevalueatro>thebeginning ofthepath. Ifthepathbeclosed Jou\!v0dr= -10v\!u:dr. (14)' Second \!x(uv)=u\!x v+\!uxv. Hence JJa"Vux v 0da=JoUv0dr-JJau"VXv0da,(15) or JJau\7Xv0da=JoUv0dr -JJa\7ttX V 0da,(1Sy r--Third"Vx(u"Vv)::;:tt"Vx"Vv+"Vux"Vv. But \!x\7v=0 Hence "Vx(u\!v)="Vux\7v, THEINTEGRAL CALCULUS OFVECTORS 199 JJsVx(U"Vv)0da=fJaVUxVv0da. ,\j;~,xv.'da= 10,v•.dr~~Io.V•.dr,(16) -Iourth '10(uv)=u'Vov+VuoV.'r!/.JJJvo(uv)dv=JIJuV"vdv+JJVuovdv. Hencei JJJuvovdv-J Ja~voda--JJJvuovdv, UIIv•.•d·~1Is••.da-III'V·.d., FifthV0(V1£Xv)='VxVu0v -Vu0Vxv. V0(Vuxv)= -Vl'•VXv,(17) (17)' JJfv·(Vuxv}dv= -JJJ Vu·Vx vdv. HenceJJaVUXv·da= -JJJVu·Vx vdv'(18) Inalltheseformulre whichcontainatripleintegral the surfaceSistheclosedsurfacebounding thebodythroughout whichtheintegration is-performed. Examples ofintegration bypartslikethoseabovecanbe, multiplied almostwithout limit.Onlyonemorewillbe givenhere.ItisknownasGreen'sTheorem andisperhaps themostimportant ofall.Ifuandvareanytwos96lar functions ofposition, 200 VECTOR ANALYSIS \70(U'Vv)=\7U0\7v+U\70\Iv, \70(v\7U)=\7U0\7v+v\70\7u, \7U0\Iv=\10(U'Vv)-U\l0\Iv=\10(v\1U)-v\70\IU, JJJ\7uo\1v!lv=JJJ\70(U\lv)dv-JJJU\7o\7vdv, = JJJ\7o(v\1U)dv-JJJv\lo\lUdv. Hence JJJ\1U0\l1Jdv=JJU\Iv0da-JJJ U\10\lvdv, = JJv\7U0da -JJJv\10\lU dv.(19) Bysubtracting theseequalities theformula .(20) JJJ(u'Vo'Vv-v\l0'Vu)dv= JJ(u'Vv-V\lu)oda. isobtained. Byexpanding theexpre88ion intermsofi,j,It theordinary formofGreen'stheorem maybeobtained. A furthergeneralization duetoThomson (LordKelvin)isthe following: JJJw\lUo\lvdv=JJuw\lvoda-JJJU\lo[W\lv]dv, = JJvW\lU0da -JJJv\l0[w\lu]dv,(21) wherewisathirdscalarfunction ofposition. Theelementofvolumedvhasnothingtodowiththescalar function vintheseequations orinthosethatgobefore.The useofvinthesetwoditIel'ent sensescanhardlygiveriseto anymisunderstanding. •85.JInthepreceding articlesthescalarandvectorfunc tionswhichhavebeensubjecttotreatment havebeensup- THEINTEGRAL CALCULUS OFVECTORS 201 posedtobecontinuous, single-valued, possessing derivatives ofthefirsttwoordersateverypointofspaceunderconsider ation.Whenthefunctions arediscontinuous ormultiple valued,orfailtopossessderivatives ofthefirsttwoorders incertainregionsofspace,somecautionmustbeexercised in applying theresultsobtained. Suppose forinstance ~Vy.x.v=---I+--J. X2+y2 X2+.'12 Theliueintegral !V d!Xdy-ydx'V•r= • x2+.'12 Introducing polarcoljrdinates x=rcos0, y=rsin0, xdy-.'Id x=rdO, Formthelineintegral fromthepoint(+1,0)tothepoint (-1,0)alongtwodifferent paths.Letonepathbeasemi circlelyingabovetheX-axis; andtheother,a.semicircle lyingbelowthataxis.Thevalueoftheintegral alongthe firstpathis 1f"-dO='1l'"; r0 alongthesecondpath,1["-dO='1l'".r0 Fromthisitappearsthattheintegraldoesnotdependmerely uponthelimitsofintegration, butuponthepathchosen, 202 VECTOR ANALYSIS thevaluealongonepathbeingthenegative ofthevalue alongtheother.Theintegralaroundthecirclewhichisa closedcurvedoesnotvanish,butisequalto±2Tr. Itmightseemtherefore theresultsofArt.79werefalse andthatconsequently theentirebottomoftheworkwhich followsfellout.Thishowever isnotso.Thedifficulty is thatthefunction -1YV=tanx isnotsingle-valued. Atthepoint(1,1),forinstance, the functionVtakesonnotonlythevalue -1 Tr V=tan1=4' butawholeseriesofvalues Tr "4+kTr, wherekisanypositiveornegative integer. Furthermore at theorigin,whichwaaincluded between thetwosemicircular pathsofintegration, thefunction Vbecomes whollyinde terminate andfailstopossessaderivative. Itwillbeseen therefore thattheoriginisapeculiarorsingularpointofthe function V.Ifthetwopathsofintegration from(+I,0)to (-1,0)had.notincluded theoriginthevaluesoftheintegral wouldnothavediffered. Inotherwordsthevalueofthe integral aroundaclosedcurvewhichdol'Snotincludethe originvanishes asitshould.. Inaamuch aatheoriginappearstobethepointwhich vitiatestheresultsobtained, letitbeconsidered asmarked byanimpassable barrier. Anyclosedcurveawhichdoes notcontaintheoriginmaybeshrunkuporexpanded atwill; butaclosedcurveCwhichsurrounds theorigincannotbe sodistorted asnolongertoenclosethatpointwithoutbreak ingitscontinuity. Thecurveanotsurrounding theorigin THEINTEGRAL CALCULUS OFVECTORS 203 mayshrinkuptonothingwithoutabreakinitscontinuity; butCcanonlyshrinkdownandfitcloserandcloserabout theorigin.Itcannotbeshrunkdowntonothing.Itmust alwaysremainencircling theorigin. ThecurveCissaidto bereducible ja;irreduc1'ble. Incaseofthefunction V,then, itistruethattheintegraltakenaroundanyreducible circuit Cvanishes; buttheintegralaroundanyirreducible circuit(j doesnotvanish. Suppose nextthatVisanyfunction whatsoever. Letall thepointsatwhichVfailstobecontinuous ortohavecon tinuous firstpartialderivatives bemarked asimpassable barriers. ThenanycircuitCwhichcontaill8 withinitno suchpointmaybeshrunkuptonothing andissaidtobe reducible,. butacircuitwhichcontains oneormoresuch pointA ca~notbesoshrunkupwithoutbreaking itscontinuity anditissaidtobeirreducible. Thetheorem maythenbe stated: Thelineintegralofthederivative 'VVofanyfunction Vvanishes aroundanyreducible circuitC.Itmayormaynot vanisharoundanirreducible cil'Cuit. Incaseoneirreducible ~.ircuitCmaybedistorted soastocoincide withanother Irreducible circuit(jwithout passing through anyofthe singular pointsofVandwithout breaking itscontinuity, thetwocircuitsaresaidtobereconcilable andthevaluesof thelineintegralof'VVabout them arethesame. AregionsuchthatanyclosedcurveCwithinitmaybe shrunkuptonothingwithout passingthrough anysingular pointofVandwithout breaking itscontinuity, thatis,a regioneveryclosedcurvein which isreducible, issaidtobe acyclic. Allotherregionsarecyclic. Bymeansofasimpledeviceanycyclicregionmayberen deredacyclic. Consider, forinstance, theregion(Fig.34)en closedbetween thesurfaceofacylinder andthesurfaceofa cubewhichcontains thecylinder andwhosebasescoincide withthoseofthecylinder. Sucharegionisrealizedinaroom 204 VECTOR ANALYSIS FIG.84.inwhichacolumn reache~fromthefloortotheceiling.It isevidentthatthisregioniscyclic.Acircuitwhichpasses aroundthecolumnisirreducible. Itcannotbecontracted to nothingwithoutbreaking itscontinuity. If nowadiaphragm beinserted reaching from thesurfaceofthecylinder orcolumntothe surfaceof_thecubetheregionthusformed bounded bythesurfaceofthecylinder, the surfaceofthecube,andthetwosidesofthe diaphragm isacyclic. Owingtotheinser tionofthediaphragm itisnolongerpossible todrawacircuitwhich shall passcompletely aroundthecyl inder-thediaphragm prevents it.Henceeveryclosedcir cuitwhichmaybedrawnintheregionisreducible andthl' regionisacyclic. Inlikemanner anyregionmayberendered acyclicby inserting asufficient numberofdiaphragms. Thebounding surfaces ofthenewregionconsistofthebounding surfaces of thegivencyclicregionandthetwofacesofeachdiaphragm. Inacyclicregionsorregionsrendered acyclicbythefore goingdevicealltheresultscontained inArts.79etseq. holdtrue.Forcyclicregionstheymayormaynothold true.Toenterfurtherintothesequestions atthispointis unnecessary. Indeed,evenasmuchdiscussion ashasbeen giventhemalreadymaybesuperfluous. Fortheyareques tionswhichdonotconcernvectormethods anymorethanthe corresponding Cartesian ones.Theybelongproperly tothe subjectofintegration itself,ratherthantotheparticular notation whichmaybeemployed inconnection withitand whichistheprimary objectofexposition here.Inthis respectthesequestions aresimilartoquestions ofrigor. THEINTEGRAL CALCULUS OFVECTORS 205 TheIntegrating Operators. ThePotential 86.]Hitherto therehavebeenconsidered line,surface, andvolumeintegrals offunctions bothscala.randvector. Thereexist,however, certainspecialvolumeintegrals which, owingtotheirintimate connection withthedifferentiating operators "il,"il","ilx,andowiI!gtotheirespecially frequent occurrence andgreatimportance inphysics, meritespecial consideration. Suppose that V(x2,Y2'%2) isascalarfunction of thepositioninspaceofthepoint (x2,Y2'%2)' Forthesakeofdefiniteness Vmayberegarded asthe densityofmatteratthepoint(x2,Y2'~).Inahomogeneous bodyVisconstant. Inthoseportionsofspaceinwhichno matterexistsVisidentically zero.Innon-homogeneous dis tributions ofmatterVvariesfrompointtopoint;butat eachpointithasadefinitevalue. Thevector drawnfromanyassumed origin,maybeusedtodesignate thepoint(x2,Y2'%2)'Let (xl'Yl'%1) beanyotherfixedpointofspace,represented bythevector rl=xli+Ylj +%lk drawnfromthesameorigin. Then r2- rl=(x2-Xl)i +(Y2-Yl)j +(%2-%1)k isthevectordrawnfromthepoint(xl'Yl'%1)tothepoint (x2,Y2'%2)'Asthisvectoroccursalargenumberoftimes inthesectionsimmediately following, itwillbedenotedby 206 VECTOR ANALYSIS FIG.85.ThelengthofrIIIisthenrIIIandwillbeassumed tobe positive. rIll=V'rIll·rIll=v'"(X;-zlya+(Yll-YI)lI+(zli-%1)1. Consider thetripleintegral f.ffV(xlI'YlI'%1)I(zl'YI'%1)=.TIll dZlIdYlldzlI· Theintegration isperformed withrespecttothevariables ZlI'YlI'%1-thatis,withrespecttothebodyofwhichV represents thedensity(Fig.35).During theintegration thepoint(zl'YI'%1)re mainsfixed.TheintegralIhasadefinite valueateachdefinitepoint(zl'YI'%1)' Itisafunction ofthatpoint.Thein terpretation ofthisintegralIise88Y,if thefunction Vberegarded 88thedensityofmatterinspace. Theelementofmassdmat(zlI'YlI'%1)is dm=V(ZlI'YlI'%1)dZlIdYlld%lI=JTdv. TheintegralIistherefore thesumoftheelements ofmass inabody,eachdividedbyitsdistance fromafixedpoint (zl'?/I'%1)' Thisiswhatistermedthepotential atthepoint(zl'YI'zl) duetothebodywhosedensityis V(zlI'YlI'%1)' Thelimitsofintegration intheintegralImaybelookedat ineitheroftwoways.Inthefirstplacetheymay·be regarded 88coincident withthelimitsofthebodyofwhich Visthedensity. Thisindeedmightseemthemostnatural setoflimits.OntheotherhandtheintegralImaybe THEINTEGRAL C.HCULUS OFVECTORS 207 regarded astakenoverallspace.Thevalueoftheintegral isthesameinbothcases.Forwhenthelimitsareinfinite thefunctionVvanishes identically ateverypoint(x"'!II'%1) situated outsideofthebodyandhencedoesnotaugment thevalueoftheintegralatall.Itisfoundmostconvenient toconsider thelimitsasinfiniteandtheintegralasextended overallspace.Thissavesthetroubleofwritinginspecial limitsforeachparticular case.Thefunction Vofitselfthen practically determines thelimitsowingtoitsvanishing iden ticallyatallpointsunoccupied bymatter. 87.]Theoperation offindingthepotential isofsuch frequent occurrence thataspecialsymbol,Pot,isusedforit. Thesymbolisread"thepotential ofV."Thepotential, Pot~isafunction notofthevariables xi''!II'%twith regardtowhichtheintegration isperformed butofthepoint (xt,'!It'%t)whichisfixedduringtheintegration. These variables enterintheexpression forTtl'Thefunction V andPotVtherefore havedifferent setsofvariables. Itmayhenecessary tonotethatalthough Vhashitherto beenregarded asthedensityofmatterinspace,suchan interpretation forVisentirelytoorestricted forconvenience. Whenever itbecomes necessary toformtheintegral (22)' ofanyscalarfunction V,nomatterwhatVrepresents, that integraliscalledthepotential ofV.Thereasonforcalling suchanintegral thepotential evenincasesinwhichithas noconnection withphysical potential. isthatitisformed according tothesameformallawasthetruepotential and 208 VECTOR ANALYSIS (23)byvirtueofthatformation hascertainsimplerulesofopera tionwhichothertypesofintegrals donotpossess. Pursuant tothisideathepotential ofavectorfunction W(x2'Y2'%2) maybewrittendown. ff.'fW(x2,Y2'%2)PotW=J rIll dX2dY2d%2' Inthiscasetheintegral isthesumofvectorquantities andisconsequently itselfavector. Thusthepotential ofa vectorfunction Wisavectorfunction, justasthepotential ofascalarfunctionVwasseentobeascalarfunction ofposi tionillspace.IfWberesolved intoitsthreecomponents W(x2'Y2'%2)= iX(X2,'#-2'%2)+jY(X2'Y2'~) +kZ(X2'Y2'%2) PotW=iPotX+jPotY+kPotZ.(24) Thepotential ofavectorfunction Wisequaltothevector_ sumofthepotentials ofitsthreecomponents.x; Y,Z. Thepotential ofascalarfunction Vexistsatapoint (Xl'Yl'%1')whenandonlywhentheintegral PotV=IfI:::dv2, takenoverallspaceconverges toadefinitevalue.If, forinstance, Vwereeverywhere constant inspacethein tegralwouldbecomegreaterandgreaterwithoutlimitas thelimitsofintegration wereextended fartherandfarther outintospace.Evidently therefore ifthepotential istoexist Vmustapproach zeroasitslimitasthepoint (~,Y2'~) recedesindefinitely. Afewimportant sufficient conditions fortheconvergence ofthepotential maybeobtained by transforming topolarcoordinates. Let orsimplyTHEINTEGRAL CALCULUS OFVECTORS 209.· x=rsin0cosif>, y=rsin0sinif>, %=rcos0, dv=r2sin0drdOdif>. Letthepoint(xl'Y1'%1)whichisfixedfortheintegration bechosenattheorigin. Then andtheintegralbecomes JJJ~: dv2=JJJ;r2sinOdrdOdif>,(22) PotV=J J J V r sin0drdOdif>. IfthefunctionVdecrease 80rapidlythattheproduct Vr8 remains finiteasl'increases indefinitely, thentheintegralcon 'fJergesasfarasthedistantregionsofspaceareconcerned. Forlet r=00 J J JVrsin0drdOdif> r=Rr=OO <JJf~ drdOdif> r=R r=oo fJf~dr r=RKdOd if>=2~-.R Hencethetripleintegral takenoverallspaceoutsideofa sphereofradiusR(whereRissupposed tobealargequan tity)islessthan271"2K/R,andconsequently converges asfar asregionsdistantfromtheoriginareconcerned. U 210 VECTOR ANALYSIS IftheJunctionVremainfiniteorifitbecomeinfiniU so weaklythattheproduct Vr remainsfinitewhenrapproaches uro,thentheintegral convergu asfarasregionsneartotheoriginareconcerned. Forlet Vr<K r=R J J JVrsin0drdOd e/> r=Or=R <J JJKdr dOde/>. r=O r=R J J J K d r dOd",=2'7r'KR. r=O Hencethetripleintegral takenoverallspaceinsideasphere ofradiusR(whereRisnowsupposed tobeasma.llquantity) islessthan2'Ii"K Randconsequently converges 8.8faras regionsneartotheoriginwhichisthepoint(Xl'YI'%1)are concerned. . Ifatanypoint(x2,y"z,)notcoincident withtheorigin, i.e.thepoint(xl'YI'ZI)'thefunctionVbecomesinfinite 80 weaklythattheproductofthevalue0/VatapointnearUJ (x"Y"z,)bythesquareofthedistanceofthatpointfrom (x2'Y2'Z2)remains finiteasthatdistance approaches uro,then theintegralconverges asfarasregionsneartothepoint(x"Y"z2) areconcerned. Theproofofthisstatement islikethosegiven before. Thesethreeconditions fortheconvergence ofthe integral PotVaresufficient. Theyarebynomeansneces sary.Theintegral mayconverge whentheydonothold. Itishowever indispensable toknowwhetherornotanintegral underdiscussion converges. Unlessthetestsgivenabove showtheconvergence, morestringent onesmustberesorted to.Such,however, willnotbediscussed here.Theybelong tothetheoryofintegration ingeneralratherthantothe FIG.36.THEINTEGRAL CALCULUS OFVECTORS 211 theoryoftheintegrating operator Pot.Thediscussion of theconvergence ofthepotential ofavectorfunction Wre ducesatoncetothatofitstkrucomponents whicharescalar functions andmaybetreatedasabove. 88.]Thepotential isafunction ofthevariables Xl'!h.zl whichareconstant withrespecttotheintegration. Letthe valueofthepetential atthepoint(Xl'!h,Zl)bedenotedby [PotVJz.:v"~. Thefirstpartialderivative ofthepotential withrespecttoXl istherefore dPotV=LI~S[PotV]%,+.\z..y"~-[PotV]""v,••,~(25) dX14X1=O( aX1 5 Thevalueofthislimitmaybedetermined byasimple device(Fig.36).Consider thepotential atthepoint (Xl+4Xl'Y1'zl) duetoacertainbodyT.This isthesameasthepotential at thepoint 1,,-,. (Xl'YlIzl) due tothesamebodyTdisplaced inthenegative direction by theamount4xl"Forinfindingthepotential atapointP duetoabodyTtheabsolute positions inspaceofthebody TandthepointPareimmaterial. Itisonlytheirpositions relativetoeackotherwhichdetermines thevalueofthepoten tial.Ifbothbodyandpointbetranslated bythesame amountinthesamedirection thevalueofthepotential isun changed. ButnowifTbedisplaced inthenegative direction bytheamount4x,thevalueofVateachpointofspaceis changed from V(x2'Y2'z2)toV(X2+4x2'Y2'Z2)' where4x2=axl" 212 HenceVECTOR ANALYSIS Hence Itwillbefoundconvenient tointroduce thelimitsof integration. Lettheportionofspaceoriginally filledbythe bodyTbedenoted byM;andlettheportionfilledbythe bodyafteritstranslation inthenegative direction through thedistance .6.x1bedenoted byM'.TheregionsMand..V' overlap. Lettheregioncommon tobothbeM,.andletthe remainder ofMbe171,,.theremainder ofM', 171,I .Then M=M+m, M'=M+171,'. Hence(25)becomes, when.6. Xlisreplaced byitsequal.6.x2' tAsallthefollowing potentials areforthepoint%1.Yl'%1thebracketand indiceshavebeendropped. THEINTEGRAL CALCULUS D.FVECTORS 218 =JJf_~ ~V(xi'Yi'Zi)dvi.tJiLT1i~Xi when~oX1approaches zeroasitslimittheregionsmandm', whichareatnopointthickerthan~x,approach zero;M' andMbothapproach Masalimit. tTherearecasesinwhichthisreversal oftheorderinwhichthetwolimits aretakengivesincorrect results. Thisisaquestiou ofdoublelimitsandleadsto themazesofmodernmathematical rigor. tIfthederivative ofVistoexi~tatthesurfacebounding Tthevaluesofthe functionVmuatdiminish continuously tozeronponthesurface.IfVchanged suddenly fromafinitevaluewithinthesurfacetoazerovalueoutsidethede rivative9V19Xlwouldnotexistandthetripleintegralwouldbemeaningless. ForthesamereasonVissupposed tobefiniteandcoutinuous ateTerypoint withintheregionT. 214 VECTOR ANALYSIS ‘Then ifitbeassumed that theregion isfinite andthat 7 vanishes upon thesurface bounding 7” Lo VegtBayypt5) anoSS.ere eaeia Line VeyYnts) =annoSS.“rgAz,2% Consequently theexpression forthederivative ofthepoten- tialreduces tomerely 3Pot 1a avSPot 1BPay,=Pot2%.(26: 32,SSL.FgDag02=PotgzOD ‘ThepartialderivativeofthepotentialofascalarfunctionV tsequal tothepotential ofthepartial derivative ofV. ThederivativeVofthepotentialofVisequaltothepotential ofthederivative VV. VPot V=Pot VV. @) *This statement follows immediately from the former. As theVupon the left-hand’ side applies tothe setofvari- ables 2,¥ys% itmay bewritten V,. Inlike manner the 'Vupon therighthand side may bewritten V,tocallatten- tion tothefactthat itapplies tothevariables zyyyy4ofV- ‘Then ¥V,PotV=PotVV ry Todemonstrate thisidentity Vmay beexpanded interms of ijkaPotY,dPotV aPotl 132, 139y, on, av av ov =iPot——+jPot<—+k Pot=—. iPod ‘otont Pots THEINTEGRAL CALCULUS OFVECTORS 215 Asi,j,kareconstant vectorstheymaybeplacedunder thesignofintegration andthetermsmaybecollected. Then bymeansof(26) 'V"lPotV=Pot'V"IIv: (29)(28)or orandThecuTl"VXanddiveTgence "V"oftMpotential ofavector functionWaTeequalTupectively tothepotential oftMC'UTland di1JeTgence ofthatfunction. "V1XPotW =Pot"VIIXW, curlPotW=PotcurlW "Vl"PotW=Pot"VII"W, divPotW=PotdivW. 216 VECTOR ANALYSiS andmayberemoved bymakinguseofasurfaceintegral. Thederivative ofthepotential wasobtained (page213)in essentially theform dPotVIff 1dVddXl= JIT12•X2VI Letdabeadirected element ofthesurfaceSbounding the regionM.Theelement ofvolumedV2intheregionm'is therefore equalto Hence Theelementofvolumedv70intheregionmisequalto Hence Consequently dPotV-ffr~C)Vdv+fJ~i.dL(34)aoX1 -JJIr12C)X22 8r12 Vr3<K.THEINTEGRAL CALCULUS OFVECTORS 217 Thevolumeintegralistakenthroughout theregionMwith theunderstanding thatthevalueofthederivative ofVat thesurfaceSshallbeequaltothelimitofthevalueofthat derivative whenthesurfaceisapproached fromtheinterior ofM.Thisconvention avoidsthedifficulty thatarisesin conn'ection withtheexistence ofthederivative atthesurface SwhereVbecomes discontinuous. Thesurfaceintegral is takenoverthesurfaceSwhichboundstheregion. Suppose thattheregionMbecomes infinite. Byvirtueof theconditions imposed uponVtoinsuretheconvergence of thepotential Letthebounding surfaceSbeasphereofradiusR,aquan titywhichislarge. i .da<r2d(Jdt/J. Thesurfaceintegral becomes smallerandsmallerandap proaches UTOasitslimitwhentheregionMbecomes infinite. Moreover thevolumeintegral remainsfiniteasMbecomes infinite. Consequently provided Vissuchafunction thatPotVexistsasfarastheinfinite regionsofspaceareconcerned, thentheequation (iPotV=Pot(iV (iXl (ix2 holdsasfarasthoseregionsofspaceareconcerned. Suppose thatVceasestobecontinuous orbecomes infinite atasinglepoint(Xl'YI'%1)withintheregionT.Surround 218 VECTOR ANALYSIS /thispointwithasmalltlphereofradiusR.LetSdenotethe surfaceofthissphereandMalltheregionTnotincluded withinthesphere. Then Bytheconditions imposeduponV Vr<K Consequently whenthesphereofradiusRbecomes smaller andsmallerthesurfaceintegralmayormaynotbecomezero. Moreover thevolumeintegral mayormaynotapproach alimitwhenRbecomes emaller andsmaller. Hencetheequation BPotV=PotBV Bxl Bx, hasnotalwaysadefinitemeaning atapointoftheregion TatwhichVbecomes infiniteinsuchamannerthatthe productVrremainsfinite. If,however, Vremainsfiniteatthepointinquestion so thattheproductVrapproaches zero,theconstant Kiszero andthesurfaceintegral becomes smallerandsmallerasR approaches zel'?'Moreover thevolumeintegral THEINTEGRAL CALCULUS OFVECTORS 219 approaches adefinitelimitasRbecomes infinitesimal. Con- sequently theequation "" dPotV;;V--=---=Pot- ~xl ~Xs holdsintheneighborhood ofallisolatedpointsatwhichV remainsfiniteeventhoughitbediscontinuous. Suppose thatVbecomes infiniteatsomesinglepoint (~,'!Is'~)notcoincident with(Xl''!II'%1)'According tothe conditions laiduponV VlS<K, wherelisthedistance ofthepoint(xs''!Is'~)fromapoint neartoit.Thenthesurfaceintegral neednotbecomezeroandconsequently theequation neednotholdforanypoint(Xl''!II'%1)oftheregion. But ifVbecomes infiniteatxS''!Is'%sinsuchamannerthat Vl<K, thenthesurfaceintegralwillapproach zeroasitslimitand theequation willhold. Finallysuppose thefunction Vremains finiteuponthe surfaceSbounding theregionT,butdoesnotvanishthere. Inthiscasethereexistsasurfaceofdiscontinuities of~ WithinthissurfaceVisfinite;without, itiszero.The surfaceintegral 220 VECTOR ANALYSIS doesnotvanishingeneral. Hencetheequation cannothold. Similarreasoning maybeappliedtoeachofthethree partialderivatives withretlpecttoxl'111,z1"Bycombining theresultsitisseenthatingeneral V"lPotV=PotV"2V+!' r~da.(35)J8r12 LetVbeanyfunction inspace,andletitbegrantedthat PotVexists.Surround eachpointofspaceatwhichV ceasestobefinitebyatlmallsphere. Letthesurfaceofthe spherebedenoted byS.Drawinspaceallthosesurfaces whicharesurfaces ofdiscontinuity ofV.Letthesesur facesalsobedenoted byS.Thentheformula(35)holds wherethesurfaceintegral istakenoverallthesurfaces whichhavebeendesignated byS.Iftheintegraltaken overallthesesurfaces vanishes whentheradiiofthespheres abovementioned becomeinfinitesimal, then (27)' Thisformula willsurelyholdatapoint(Xl'Yl'Zl)ifVremains always finiteorbecomesinfiniteatapoint(x2'Y2'Z2)80thattlu product V Iremains finite,andifVpossesses nosurfacesof discontinuity, andiffurthermore theproductVr8remai'TUIfiniu asrbecomesinfinite.lInothercasesspecialtestsmustbe appliedtoascertain whether theformula (27)Icanbeused or themorecomplicated one(35)mustberesortedto.. 1Forextensions andmodifications ofthistheorem, seeexera-. THEINTEGRAL CALCULUS OFVECTORS 221 Therelation (27)issosimpleandsoamenable totrans formation thatVwillingeneralbeassumed tobesucha function that(27)holds.IncasesinwhichVpossesses a surfaceSofdiscontinuity itisfrequently foundconvenient toconsider Vailreplaced byanotherfunction Vwhichhas ingeneralthesamevaluesasVbutwhichinsteadofpossess ingadiscontinuity atSmerelychanges veryrapidlyfrom onevaluetoanotherasthepoint(x2,Y2'z2)passel!fromone sideofStotheother.Suchadevicerendersthepotential ofVsimplertotreatanalytically andprobably conforms to actualphysical statesmorecloselythanthemoreexact conception ofasurfaceofdiscontinuity. Thisdeviceprac ticallyamounts toincluding thesurfaceintegral inthe symbolPot\lv: Infactfromthestandpoint ofpuremathematics itis bettertostatethatwherethereexistsurfaces atwhichthe functionVbecomes discontinuous, thefullvalueofPot\lV shouldalwaysbeunderstood asincluding thesurfaceintegral IfV-da 8T12 inaddition tothevolumeintegral InlikemannerPotV'.W,PotV'XW,NewV'.Wandother similarexpressions tobemetinthefuturemustberegarded asconsisting notonlyofavolumeintegral butofasurface integralinaddition, whenever thevectorfunction Wpossesses asurfaceofdiscontinuities. Itisprecisely thisconvention intheinterpretation of formulre whichpermitssuchsimpleformulre as(27)tohold ingeneral,andwhichgivestothetreatment oftheintegrat ingoperators anelegance oftreatment otherwise unobtainable. 222 VECTOR ANALYSIS Theirregularities whichmayarisearethrownintotheinter pretation, notintotheanalytic appearance oftheformulm. ThisistheessenceofProfessor Gibbs'smethodoftreatment. 90.]Thefirstpartialderivatives ofthepotential mayalso beobtained bydifferentiating underthesignofintegration.1 InlikemannerforavectorfunctionW Or and(37)' (38)' •dPotV.ClPotV 'L.dPotV VPotV==1S+J1'\+.."- Xl "'III "ZI fffli(X2-XI)Vj(Y2-?h)V+k(Z2-ZI)V!d 8+ 8 8 VI"r12 rIS rII Buti(x2-xl)+i(Y2-YI)+k(Z2-Z1)=r12. 1Ifanattemptweremadetoobtaint1le_dpartialderintives inthesame manner,itwouldbeseenthatthevolumeintegrals nolongerconverged. THEINTEGRAL CALCULUS OFVECTORS 223 JJJrlZvHence '\1PotV= rB11dvI_ Inlikemanner '\1xPotW=JJJ r19'r~lw dv'.l'(39) (40) (41) and '\1.PotW =JJJr1;:I:d111_ Thesethreeintegrals obtained fromthepotential bythe differentiating operators areofgreatimportance inmathe maticalphysics_ Eachhasitsowninterpretation. Conse quentlyalthough obtained sosimplyfromthepotential each isgivenaseparate name.Moreover inasmuch asthese integrals mayexistevenwhenthepotential isdivergent, theymustbeconsidered independent ofit.Theyareto belookeduponasthreenewintegrating operators defined eachuponitsownmeritsasthepotential wasdefined. Let,therefore, (42) (43) (44) IItkepotential exists,then '\1PotV=NewV '\1xPotW=LapW '\1-PotW=MaxW.(45) ThefirstiswrittenNewVandread"TheNewtonian ofV." 224 VECTOR ANALYSIS ThereasonforcallingthisintegraltheNewtonian isthatif Vrepresent thedensityofabodytheintegralgivestheforce ofattraction atthepoint(Xl'Y1'%1)duetothebody.This willbeprovedlater.ThesecondiswrittenLapWand read"theLaplacian ofW."Thisintegral wasusedtoa considerable extentbyLaplace.Itisoffrequent occurrence inelectricity andmagnetism. IfWrepresent thecurrent CinspacetheLaplacian ofCgivesthemagnetic forceatthe point(Xl'Y1'%1)duetothecurrent. Thethirdiswritten llfaxWandread..theMaxwellian ofW."Thisintegralwas usedbyMaxwell. It,too,occursfrequently inelectricity andmagnetism. Forinstance ifWrepresent theintensity ofmagnetization 1,theMaxwellian ofIgivesthemagnetic potential atthepoint(Xl'Y1'%1)duetothemagnetization. ToshowthattheNewtonian givestheforceofattraction according tothelawoftheinversesquareofthedistance. Letdm2beanyelement ofmasssituated atthepoint (x?'Y2'%2).Theforceat(Xl'Y1'%1)duetodmisequalto inmagnitude andhasthedirection ofthevectorrufromthe point(Xl'Y1'%1)tothepoint(x:a'Y?,22).Hencetheforceis Integrating overtheentirebody,oroverallspaceaccording totheconvention hereadopted, thetotalforceis whereVdenotesthedensityofmatter. THEINTEGRAL CALCULUS OFVECTORS 225 Theinregralmaybe~xpanded inrerIllSofi,j,k, NewV=iJII(x2~3l:l)VdV2+jfff (Y2~3~21)VdV2 '- +kffI(Z2~3;2l)Vdv2• Thethreecomponents maybeexpressed intermsofthepo tential(ifitexists)as ItisinthisformthattheNewtonian isgenerally foundin books. ToshowthattheLaplacian givesthemagnetic forceper unitpositivepoleatthepoint(Xl'YI'zl)duetoadistribution W(x2,Y2'Z2)ofelectricflux.Themagnetic forceat(Xl'Yl'Zl) duetoanelement ofcurrent dC2isequalinmagnitude to themagnitude dO2ofthatelementofcurrentdividedbythe squareofthedistance ra;thatis dO'l,. r\:l Thedirection oftheforceisperpendicular bothtothevector elementofcurrentdC2andtothelinerajoiningthepoints. Thedirection oftheforceistherefore thedirection ofthe vectorproductofr12anddC2•Theforceistherefore raXdC2 r3u 15 226 VECTOR .ANALYSIS Integrating overallspace,thetotalmagnetic forceactingat thepoint(Xl''111'%1)uponaunitpositivepoleis Thisintegralmaybeexpanded intermsofi,j,k.Let W(X2''112'%2)=iX(X2''112'%2)+jY(Xli''112';) +kZ(X2,'112';). r12=(Xli-Xl)i+(Y,I-111)j+(%,1-%1)k. Thei,j,kcomponents ofLapWarerespectively . L W-fff ('112-111)Z-(%2-%1)Y d1·ap- r8 "'I 12 (48)' JLW-Jff(%2-%I)X-(X2-Xl)Zd•ap- 8' 'VIr12 t.LapW=fJf(x2-Xl)Yr~2(Y2 -'111)Xd'Vl Intennsofthepotential (ifoneexists)thismaybewritten .LaW(iPotz·(iPotY 1 •P= - ----..--(iYl (i%l • L W ~PotXJ.ap=" 0'Zl(48)" k•LapW=~PotY~PotX. ~xl ~Yl ToshowthatifIbetheintensity ofmagnetization atthe point(X2'1/2'%2)'thatis,ifIbeavectorwhosemagnitude is equaltothemagnetic moment perunitvolumeandwhose THEINTEGRAL CALCULUS OFVECTORS 227 direction isthedirection ofmagnetization oftheelementd11, fromsouthpoletonorthpole,thentheMaxwellian ofIisthe magnetic potential duetothedistribution ofmagnetization. Themagnetic momentoftheelementofvolume,d'VIIisIdvll• Thepotential at(Xl'Yl'zl)duetothiselementisequaltoits magnetic moment dividedbythesquareofthedistance rIll andmultiplied bythecosineoftheangle between thedirec tionofmagnetization Iandthevector rIll'Thepotential is therefore ~l~Idv,. r3 I11 Integrating, thetotalmagnetic potential isseentobe Thisintegralmayalsobewrittenoutintermsofx,Y,z. Let I(x\\'YII'%11)=iA(x,.Y\\,%11)+jB(XII'YII'ZII)+kC(XII'Y\\,%11) rIll·I=(XII-xl).A+(YII-Yl)B+(ZII-%1)C. Ifinsteadofxl'Yl'%1thevariables x,y,z;andinsteadof XII'YIl'ZIIthevariablesf,'I.~beusedItheexpression takes oqtheformgivenbyMaxwell. MaxI=ffftA(f-x)+B(71-Y)+C(~-z>l:3d'/}. According tothenotation employed fortheLaplacian 1Maxwell: Electricity andMagnetism, Vol.II.p.9. 228 VECTOR ANALYSIS TheMaxwellian ofavectorfunction isascalarquantity. Itmaybewrittenintermsofthepotential (ifitexists)as ~PotXaPotY~PotZ MaxW = ~+~+ (44)"xl YlaZl Thisformofexpression ismuchusedinordinary treatises uponmathematical physics. TheNewtonian, Laplacian, andMaxwellian, however, should notbeassociated indissolubly withtheparticular physical interpretations giventothemabove.Theyshouldbelooked uponasintegrating operators whichmaybeapplied, asthe potential is,toanyfunctions ofpositioninspace.TheNew tonianisappliedtoascalarfunction andyieldsavector function. TheLaplacian isapplied toavectorfunction andyieldsafunction ofthesamesort.TheMaxwellian isappliedtoavectorfunction andyieldsascalar funct~n. Moreover, theseintegrals shouldnothelookeduponasthe derivatives ofthepotential. Ifthepotential existsthey areitsderivatives. Buttheyfrequently existwhenthe potential failstoconverge. 9L]LetVandWbesuchfunctions thattheirpotentials existandhaveingeneraldefinitevalues. Thenby(27)and (29) V.VPotV=V.PotVV=PotV.VV. Butby(45) and HenceVPotV= NewV; V.PotVV=MaxVv: V·VPotV=V.NewV= MaxVV' =PotV.VV (46) By(27)and(29)VV.PotW=VPot V.W=PotVV.W. Butby(45) V.PotW =MaxW, andby(45) VPotV.W =NewV.W. HenceTHEINTEGRAL CALCULUS OFVECTORS 229 'l'l,PotW='lMaxW=New\lW =Pot'l'l.W (47) By(28)'lX'lxPotW='lxPot'lxW =Pot'lx'lxW. Butby(45) 'lxPotW=LapW, and 'lxPot\lXW=Lap'lxW. Hence'lx'lxPotW='lxLapW=Lap'lXW =Pot'lx'lxW. (48) By(56),Chap.III.'l•'lxPotW=0, or 'l•Pot'lxW=O. Hence 'l.LapW=Max'lxW=O. (49) Andby(52),Chap.III.'lx'lPotV=0, or 'lxPot\lV=O. Hence 'lxNewV= Lap'lV=O. (50) Andby(58),Chap.III.'lx'lxW='l'l.W-'l.'lW, 'l•'lW='l'l•W -'lx'lxW. Hence or'l•'lW=Nf;W'l•W -Lap'lxW, 'l•VW ='lMaxW -'lxLapW.(51) Theseformulre maybewrittenoutintermsofcurland divifdesired. Thus divNewV= Max'lV, (46)' 'lMaxW=NewdivW (47)' curlLapW=LapcurlW (48)' divLapW=MaxcurlW=0 (49)' curlNewV=Lap'lV=0 (50)' 'l.'lW=NewdivW-LapcurlW.(51)' 230 VECTOR ANALYSIS PoisMm's Equation 92.JLetVbeanyfunction inspacesuchthatthepotential PotV hasingeneraladefinitevalue.Then '1.'1PotV= -4'1rV; (52) or ButThisequation isknownasPoisson's Equation. Theintegralwhichhasbeendefinedasthepotential isa solutionofPoisson's Equation. Theproofisasfollows. PotV=IIIr:dv2• Thesubscripts 1and1!havebeenattached todesignate cl,earlywhatarevariables withrespecttowhichthedifferen tiationsareperformed. '11•'\71PotV=V'1·Newv-III'llr~2•V'2Vdv2• 1 1V'1-=-'\72- r12 r12 THEINTEGRAL CALCULUS OFVECTORS 231 Hence-'V2-.!.- 0'V2V=V\720'V2..!..--'V20(V'VI~) r~ r~ r~ Integrate: III'Vlr~2 0'V2Vdv2=IIIV'V20'V2r~2 d"l But+III'V20 (V'VIr~2)dv2° 1'\72o'V2-=0. r12 HenceThatistosay.!..satisfiesLaplace's Equation. Andby(8)r III'V20(v'V10r~2)dV2~IiV\71r~2 0dL 'VI0'VIPotV=III'VIr~2 0'V2Vdv2(53) Thesurfaceintegralistakenoverthesurfacewhichbounds theregionofintegration ofthevolumeintegral. Thisis taken"overallspace." Hencethesurfaceintegral mustbe takenoverasphereofradiusR,alargequantity, andRmust beallowedtoincreasewithoutlimitAtthepoint(XI'YI'%1)' however, theintegrand ofthesurfaceintegral becomes in finiteowingtothepresence oftheterm 1'V 1_ 0 ra 232 VECTOR ANALYSIS HencethesurfaceSmustincludenotonlythesurfaceofthe sphereofradiusR,butalsothesurfaceofasphereofradius R',asmallquantity, surrounding thepoint(xl'9'1'zl)andRI mustbeallowed toapproach zeroasitslimit. Asithasbeenassumed thatthepotential ofVexists,itis assumed thattheconditions given(Art.87)fortheeXUltence ofthepotential hold.Thatis V1'3<K,whenl'islarge Vl'<K,whenl'issmall. Introduce (xl'9'1'zl)· andpolarcoordinates withtheoriginatthepoint Then 1'12becomes simplyr 1 1 rv\-=-V'II-=a· rill ·rIIIl' ThenforthelargesphereofradiusR V'1~•da=-;I'llsineded4J. 1'12 r Hencethesurfaceintegral overthatsphereapproaches zero asitslimit.For HencewhenRbecomes infinitethesurfaceintegral overthe largesphereapproaches zeroasitslimit. Forthesmallsphere Hencetheintegral overthatspherebecomes -ffVsineded4J. THEINTEGRA.L CALCULUS OFVECTORS 233 LetVbesupposed tobefiniteandcontinuous atthepoint (xl'~h'%1)whichhasbeenselected asorigin. Thenforthe surfaceintegral Vispractically constant andequaltoits value atthepointinquestion. JJSinOdOdep=47r. Hence - J J V sin0dOdep==-47rV whentheradiusR'ofthesphereofintegration approaches zeroasitslimit.Hence and "V•"VPotV= -47rV. (52) InlikemannerifWisavectorfunction whichhasin generaladefinitepotential, thenthatpotential satisfiesPois son'sEquation. "V•"VPotW= -47rW. (52)' Theproofofthisconsistsinresolving Wintoitsthreecom ponents. Foreachcomponent theequation holds.Let W=Xi+Yj+Zk, 'Sl•"VPotX= -47rX, "V•"VPotY= -47r~ "V•"VPotZ=-47rZ. Consequently "V."VPot(Xi+Yj+Zk)= -47r(Xi+Yj+Zt). 234 VECTOR ANALYSIS Theorem: IfVandWare8uchfunctions ofpositionin8paa thattheirpotential, existingeneral,thenforallpointsatwhick VandWarefiniteandcontinuous thosepotentials satisfy P0i8IJ()7/," Equation, 'il•'ilPotV= -4'7r~ (52) 'il•'ilPotW= -4'7rW. '(52)' (53) HenceThemodifications inthistheorem whicharetobemadeat pointsatwhichVandWbecome discontinU0U8 willnotbe takenuphere. 93.]Itwaaseen(46)Art.91that 'il.'ilPotV='il.NewV=Max'ilY. 'il•NewV= -4'7rV or Max'ilV= -4'7rV. Inasimilarmanneritwasseen(51)Art91that 'il•'ilPotW ='ilMaxW -'ilxLapW =New'il.W -Lap'ilxW. Hence or'ilMax:W -'ilxLapW= -4'7rW, New'il.W -Lap'ilxW= -4'7rW.(54) (54)' Byvirtueofthisequality Wisdividedintotwoparts. 1 1W=4'7rLap'ilx W -4'7rNew'il.W.(55) Let W=W1+W" where1 1 W1=4'7rLap'ilx W=4'7rLapcurlW(56) andW,= -41 '7rNew'il.W=-41 '7rNewdivW.(57) THEINTEGRAL CALCULUS OFVECTORS 235 _Equation (55)statesthatanyvectorfunction Wmultiplibd by4.".i8equaltothedifference oftheLaplacian ofitscurl andtheNewtonian ofitsdivergence. Furthermore 1 1 'V•WI=4.".'V.Lap'VxW=4.".'V.'VxLapWI' Butthedivergence ofthecurlofavectorfunction iszero. Hence 'V•WI=divWI=0 (58) 1 1 'VXW2=-47r'VXNew'V•W2=-4.".'VX'VMaxW2' Butthecurlofthederivative ofascalarfunction iszero. Hence 'VXW2=curlW2=O. (59) LetConsequently anyvectorfunction Wwhichhasapotential maybedividedintotwopartsofwhichonehasnodivergence andofwhichtheotherhasnocurl.ThisdivisionofWinto twosuchpartsisunique. Incaseavectorfunction hasnopotential butbothitscurl anddivergence possesspotentials, thevectorfunction maybe dividedintothreepartsofwhichthefirsthasnodivergence; thesecond,nocurl;thethird,neitherdivergence norcurl. 1 1 W=47rLap'VXW-4.".New'V.W+WS'(55)' Asbefore and1 1 ' 47r'V.Lap'VXW=4.".'V.'VXPot'VXW=0 -1 -1 4.".'VXNew'V.W=4.".'VX'VPot'V.W=O. Thedivergence ofthefirstpartandthecurlofthesecond partofWaretherefore zero. 236 forVECTOR ANALYSIS 1 1-'VxLap'VxW=4-'Vx'VxPot'VxW 4'71'" '71'" 1 1= -'V'V•Pot'VXW--4'V•'VPot'VxW.4'71'" . '71'" 1 1-'V'V•Pot'VxW=- 'VPot'V.'VxW=O, 4'71'" 4'71'" 'V.'VXW=O. Hence Hence-1-'V.'VPot'VxW='VXW. 4'71'" 1 4'71'"'VxLap'VxW='VxW='VXWI' Thecurlof'Viisequaltothecurlofthefirstpart 1 4'71'"Lap\1XW intowhichWisdivided. Henceasthesecondparthasno curl,thethirdpartcanhavenone.Moreover 1--\lNew\l.W='V.W=\l· WI'4'71'" Thusthedivergence ofWisequaltothedivergence of thesecondpart -1 4'71'"New\l.W. intowhichWisdivided. Henceasthefirstparthasno divergence thethirdcanhavenone.Consequently thethird partW8hasneithercurlnordivergence. Thisprovesthe statement. BymeansofArt.96itmaybeseenthatanyfunction Ws whichpossesses neithercurlnordivergence, musteither (61)THEINTEGRAL CALCULUS OFVECTORS 237 vanishthroughout allspaceormustnotbecomezeroat infinity. Inphysicsfunctions generally vanishatinfinity. Hencefunctiollil whichrepresent actualphenomena maybe dividedintotw~parta,ofwhichonehasnodivergence and theothernocurl. 94.]lJefinition: Avectorfunction thedivergence ofwhich vanishes ateverypointofspaceissaidtobesoleMidal. A vectorfunction thecurlofwhichvanishes ateverypointof spaceissaidtobeirrotational. Ingeneralavectorfunction isneithersolenoidal norirrota tional.Butithasbeenshownthatanyvectorfunction which possesses apotential maybedividedinoneandonlyone wayintotwopartBWI'W2ofwhichoneissolenoidal and theotherirrotational. Thefollowing theorems maybestated. ~have allbeenprovedintheforegoing sections. Withrespect.to asolenoidal function WI'theoperators 1 4'1f'Lapand"VXorcurl areinverseoperators. Thatis 1 1 4'1f'Lap"VXWI="VX4'1f'LapWI=WI' (60) Applied toanirrotational function W2eitheroftheseopera torsgivesuro.Thatis 1-LapW2=0,"VXW2=O.4'1f' Withrespecttoanirrotational function W2'theoperators -.!.-Newand-"V.or-div4'1f' areinverseoperators. Thatis 1 1 -4'1f'New"V.W2=-"V.4'1f'NewW2=W2•(62) 238 VECTOR ANALYSIS Withrespecttoascalarjunction Vtkeoperators , 1-"V.or-diVand47rNew, andalso areinverseoperators.1- -Maxand"V4'7T Thatis a.nd1-"V•-NewV=V4'7T 1--Max"VV=V.4'7T(63) Withrespecttoasolenoidal junction Witkeoperators 147rPotand"VX"VXorcurlcurl areinverseoperators, Thatis 1 1 4'7TPot"VX"VXWi="VX"VX47rPotWi=Wi'(64) Withrespecttoanirrotational junction W2theoperators 1-Potand-"V"V•4'7T areinverseoperators. Thatis 1 1 -4'7TPot"V"V•W2= -"V"V•4'7TPotW2=W2'(65) Withrespecttoanyscalarorvectorjunction V,Wthe operators 1-Potand-"V•"V47r areinverseoperators. Thatis THEINTEGRAL CALCULUS OFVECTORS 239 1 1- -PotV0VV= -V0V-PotV=V 4~ 4~ and1 1- -PotV0VW = -V0V-Potw=W.(66) 4~ 4~ (68) 1--LapLap 4~Withrespecttoasolenoidal functionWithedifferentiating operators ofthesecondorder -V0Vand\lxVx areequivalent -VoVW1=Vx\lxWl' (67) Withrespecttoanirrotational function W2thedifferentiat ingoperators ofthesecondorder \l0ValidVV0 areequivalent. Thatis V0VW:1=VV' W3' Byintegrating theequations 4~V=-V0NewV and 4~W =\lxLapW -VMaxW bymeansofthepotential integral Pot 4~PotV=-PotV0NewV=-MaxNewV(69) 4~PotW =PotVxLapW -PotVMaxW 4~PotW=LapLapW -NewMaxW.(70) Henceforscalarfunctions andirrotational vectorJunctions 1- -NewMax 4~ isanoperator whichisequivalent toPot.Forsolenoidal vect01' functions theoperator 240 VECTOR ANALYSIS Oneoftheintegrations maybeperformedgivesthepotential. FOTanyvectorfunction thefirstoperator givesthepotential oftheirrotational partjthesecond,the potential ofthesolenoidal part. -95.]Thereareanumberofdoublevolumeintegrals which areofsuchfrequent occurrence inmathematical physicsas tomeritapassingmention, although thetheoryofthemwill notbedeveloped toanyconsiderable extent. Thesedouble integrals areallscalarquantities. Theyarenotscalarfunc tionsofposition inspace.Theyhavebutasinglevalue. Theintegrations intheexpressions maybeconsidered for convenience asextended overallspace.Thefunctions by vanishing identically outsideofcertainfinitelimitsdeter mineforallpractical purposes thelimitsofintegration in casetheyarefinite. Giventwoscalarfunctions U,Vofposition inspace. Themutualpotential orpotential product, asitmaybecalled, ofthetwofunctions isthesextuple integml Pot(U,V)=ff f Jf JU(Xl'Yl,ZI;I~ (x2'Y"~)dV1dv2· (71) Pot(u,V)=JffU(x1,Yl'%1)PotVdV1 =fJfvex"~y"%,)PotU dv,. (72) Inasimilarmannerthemutualpotential orpotential product oftwovectorfunctions W',W"is Pot(W',W")=fJffffW'(Xl'Yl'~;.W"(X"y,,%,) dV1dvs' I' (71)' Thisisalsoascalarquantity. OnE"integration maybecar riedout Lap(W',W")THEINTEGRAL CALCULUS OFVECTORS 241 Pot(W',W").....;JJJW'(xl'Yl'%l>•PotW"dV1 Themutual Laplacian orLaplacian product oftwo vectorfunctions W',W"ofposition inspaceisthesextuple integral Oneintegration maybeperformed. Lap(W',W")=JJJW"(x2,'!I2'%,)•LapW'dv, (74) .....;JJJW'(xl'Yl'%1)•LapW"dvl· TheNewtonian productofascalarfunction V,andavector function Wofpositioninspaceisthesextuple integral Byperforming oneintegration InlikemannertheMaxwellian productofavectorfunction Wand ascalarfunction Vofposition inspaceisthe integral Max(W,V)=JJJJJJV(xl'Yl'%l) ~;l~.W(x,,'!1,,~)dvldv,. (77) 16 242 VECTOR ANALYSiS Oneintegration yields Max(W,V)=J JJV(Xl'Yl'~)MaxWdvl=-New(v,W). (78) By(53)Art.93. 47rUPotV= -(\1•NewU)PotV. \1.[NewUPotV]=('\1•NewU)PotV+(NewU)•'\1PotV. -('\1.NewU)PotV=- '\1.[NewUPotV]+ NewU.NewY'. Integrate : 47rJ J JUPotV dv= -J JJ'\1•[NewUPotV]dv +J J J NewU•NewV dv. 47rPot(U,V)=J JJNewU. NewVd.., - JfaPotVNewU. da. (79) Thesurfaceintegral istobetakenovertheentiresurfaceS bounding theregionofintegration ofthevolumeintegral. Asthisregionofintegration is..allspace,"thesurfaceSmay belookeduponasthesurfaceofalargesphereofradiusR. Ifthefunctions UandVvanishidentically forallpointsout sideofcertainfinitelimits,thesurfaceintegral mustvanish. Hence 47rPot(U,V)=J JJNewu. NewVdv. (79)' By(54)Art.93, 47rW".PotW'='\1xLapW".PotW' -'\1MaxW"•PotW'. THEINTEGRAL CALCULUS OFVECTORS 243 But\l 0[LapWI!XPotW']=PotW' 0\lXLapWI! -LapWI! 0\lXPotW', and\l0[MaxWI!PotW']=PotW' 0\lMaxWI! +MaxW""il0PotW'. HenceVXLapWI! 0PotW'=\l0[LapW"XPotW'] +LapW" 0LapW', and\lMaxW" 0PotW'=\l0[MaxWI!PotW'] -MaxWI!MaxW'. Hencesubstituting: 47TW" 0PotW'=LapW' 0LapW'+MaxW'MaxW" +\l0[LapWI!XPotW'] -\l 0[MaxWI!PotW']. Integrating: 47TPot(W', WI!)=JJJLapW'oLapW"dv +JJJMaxW' MaxW"dv(80) -JJaPotW'XLapWl!da-JJaMaxW"PotW'oda. IfnowW'andW"existonlyinfinitespacethesesurface integrals takenoveralargesphereofradiusRmustvanish andthcn 47TPot(W',WI!)=JJJLapW' 0LapWI!dv +JJJMaxW'MaxWI!dv.(80)' •96.]Thereareanumberofusefultheorems ofafunction theoretic nature whichmayperhapsbementioned hereowing 244 VECTOR ANALYSIS totheirintimate connection withtheintegral calculus of vectors. Theproofsofthemwillinsomeinstances begiven andinsomenot.Thetheorems areoftenusefulinpractical applications ofvectoranalysistophysicsaswellasinpurely mathematical work. Theorem:IfV(x,y,z)beascalarfunction ofposition inspacewhichpossesses ingeneraladefinitederivative "VV andifinanyportionofspace,finiteorinfinitebutnecessarily continuous, thatderivative vanishes, thenthefunction Vis constant throughout thatportionofspace. Given Toshow"VV=O. V=const. Chooseafixedpoint(Xl'Yl'zl)intheregion. By(2)page 180 But HenceJ"VV.dr=J0 •dr=O. Theorem:IfV(x,y,z)beascalarfunction ofposition inspacewhichpossesses ingeneraladefinitederivative "VV; ifthedivergence ofthatderivative exists and iszerothrough outanyregionofspace,!finiteorinfinitebutnecessarily continuous; andiffurthermore thederivative "VVvanishes ateverypointofanyfinitevolumeorofanyfiniteportionof surfaceinthatregionorbounding it,thenthederivative vanishes throughout allthatregionandthefunction Vre ducestoaconstant bythepreceding theorem. 1Thetermt},rollgT,ollt anyregionofspacemORtberegarded asincloding the boundaries oftheregionas.wellastheregionitself. THEINTEGRAL CALCULUS OFVECTORS 245 Given and Toshow'l•'lV=0foraregionT, 'lV=0forafiniteportionofsurfaceS. V=const. Since'lVvanishes fortheportionofsurfaceS,Viscertainly constant inS.Suppose that,upononesideofSandinthe regionT.Vwerenotconstant. Thederivative 'lVupon thissideofShasinthemainthedirection ofthenormalto thesurfaceS.ConMider aspherewhichliesforthemost partupontheoutersideofSbutwhichprojects alittle through thesurfaceS.Thesurfaceintegral of'lVover thesmallportionofthespherewhichprojects through the surfaceScannotbezero.For,as'lVisinthemainnormal toS,itmustbenearlyparallel tothenormaltothepoltion ofspherical surfaceunderconsideration. Hencetheterms 'lV.da inthesurfaceintegral allhavethesamesignandcannot canceleachotherout.Thesurface integral of'lVover thatportionofSwhichisintercepted bythespherical sur facevanishes because'lViszero.Consequently thesurface integralof'lVtakenovertheentiresurfaceofthespherical segment whichprojectsthrough Sisnotzero. But Hencef f'lV.da=f ff'l•'lVdv=o. ff'lV•da=O. Ittherefore appears thatthesupposition thatVisnot constant upononesideofSleadstoresultswhichcontradict thegiven relation 'l•'lV=O.Thesupposition mustthere forehavebeenincorrect andVmustbeconstant notonlyin 8butinallportions ofspaceneartoSintheregionT.By 246 VECTOR ANALYSIS anextension ofthereasoning Visseentobeconstant throughout theentireregionT. Theorem:IfV(x,y,z)beascalarfunction ofpositionin spacepossessing ingeneraladerivative \1Vandifthrough outacertainregion 1Tofspace,finiteorinfinite,continuous ordiscontinuous, thedivergence \1.\1Vofthatderivative existBandiszero,andiffurthermore thefunction Vpossesses aconstant valuecinallthesurfaces bounding theregion andV(x,y,z)approaches casalimitwhenthepoint(x,y,z) recedestoinfinity,thenthroughout theentireregionTthe function Vhasthesameconstant valuecandthederivative "ilVvanishes. Theproofdoesnotdifferessentially fromtheonegiven inthecaseofthelasttheorem. Thetheorem maybegen eralizedasfollows: Theorem:IfV(x,y,z)beanyscalarfunction ofposition inspacepossessing ingeneraladerivative "ilV,.ifU(x,y,z) beanyotherscalarfunction ofposition whichiseitherposi tiveornegative throughout andupOntheboundaries ofa regionT,finiteorinfinite, continuous ordiscontinuous; if thedivergence "il.[U"ilVJoftheproduct ofUand"ilV existsandiszerothroug?out andupon the boundaries ofT andatinfinity; andiffurthermore Vbeconsta.nt andequal tocuponalltheboundaries ofTandatinfinity; thenthe function Visconstant throughout theentireregionTand isequaltoc. Theorem:IfV(x,y,z)beanyscalarfunction ofposition inspacepossessing ingeneraladerivative "ilV,.ifthrough outanyregionTofspace,finiteorinfinite, continuous or discontinuous, thedivergence "il'."ilVofthisderivative exists andiszero;andifinallthebounding surfaces oftheregion Tthenormalcomponent ofthederivative "ilVvanishes and atinfinitedistances inT(ifsuchtherebe)theproduct 1Theregioninclndesitsbonndaries. THEINTEGRAL CALCULUS OFVECTORS 247 r2~V/~rvanishes, whererdenotes thedistance measured fromanyfixedorigin;thenthroughout theentireregionT thederivative '\7Vvanishes andineachcontinuous portion ofTVisconstant, although fordifferent continuous portions thisconstant maynotbethesame. Thistheorem maybegeneralized asthepreceding one wasbythesubstitution oftherelation\1.(U\1V)=0for \1•'\7V=0andUr2~V/~r=0forr2C)V/C)r=O. Ascorollaries oftheforegoing theorems thefollowing statements maybemade.Thelanguage isnotsoprecise asinthetheorems themselves, butwillperhapsbeunder stoodwhentheyareborneinmind. If'\7U='\7V;thenUandVdifferatmostbya constant. If'\7.\1U=\1•\1Vandif'\7U='\7Vinanyfinite portionofsurfaceS,then'\7U='\7VatallpointsandU differsfromVonlybyaconstant atmost. If\1.\1U='\7•'\7VandifU=Vinallthebounding surfaces oftheregionandatinfinity(iftheregionextend thereto), thenatallpointsUandVarcequal. If\1.\1U='\7•'\7Vandifinallthebounding surfaces oftheregionthenormalcomponents of'\7Uand'\7Vare equalandifatinfinitedistances r2(~U/C)r-~ V/C)r)is zero,then\1Uand\1Vareequalatallpointsoftheregion andUdiffersfromVonlybyaconstant. Theorem:IfW'andW"aretwovectorfunctions ofposition inspacewhichingeneral possesscurlsanddivergences; if foranyregionT,finiteorinfinitebutnecessarily continuous, thecurlofW'isequaltothecurlofW"andthedivergence ofW'isequaltothedivergence ofW";andifmoreover thetwofunctions W'andW"areequaltoeachotherat everypointofanyfinitevolumeinTorofanyfinitesurface inTorbounding it;thenW'isequaltoW"ateverypoint oftheregionT. 248 VECTOR ANA.LYSIS Since"i1XW'="i1xW","i1X(W'-W")=o.Avec torfunction whosecurlvaIllilhes isequaltothederivative 1 ofascalarfunction V(page197).Let"i1V=W'-W". Then"i1•"i1V=0owingtotheequality ofthedivergences. Thetheorem therefore becomes acorollary ofapreceding one. Theorem:IfW'andW"aretwovectorfunctions ofposi tionwhichingeneralpossessdefinitecurlsanddivergences ; ifthroughout anyaperiphractic 2regionT,finitebutnot necessarily continuous, thecurlofW'isequaltothecurlof W"andthedivergence ofW'isequaltothedivergence of W";andiffurthermore inallthebounding surfaces ofthe regionTthetangential components W'andW"areequal; thenW'isequaltoW"throughout theaperiphractic regionT. Theorem:IfW'andW"aretwovectorfunctions ofposi tioninspacewhichingeneral possess definite curlsand divergences; ifthroughout anyacyclicregionT,finitebutnot necessarily continuoUl:~, thecurlofW'IIIequaltothecurl W"andthedivergence ofW'isequaltothedivergence of W";andifinallthebounding surfaces ofthereg-ionTthe normalcomponents ofW'andW"areequal;thenthefunc tionsW'andW"areequalthroughout theregionacyclicT. Theproofsofthesetwotheorems arecarriedoutbymeans ofthedevicesuggested before. Theorem:IfW'andW"aretwovectorfunctions such that"i1•"i1WIand"i1•"i1W"haveingeneraldefinitevalues inacertainregionT,finiteorinfinite,continuous ordiscon tinuous; andifinallthebounding surfaces oftheregion andatinfinitythefunctions W'andW"areequal;thenW' isequaltoW"throughout theentireregionT. Theproofisgivenbytreating separately thethreecom ponentsofW'andW". 1TheregionTmayhavetobemadeacyclicbytheinl!ertioD ofdiaphragms. tAregionwhichencloses withinitllelfanotherregioni.aaaidtobeperiphrac tic.Hitencloses noregioniti.aaperiphractic. THEINTEGRAL CALCULUS OFVECTORS 249 SUMMARY OFCHAPTER IV Thelineintegralofavectorfunction WalongacurveCis definedas Thelineintegral ofthederivative "i1Vofascalarfunction ValongacurveCfromrotorisequaltothedifference between thevaluesofVatthepointsrandroandhencethe lineintegral takenaroundaclosedcurveiszero;andcon verselyifthelineintegral ofavectorfunction Wtaken· aroundanyclosedcurvevanishes, thenWisthederivative "i1Vofsomescalarfunction V. andifJoW•dr=0,thenW="i1v: Illustration ofthetheorem byapplication tomechanics. Thesurfaceintegralofavectorfunction Woverasurface Sisdefinedas JJaW.da-JJ/W 1dydz+W2dzdx+Wadxdy]. Gauss'sTheorem: Thesurfaceintegral ofavectorfunc tiontakenoveraclosedsurface isequaltothevolume integral ofthedivergence ofthatfunction takenthroughout thevolumeenclosed bythatsurface 250 orVECTOR ANALYSIS =ffs[Xdydz+Ydzdx+Zdxdy], (8) ifX,Y,Zbethethreecomponents ofthevectorfunction W. Stokes's Theorem: Thesurfaceintegral ofthecurlofa vectorfunction takenoveranysurfaceisequaltotheline integral ofthefunction taken around thelinebounding the surface. Andconversely ifthesurfaceintegral ofavector function Utakenoveranysurfaceisequaltothelineintegral ofafunction Wtakenaroundtheboundary, thenUisthe curlofW. ffs'Vx W 0da=foWodr, (11) andifffsU 0da=1:W 0dr,thenU="VxW.(12) Application ofthetheorem ofStokestodeducing the equationsoftheelectro-magnetic fieldfromtwoexperimental factsduetoFaraday. Application ofthetheorems ofStokes andGausstotheproofthatthedivergence ofthecurlof avectorfunction iszeroandthecurlofthederivative of ascalarfunction iszero. Formulre analogous tointegration byparts f1t"Vvodr=[uv(- Jv"Vuodr, (14) ffs"Vuxv'da=!ouv0dr -JJsu"Vxv0da,(15) ffs"Vux"Vvoda=!cu"Vvodr=-!ov"Vuodr, (16) THEINTEGRAL CALCULUS OFVECTORS 251 fJJU\7oVdv=J Jauvoda-JJJ'V1toVdv,(17) JJs\7uxv0da= -ffJ\7u0\7xvdv.(18) Green'sTheorem: - IJJJ\7uo\7vdv=JJsu\7voda-JJJu\7o\7vdv -JJ/\7u0da-JJJv\70\7itdv,(19) JJJ(u\7o\7v-v\7o\7u)dv=JJ/u\7v-v\7u) 0da.(20) Kelvin's generalization: JJJw\7uo\7vdv=JLuw\7voda-JfJu 0\7[w\7v]dv -JJsvw\7uoda-JJJ'O\7o[W\7U]dl" (21) Theintegrating operator knownasthepotential isdefined b.ytheequation PotV=JJJ V(X2~~2'Z2)dX2dY2dz2"(22) PotW=JJJW(X~1~2'Z2)dx,,2dY2dz2·(23) \7PotV=Pot'VV; (27) \7xPotW=Pot\7xW, (28) \7•PotW=Pot\7•W, (29) \70\7PotV=Pot\70\7~ (30) 252 VECTOR ANALYSIS v.VPotW=PotV.VW, (81) VV•PotW=Pot'V'V•W, (82) VxVxPotW=Pot'lx'lxW. (33) Theintegrating operatorPotandthedifferentiating operator 'larecommutative. Thethreeadditional integrating operators knownasthe Newtonian, theLaplacian, andtheMaxwellian. NewV=JJJfaV~:~Y2'~)dx'JdY2d~. (42) LapW -JJJf12XWjl:2'Y2'~)dX2dY2d~.(43) MaxW=JJJf12•W~::'Y'J'%2)dX2dy'Jd%2.(44) Ifthepotential existstheseintegrals arerelatedtoitasfol lows: VPotV=NewV, 'lxPotW=LapW, 'l•PotW=MaxW.(45) Theinterpretation ofthephysicalmeaning oftheNewtonian ontheassumption thatVisthedensityofanattracting body,oftheLaplacian ontheassumption thatWiselectric flux,oftheMaxwellian ontheassumption thatWisthe intensity ofmagnetization. Theexpression oftheseintegrals ortheircomponents intermsofx,y,%,.formulre (42)',(43)', (44)'and(42)",(43)",(44)". 'l.NewV=Max'lV, 'lMaxW=New'l.W, VxLapW=Lap'lxW,(46) (47) (48) THEINTEGRAL CALCULUS OFVECTORS 253 'V•LapW=:Max'VXW=0, (49) 'VXNewV=Lap'VV=O, (50) 7•'VW=New'V•W -Lap'VXW ='VMaxW -'VXLapW.(51) Thepotential isasolution ofPoisson's Equation. Thatis, and'V•'VPotV= -4"17'V; 'V.'VPotV=-4"17'W.(52) (52)' -1V=-'V.NewV; (53)4"17' 1 1W=4"17'Lap'VXW -4"17'New'V•W.(55) HenceWisdivided into.twopartsofwhichoneis solenoidal andtheotherirrotational, provided thepotential exists.Incasethepotential doesnotexistathirdtermWs mustbeaddedofwhichboththedivergeace andthecurl vanish. Alistoftheorems whichfollow i~mediately from equations (52),(52)'.(53),(55)andwhichstatethatcertain integrating operators areinversetocertaindifferentiating operators. LetVbeascalarfunction, W,asolenoidal vector function, andW:aanirrotational vectorfunction. Then 1 1 4"17'Lap'VXWI='VX4"17'LapWI=WI'(60) 1 4"17'LapW:a=0,'VXW:a=0 (61) 1 1 -4"17'New'V.W:a=-'V.4"17'NewW:a=W:a'(62) VECTOR ANALYSIS 1 {-"VI"471'NewV=y(68) -471'Max"VV=Y. 1 471'Pot"VXVXWI="VX"VXPotWI=WI(64) 1 1 -471'Pot"V"V"W2= -"V""V471'PotW2=W2•(65) 1 1 {---Pot"V""VV=-"V""V-PotV=V471' 471' 1 1 (66) --Pot"V"\lW= -V•"V-PotW=W.471' 471' -"V""VWI="VX"VXWI (67) "V"\lW2="V"V•W2 (68) 471'PotV=-MaxNewV (69) 471'PotW=LapLapW -NewMaxW.(70) Mutualpotentials Newtonians, Laplacians, andMaxwellians maybeformed. Theyaresextuple integrals. Theintegra tionscannotallbeperformed immediately jbutthefirstthree maybe.Formulre (71)to(80)inclusive dealwiththeseinte grals.Thechaptercloseswiththeenunciation ofanumber oftheorems ofafunction-theoretic nature. Bymeansof thesetheorems certainfactsconcerning functions maybe inferredfromtheconditions thattheysatisfyLaplace's equa tionandhavecertainboundary conditions. Amongtheexercises number6isworthyofespecial atten tion.Theworkdoneinthetexthasforthemostpartassumed thatthepotential exists.Butmanyoftheformulm cO'lI,'Mcting Newtonians, Laplacians, andMaxwellian.~ holdwhenthepoteTJr tialdoesnotexist.ThesearetakenupinExercise 6referredto. JcVdrTHEINTEGRAL CALCULUS OFVECTORS 255 EXERCISES ONCHAPTER IV 1.1IfVisascalarfunctionofpositioninspacetheline integral isavectorquantity. Showthat Thatis;thelineintegral ofascalarfunction arounda closedcurveisequaltotheskewsurfaceintegralofthederiv ativeofthefunction takenoveranysurfacespanned into thecontourofthecurve.ShowfurtherthatifVisconstant theintegral aroundanyclosedcurveiszeroandconversely iftheintegralaroundanyclosedcurveiszerothefunction V isconstant. Hint:Insteadoftreatingtheintegralasitstandsmultiply it(withadot)byanarbitrary constant unitvectorandthus reduceittothelineintegralofavectorfunction. 2.IfWisa.vectorfunction thelineintegral isa.vectorquantity. Itmaybecalledtheskewlineintegral ofthefunction W.If0isanyconstant vector,showthatif theintegralbetakenaroundaclosedcurve H·c=Jfa(0\1.W -o·\lW).da.=0 •faWXdr, 1Thefirstfourexercill8ll aretakeufromFoppl'sEinfiihrung indieMax weU'lIChe TheoriederElectricitiit wheretheyareworkedout. 256 VECTOR AN..4LYSIS andHoc=C0~J1:V..Wda-JJ>J(W 0dad +J1:wo[coV(da)]. Incasetheintegral istakenoveraplanecurveandthe surfaceSistheportionofplaneincluded bythecurve Showthattheintegral takenoveraplanecurvevanishes whenWisconstant andconversely iftheintegral overany planecurvevanishes Wmustbeconstant. 3.ThesurfaceintegralofascalarfunctionVis Thisisavectorquantity. Showthatthesurfaceintegral ofVtakenoveranyclosedsurface ~equaltothevolume integral ofVVtakenthroughout thevolumebounded by thatsurface. Thatis JJsVda=-IJJVVdv. Henceconclude thatthesurfaceintegral overaclosedsur facevanishes ifVbeconstant andconversely ifthesurface integraloveranyclosedsurfacevanishes thefunctionVmust beconstant. .., 4.IfWbeavectorfunction, thesurfaceintegral maybecalledtheskewsurfaceintegral. Itisavector quantity. Showthattheskewsurfaceintegralofavector THEINTEGRAL CALOULUS OFYEOTORS 257 function takenoveraclosedsurfaceisequaltothevolume integralofthevectorfunction takenthroughout thevolume bounded bythesurface. Thatis JJada xW=JJJ'VxWdv. Henceconclude thattheskewsurfaceintegral takenover anysurfaceinspacevanishes whenandonlywhenWisan irrotational function. Thatis,whenandonlywhentheline integralofWforeveryclosedcircuitvanishes. 5.Obtainsomeformuloo fortheseintegrals whichare analogous tointegrating byparts. 6.Theworkinthetextassumes forthemostpartthatthe potentials ofVandWexist.Manyoftherelations, however, maybedemonstrated withoutthatassumption. Assumethat theNewtonian, theLaplacian, theMaxwellian exist.For simplicity inwritinglet 1 112 Pa=-, VIPl~=-8- Tl~ Tl~ ThenNewv=JJJ'V1P12V(X2d/t,Zt)dVt, (81) LapW=JJJ'VIPltXW(x~,Yt'z~)dVt•(8:&) Maxw=JJJ'V1Pa OW(x2,Y2'z2)dv2,(83) 'V1P12= -'VtP12 (84) 'Vt(PItV)=V2P12V+P12'V2~ JJJVtPltVdvt-JJJVt(P12V)dVt -JJJ PItVtVdVt· 17 258 VECTOR ANALYSIS Byexercise(8)f f fV't(PlllV)d'Dli=f fPlllVdL Itcanbeshownthatifvissuchafunction thatNewP' exists,thenthissurfaceintegral takenoveralargesphereof radiusRandasmallsphereofradiusR'approaches zero whenRbecomes indefinitely great;andR',indefinitely small.Hence or NewV=PotV'V: (85) Proveinasimilarmannerthat LapW =PotV'XW, MaxW=PotV'•W.(86) (87) Bymeansof(85), (86), (87)itispossibletoprovethat V'XLapW=LapV'XW, V'0NewV=MaxV'V, V'MaxW=NewV'0W. Thenprove V'XLapW=fffPltV'V'oWd'Dli-f ffPltV'oV'Wdv s and V'MaxW =f f fP ItV'V'•WdvlI' HenceV'XLapW -V'MaxW = -fffPltV'0V'Wd'Vs- Hence V'XLapW -V'MaxW = 4 7T'W. (88) '1.AnintegralusedbyHelmholtz is H(V)=f f frlllVdl1p THEINTEGRAL CALCULUS OFVECTORS 259 orifWbeavectorfunction (90) Showthattheintegral converges ifVdiminishes sorapidly that Vr6<K whenrbecomesindefinitely great. "VH(V)=H(VV)=New(r2V), (91) "V•H(W)=H("V•W)=Max(r2W), (92) "VxH(W)=H("VxW)=Lap(r2W),(98) "V•"VH(V)=H("V."VV)=Max(r2"VV)=2PotV(94) "V•"VH(W)=H("V•"VW)=2PotW.(95) 1H(V)=-2."PotPotv: (96) 1H(W)= -2."PotPotW. (97) - 2W="Vx"VxH(W)+"V"V•H(W). (98) 8.GiveaproofofGauss'sTheorem whichdoesnotdepend uponthephysical interpretation ofafunction asthefluxofa fluid.Thereasoning issimilartothatemployed inArt.51 andinthefirstproofofStokes's Theorem. 9.Showthatthedivision ofWintotwoparts,page235, isunique. 10.Treat,inamanneranalogous tothatuponpage220, thecaseinwhichVhascurvesofdiscontinuities. CHAPTER V LINEAR VECTOR FUNCTIONS 97.]AFTERthedefinitions ofproducts hadbeenlaiddown andapplied, twopathsofadvance wereopen.Onewas differential andintegral calculus; theother,higheralgebra inthesenseofthetheoryoflinearhomogeneous substitutions. Thetreatment ofthefirstofthesetopicsledtonewideM andnewsymbols-tothederivative, divergence, curl,scalar andvectorpotential, thatis,to\1,\1.,\1x,andPotwiththe auxiliaries, theNewtonian, theLaplacian, andtheMaxwellian. Thetreatment ofthesecondtopicwilllikewise introduce noveltybothinconceptandinnotation -thelinearvector function, thedyad,andthedyadicwiththeirappropriate symbolization. Thesimplest example ofalinearvectorfunction isthe product ofascalarconstant andavector. Thevectorr' r'=cr (1) isalinearfunction ofr..Amoregenerallinearfunction maybeobtained byconsidering thecomponents ofrindivid ually.Leti,j,kbeasystemofaxes.Thecomponents of ,rare i .r,j.r,k•r. Leteachofthesebemultiplied byascalarconstant which maybedifferent forthedifferent components. cak•r. LINEAR VECTOR FUNCTIONS Taketheseasthecomponents ofanewvectorr'261 r'=i(c1i.r)+j (caj·r)+k (cak.r). (2) Thevectorr'isthenalinearfunction ofr.Itscomponents arealwaysequaltothecorresponding components ofreach multiplied byadefinitescalarconstant. Suchalinearfunction hasnumerous applications ingeom etryandphysics.If,forinstance, i,j,kbetheaxesofa homogeneous Rtrainandc1'ca'Ca'theelongations alongthese axes,apoint r=ix+jy+kz becomes or r'=iCIi.r+j(Jaj •r+kcak•r. Thissortoflinearfunction occursinthetheoryofelasticity andinhydrodynamics. Inthetheoryofelectricity and magnetism, theelectricforceEisalinearfunction ofthe electricdisplacement Dinadielectric. Forisotropic bodies thefunction becomes merelyaconstant E=kD. Butincasethebodybenon-isotropic, thecomponents ofthe forcealongt.hedifferent axeswillbemultiplied bydifferent constants kl,ka,ks'Thus Thelinearvectorfunction isindispensable indealingwith thephenomena ofelectricity, magnetism, andopticsinnon isotropic bodies. 98.JItispossibletodefinealinearvectorfunction, ashas beendoneabove,bymeansofthecomponents ofavector. Themostgeneraldefinition would be 262 VECTOR ,ANALYSIS Definition: AvectorrIissaidtobealinearvectorfunc tionofanother vectorrwhenthecomponents ofrIalong threenon-coplanar vectorsareexpreBSible linearlywithscalar coefficients intermsofthecomponents ofralongthosesame vectors. If r=xa+yb+za,where[aba]:t0, and r'=x'a+y'b+zlc, (8)andif xI=aIx+bIY+CIZ. y'=a2x+b'Jy+c'Jz, Zl=aax+baY+caz, thenr'isalinearfunction ofr.(Theconstants ~,bl,cI' etc.,havenoconnection withthecomponents ofa,b,apar alleltoi,j,k.)Another definition howe-ver isfoundtobe moreconvenient andfromittheforeKOing maybededuced. Definition: Acontinuous vectorfunction ofavectoris saidtobealinearvectorfunction whenthefunction ofthe sumofanytwovectorsisthesumofthefunctions ofthose vectors. Thatis,thefunctionfislinearif j(rl+r2)=j(rl)+j(r'J)' (4) Thwrem:Ifabeanypositive ornegative scalarandif/ bealinearfunction, thenthefunctionofatimesrisatimes thefunction ofr. Andhencej(ar)=aj(r). (5) jCalrl+a'Jr'J+aara+...) =alj(rt)+a'Jj(r2)+asjCra)+... (5)' Theproofofthistheorem whichappears moreorle88 obviouRisatriflelong.Itdepends uponmakingrepeated useofrelation(4). j(r+r)=jCr)+jCr)=2jCr). LINEAR VECTOR FUNCTIONS 268 Hence Inlikemannerf(2r)=2f(r). f(nr)=nf(r) wherenisanypositiveinteger. Letmbeanyotherpositiveinteger. Thenbytherelation justobtained and Bencef(r)=f(m~)=mf(~) f(~)= ~f(r). f(n~)=f(..!-r)=-~f(r).In m m Thatis,equation (5)haSbeenprovedincasetheconstant a isarational positive number. Toshowtherelationfornegative numbers notethat f(O)=f(O+0)=2f(0). Hence f(0)=O. But f(O)=f(r-r) =f(r+(-r»)=f(r)+f(-r). Hence f(r)=-f(-r). Toprove(5)forincommensurable valuesoftheconstant a,itbecomes necessary tomakeuseofthecontinuity ofthe functionf. Thatis LIM(LIM)x::::!:::.a/(xr)=f x=a(xr) . Letxapproach theincommensurable number abypassing throughasuiteofcommensurable values. Then f(xr)=xf(r). Hence LIMf(xr)=af(r) x::::!:::.a 264 HenceVECTOR ANALYSIS LIM(xr)=ar.x=a /(ar)=a/(r) whichprovesthetheorem. Thwrem: Alinearvectorfunction/ (r)isentirelydeter minedwhenitsvaluesforthree non-coplanar vectorsa,b,0are known.' Let 1'=/(a), m=/(b), n=/(0). SincerisanyvectorwhatBoever, itmaybeexpressed as r=xa+yb+zo. Hence /(r)=x1+Ym+zD. 99.]InArt.97aparticular caseofalinearfunction was expressed as r'=iC1i0r+jCzj0r+kCsk0r. Forthesakeofbrevityandtosaverepeating thevectorr whichoccursineachofthesetermsinthesamewaythis maybewritten inthesymbolic form r'=(iCli+jCzj+kCsk)0r. Inlikemannerifal'a2'as...beanygivenvectors,andbt,bl' bs'...anothersetequalinnumber, theexpression r'=albtor+azbzor+asbsor+... (6) isalinearvectorfunction ofr;forowingtothedistributive character ofthescalarproduct thisfunction ofrsatisfielti .~'-. relation (4).Forthesakeofbrevityr'maybewrittensym bolicallyintheform r'=(0.1bl+azbz+asbs+...)0r.(6)' LINEAR VECTOR FUNCTIONS 265 Noparticular physical orgeometrical significance istobe attributed atpresenttotheexpression (a1b1+allbll+a8bs+...). (7) Itshouldberegarded asanoperator orsymbolwhichcon vertllthevectorrintothe vector r'andwhichmerely affordsaconvenient andquickwayofwritingtherelation (6). Definition: Anexpression abformedbythejuxtaposition oftwovectorswithout.theintervention ofadotoracrOBBis calledadyad.Thesymbolic sumoftwodyadsiscalleda dyadicbinomial; ofthree,adyadictrinomial; ofanynum ber,adyadicpolynomial. Forthesakeofbrevitydyadic binomials, trinomials, andpolynomials willbecalledsimply dyadics. Thefirstvectorinadyadiscalledtheantecedent; andthesecondvector,theconsequent. Theantecedents ofa dyadicarethevectorswhicharetheantecedents ofthe individual dyadsofwhichthedyadiciscomposed. Inlike manner theconsequents ofadyadicaretheconsequents of theindividual dyads.Thusinthedyadic(7)Pot'~,as...are theantecedents andb1,b2,bs...theconsequents. Dyadics willberepresented symbolically bythecapital Greekletters. Whenonlyonedyadici8presenttheletter rpwillgenerally beused.Incaseseveralareunderconsid erationotherGreekcapitals willbeemployed also.With thisnotation (7)becomes rp=a1b1+allb2+asbs+..., (7)' and(6)'maynowbewrittenbrieflyintheform r'=rp.r. (8) Bydefinition rp.r=a1b1•r+a2b2•r+asba•r+... Thesymbolrp.risreadrpdotr.Iti8calledthedirect productofrpintorbecause theconsequents bI,b2,ba.•.are 266 VECTOR ANALYSIS andmultiplied intorbydirectorscalarmultiplication. The orderofthefactors fPandrisimportant. Thedirect product ofrintofPis r·fP=r·(&1bl+0.2b2+asbs+...) = r •0.1bl+r·&2b2+r·&sbs+... (9) Evidently thevectorsfP.randr.fPareingeneraldifferent. Definition: Whenthedyadic(/)ismultiplied intorasfP.r, (/)issaidtobeaprejactor tor.Whenrismultiplied infPas r·fP,(/)issaidtobeapostjactor tor. Adyadic (/)usedeithernsaprejactor orasapostjactor toa 'Vectorrdetermines alineart'ectorjunction ojr.Thetwolinear vectorfunctions thusobtained areingeneraldifferent from oneanother. Theyarecalledconjugate linear vector func tions.Thetwodyadics (/)=&1bl+&2b2+&sbs+ . fI'=bio.I+b20.2+bs&8+ , eachofwhichmaybeobtained fromtheotherbyinter changing theantecedents andconsequents, arecalledconju gatedyadics. Thefactthatonedyadicistheconjugate of anotherisdenoted byaffixingasubscriptatoeither. Thus fI'=(/)c(/)=fI'c. Theorem: Adyadicusedasapostfactor givesthesame resultasitsconjugate usedasaprefactor. Thatis r·rft=rftc'r. (9) 100.]Definition: Anytwodyadics (/)andfI'aresaidto beequal when orwhen orwhen(/).r=fI'.r r.rft=r·fI'forall values ofr, forallvaluesofr,(10) forallvaluesof.andr. LINEAR VECTOR FUNCTIONS 267 lP.b, lP=ai+bj+ck,Thethirdrelationisequivalent tothefirst.For,ifthe vectorsrp.rand1J!.rareequal,thescalarproducts ofany vector 1intothemmustbeequal.Andconversely ifthe scalarproductofanyandeveryvector 1intothevectorslP.r and1J!.rareequal,thenthosevectorsmustbeequal.In likemanneritmaybeshownthatthethirdrelationisequiva lenttothesecond. Henceallthreeareequivalent. Theorem: AdyadiclPiscompletely determined whenthe values wherea,b,careanythreenon-coplanar vectors,areknown. Thisfollowsimmediately fromthefactthatadyadicdefines alinearvectorfunction.If r=xa+yb+zc, lP.r=lP.(xa+yb+zc)=xrp.a+ylP.b+zlP.c, consequently twodyadics lPand1J!areequalprovided equa tions(10)holdforthreenon-coplanar vectorsrandthree non-coplanar vectors I. Theorem: Anylinearvectorfunction.j mayberepresented bjadyadicrptobeusedasaprefactorandbyadyadic 1J!, whichisthE"conjugate oflP,tobeusedasapostfactor. Thelinearvectorfunction iscompletely determined when itsvaluesforthreenon-coplanar vectors(sayi,j,k)are known(page264).Let f(i)=a,f(j)=b,f(k)=c. Thenthelinearfunctionfisequivalent tothedyadiclP givenby tobeusedasapostfactor; andtothedyadic 1J! 1J!=lPc=ia+jb+kc, tobeusedasaprefactor. f(r)=lP.r=r. lPc- 268 VECTOR AN.ALYSIS Thestudyoflinearvectorfunctions therefore isidentical withthestudyofdyadics. Definition: Adyadabissaidtobemultiplied byascalar awhentheantecedent ortheconsequent ismultiplie~ by thatscalar,orwhenaisdistributed inanymannerbetween theantecedent andtheconsequent. Ifa=a'a" a(ab)= (aa)b =a(ab)= (a'a)(a"b). Adyadic fPissaidtobemultiplied bythescalarawhen eachofitsdyadsismultiplied bythatscala.r.Theproduct iswritten a(/)or(/)a. ThedyadicacPappliedtoavectorreitherasaprefactor or 118apostfactor yieldsavectorequaltoatimesthe vector obtained byapplying (/)tor -thatis (afP) 0r=a(fP 0r). Tlw:Ytvm: Thecombination ofvectorsinadya.disdistril? utive.Thatis a.nd(a+b)e=ac+ be a(b+0)=ab+ac.(11) Thisfollowsimmediately fromthedefinition of equality of dyadics(10).For [(a+b)oJ0r =(a+b)00r =a00r +beor=(ae'+b0)0r and [a(b+0)]0r =a(b+0)0r =abor +ac 0r =(ab+ae)0r. Henceitfollowsthatadyadwhichconsistsoftwofactors, eachofwhichisthesumofanumber ofvectors, maybe multiplied outaccording tothelawofordinary algebra -exceptthattheorderofthefactorsinthedyad8must ~c maintained. LINEAR VECTOR FUNCTIONS 269 (a+ b + c+...)(l+ m+ n + ...)=a1+am+an+... + b1+ bm+ bn + . ..(11)' +cl+cm+cn+ .., +.... Thedyadtherefore appearsasaproductofthetwovectorsof whichitiscomposed, inasmuch asitobeysthecharacteris ticlawofproducts -thedistributiye law.Thisisajustifi cationforwritingadyadwiththeantecedent andconse quentinjuxtap08ition asiscustomary inthecaseofproducts inordinary algebra. TJuNonionFormofaDyadic 101.]Fromthethreeunitvectorsi,j,kninedyadsmay beobtained bycombining twoatatime.Theseare ii,ij,ik, '..jj,jk, JI, k'kj,kk. 1,(12) Ifalltheantecedents andconsequents inadyadicfPbeex pressedintermsofi,j,k,andiftheresulting expression be simplified byperforming themultiplications according tothe distributive law(11)'andifthetermsbecollected, thedyadic rpmaybereducedtothesumofninedyadseachofwhichis ascalarmultiple ofoneoftheninefundamental dyadsgiven above. . fP=a11i i+a12ij+a18ik +a2di+a22jj+a28kk (13) +a31ki+a32kj+a38kk. ThisiscalledthenonionformoffP. Theorem: Thenecessary andsufficient condition thattwo dyadics rpand1Jfbeequalisthat,whenexpressed innonian 270 VECTOR ANALYSIS form,thescalarcoefficients ofthecorresponding dyadsbe \equal. IfthecoefficientB beequal,thenobviously fP·r=1J1'·r foranyvalueofrandthedyadicsby(10)must.:be equal. Conversely, ifthedyadics fPand1J1'areequal,thenby(10) forallvaluesof8andr.Let8andreachtakeonthevalues i,j,k.Then (14) i .fP.i=i.1J1'.i,i·fP.j=i.1J1'.j,i·fP.k=i.1J1'.k j •fPoi=j •1J1'oi,j •fP•j=j.1J1'.j,j.fP•k=i.1J1'.k k·fP.i=k.1J1'.i,k•fP•i=k·1J1'.i,k.fP•k=k·1J1'.k. Butthesequantities areprecisely theninecoefficients inthe expansion ofthedyadics fPand1J1'.Hencethecorresponding coefficients areequalandthetheorem isproved.! This analytic statement oftheequality oftwodyadicscansome timesbeusedtogreateradvantage thanthemorefundamental definition (10)basedupontheconception ofthedyadicas defining alinearvectorfunction. Theorem: AdyadicfPmaybeexpressed asthesumofnine dyadsofwhichtheantecedents areanythreegivennon coplanar vectors,a,b,candtheconsequents anythreegiven non-coplanar vectors 1,m,D. Everyantecedent maybeexpressed intermsofa,b,0; andeveryconsequent, intermsof1,m,D.Thedyadicmay thenbereducedtotheform fP=a11a1+a12am+a13aD +a21bl+a22bm+a23bD (15) +a31b1+a32em+a33CD. ~)1AIacorollary ofthetheoremitillevidentthattheninedyads(12)arein· dependent. Noneofthemmaybeexpressed linearlyintermsoftheothers. LINEAR VECTOR FUNCTIONS 271 Thisexpression off/Jismoregeneral thanthatgivenin (13).Itreducestothatexpression wheneachsetofvectors &,b,0and1,In,ncoincides withi,j,k. Theorem: Anydyadio f/Jmaybereduced tothesumof threedyadsofwhicheithertheantecedentll ortheconsequents, butnotboth,maybearbitrarily chosenprovided theybenon coplanar. Letitberequired toexpress f/J88thesumofthreedyads ofwhicha,b,0aretheantecedents. Let1,In,nbeanyother threenon-coplanar vectors. f/Jmaythenbeexpressed 88in (15).Hence f/J=a(all1+a12m+alBn)+b(a211+anm+atSn) +0(aSl1+anm+aStn), or f/J=aA+bB+0c. (16) Inlikemannerifitberequired toexpress f/J88thesumof threedyadsofwhichthethreenon-coplanar veotors 1,In,nare theconsequents wheref/J=Ll+Km+lIn, L=ana+atlb+a8l0, X=aua+anb+an0, 11=alSa+a23b+aaso.(16)' Theexpressions (15),(16),(16),forf/Jareunique.Twoequal dyadics whichhavethesamethreenon-coplanar ante cedents, a,b,0,havethesameconsequents A,B,C -these however neednotbenon-coplanar. Andtwoequaldyadics whichhavethesamethreenon-coplanar consequents 1,In,n, havethesamethreeantecedents. 102.]Definition: Thesymbolic productformedbythejuxta position oftwovectfrs,.,bwithouttheintervention ofadot oracrossiscalledtheindeterminate productofthetwovectors aandb. 272 VECTOR ~N~LYSIS Thereasonforthetermindeterminate isthis.Thetwo productH a.banda x bhavedefinite meanings. Oneisa certainscalar,theotheracertainvector. Ontheotherhand theproduct abisneithervectornorscalar-itispurely symbolic andacquires adeterminate physical meaning only whenusedasanoperator. Theproduct abdoesnotobey thecommutative law.Itdoeshowever obeythedistributive law(11)andtheassociative lawasfarasscalarmultiplication isconcerned (Art.100) Theorem: Theindeterminate productaboftwovectorsis themostgeneralproduct inwhichscalarmultiplication is associative. Them08tgeneralproduct conceivable oughttohavethe property thatwhentheproduct isknownthetwofactorsare alsoknown. Certainly noproduct couldbemoregeneral. Inasmuch asscalarmultiplication istobeassociative, thatis a(ab)=(aa)b=a(ab)=(a'a)(a"b), itwillbeimpo88ible tocompletely determine thevectors a. andbwhentheirproduct abisgiven.Anyscalarfactor maybetransferred fromonevectortotheother.Apartfrom thisp08sible transference ofascalarfactor,thevectorscom posingtheproductareknownwhentheproductisknown.In otherwords- Theorem:Ifthetwoindeterminate products abanda'br areequal,thevectorsaanda',bandb'mustbecollinear and theproductofthelengthsofaandb(takingintoaccountthe positiveornegative signaccording asaandbhaverespec tivelyequal oropposite directions toa'andb')isequaltothe productofthelengthsofa'andb'. Let ThenLINEAR VECTOR FUNCTIONS a'=at'i+as'j+aa'k, b'=bl'i+bs'j+ba'k. ab=alb lii+albsij+albait +aSblji+a2bsjj+a2bajt +aabltj+aab2tj+aabatk. and a'b'=ai'bl'ii+aI'b2'ij+ai'ba'it +a2'bl'j i +as'b2'jj +a2bajk +aa/bl'ki+aa'b2'kj+aa'ba'kk.278 Sinoeab=a'b'corresponding coefficients areequal.Hence al:a2:aa=at':a'J':as', whichshowsthattheveetomaanda'arecollinear. And whichshowsthatthevectorsbandb'arecollinear. But Thisshowsthattheproductofthelengths(including sign) areequalandthetheoremisproved. Theproofmaybecarriedoutgeometrioally asfollows. Sinceabisequaltoa'b' ab •r=a'b'•r forallvaluesofr.Letrbeperpendicular tob.Thenb.r vanishes andconsequently b'·ralsovanishes. Thisistrue foranyvectorrintheplaneperpendicular tob.Henceband b'areperpendicular tothesameplaneandarecollinear. In likemannerbyusingabasap08tfactor aanda'are.seen tobeparallel. Also ab • b=a'b'•b, whichshowsthattheproducts ofthelengthsarethesame. t8 274 VECTOR· ANALYSIS Theindeterminate productabimposesjiveconditions upon thevectorsaandb.Thedirections ofaandbarefixedand likewise theproduct oftheirlengths. Thescalarproduct a.b,beingascalarquantity, imposesonlyonecondition upon aandb.Thevectorproducta xb,beingavectorquantity, imposesthreeconditions. Thenormaltotheplaneofaand bisfixedandalsotheareaoftheparallelogram ofwhichthey aretheside.Thenineindeterminate products (12)ofi,j,k intothemselves areindependent. Theninescalarproducts arenotindependent. Onlytwoofthemaredifferent. i . i=j •j=k•k=1, andi . j=j • i=j •k=k•j=k•i=i .k=O. Theninevectorproducts arenotindependent either;for ixi=jxj=kxk=0, andixj= -jxi,jx k = -kxj,kxi= -ixk. Thetwoproductsa.banda x bobtained respectively from theindeterminate product b)~inserting adotandacrossbe tweenthefactorsarefunctions oftheindeterminate product. Thatistosay,whenabisgiven,a·banda x baredetermined. Fortheseproducts dependsolelyuponthedirections ofaandb andupontheproductofthelengthofaandb,allofwhich areknownwhena bisknown. Thatis ifab=a'b', a.b=a'.b'andaxb=a'xb'. (17) Itdoesnotholdconversely thatifa•banda x bareknown a bisfixed;fortakentogethera.'b.~da x bimposeuponthe vectorsonlyfourconditions, whereasa bimposesfive.Hence abappearsnotonlyastheIllostgeneral productbutasthe ,mostfundamental product. Theothersaremerelyfunctions ofit.Theirfunctional natureisbrought outclearlybythe notation ofthedotandthecross. LINEAR VECTOR FUNCTIONS 275 Definition: Ascalarknownasthescalarojf/Jmaybeob tainedbyinserting adotbetween theantecedent andconse quentofeachdya.dinadyadic. Thisscalarwillbedenoted byasubBcript Sattached tof/J.1 If f/J=albl+a~b~+a8b8+'" f/JtI=alobl+a~ob~+a8ob8+'" (18) InlikemanneraYectorknownastMvectorojf/Jmaybe obtained byinserting acrossbetween theantecedent andcon sequentofeachdyadinf/J.Thisvectorwillbedenotedby attaching asubBcript cr08Stof/J. f/J"=alxbl+a~xb~+ a8xb8+ ...(19) Iff/Jbeexpanded innonionformintermsofi,j,k, (20) fP"=(a~-a8~)i+(a8l-ala)j+(al~-au)k.(21) Or f/JtI=iof/Joi+jofPoj+kofPok, (20)' fP"=(j0f/J0k-k0f/Joj)i +(k 0fPoi- i 0fP0k)j +(i0fPoj-j 0fPoi)k. (21)' Inequations (20)and(21)thescalarandvectoroffPare expressed intermsofthecoefficients off/Jwhenexpanded inthenonionform.Henceiff/Jand'Far~twoequal dyadics, thescalaroffPisequaltothescalarof'Fandthe vectoroffPisequaltothevectorof-'F. If (22) Fromthisitappearsthatf/JtIandf/J"arefunctions offP uniquely determined whenfPisgiven.Theymaysometimes beobtained moreconveniently from(20)and(21)thanfrom (18)and(19),andsometimes not. 1ASUbscript dotmightbe1lIlIldforthescalarof.ifitweresufficiently distind andfreefromliabilitytomisinterpretation. 276 VECTOR ANALYSIS ProductsofDyadics 103.]Ingivingthedefinitions andproving thetheorems concerning products ofdyadics, thedyadismadetheunder lyingprinciple. Whatistrueforthedyadistrueforthe dyadicingeneralowingtothefactthatdyadsanddyadics obeythedistributive lawofmultiplication. .~Definition: Thedirectproduct ofthedyadabintothe v"';;l~~ 1...-L,....J .~fJ/Lad odiswritten (ab) 0(0d) I'.,....~~ ~t£,~d andisbydefinition equaltothedyad(b00)ad. (ab)o(od)=a(boo)d=boo ad.l(23) Thatis,theantecedent ofthefirstandtheconsequent ofthe seconddyadaretakenfortheantecedent andconsequent respectively oftheproductandthewholeismultiplied by thescalarproduct oftheconsequent ofthefirstandthe antecedent ofthesecond. Thusthetwovectorswhichstandtogether intheproduct (23)'fl=(alb l+a2b2+aaba+"') 7J!'=(oldl+02d2+0ada+ ) 'flo7J!'=(a1bl+a2b2+aabs+ )0 (01d1+02d2+oada+ ) =alblooldl +alblo02d2 +albloosds+'" +a2b20°1d1+a2b20°2d2+a2b2•Osds+... +aSbaooldl +ssb So02d2+aabaocsda+'"and(ab)0(0d) aremultiplieq astheystand.Theothertwoarelefttoform anewdyad.Thedirectproduct oftwodyadicsmaybe definedastheformalexpansion (according tothedistributive law)oftheproductintoasumofproducts ofdyads.Thus if~"-~II~\_,.. U,\\..tC, ."1.{~t,,(/'" ~ \ II\.(.)c:::\..t ~ T. 1Theparentheses maybeomittedineachofthesethreeexprellllioni. LINEA.R VECTOR FUNCTIONS ~OW=bloo laldl+blooialdi+blooSalds+··· +bll001alldl+bi0°11aidi+bi0Os~ds+... +bSoolaSdl+bSociasdll+bsocsasds+'"277 +.. (23)" Theproductoftwodyadics rpand1JTisadyadicrp01JT. Theorem: Theproduct rp01JToftwodyadics rPand1JTwhen regarded asanoperator tobeusedasaprefactor isequiva lenttotheoperator 1JTfollowed bytheoperator rp. Let Toshow orJJ=rp01JT. JJ0r=rp0(1JT 0r), (rp.1JT)or= rpo(1JTor). (24) Letabbeanydyadofrpand0danydyadof'T. (ab0cd) 0r = b 0c(ad0r)=(b00)(d0r)a, abo(cd0r)=abo0(d0r)=(b0c)(d0r).. Hence (abocd)or= abo(cdor). Thetheorem istruefordyads. Consequently byvirtueof thedistributive lawitholdstruefordyadicsingeneral. Ifrdenotethepositionvectordrawnfromanassumed origin toapointPinspace,r'=1JT0rwillbetheposition vectorof another pointpI,andr"=rp0(IJT 0r)willbetheposition vectorofathirdpointP".Thatistosay,'Tdefinesatrans formation ofspacesuchthatthepointsPgooverintothe pointsPl.rpdefinesatransformation ofspacesuchthatthe pointspIgooverintothepointsPII.Hence'Tfollowed by rpcarriesPintoPII.Thesingleoperation rp0IJTalsocarries PintoP". Throrem: Directmultiplication ofdyadicsobeysthedis tributive law.Thatis 278 VECTOR ANALYSIS andcP0 (lJf+lJf)=cP0lJf+cP0lJf' «(P'+cP) 0IJI'=cP' 0lJf+(p01J"o (25) (26)Henceingeneraltheproduct (cP+(p'+cP"+...)o(lJf+lJf'+lJf"+...) maybeexpanded formally according tothedistributive law. TJuorem: Theproductofthreedyadics (p,lJf,Qisassocia tive.Thatis (26)' cP0lJf0Q.andconsequently eitherproduct maybewrittenwithout parentheses, as Theproofconsists inthedemonstration ofthetheorem for threedyadsab,cd,eftakenrespectively fromthethree dyadics cP,lJf,fJ. (abocd)oef= (boc)adoef= (boc)(doe)af, abo(cdoef)= (doe)abocf= (doe)(boc)af. Theproofmayalsobegivenbyconsidering cP,IJI',and!J asoperators Let 12or=r' {(cP 01J") 0.Q} 0r =(cP 0lJf) 0r'=cP0 (lJf0r'). Let IJI'0r'=r", I«(polJf) 0121or=cP0r"=rIll. Again {(P 0(lJf 0Q)} 0r=(p0[(lJf0Q) 0r]. (lJfofJ)or= lJI'o(fJor) =lJI'or'=r". {It> 0 (IJI'0Q)} 0r=(p0 [IJI'0r']=It>0r"=I'" Hence {(It>0IJI') 0QI 0r={(P 0(IJI' 0Q)I 0r forallvaluesofr.Consequently (It> 0lJf) 0Q=It>0(lJfoJ/). LINEAR VECTOR FUNCTIONS 279 Thetheorem maybeextended bymathematical induction tothecaseofanynumber ofdyadics. Thedirectproduct ofanynumberofdyadicsisassociative. Parentheses may beinsertedoromittedatpleasure withoutalteringtheresult. Itwasshownabove(24)that (,p.lJI).r=,p.(W.r)=,p. 7Jf.r. (24)' Hericetheproductoftwodyadicsandavectorisassociative. Thetheorem istrueincasethevectorprecedes thedyadics andalsowhenthenumber ofdyadicsisgreaterthantwo. Butthetheorem isuntruewhenthevectoroccursbetween thedyadics. Theproductofadyadic,avector,andanother dyadicisnotassociative. (,p.r).W*,p.(r.lJI). Letabbeadyadof,p,andcdadyadofW. (ab·r)•cd = b •r (a.cd)=(b.r)(a.c)d, ab.(r.cd) =ab.d(r.c) =b.d(r.c)a Hence (ab.r).cd*ab.(r.cd).(27) Theresultsofthisarticlemaybesummed upasfollows: Theorem: Thedirectproduct ofanynumber ofdj'adics orofanynumberofdyadicswithavectorfactorateither endoratbothendsobeysthedistributive andassociative lawsofmultiplication -parentheses maybeinserted or omittedatpleasure. Butthedirectproductofanynumber ofdyadicswithavectorfactoratsomeotherpositionthanat eitherendisnotassociative -parentheses arenecessary to givetheexpression adefinitemeaning. Lateritwillbeseenthatbymakinguseoftheconjugate dyadicsavectorfactorwhichoccursbetween otherdyadics maybeplacedattheendandhencetheproduct maybe madetoassumeaforminwhichitisassociative. 280 VECTOR ANALYSIS (28)104.]Definition: Theskewproducts ofadyad8binto avectorrandofavectorrintoadyad 8baredefined respectively bytheequations (ab)x r=8(bxr), r x(8b)=(rx8)b. (29) butTheskewproductofadyadandavectorateitherendisa dyad.Theobviousextension todyadicsis rpx r=(81b1+8,b,+aaba+...) x r =81b1x r +a2b,x r + 8abax r +... rxrp=rx(a 1b1+a,b,+8 ab8+···) (28)' =r x81b1+rX",b,+r x8aba+.., Th.eOTem: Thedirectproduct ofanynumber ofdyadics multiplied ateitherendoratbothendsbyavectorwhether themultiplication beperformed withacrossoradotis associative. Butincasethevectoroccursatanyother position thantheendtheproductisnotassociative. Thatis, (rX(/J).fJI"=rX(rp.fJI")=rXrp.fJI", (rp.fJI")xr=(/J.(fJI"Xr)=rp.fJI"xr, (rX(/J).1=rx(rp.I)=rxrp... r •(rpXI)=(r•rp)x1=r •rpx8, r x(rpXI)=(rxrp)x1= r xrpxI, IJT'•(rxrp)t:-(fJI".r)xIJT'. Furthermore theexpressions 1 •r xrpaudrpxr•• canhavenoothermeaning than I·r xrp=I·(rxfP). rpx r •1=(rpxr)•I,(30) LINEAR VECTOR FUNCTIONS 281 sincetheproductofadyadic (/Jwithacrossintoascalara.r ismeaningless. Moreover sincethedotandthecrossmay beinterchanged inthescalartripleproductofthreevectors itappearsthat anda•r x(/J=(axr)•(/J, (/Jx r •a=(/J.(rxa), (/J.(rxfJf)=«(/Jxr).iF.(31) Theparentheses inthefollowing expressions cannotbe omitted without incurring ambiguity. (/J.(rxa)=t-«(/J.r)xI, (axr).(/J-:tax(r.(/J), «(/J.r)xfJf=t-(/Jx(r.fJf).(31)' Thefonna!skewproductoftwodyadsabandcdwouldbe (ab)x(cd)=a(bxc)d. (32) ratInthisexpression threevectorsa,b xc,dareplacedside bysidewithnosignofmultiplication unitingthem.Such anexpression iscalledatriad,.andasumofsuchexpre!liions. atriadic. Thetheoryoftriadicsisintimately connected withthetheory oflineardyadicfunctions ofavector,justasthetheoryof dyadicsisconnected withthetheoryoflinearvectorfunctions ofavector.Inasimilarmanner bygoingastephigher tetradsandtetradic8 maybeformed,andfinallypolyadsand polyadics. Butthetheoryofthesehighercombinations of vectorswillnotbetakenupinthisbouk.Thedyadic furnishes aboutasgreatagenerality asisevercalledforin practica.l applications ofvectormethods. 282 VECTOR ANALYSIS Degrees01Nullity01Dyadics 106.]Itwasshown(Art.101)thatadyadiccouldalways bereducedtoasumofthreetermsatmost,andthisreduction canbeaccomplished inonlyonewaywhentheantecedenta ortheconsequentd arespecified. Inparticular casesitmay ',',~,~~~,,~bepossible toreducethedyadicfurthertoasumoftwo ....'.'{,~"-.~termsortoasingletermortozero.Thuslet .~. ~=aI+bm+cD. If1,m,narecoplanar oneofthethreemaybeexpressed intermsoftheothertwoas I=xm+yn. Then ~= axm + ayn+ bm +en, ~=(ax+b)m+(ay+c)11. Thedyadichasbeenreduced totwoterms.If1,m,nwere allcollinear thedyadicwouldreducetoasingletermandif theyallvanished thedyadicwouldvanish. Theorem:Ifadyadic ~beexpressed asthesumofthree terms ~=aI+bm+cn ofwhichtheantecedents a,b,careknowntobenon~oplanar, thenthedyadicmaybereduced tothesumoftwodyads whenandonlywhentheconsequents arecoplanar. Theproofof thefirstpartofthetheoremhasjustbeen given.Toprovethesecondpart,suppose thatthedyadic couldbereduced toasumoftwoterms ~=dp+eq andthattheconsequents 1,m,nof~were non~oplanar. Thissupp08ition leadstoacontradiction. ForletI',m',n' bethesystemreciprocal to1,m,11.Thatis, ,l\....... ~... 1\.r'f' 1'=mxnm'=~, n'=Ixm• [1mn]' [1mn] [1mn] LINEAR VECTOR FUNCTIONS 283 ThevectorsI',m',n'existandarenon-coplanar because 1,m,nhavebeenassumed tobenon-coplanar. Anyvectorr maybeexpressed intermsofthemas r=xI'+'!Im'+z0,' ~•r =(a1+ bm + c n)•(xI'+'!Im'+zn'). But 1•I'=m •m'=n•n'=1, and1•m'=I'• m = m • n'=m'•n=n•I'=n'• 1=O. Hence rp•r=xa+'!Ib+zc. Bygivingtorasuitablevaluethevectorrp.rmaybemade equaltoanyvectorinspace. Butrp.r=(dp+eq).r=d(p•r)+e(q.r). Thisshowsthatr/).rmustbecoplanar withdande.Hence ~.rcantakeononlythosevectorvalueswhichlieinthe planeofdande.Thustheassumption that1,m,narenon coplanar leadstoacontradiction. Hence1,m,nmustbe coplanar andthetheorem isproved. Theorem:Ifadyadicrpbeexpressed asthesumofthree terms rp=al+bm+co, ofwhichtheantece~ents a,b,ca.reknowntobenon-coplanar, thedyadicrpcanbereducedtoasingledyadwhenandonly whentheconsequents 1,m,narecollinear. Theproofofthefirstpartwasgivenabove.Toprove thesecondpartsuppose rpcouldbeexpressed as Letrp=dp. 'F=r/)x p = dp x p = dO=0, 'F=a 1x p+bm x p+cnxp. 284 VECTOR ANALYSIS Fromthesecondequation itisevidentthat1J!usedasa postfactor foranyv~ctor r=xa'+yb'+zo', wherea',b',0'isthereciprocal systemtoa,b,0gives r •1Jl'=x1XP+YmXP+znXp. Fromthefirstexpression r.0=O. Hence xlxp+ymxp+znxp mustbezeroforeveryvalueofr,thatis,foreveryvalueofx, y,z.Hence 1XP=0,mXP=0,nXP=O. Hence1,m,andnareallparalleltopandthetheorem has beendemonstrated. Ifthethreeconsequents 1,m,Dhadbeenknowntobenon coplanar insteadofthethreeantecedents, thestatement of thetheorems wouldhavetobealteredbyiBterchanging the wordsantecedent andconsequent throughout. Thereisafur thertheorem dealingwiththecaseinwhich bothantecedents andconsequents ofrparecoplanar. Thenrpisreducible to thesumoftwodyads. 106.]D,jinition: Adyadicwhichcannotbereducedto thesum'offewerthanthreedyadsissaidtobecompkte. A dyadicwhichmaybereduced tothesumoftW?dyads,but cannotbereduced toasingledyadissaidtobeplanar. In casetheplaneoftheantecedents andtheplaneofthecon sequents coincide whenthedyadicisexpressed asthesumof twodyads,thedyadicissaidtobeuniplanar. Adyadic whichmaybereduced toasingledyadissaidtobelinear. Incasetheantecedent andconsequent ofthatdyadarecol. LINEAR VECTOR FUNCTIONS 285 linear,thedyadicissaidtobeunilinear. Ifadyadicmaybe IilOexpressed thatallofitstennsvanishthedyadicissaidto bezero.Inthigcasetheninecoefficients ofthedyadicas expressed innonionformmustvanish. Theproperties ofcomplete, planar,uniplanar, linear,and unilinear dyadicswhenregarded asoperators areasfollows. Let • =(/J•randt=r •(/J. If(/Jiscomplete IandtmaybemtUl.etotakeonanydesired valuebygi1l'ingrasuitablevalue. (/J=aI+bm+cn. As(/Jiscomplete 1,m,narenon-coplanar andhencehavea reciprocal systemI',m',n'. I=(/J•(xI'+Ym'+zn')=xa+yb +zc. Inlikemanner a,b,cpossessasystemofreciprocals a',b',c'. t=(xa'+yb'+zc'). (/J=xI+ym+zn. Acomplete dyadic (/Jappliedtoavectorrcannotgivezero unlessthevectorritselfiszero. If(/JisplanarthevectorImaytakeonanyvalueintheplane oftheantecetknts andtanyvalueintheplaneoftheconsequents of(/J;butno."aluuoutofthoseplanes. Thedyadic (/Jwhen usedasaprefactor reduceseveryvectorrinspacetoavector intheplaneoftheantecedents. Inparticular anyvectorr perpendicular totheplaneoftheconsequents of(/Jisreduced tozero.Thedyadic (/Jusedasapostfactor reduces every vectorrinspacetoavectorintheplaneoftheconsequents of(/J.Inparticular avectorperpendicular totheplaneof theantecedents of(/Jisreduced tozero.Incasethedyadic it!uniplanar thesamestatements hold. If(/Jislinearthe."ector Imaytakeonanyvaluecollinear withtheantecedent of(/Jandtanyvaluecollinearwiththecon- 286 VECTOR ANALYSIS uqtunt0/$jbutnootlurvalues.Thedyadic (j)usedasa pre£actor reduces anyveetorrtothelineoftheantecedent of$.Inparticular anyvectorsperpendicular tothecon sequent of(j)arereduced tozero.Thedyadic f/Jusedasa postfactor reduces ILnyvectorrtothelineoftheconsequent off/J.Inparticular anyvectorsperpendicular totheante cedentoff/Jarethusreduced tozero. II(j)isaurodyadicthevectors•andtarebothurono matterwhatthevalue01rmaybe. Definition: Aplanardyadici8saidtopossessonetkgru0/ nullity. Alineardyadicissaidtopossesstwodegrees0/ nullity.Azerodyadicissaidtopossessthrudegrees01nul lityorcomplete nullity. 107.]Theorem: Thedirectproductoftwocomplete dyadics iscomplete; ofacomplete dyadicandaplanardyadic, planar; ofILcomplete dyadicandalineardyadic,linear. Theorem: Theproduct oftwoplanardyadics isplanar exceptwhentheplaneoftheconsequent ofthefirstdyadic intheproduct isperpendicular totheplaneoftheantece dentoftheseconddyadic. Inthiscasetheproduct reduces toalineardyadic-andonlyinthiscase. Let (j)=a1b1+aibi, 'I"=01d1+°idi' JJ=(j).'1". Thevector•='1".rtakesonall values intheplaneof01 and0i•=x01+Y0i· Thevector.'=(j)••takesonthevalues .'=(j)••=X(b1•01)a1+Y(b1.°2)a1 +x(bi•01)a2+Y(bi•0)ai' .'=IX(b1.°1)+Y(bl.02)}a1+{x(bi·01)+Y(b,.~)}at. Let where andLINEAR VECTOR FUNCTIONS .'=x'a1+y'ai, x'=x(b1•01)+y(b1•02)' y'=x(bi•01)+y(b2•02)'287 Theseequations mayalwaysbesolvedforxandywhen anydesiredvaluesx'andy'aregiven-that is,when.'has anydesired valueintheplaneof~and~-unlessthe determinant Ib1 •°1b1 •°iI=O. bi•01b2•02 Butby(25),Chap.II.,thisismerelytheproduct (b1Xb2) •(01X02)=O. Thevectorb1Xb2isperpendicular totheplaneofthecon sequents ofrp;and01Xai'totheplaneoftheantecedents of lJT.Theirscalarproduct vanishes whenandonlywhenthe vectorsareperpendicular -thatis,whentheplanesareper pendicular. Consequently.' maytakeonanyvalueinthe planeofa1anda,andrp.1Jl'istherefore aplanardyadic unlesstheplanesofb1andb2•01and02areperpendicular. Ifhowever b1andbi'01and0iareperpendicular .'cantake ononlyvaluesinacertainlineoftheplaneofa1and~.and hencerp•1J!'islinear.Thetheorem istherefore proved. Theorem: Theproduct oftwolineardyadics islinear exceptwhentheconsequent ofthefirstfactorisperpen diculartotheantecedent ofthesecond. Inthiscasethe product iszero-andonlyinthiscase. Theorem: Theproduct ofaplanardyadicintoalinearis linearexceptwhentheplaneoftheconsequents ofthe planardyadicisperpendicular totheantecedent ofthelinear dyadic. Inthiscasetheproduct iszero-andonlyinthis case. Theorem: Theproduct ofalineardyadicintoaplanar dyadicislinearexceptwhentheconsequent ofthelinear 288 VECTOR ANALYSIS dyadicisperpendicular totheplaneoftheantecedenta of theplanardyadic.Inthiscasetheproduct iszero-and onlyinthiscase. Itisimmediately evidentthatinthecasesmentioned the producta doreducetozero.Itisnotquitesoapparent that theycanreducetozeroinonlythosecases.Theproofsare similartotheonegivenaboveinthecaseoftwoplanar dyadic$~ They·arelefttothereader. Theproofofthe firsttheorem stated,page286,isalsolefttothereader. TheIdem/actor,. 1Reciprocals andConjugatu0/Dyadic& 108.]Definition: Ifadyadicappliedasaprefactor oras apostfactor toanyvectoralwayllyieldsthatvectorthe dyadicissaidtobeanidem/actor. Thatis if orif(/J.r=rforallvaluesofr, r •(/)=rforallvaluesofr, then(/Jisanidemfactor. ThecapitalIisusedasthesym bolforanidemfa.ctor. Theidemfa.ctor isacompkte dyadic. Fortherecanbenodirection inwhichI.rvanishes. Thw1'em: Whenexpressed innonionformtheidemfactor is I=ii+ii+ kk. (33) Henceallidemfactors areequal. .Toprovethattheidemfa.ctor takestheform(33)itis merelynecessary toapplytheidemfa.ctor Itothevectors i,i,krespectively. Let I=allii+a12ij+aISik +a21ji+a22jj+a2sjk + aS1ki+aS2kj+aSSkk. 1Inthetheoryofdyadiestheidemfaetor Iplay.anileanalogoue tounityill ordinary algebra. Thenotationisintended to8uggestthisanalogy. IfLINEAR VECTOR FUNCTIONS Ioi=alli+a2ti+aS1k. Ioi=i,289 Inlikemanneritmaybeshownthatallthecoefficients vanishexceptall'a22,assallofwhichareunity.Hence I=ii+jj+kk. (33) Theorem: Thedirectproductofanydyadicandtheidem factoristhatdyadic. Thatis, fP°I=fPandI°fP=fP. For «/)oI)or=fPo(Ior)=fPor, nomatterwhatthevalueofrmaybe.Hence,page266, InlikemanneritmaybeshownthatI°fP=fP. Theorem:Ifa',b',0'anda,b,0betworeciprocal systems ofvectorstheexpressions I=aa'+bb'+oo', I=a'11.+b'b+0'0 areidemfactors. Forby(30)and(31)Chap.II.,(34) andr =roll.a'+robb'+roo0', r =roll.'11.+rob'b +roo'o. Hencetheexpressions mustbeidemfactors bydefinition. Theorem: Conversely iftheexpression (/)=al+bm+on isanidemfactor 1,m,nmustbethereciprocal systemof a,b,c. 19 290 VECTOR ANALYSIS In:thefirstplacesincefPistheidemfa.ctor, itisacomplete dyadic. Hencetheantecedents a,b,0arenon-<:oplanar and possessasetofreciprocals a',b',0'.Let r=xa'+yb'+zo'. Byhypothesis r.fP=r. Then r •fP=xl+ym+·zn=xa':I-yb'+zc forallvaluesofr,thatis,forallvaluesofx,y,z.Hencethe corresponding coefficients mustbeequal.Thatis, l=a',m=b', n=o'. Theorem:IffPand1J!beanytwodyadics,andiftheproduct fP•1J!isequaltotheidemfa.ctor; 1thentheproduct1J!.fP, whenthefactorsaretakeninthereversed order,isalso equaltotheidemfactor. Let ToshowfP•1J!=I. 1J!.fP=I. r •(fP•1J!)=r •I=r, r·(fP•1J')•fP=r •fP, r •(fP•1J!)•fP=(r•fP)•(1J!•fP)=r •(fl. Thisrelationholdsforallvaluesofr.AsfPiRcomplete r •(/J musttakeonalldesiredvalues. Hencebydefinition 1J!.fP=I. Iftheproductoftwodyadicsisanidemfa.ctor, thatproduct maybetakenineitherorder. 109.]Definition: Whentwodyadicsaresorelatedthat theirproduct isequaltotheidemfactor,theyaresaidtobe 1Thisntee88itate8 boththedyadic8•and.tobecomplete. Fortheproduct oftwoincomplete dyadicsisincomplete andhencecouldnotbeequaltothe. ldemfactor. LINEAR VECTOR FUNCTIONS 291 reciprocals.1Thenotation usedforreciprocals inordinary algebraisemployed todenotereciprocal dyadics. Thatis, ifrp.IJI'=I,I Irp=IJI'-l= -andIJI'=rp-l= - . IJI' rp(35) Theorem: Reciprocals ofthesameorequaldyadicsare equal. LetrpandIJI'betwogivenequaldyadics, rp-landIJI'-I theirreciprocals asdefinedabove.Byhypothesis r;-(~~."u=;., and Toshow As Hencerp=IJI', rp.rp-l=I, 1JT.IJI'-I=I. rp-I=IJI'-I. rp.rp-I=I=IJI'.IJI'-I. rp=1Jf,rp.rp-l=rp.IJI'-l, rp-I.rp•rp-l=(fr-I.rp.IJI'-I, rp-l.rp=I, I.rp-l=rp-l=I.IJI'-l=IJI'-I. rp-l=IJI'-I.r }"'--t...L-f...._'_I-c...L .... '''-'-'-'-c'r~ V Thereciprocal ofrpisthedyadicwhoseantecedents arethe reciprocal systemtotheconsequents ofrpandwhoseconse quentsarethereciprocal systemtotheantecedents ofrp. Hacomplete dyadicrpbewrittenintheform rp=al+bm+0n, itsreciprocal isrp-l=I'a'+m'b'+n'o'. (36) For(aI+bm+on). (I'a'+n'b'+n'o')=aa'+bb'+00'. Theorem:Ifthedirectproducts ofacomplete dyadicrp intotwodyadics IJI'and!JareequalasdyadicsthenIJI'and!J 1Anincomplete dyadichasno(finite)reciprocal. 292 VECTOR ANALYSIS t-lxr=t.JxI,areequal.Iftheproduct ofadyadic fPintotwovectoI"IJ randI(whether themultiplication beperformed withadot oracr08ll)a.reequal,thentheve~I'8randIareequal. Thatis, if fP.1Jl'=fP.fJ, then1Jl'=g, andif fP.r=fP•.,thenr=., (37) andiffPxr=fPx., thenr=.. Thismaybeseenbymultiplying eachoftheequa.tions through bythereciprocal offP, fP-1•fP.1Jl'=1Jl'=rp-l•rp.g=g, rp-l•rp.r = r = fP-1•fP.I =., fP-1•fPX r =IX r =rp-l•fPX I =IXI. Toreducethelastequation proceed asfollows. LettDe anyvector, Hence txr=tXI. Astisanyvector,risequaltoI. Equations (37)givewhatisequivalent tothelawofcan celation forcomplete dyadics. Complete dyadics maybe canceled fromeitherendofanexpression justasifthey werescalarquantities. Thecancelation ofanincomplete dyadicisnotadmissible. Itcorresponds tothecancelation ofazerofactorinordinary algebra. 110.]Theorem: Thereciprocal oftheproduct ofany numberofdyadicsisequaltotheproductofthereciprocals takenintheoppos-ite order. Itwillbesufficient togivetheproofforthecaseinwhich theproductconsistsoftwodyadics. Toshow LlNEAR VECTOR FUNCTIONS 293 ((/).1j!')-1=1j!'-1.(/)-1, (/).1j!'.1j!'-1•(/)-1=(/).(1j!'.1j!'-1)•(/)-1=(/).(/)-1=I. Hence ((/).7Jf)• (1j!'-1•(/)-1)=I. Hence (/).7Jfand1j!'-1.(/)-1mustbereciprocals. Thatis, ((/).7Jf)-1=1j!'-1.(/)-1. Theproofforanynumber ofdyadicsmaybegiveninthe samemannerorobtained bymathematical induction. Definition: Theproducts ofadyadictP,takenanynumber oftimes,byitselfarecalledpowersoftPandaredenoted in thecustomary manner. (/).tP=(/)2, tP.tP.tP=tP.(/)2=(/)8, and80forth. Theorem: Thereciprocal ofapowerof(/)isthepowerof thereciprocal of(fl. (tPII)-1=«/)-1)11=tP-II• (37) Theprooffollowsimmediately asacorollary ofthepreced ingtheorem. Thesymbol (/)-11maybeinterpreted asthe nthpowerofthereciprocal oftPorasthereciprocal of thenthpoweroftP. If(/)beinterpreted asanoperator determining atrans formation ofspace,thepositive powersof(/)correspond to repetitions ofthetransformation. Thenegative powersoftP correspond totheinversetransformations. Theidemfactor corresponds totheidentical transformation -thatis,notrans formation atall.Thefractional andirrational powersof(/) willnotbedefined. Theyareseldomusedandarenot single-valued. Forinstance theidemfactor Ihasthetwo squareroots±I.Butinaddition totheseithasadoubly infiniteI;ystemofsquarerootsoftheform (/)=-ii+jj+kk. 294 VECTOR ANALYSIS Geometrically thetransformation r'=(/).r isareflection ofspaceinthejk-plane. Thistransformation replaces eachfigurebyasymmetrical figure,symmetrically situated upontheopposite sideofthejk-plane. Thetrans formation issometimes calledperversion. Theidemfactor hasalsoadoublyinfinitesystemofsquarerootsoftheform W=ii-jj-kk. Geometrically thetransformation r'=IJ'.r isareflection inthei-axis.Thistransformation replaceseach figurebyitsequalrotatedaboutthei-axisthrough anangle of180°.Theidemfactorthuspossesses notonlytwosquare roots;butinaddition twodoublyinfinitesystemsofsquare rootS;anditwillbeseen(Art.129)thatthesearebyno meailSall. l11.JTheconjugate ofadyadichasbeendefined(Art.99) asthedyadicobtained byinterchanging theantecedents and consequents ofagivendyadicandthenotation ofasubscript Chasbeenemployed. Theequation (9) hasbeendemonstrated. Thefollowing theorems concerning conjugates areuseful. Them'em: Theconjugate ofthesumordifference oftwo dyadicsisequaltothesumordifference oftheconjugates, Theorem: Theconjugate ofaproductofdyadics isequal totheproductoftheconjugates takenintheoppositeorder. LINEAR VECTOR FUNCTIONS 295 Itwillbesufficient todemonstrate thetheorem incase theproductcontains twofactors. Toshow (fP•IJI)c=IJIc•fPc. (40) (fP•1JI)c'r=r •(fP•IJI)=(r•fP)•IJI, Hence(r•fP)•IJI=IJIc'(r•fP)=IJIc•fPc'r. (fP•IJI)c=IJIc•fP(Jo Theorem ..Theconjugate ofthepowerofadyadicisthe poweroftheconjugate ofthedyadic. (41) Thisisacorollary oftheforegoing theorem. Theexpression t/J'Cmaybeinterpreted ineitheroftwoequalways. Theorem: Theconjugate ofthereciprocal ofadyadicis equaltothereciprocal oftheconjugate ofthedyadic. For(fP-1)c=(fPc)-1=fPc1. (fP-1)c'fPc=(fP. fP-1)C=Ic=1.(42) Theidemfactor isitsownconjugate asmaybeseenfrom thenonionform. Hence HenceI=ii+ii+kk (fPc)-1.fPc=J. (fPc)-1•fPc=(f!r1)c' fPC" «(/)c)-1=(fP-1)(Jo Theexpression (/)c-1maytherefore beinterpreted ineither oftwoequivalent ways-asthereciprocal oftheconjugate orastheconjugate ofthereciprocal. ])efinition: Ifa.dyadicisequaltoitsconjugate, itissaid tobeself-conjugate. Ifitisequaltothenegative ofitscon- 296 VECTOR ANALYSIS jugate,itissaidtobeanti-self-conjugate. Forself-conjugate dyadics. r.fP=fP.r, fP=fPo- Foranti-self-conjugate dyadics r.r[J=-fP.r, fP=-fPc- Theorem: Anydya.dicmaybedividedinoneandonlyone wa.yintotwopartsofwhichoneisself-eonjugate andthe otheranti-self-eonjugate. For But and1 1fP=1.(fP+fPc)+i(fP-fPc)· (fP+fPc)c=fPc+fPcc=fPc+fP, (fP-fPc)c=fPc-fPcc=fPc-fP.(43) andHencethepart!(fP+fP-C>isself-eonjugate; andthepart HfP-fPd,anti-self-conjugate. Thusthedivision hasbeen accomplished inoneway.Let ~(r[J+fPc)=fP' ~(fP-fPc)=fP". fP=fP'+fP". Suppose itwerepossible todecompose fPinanotherway intoaself-eonjugate andananti-self-eonjugate part.Let then r[J=(r[J'+Q)+(r[J"-Q). Where (r[J'+Q)=(r[J'+Q)e=r[J'e+Qe=fP'+Q(J' Henceif(r[J'+Q)isself-conjugate, Qisself-conjugate. -(r[J"-Q)=(r[J"-Q)e=r[J"e-Qe= -r[J"-Qe- Henceif(r[J"-Q)isanti-self-conjugate Qisanti-self conjugate. LINEAR VECTOR FUNCTIONS 297 , Anydyadicwhichisbothself-conjugate andanti-self-conju- gateisequaltoitanegative andconsequently vanishes. HenceQit!zeroandthedivision ofrpintotwopartais unique. Anti-8elf-conjugate Dyadics. TheVecturProduct 112.]Incaserpisanydyadictheexpression rpl!=!(rp-rp) ~ c givestheanti-self~onjugate partofrp.Ifrpshouldbeen tirelyanti-fJelf-conjugate rpisequaltorp".Lettherefore rp" beanyanti-self~onjugate dyadic, rpl!=~(f/J-rpc). Suppose rp=al+bm+en, (fj_¢c.).r-rp-rpc=a1- 1a+bm - mb+0n -no,. J/lf'•r = a1•r - 1a •r +bm•r - mb •r +en·r -no.r. But al.r-Ia.r=-{axI)xr, bm·r-mb.r=- (bxm)xr, on·r -no.r= -(0xn)xr. Hence rpl!•r= -~(ax1+b x m+0xn)xr. Butbydefinition Hencerp"= a x 1+b x m+c xn. rpl!•r - -!f/Jx r-2" , r.rpll=rpl!c·r=-f/JI!.r=~f/J" xr=-~r Xrp". Theresultamaybestatedinatheorem DBfollows. Theorem: Thedirectproduct ofany anti-self~onjugate dyadicandthevectorrisequaltothevectorproduct 01 minusonehalfthevectorofthatdyadicandthevectorr. 298 VECTOR ANALYSIS 1 12r.(fP-fPC)=-2rx fP".(44) (45)Theorem: Anyanti-self-conjugate dyadicfP!'possesses one degreeofnullity.Itisauniplanar dyadictheplaneof whoseconsequents andantecedents isperpendicular tofP,,", thevectoroffP. Thistheorem followsasacorollary fromequations (44). Theorem: AnydyadicfPmaybebrokenupintotwoparts ofwhichoneisself-conjugate andtheotherequivalent to minusonehalfthevectoroffPusedincrossmultiplication. fP.r=rp'.r-~(/J" Xr, fP.=fP'.-~rpX. l!"orsymbolically 113.]Anyvector 0usedinvectormultiplication definesa linearvectorfunction. For oX(r+I)=0Xr+0X•• Henceitmustbepossible torepresent theoperator 0Xasa dyadic. Thisdyadicwillbeuniplanar withplaneofits antecedents andconsequents perpendicular to0,sothatit willreduceallvectorsparallelto0tozero.Thedyadicmay befoundasfollows oXr=I •0Xr=Ix0•r=(Ix0)•r. By(31) I.(0xI)=(Ix0)•I, (Ix0)•r=!(Ix0)•I}•r={I•(0xI)}•r =I.(0xI).r=(0XI).r. Hence 0x r=(Ix0)•r=(0xI)•r, and r x0=r·(IX0)=r •(0XI). (46) Thismaybestatedinwords. LINEAR VECTOR FUNCTIONS 299 Theorem: Thevector 0usedinvectormultiplication with avectorrisequaltothedyadic1 x0or0 XIusedindirect multiplication withr.If0precedes rthedyadicsaretobe usedasprefactors;if0followsr,aspostfactors. Thedyadics I x0and0x 1areanti-aelf-conjugate. Incasethevector 0isaunitvectortheapplication ofthe operator 0xtoanyvectorrinaplaneperpendicular to0is equivalent toturningrthrough apositive rightangleabout theaxiso.Thedyadic 0x 1or1x0where 0isaunitvector therefore turnsanyvectorrperpendicular to0through a rightangleabouttheline0asanaxis.Ifrwereavector lyingoutofaplaneperpendicular to0theeffectofthedyadic I x0or0x Iwouldbetoannihilate thatcomponent ofrwhich isparallelto0andtumthatcomponent ofrwhichisperpen dicularto0through arightangleabout 0asaxis. Ifthedyadicbeappliedtwicethevectorsperpendicular to rarerotatedthrough tworightangles. Theyarereversed in direction. Ifitbeappliedthreetimestheyareturnedthrough threerightangles. Applying theoperator I x0or0 XIfour timesbringsavectorperpendicular to0backtoitsoriginal position. Thepowersofthedyadicaretherefore (IX0)2=(0X1)2=-(I-00), (Ix0)3=(0X1)3= - 1x0 = - 0xI, (Ix0)·=(0xI)·=I -00, (IX0)6=(0X1)6= 1x0 = 0xI.(47) ItthusappearsthatthedyadicI x0or0 XIobeysthesame lawasfarasitspowersareconcerned asthescalarimaginary v=tinalgebra. ThedyadicI x0or0x 1isaquadrantal versoronlyfor vectorsperpendicular too.Forvectorsparallelto0itacts asanannihilator. Toavoidthiseffectandobtainatrue 800 VECTOR ANALYSIS quadrantal versorforallvectorsrinspaceitillmerelyneces sarytoaddthedyadcctothedyadicI x corc x1. (48)x=IXc+co=cXI+C.o, XJ=-I, XS=-X, X·=I, X'=X. ThedyadicXtherefore appears asafourthrootofthe idemfactor. Thequad~tal versorXisanalogous tothe imaginaryv'-1ofascalaralgebra.. ThedyadicXiscom pleteandconsists oftwopartsofwhichIXcisanti-Belf conjugate; and00,self-conjugate. 114.]Ifi,j,karethreeperpendicular unitvectorsIf Ixi=ixI=kj-jk, Ixj=jxI=ik-ki, I xk= k x I = j i- ij, asmaybeseenbymultiplying theide'mfactor I=ii+jj+kk(49) intoi,j,andksuccessively. Theseexpressions represent quadrantal versorsabouttheaxisi,j,krespectively combined withannihilators alongthoseaxes.Theyareequivalent, whenusedindirectmultiplication, toix,jx,kxrespectively, (Ixk)2=(kxIr~= -(ii+jj), (IXk)3=(kxI)S= -(ji -ij), (Ixk)·=(kXI)i=ii+jj, Theexpression (Ixk)iisanidemfactor fortheplaneofiand j,butanannihilator forthedirection k.Inasimilarman nerthedyadkkisanidemfactor for,thedirection k,butan LINEAR VECTOR FUNCTIONS 301 iloilo'+bb'an~hilator fortheplaneperpendicular tok.Thesepartial idemfactors arefrequently useful. Ifa,b,0areanythreevectorsanda',b',0'thereciprocal system, medasa.prefactorisanidemfactor forallvectorsinthe planeofaandb,butanannihila.tor forvectorsinthedirec tiono.Usedasapostfactor itisanidemfactorforallvectors, intheplaneofa'andb',butanannihilator forvectorsinthe direction 0'.Inlikemannertheexpression 00' usedasaprefactor isanidemfactor forvectorsinthedirection 0,butforvectorsintheplaneofaandbitisanannihilator. Usedasapostfactor itisanidemfactor forvectorsinthe direction 0',butanannihilator forvectorsintheplaneofa' andb',thatis,forvectorsperpendicular ofo. Ifaandbareanytwovectors For(axb)x I = I x (axb)= ba- ab.(50) (ClioXb)xI}.r=(a xb)xr=ba.r-ab.r=(ba-ab).r. Thevectora x bincrossmultiplication istherefore equalto thedyadic(ba- ab)indirectmultiplication. Ifthevector isusedasaprefactor thedyadicmustbesoused. (axb)x r =(ba - ab)•r, rx(axb)=r.(ba-ab). (51) Thisisasymmetrical andeasyforminwhichtoremember theformulaforexpanding atriplevectorproduct. 802 VECTOR ANALYSIS r'=rp.r.Reduction ojDyad-icstoNormalForm 115..]Letrpbeanycomplete dyadicandletrbeaunit vector. Thenthevectorr' i-'=rp.r isalinearfunction ofr.Whenrtakesonallvaluesconsis tentwithitsbeingaunitvector-thatis,whentheterminus ofrdescribes thesurfaceofaunitsphere,-thevectorr' variescontinuously anditsterminWl describes asurface. This surfaceisclosed.Itisinfactanellipsoid.l Theorem: Itisalwayspossibletoreduceacomplete dyadic toasumofthreetermsofwhichtheantecedents among themselves andtheconsequents amongthemselves aremutu allyperpendicular. Thisiscalledthenormaljormofrp. rp=ai'i+bj'j+ak'k. Todemonstrate thetheorem consider thesurfacedescribed by Asthisisaclosedsurfacetberemustbesomedirection ofr whichmakesr'amaximum oratanyl""cl.tegivesr'asgreat avalueasitispossibleforr'totakeon.Letthisdirection ofrbecalledi,andletthecorresponding direction ofr' thedirection inwhichr'takesonavalueatleastasgreatas any-becalleda.Consider nextallthevaluesofrwhich lieinaplaneperpendicular toi.Thecorresponding values ofr'lieinaplaneowingtoIifactthatrp.risalinearvector 1Thismaybeproved 88folloWII: r'=••r r =.-1.r'= r•••-1. Hence r.r=l= r'.(••-1.•-1).r'=r' .....r'. ByexpreesingY innonionform,theequationr'.....r'=1isseentobeofthelIeCond degree. Hencer'describea aqnadricInr£ace. TheonlycloaedquadricInrface i.atheelliJllOicL UNEAR VEC'TOR FUNCTIONS 303 function. Ofthesevaluesofr'onemustbeatleastll.Bgreat asanyother.Callthisbandletthecorresponding direction ofrbecalledj.Finallychoosekperpendicular toiandj uponthepositive side ofplaneofiandj.Let0bethe valueofr'whichcorresponds tor =k.SincethedyadicfP changesi,j,kintoa,b,0itmaybeexpreBBed intheform fP=ai+bj+ok. Itremainstoshowthatthevectorsa,b,casdetermined abovearemutually perpendicular. r'=(ai+bj+ok).r, dr~=(ai+bj+ok).dr, r'.dr'=r'.ai.dr +r'.bj.dr +r'.ok.dr. Whenrisparalleltoi,r'isamaximum andhencemustbe perpendicular todr'.Sincerisaunitvectordrisalways perpendicular tor.Hencewhenrisparalleltoi r'.bj.dr+r'.ok.dr=O. Iffurtherdrisperpendicular toj,r'.0vanishes, andif drisperpendicular tok,r'•bvanishes. Hencewhenria parallel toi,r'isperpendicular tobothbandc.Butwhen risparalleltoi,r'isparallelto&.Hence &isperpendicular tobandc.Consider nexttheplaneofjandkandthe planeofbando.Letrbeanyvectorintheplaneofjandk. r'=(bj+ok)•r, dr'=(bj+ok)•dr, r'.dr'=r'•bj.dr+r'·0k·dr. Whenrtakesthevaluej,r'isamaximum inthisplaneand henceisperpendicular todr'.Sincerisaunitvectoritis 894 VECTOR ANALYSIS perpendicular todr.Hencewhenr18parallel toj,dr isperpendicular toj,and r'.dr'=O=r'.c t·dr. Hencer'.0iszero.Butwhenrisparalleltoj,r'takesthe valueb.Consequently bisperpendicular too. Ithastherefore beenshownthataisperpendicular toband 0,andthatbisperpendic:ular too.Consequently thethree antecedents offParemutually perpendicular. Theymaybe denotedbyi',j',k'.ThenthedyadicfPtakestheform fP=ai'i+hj'j+ck'k, (52) wherea,h,carescalarconstants positiveornegative. 116.]Theorem: Thecomplete dyadicfPmayalwaysbe reduced toasumofthreedyadswhoseantecedents and whoseconsequents formaright-handed rectangular system ofunitvectorsandwhosescalarcoefficients areeitherall positiveorallnegative. fP=±(ai'i+hj'j+ct't). (53) Theproofofthetheorem depends uponthestatements madeonpage20thatifoneorthreevectorsofaright-handed systembereversed theresulting systemisleft-handed, but iftwobereversed thesystemremainsright-handed. Ifthen oneofthecoefficients in(52)isnegative, thedirections ofthe othertwoaxesmaybereversed. Thenallthecoefficients arenegative.Iftwoofthecoefficients in(52)arenegative, thedirections ofthetwovectorstowhichtheybelongmay bereversed andthenthecoefficients infPareallpositive. Henceinanycasethereduction totheforminwhichall thecoefficients arepositive orallarenegative hasbeen performed. Asalimiting casebetween thatinwhichthecoefficients areallpositiveandthatinwhichtheyareallnegative comes LINEAR VECTOR FUNCTIONS 305 thecaseinwhichoneofthemiszero.Thedyadicthen takestheform d>=ai'i+bj'j (54) (55) d>=aii+bjj+ckkandisplanar. Thecoefficients aandbmayalwaysbetaken positive. Byaproofsimilartotheonegivenaboveitis possible toshowthatanyplanardyadicmaybereduced to thisform.ThevectorsiIandjIareperpendicular, andthe vectorsiandjarelikewise perpendicular. Itmightbeaddedthatincasethethreecoefficients a,b,c inthereduction (53)arealldifferent thereduction canbe performed inonlyoneway.Iftwoofthecoefficients (say aandb)areequalthereduction maybeaccomplished inan infinitenumberofwaysinwhichthethirdvectork'isalways thesame,butthetwovectorsi',j'towhichtheequalcoeffi cientsbelongmaybeanytwovectorsintheplaneper pendicular tok.Inallthesereductions thethreescalar coefficients willhavethesamevaluesasinanyoneofthem. Ifthethreecoefficients a,b,careallequalwhenrpisreduced tothenormalform(53),thereduction maybeaccomplished inadoublyinfinite number ofways.Thethreevectors i',j',k'maybeanyright-handed rectangular system iIi space.Inallofthesereductions thethreescalarcoefficients arethesameasinanyone ofthem.Thesestatements will notbeproved. Theycorrespond tothefactthattheellipsoid whichisthelocusoftheterminus ofr'mayhavethree different principal axesoritmaybeanellipsoid ofrevolution, orfinallyasphere. Thwrem: Any self~onjugate dyadicmaybeexpressed in theform wherea,b,andcarescalars,positiveornegative. Let d>=ai'i+bj'j+ck'k, d>0=aii'+bjj'+ckk', 20(52) 306 VECTOR ANALYSIS ~.~c=a~i'i'+b~j'j'+cllk'k', ~o·~=a2ii +b~jj +c~kk. Since ~=~OI ~·~c=~o·~=~~· I=ii+jj+kk=i'i'+j'j+k'k', ~~-a~I=(b~-a~)j'j'+(c2-a2)k'k', (~~-a~I).i'=O ~~-a2J=(b2-a2)jj+(c~-a~)kk, (~~-a~I)oi=O. Ifiandi'werenotparallel (~~-a~I)wouldannihilate twovectorsiandiIandhenceeveryvectorintheirplane. (~~-a~I)wouldtherefore possesstwodegreesofnullity andbelinear.Butitisapparent thatifa,b,caredifferent thisdyadicisnotlinear.Itisplanar. HenceiandiImust beparallel. InlikemanneritmaybeshownthatjandjI. kandk'areparallel. Thedyadic ~therefore takestheform ~=aii+bjj+ckk wherea,b,carepositiveornegative scalarconstants. (56) ab:cd=a·c b·d. ,,.,..~Thisproductevidently obeysthecommutative law,, • 'l•lt~(.1 ," ab :cd=cd:ab,J':.[~~'~)"'l":,7><cJ DoubleMultiplication1 e-l-IOo.....fl ~('\tl,,\ vJI .117.]Definition: Thedoubledotproductoftwodyadsis ':::. ';~~~('"thescalarquantity obtained bymultiplying thescalarproduct I•••\h(-f-_oftheantecedents bythescalarproduct oftheconsequents. ~,,".'1"1"",Theproductisdenoted byinserting twodotsbetween the I').",I..;J ~''~)I ~dyads. 1Theresearches ofProfllll8Or GibbsuponDoubUMultiplicaliOfl arehere printedforthefi",ttime. LINEAR VECTOR FUNCTIONS 307 andthedistributive lawbothwithregardtothedyadsand withregardtothevectorsinthedyads.Thedoubledot product oftwodyadicsisobtained bymultiplying theprod uctoutformally according tothedistributive lawintothe sumofanumberofdoubledotproducts ofdyads. If andqJ=&1b1+&2b2+&SbS+..• 7J!'=01d1+02d2+OsdS+•.. qJ:7J!'=(&1b1+ a 2b2+ &Sb3 + ...) :(01d1+ °2dS +°ada+...) = a1b1:01d1+ a1b1:02d2+ a1b1:Osda+ . +&2b2:01dl +a2b2:02d2+&2b2:0sda +(56)' +asba:01d1+asbs:02d2+&abs:Osds+ . + . qJ:7J!'=&tOOIb10dt+ a10°2blod2+&lo0S b1ods+··· +a2001b2od1+d2°02b20d2+&20Osb20d3+.•. +as001bs0d1+as002bs0dt+ a30Osba0ds+... + . (,1>6)" Definition: Thedoublecrossproduct oftwodyadsisthe dyadofwhichtheantecedent isthevectorproductofthe antecedents ofthetwodyadsandofwhichtheconsequent is thevectorproductoftheconsequent ofthetwodyads.The product isdenoted byinserting twocrossesbetween the dyads &b~od=&xo bxd. (57) Thisproductalsoevidently obeysthecommutative law &b:od=od:ab, 808 VECTOR ANALYSIS andthedistributive lawbothwithregardtothedyadsand withregardtothevectorsofwhichthedyadsarecomposed. Thedoublecrossproductoftwodyadicsistherefore defined astheformalexpansion oftheproduct according tothe distributive lawintoasumofdoublecrossproducts of dyads. If andrp=&1b1+&1bl+&sbs+ . 1Jf=°1d1+°1dl+Osds+ . rp=lJl'=(&1b1+albl+asbs+...):(Oldl +Old, +osds+"') =&1b1:01d1+alb1:°1dl+&1b1:Osds+ . +albl=01d1+albl:°1dl+a,bl:Osds+(57)' +asbs:0ldl+asbs:0ldl+asbs:°sds+ . + . rp:lJl'=a1X01b1Xd1+&1X°1b1Xdz+&1XOsb1Xda+ . +&zX01blXd1+&zXCzb,Xdz+&zXOsblXda+ . +asXOsbsXd1+asXCzbsXd1+asXOsbsXda+ . +. (57)" Theorem: Thedoubledotanddoublecrossproducts of twodyadicsobeythecom~utative anddistributive lawSof multiplication. Butthedoubleproducts ofmorethantwo dyadics(whenever theyhaveanymeaning) donotobeythe associative law. rp:lJl'=lJl':rp (rp:lJT):.Q-:trp:(lJl':!J).(58) Thetheorem issufficiently evidentwithoutdemonstration. LINEAR VECTOR FUNCTIONS 809 Theorem: Thedoubledotproduct oftwofundamental dyadsisequaltounityortozeroaccording asthetwo dyadsareequalordifferent. ij:ij=i.i j.j=l ij:ki=i.k joi=O. Theorem: Thedoublecrossproduct oftwofundamental dyads(12)isequaltozeroifeithertheIUltecedents orthe consequents areequal.Butifneitherantecedents norcon sequents areequaltheproduct isequaltooneofthefunda mentaldyadstakenwithapositive oranegative sign. Thatis ij=ik=ixi jxk= 0 ij=ki =ixkjxi=+jk. Thereexistsascalartripleproduct ofthreedyadsin whichthemultiplications aredouble. LetfP,1JT,/Jbeany threedyadics. Theexpression fP:7J!':/J (59) ab:od:ef=[ace] [bdf].isascalarquantity. Themultiplication withthedouble crossmustbeperformed first.Thisproduct isentirelyin dependent oftheorderinwhichthefactorsarearranged or theposition ofthedotandcrosses. Letab,cd,andefbe threedyads, Thatis,theproductofthreedyadsunitedbyadoublecross andadoubledotisequaltotheproductofthescalartriple productofthethreeantecedents bythescalartripleproduct ofthethreeconsequents. Fromthisthestatement made abovefollows. Forifthedotsandcro88esbeinterchanged' oriftheorderofthefactorsbepermuted cycliclythetwo scalartripleproducts arenotaltered.Ifthecyclicorderof 810 VECTOR ANALYSIS thefactorsisreversed eachscalartripleproduct changes sign.Theirproduct therefore isnotaltered. 118.JAdyadic ~maybemultiplied byitselfwithdouble cross.Let ~=al+bm+on ~:~=(al+bm+on):(al+bm+on) =axaIxl+axb Ixm+axc Ixn +bxamxl+bxb mxm+bxc mxn +cxanxl+oxb nxm+cxc nxD. Theproducts inthemaindiagonal vanish. Theothersare equalinpairs.Hence ~~~= 2(bx0m x n+ 0x a n x I+ a x b I x m).(60) 1£a,b,0and1,m,narenon~oplanar thismaybewritten 2'~)(~= (a'I'+b'm'+o'n'). (60)' )([aboJ[1mn] Theproduct ~~~isaspeciesofpower of~.Itmaybere gardedasasquareof~.Thenotation ~2willbeemployed torepresent thisproductafterthescalarfactor2hasbeen stricken out. ~)(~ ~2=T=(bx c mx n + c x a n XI+ aXb I xm)(61) Thetripleproductofa.dyadic ~expressed asthesumof threedyadswithitselftwicerepeated is ~:~:~=2~2:~ ~2:~=(bx c m x 'n+ c x a n XI + a x b I x m) :(al+bm+on). Inexpanding thisproducteveryterminwhichaletteris repeated vanishes. Forascalartripleproductofthreevee- (63)LINEAR VECTOR FUNCTIONS 311 torstwoofwhichareequaliszero.Hencetheproduct reducEls tothreetermsonly (/)2:(/)=[bea][mnl]+[eab][nlm]+[abe][lmn] or (/)2:(/)=3[abe][lmn] (/)~(/):(/)=6[abe][lmn]. Thetripleproduct ofadyadicbyitselftwicerepeated is equaltosixtimesthescalartripleproductofitsantecedents multiplied bythescalartripleproduct ofitsconsequents. Theproduct isaspeciesofcube.Itwillbedenotedby(/)8 afterthescalarfactor6hasbeenstricken out. (/):(/):(/) (/)8=6=[abe][lmn]. (62) 119.]If(/)2becalledthesecondof(/);and(/)8'thethirdof (/),thefollowing theorems maybestatedconcerning the secondsandthirdsofconjugates, reciprocals, andproducts. TMorem: Thesecondoftheconjugate ofadyadicisequal totheconjugate ofthesecondofthatdyadic. Thethirdof theconjugate isequaltothethirdofthedyadic. «/)2)C=«/)0)2 (/)8=«/)c)8' TMorem: Thesecondandthirdofthereciprocal ofa dyadicareequalrespectively tothereciprocals ofthesecond andthird. Let«/)-1)2=«/)2)-1=(/)2-1 «/)-1)8=«/)8)-1=(/)3-:1 (/)=al+bm+en (/)-1=l'a'+m'b'+n'e' a'l'+b'm'+0'n'(/)--- 2-[ab0][1mn](64) (36) (60)' 812 VECTOR ANALYSIS (~11)-1=[abo][1mn](Ia+mb+no) -1_1a+mb+no. (~)II-[a'b'0'][1'm'n'] But(a'b'0'][ab0]=1and[1'm'n'][1mn]=1. Hence (~11)-1=(~-1),=~,-l. fl.=[abo][1mn], (~-1_ 1 a)-[abo][lmn]' «(J)-I)a=[a'b'0'][1'm'n']. Hence (~a)-1=(~-l)a=iPa-1. .Theorem: Thesecondandthirdofaproduct areequal respectively totheproductofthesecondsandtheproductof thethirds. «(J).gJ),=fill'F, (fl.F)a=~aFa·(65) ChooselUlythree non~op1lUlar vectors1,m,nasconsequents of(J)andlet1',m',n'betheantecedents ofF. ~=al+b m+oD, 7Jf=I'd+m'e+n'f, (J).F=ad+be+of, ((J).'111= bXceXf + cXa fXd + aXb dXe, fill=bXc mXn+0 Xa nX1+aXb 1Xm, F,=m'Xn'eXf+n'Xl'fXd+l'Xm'dXe. Hence ~"1J.!',=bXceXf+0 XafXd+aXb dXe. Hence (iP.gJ)1I=fI,'1J.!'1I' «(J).F)a=[abo][def] LINEAR VECTOR FUNCTIONS 818 Hence HenceiP.=[ab0][lmn], ".=[I'm/n'][def]. iP.".=[abo][def]. (iP.")s=iPs"•• (66)(iP-)'J=(iP'J)-=iP'J (iP-)s=(iPs)-=iPs-'Thwrf/T1l,: Thesecondandthirdofapowerofadyadicare equalrespectively tothepowersofthesecondandthirdof thedyadic. (67)Thwrem: Thesecondoftheidemfactor istheidemfactor. Thethirdoftheidemfactor isunity. I'J=I Is=1. Thwrem: Theproduct ofthesecondandconjugate of adyadicisequaltotheproduct ofthethirdandthe idemfactor. iP'J•iPc=iPsI, (68) iP2=bx0mXn+0 Xa n x1+aXb 1xm, iP0=1a+mb+nc, iP'J.iPo=[lmn](bx0a +0 Xa b + a Xb0). Theantecedents a,b,0ofthedyadic tPmaybeassumed to benon-coplanar. Then (bxoa+oxa b+axb 0)=[abo](a'a+b'b+o'o) =[abo]I. Hence tP2•iPs=tP8I. 120.]LetadyadictPbegiven.Letitbereducedtothe Bumofthreedyadsofwhichthethreeantecedents are non-coplanar. 814 VECTOR ANALYSIS ~=al+b m+0n, ~t=bX 0-mXn+0 Xa nX1+a xb 1xIll, tPs=[abo][lmn]. Theorem: Thenecessary andsufficient condition thata dyadictPbecomplete isthatthethirdoffIbedifferent from zero. Foritwasshown(Art.106)thatboththeantecedents and theconsequents ofacomplete dyadicarenon-<Joplanar. Hencethetwoscalartripleproducts whichoccurin~, cannotvanish. Theorem: Thenecessary andsufficient condition thata dyadicfIbeplanaristhatthethirdof~shallvanishbutthe secondoffPshallnotvanish. Itwasshown(Art.106)thatifadyadic ~beplanaritscon sequents 1,In,nmustbeplanarandconversely iftheconse quentsbecoplanar thedyadicisplanar. Henceforaplanar dyadic ~smustvanish. But~" cannotvanish. Sincea, b,0havebeenassumed non-<Joplanar, thevectorsbx0,0xa, a x barenon-<Joplanar. HenceiffP"vanishes eachofthe vectorsm xn,D X1,1x mvanishes -thatis,1,m,narecol linear.Butthisisimpossible sincethedyadictPisplanar andnotlinear. Theorem: Thenecessary andsufficient condition thata non-vanishing dyadicbelinearisthatthesecondoftP,and consequently thethirdof~,vanishes. Forif~belineartheconsequents 1,In,n,arecollinear. Hencetheirvectorproducts vanishandtheconsequents of fP"vanish.Ifconversely tP"vanishes, eachofitsconsequents mustbezeroandhencetheseconsequents oftParecollinear. Thevanishing ofthethird,unaccompanied bythevanish ingofthesecondofadyadic,impliesonedegreeofnullity. Thevanishing ofthesecondimpliestwodegreesofnullity. LINEAR VECTOR FUNCTIONS 315 Thevanishing ofthedyadicitBelfiscomplete nullity. The resultBmaybeputintabularform. tPa:t0,tPiscomplete. tPa=0,tP,,:t0,fPisplanar. (69) tPa=0,tP"=0,fP:t0,tPislinear. Itfollowsimmediately thatthethirdofanyanti~elf-conjugate dyadicvanishes; buttheseconddoesnot.Foranysuch dyadicisplanarbutcannotbelinear. NonionForm.Determinants.lInvariants ofaDyadic 121.]IffPbeexpressed innonionform tP=anii+al'Jii+alait (13) +a"di+at"jj+a"ajt +aalti+a8'Jtj+aaatt. Theconjugate oftPhasthesamescalarcoefficientB asfP,but theyarearranged symmetrically withrespecttothemain diagonal. Thus tPc=anii+at1ij+aalit, +al"ji+at"jj+a3"jt, (70) +alati+asait+a83tt. ThesecondoftPmaybecomputed. Take,forinstance, one term.Letitberequired tofindthecoefficient ofijiniPs' WhattermsinfPcanyieldadoublecrossproductequalto ij?Thevectorproductoftheantecedents mustbeiand thevectorproductoftheconsequentB mustbej.Hencethe anteced,ents fIlustbejandt;andtheconsequentB, tandi. Thesetermsare a"di~aaatt= -a"la83ij a31ti~atait=aalat3ij. 1Theresultllholdonlyfordeterminants ofthethirdorder.Theextension to determiDantll ofhigherordersisthroughMultiple Algebra. 816 VECTOR ANALYSIS Hencetheterminijin~sis (aSIass-auaaa)ij. ThisisthefirstminorofalSinthedeterminant Thisminoristakenwiththenegative sign.Thatis,the coefficient ofijin~siswhatistermedthecofactorofthe coefficient ofijinthedeterminant. Thecofactorismerely thefirstminortakenwiththepositive ornegative sign according asthesumofthesubscripts ofthetermwhose firstminorisunderoonsidemtion isevenorodd.Theco efficientofanydyadin~siseasilyseentobethecofactorof thecorresponding terminfP.Thecofactors aredenoted generally bylargeletters. assIisthecofactorofau· aaa aSSIisthecofactorofalS• aaa alSIisthecofactorofan.au Withthisnotationthesecondof~becomes ~s=.Auii+.A12ij+.A18ik +.Astii +.Assjj+Asskk (71) +.A81ki+.Asskj+.AaakIL Thevalueofthethirdof~maybeobtained bywriting ~ asthesumofthreedyads ~=(aui +as!j +a81k)i +(a1lli +assj+ank)j +(alsi+assj+aaak)k LINEAR VECTOR FUNCTIONS iPs=[(ani+a~uj+0Slk) (a~ui+a22j+assk) (alSi+ a2Sj+ as8k)][ijk] Thisiseasilyseentobeequaltothedeterminant317 (72) ForthisreasoniPsisfrequently calledthedeterminant ofiP andiswritwn (72)' Theideaofthedeterminant isverynatural wheniPis regarded asexpreBBed innonionform.Ontheotherhand unlessiPbeexpressed inthatformtheconception ofiPs' thethirdofiP,ismorenatural. Thereciprocal ofadyadicinnonionformmaybefound mosteasilybymakinguseoftheidentity or or HenceAI-l1AI "=iP"2C· S {Allii+A21ij+ASlik} +A12ji +Anjj +AS2jk ,p-l=+ A1Ski+A23kj+A33kk ana12a13 a2la22a23 a3la32a33(68) (73) 818 VECTOR ANALYSIS Ifthedetermina.nt bedenotedbyD ,11-1_Au..+A~n..+AS1l'k\I'--11-lJ -D D J) Au..A2t• •An.k+-Jl+-JJ+-JJ)D D If1Jfisaseconddyadicgiveninnonionformas 1Jf=bllii+bIllij+blSik, +b21ji+bnjj+bllsjk, +bSIki+bnkj+bS3kk, theproduct (J.1jI'ofthetwodyadicsmayreadilybefound byactually performing themultiplication ~.7Jf=(aubu+a12bill+a18bSl) ii+(aubl2+al2bn + alSbu)ij+(anblS+a12b23+ al3b33)ik +(aliIbu+anb21+a28bSl)ji +(a21bl2+a22bn +a2sbss)jj+(a21blS+a2tb2S+a28baa)jk +(aSIbll+QStb21+aaabSI)ki+(aSIb12+a82bn +Qaabss)kj+(aSIbl2+ aS2b23+aaab3S)kk. ~:7JI'=allbll+al2b12+olSb18 +aliIb21+all2htll+ailSb23 (75) +aSIbSI+assb82+aaabss' Sincethethirdordeterminant ofaproductisequaltothe productofthedeterminants, thelawofmultiplication of determinants followsfrom(65)and(74). LINEAR VECTOR FUNCTIONS 819 aua12al3 a2la22a2S aSlaS2assbllb12blSanbn+a12b2l+ al3bSl b2lbub2S-aubu+a22b21+ a2SbSl bSlbS2bssaSlbu+aS2b2l+assb31 aub12+a12b22+alSbS2 aub12+a22b22+a2SbS2 aSlb12+aS2b22+assbS2aublS+a12b2S+aISb83, aublS+a22b2s+a23b83,(76) a8lb13+aS2b2S+083b83• Therulemaybestatedinwords.Tomultiply twodeter minants formthedeterminant ofwhichtheelement inthe mthrowandnthcolumnisthesumoftheproducts ofthe elements inthemthrowofthefirstdeterminant andnth columnofthesecond. If fP=al+bm+cn, '12=bxcmXn+cXa nxl+aXb 1Xm. Then I'12I= (4>2)S=[bXc cXa aXb][mXn nX1IXm1 Hence 14>21=(fP2)s=[abC]2[lmn]2=fPs2. Hence AllA12AlSaua12alS2 14>21=A21A22A2S-a21a22a2S (77) ASIAS2A88aSIaS2aSS Thedeterminant ofthecofactors ofagivendeterminant of thethirdorderisequaltothesquareofthegivendeterminant. 122.]Adyadic 4>hasthreescalarinvariants -thatis threescalarquantities whichareindependent oftheformin which'Iisexpressed. Theseare 4>8'(4)2)8' 4>s, thescalarof'I,thescalarofthesecondof4>,andthethird ordeterminant of'fl.Ifrpbeexpressed innonionformthese quantities are 320 VECTOR ANALI'SIS (/Is=all+a22+a28 (iP2)Jf=All+A22+AS3 (78) aualS a22a2S aS2ass Nomatterintermsofwhatright-qanded rectangular system oftheseunitvectors fPmaybeexpressed thesequantities are thesame.ThescalaroffPisthesumofthethreecoefficients inthemainaiagonal. Thescalarofthe.secondofiPisthe sumofthefirstminorsorcofactors ofthetermsinthe maindiagonal ThethirdofiPisthedeterminant ofthe coefficients. Thesethreeinvariants arebyfarthemost important thatadyadic fPpossesses. Theorem: Anydyadicsatisfies acubicequation ofwhich thethreeinvariants fP/JliP28'fPsarethecoefficients. By(68) an-x012alS (fP-xIh=al2an-xa28 al3aS2ass-x Hence asmaybeseenbyactually performing theexpansion. (fP-X1)2•(fP-xI)c=iPs-XfP2s+x2iPs-x3• Thisequation isanidentity holding forallvaluesofthe scalarx.Ittherefore holds,ifinplaceofthescalarx,the dyadic fPwhichdepends uponninescalarsbesubstituted. Thatis (fP-iPoIh.(fP-fP.I)s=IiPs-iPiP2s+iP2iPs-iPs. B.utthetermsupontheleftareidentically zero.Hence (/)2_fPSfP2+fP28fP-(/)sI=O. (79) LINEAR VECTOR FUNCTIONS 321 Thisequation maybecalledtheHamilton-Cayley equation. Hamilton showed thataquaternion satisfied anequation analogous tothisoneandCayleygavethegeneralization to matrices. Amatrixofthenthordersatisfies analgebraic equation ofthenthdegree. Theanalogy between thetheory ofdyadicsandthetheoryofmatrices isveryclose.Infact, adyadicmayberegarded asamatrixofthethirdorderand conversely amatrixofthethirdordermaybelookeduponas adyadic. Theaddition andmultiplication ofmatrices and dyadicsarethenperformed according tothesamelaws.A generalization oftheideaofadyadictospacesofhigher dimensions thanthethirdleadstoMultiple Algebra andthe theoryofmatrices ofordershigherthanthethird. SUMMARY OFCHAPTER V Avectorr'issaidtobealinearfunction ofavectorr whenthecomponents ofr'arelinearhomogeneous functions ofthecomponents ofr.Orafunction ofrissaidtobea linearvectorfunction ofrwhenthefunction ofthesumof twovectorsisthesumofthefunctions ofthosevectors. (4) Thesetwoideasofalinearvectorfunction areequivalent. Asumofanumber ofsymbolic products oftwovectors, whichareobtained byplacing thevectorsinjuxtaposition without intervention ofadotorcrossandwhicharecalled dyads,iscalledadyadicandisrepresented byaGreek capital. Adyadicdetermines alinearvectorfunction of avectorbydirectmultiplication withthatvector f/J=a1b1+a2b2+asbs+... (7) f/J.r=a1b1•r+a2b2•r+a3bs• r+.,.(8) 21 822 VECTOR ANALYSIS Twodyadicsareequalwhentheyareequalasoperatol'8 uponallvectorsoruponthreenon-eoplanar vectors. That is,when f/J•r=1j!.rforallvaluesorforthreenon- coplanar valuesofr, (10) orr •f/J=r.1j!forallvaluesorforthreenon- coplanar valuesofr, orI •f/J•r=I.1j!.rforallvaluesorforthreenon coplanar valuesofrandI. Anylinearvectorfunction mayberepresented byadyadic. Dyadsobeythedistributive lawofmultiplication wit~ regardtothetwovectorscomposing thedyad (a+ b+ c+...)(1+ m+ n + ...)=a1+am+an+ . +bl+bm+bn+ . +cl+cm+cn+··· + (11)' Multiplication byascalarisassociative. InvirtUeofthese twolawsadyadicmaybeexpanded intoaS'Ullofnineterms. bymeansofthefundamental dyads, (12) (13)ii,ii,Lt, ii,ii,Jk, ki,ki,kk, f/J=allii+a12ij+a18ik, =a21ji+a22j j+a28ik, =a81ki+a82kj+a88kk. Iftwodyadicsareequalthecorresponding coefficients in theirexpansions intononionformareequalandconversely_as LINEAR VECTOR FUNCTIONS 323 Anydyadicmaybeexpressed asthesumofthreedyadsof whichtheantecedents ortheconsequents areanythree givennon-coplanar vectors. Thisexpression ofthedyadicis unique. Thesymbolic product &bknownasadyadisthemost generalproduct oftwovectorsinwhichmultiplication bya scalarisassociative. Itiscalledtheindeterminate product. Theproductimposes fiveconditions uponthevectorsaand b.Theirdirections andtheproduct oftheirlengthsare determined bytheproduct. Thescalarandvectorproducts arefunctions oftheindeterminate product. Ascalarand avectormaybeobtained fromanydyadicbyinserting adot andacrossbetween thevectorsineachdyad.Thisscalar andvectorarefunctions ofthedyadic. ~JI=&1·bl+&2•b2+&a•ba+... (18) ~x=&1Xbl+&2Xb2+&aXba+... (19) ~JI=i •tP•i+j •tP•j+k•tP•k(20) ~lt=(j•tP•k-k•tP•j)i+(k•tP•i - i .tP•k)j +(i.tP•j - j •tPoi)k (21) =(a2S-a82)i+(aal-ala)j+(a12-a2l)k. Thedirectproductoftwodyadsisthedyadwhoseante cedentandconsequent arerespectively theantecedent ofthe firstdyadandtheconsequent ofthesecondmultiplied by thescalarproduct oftheconsequent ofthefirstdyadand theantecedent ofthesecond. (ab)•(cd)=(b•c)&b. (23) Thedirectproduct oftwodyadicsistheformalexpansion, according tothedistributive law,oftheproduct intothe 824 VECTOR ANALYSIS 8umofproducUl ofdyads.Directmultiplication ofdyadics orofdyadicsandavectorateitherendoratbothendsobeys thedistributive andassociative lawsofmultiplication. Con sequently suchexpressions as (/J.'F.r,I.(/J.1J!,I.(/J.1J!.r,(/J.1J!.!J(24)-(26) maybewrittenwithout parentheses; forparenthe8P.8 may beinserted atpleasure without altering thevalueofthe product. Incasethevectoroccursatotherpositions than attheendtheproductisnolongerassociative. Theskewproduct ofadyadandavectormaybedefined bytheequation (ab)x r=abxr, r x(ab)=rXab. (28) Theskewproduct ofadyadicandavectorisequaltothe formalexpansion ofthatproduct intoasumofproducts of dyadsandthatvector. Thestatement madeconcerning the associative lawfordirectproducts holdswhenthevectoris connected withthedyadicsinskewmultiplication. The expressions r x(/J•'F,(/J•1J!xr,r x(/J•I,r •(/JXI,r xiPxI(29) maybewrittenwithout parentheses andparentheses maybe insertedatplel\Sure withoutalteringthevalueoftheproduct. Moreover I.(rX(/J)=(IXr)•(/J,((/Jxr)•I=(/J•(rXI), (/J.(rX'F)=«(/JXr)•1J!. (81)' Buttheparentheses cannotbeomitted. Thenecessary andsufficient condition thatadyadicmay bereduced tothesumoftwodyadsortoasingledyador tozeroisthat,whenexpressed asthesumofthree dyadsofwhichtheantecedents (orconsequents) areknown LINEAR VECTOR FUNCTIONS 825 tobenon-coplanar, theconsequents (orantecedents) shall berespectively coplanar orcollinear orzero.Acomplete dyadicisonewhichcannotbereduced toasumoffewer thanthreedyads. Aplanardyadicisonewhichcanbe reduced toasumofjusttwodyads. Alineardyadicisone whichcanbereduced toasingledyad. Acomplete dyadicpossesses nodegreeofnullity. There isnodirection inspaceforwhichitisanannihilator. A planardyadicpossesses onedegreeofnullity. Thereisone direction inspaceforwhichitisanannihilator whenusedas .aprefactor andonewhenused 808apostfactor. AlinAar dyadicpossesses twodegrees ofnullity. Therearetwo independent directions inspaceforwhichitisanannihilator whenusedasaprefactor andtwodirections whenusedasa postfactor. Azerodyadicpossesses threedegreesofnullity orcomplete nullity.Itannihilates everyvectorinspace. Theproducts ofacomplete dyadicandacomplete, planar, orlineardyadicarerespectively complete, planar,orlinear. Theproducts ofaplanardyadicwithaplanarorlineardyadic arerespectively planarorlinear,exceptincertaincaseswhere relatiops ofperpendicularity between theconsequents ofthe firstdyadicandtheantecedents ofthesecondintroduce one moredegreeofnullityintotheproduct. Theproduct ofa lineardyadicbyalineardyadicisingenerallinear;butin casetheconsequent ofthefirstisperpendicular totheante cedentofthesecondtheproduct vanishes. Theproductof anydyadicbyazerodyadicillzero. Adyadicwhichwhenappliedtoanyvectorinspacere produces thatvectoriscalledanidemfactor. Allidemfactors areequalandreducible totheform OrI=i i-+;jj+kk. 1=aa'+bb'+ee'.(33) (34) Theproduct ofanydyadicandanidemfactor isthatdyadic. 326 VECTOR ANALYSIS Iftheproductoftwocomplete dyadicsisequaltotheidem factorthedyadics arecommutative andeitheriscalled thereciprocal oftheother. Acomplete dyadicmaybe canceled fromeitherendofaproductofdyadicsandvectors asinordinary algebra; forthecancelation isequivalent to multiplication bythereciprocal ofthatdyadic. Incomplete dyadics possessnoreciprocals. Theycorrespond tozeroin ordinary algebra. Thereciprocal ofaproductisequaltothe productofthereciprocals takenininverseorder. (38) Theconjugate ofadyadicisthedyadicobtained byinter changing theorderoftheantecedents andconsequents. The conjugate ofaproduct isequaltotheproduct ofthecon jugatestakenintheopposite order. (40) Theconjugate ofthereciprocal isequaltothereciprocal of theconjugate. Adyadicmaybedivided inoneandonly onewayintothesumoftwopartsofwhichoneisse1£ conjugate andtheotheranti-self-conjugate. (43) Anyanti-self-conjugate dyadicortheanti-se1£-conjugate partofanydyadic,usedindirectmultiplication, isequivalent tominusone-half thevectorofthatdyadicusedinskew multilJlication. 1 1f/J2(f/J-f/Jo)·r=-2"xr, 1 1 ,112r •(f/J-f/J0)= -2r x""". (44) AdyadicoftheformcXIorIXcisanti-se1£-conjugate and usedindirectmultiplication isequivalent tothevector 0 usedinskewmultiplication. AlsoLINEAR VECTOR FUNCTIONS oXr=(IX0)•r=(0XI)•r, oX(/J=(IX0)•(/J=(0XI)•f/J.32; (46) Thedyadic 0 XIorIX0,where 0illaunitvectorisaquad rantalversorforvectorsperpendicular to0andanannihilator forvectors parallel too.ThedyadicIXc+00isatrue quadrantal versorforallvectors. Thepowersofthesedyadics behavelikethepowersoftheimaginary unit,..1-1,asmay beseenfromthegeometric interpretation. Applied tothe unitvectorsi,j,t IXi=iXI=tj-jt,etc. (49) ThevectoraXbin~kewmultiplication isequivalent to (aXb)XIindirectmultiplication. (axb)xI=lx(axb)=ba-ab (50) (aXb)x r=(ba-ab)• r r x(aXb)=r •(ba- ab). (51) Acomplete dyadicmaybereduced toasumofthree dyadsofwhichtheantecedents amongthemselves andthe consequents alllongthemselves eachformaright-handed rectangular systemofthreeunitvectorsandofwhichthe scalarcoefficients areallpositive orallne~ative. f/J=±(ai'i+bj'j+ct't). (53) Thisiscalledthenormalformofthedyadic. Anincom pletedyadicmaybereduced tothisformbutoneormoreof thecoefficients arezero.Thereduction isuniqueincase theconstants a,b,caredifferent. Incasetheyarenot different thereduction maybeaccomplished inmorethan oneway.Anyself-conjugate dyadicmaybereduced to thenormalform (/J=aii+bjj+ctt, (55) inwhichtheconstants a,b,carenotnecessarily positive. 328 VECTOR ANALYSIS Thedoubledotanddoublecrossmultiplication ofdyads isdefinedbytheequations ab:cd=a·c b.d, ab~cd=axc bxd.(56) (57) (63) (64) (65)Thedoubledotanddoublecr08Smultiplication ofdyadic~ isobtained byexpanding theproduct formally, according to thedistributive law,intoasumofproducts ofdyads. The doubledotanddoublecrossmultiplication ofdyadicsiscom mutative butnotassociatiye. One-half thedoublecrossproductofadyadic (/Jbyitself iscalledthesecondof(/).If (/J=al+bm+cn, (/J2=j(/Jx(/J=bxc mxn+cxa nxl+axb lxm.(61) One-third ofthedoubledotproductofthesecondof(/)and(/) iscalledthethirdof(/Jandisequaltotheproduct ofthe scalartripleproductoftheantecedents of(/Jandthescalar tripleproductoftheconsequent of(/J. I (/Ja=6(/)~ (/J:(/J=[abo] [lmn]. (62) Thesecondoftheconjugate istheconjugate ofthesecond. Thethirdoftheconjugate isequaltothethirdofthe original dyadic. Thesecondandthirdoftheteciprocal are thereciprocals ofthesecondandthirdofthesecondand thirdofadyadic. Thesecondandthirdofaproductarethe products ofthesecondsandthirds. «(/)C)2=«(/)2)'" (rpc)a=(/)a' (~lh=«(/)2)-1, (~1)3=«(/Ja)-I. ((/).1JT)2=(/)2•1JT2 ((/).IJT)3=f/J3lJTa• LINEAR VECTOR FUNCTIONS 329 Theproductofthesecondandconjugate ofadyadicisequal totheproductofthethirdandtheidemfactor. (68) Theconditions forthevariousdegrees ofnullitymaybe expressed intermsofthesecondandthirdoffP. fP8=t-0,(/)iscomplete (/)8=0,fP2=t-0,fPisplanar (69) (/)3=0,fP2=0,fP=t-0,fPislinear. Theclosingsections ofthechaptercontaintheexpressions (70)-(78) ofanumberoftheresult.:;innonionformandthe deduction therefrom ofanumber oftheorems concerning determinants. Theyalsocontainthecubicequation whichis satisfied byadyadic (/). (/)8-fP8(/)2+fP28(/)3+fP3I=O.(79) ThisiscalledtheHamilton-Cayley equation: Thecoeffi cients(/)8'fP28,andfP3arethethreefundamental scalarin variants offP. EXERCISES ONCHAPTER V 1.Showthatthetwodefinitions giveninArt.98for alinearvectorfunction areequivalent. 2.Showthatthereduction ofadyadicasin(15)canbe accomplished inonlyonewayifa,b,c,1,m,n,aregiven. 3.Show (fPxa)c= -ax(/)e- 4.Showthatif(/)x r=1J!'x rforanyvalueofrdifferent fromzero,then(/)mustequal 1J!'-unlessboth(/)and1J!'are linearandthelineoftheirconsequents isparalleltor. 5.ShowthatiffP•r=0foranythreenon-coplanar values ofr,thenfP=O. 880 VECTOR ANA.LYSIS 6.Provethestatements madeinArt.106andthecon· verseofthestatements. 7.Showthatif!Jiscomplete andif~•!J='F.!J,then ~and1JTareequal.Givetheproofbymeansoftheory developed priortoArt.109. 8.Definition: Twodyadicssuchthat~•1JT=1JT.~-that istosay,twodyadicsthatarecommutative -aresaidtobe lwmologou8. Showthatifanynumberofdyadicsarehomoge neoustooneanother, anyotherdyadicswhichmaybeobtained fromthembyaddition, subtraction, anddirectmultiplication arehomologous toeachotherandtothegivendyadics. Show alsothatthereciprocals ofhomologous dyadicsarehomolo gous.Justifythestatement thatif~.1JT-1or1JT-l.~, whichareequal,becalledthequotient of~by1JT,thenthe rulesgoverning addition, subtraction, multiplication and di~ision ofhomologous dyadics areidentical withtherules governing theseoperations inordinary algebra-itbeing understood thatincomplete dyadics areanalogous tozero, andtheidemfactor,tounity.Hencethealgebraandhigher analysis oflwmologous dyadics ispractically identical with thatofscalarquantities. 9.Showthat(Ix0)•~=0x1f'and(0xI).~=0 X~. 10.Showthatwhether ornota,b,0becoplanar ab x0+boxa+0a x b=[ab0]1 and b x0a+0x a b+a x b 0=[ab0]I. 11.Ifa,b,0arecoplanar usetheaboverelationtoprove thelawofsinesforthetriangle andtoobtaintherelation withscalarcoefficients whichexistsbetween threecoplanar vectors. Thismaybedonebymultiplying theequation bya unitnormaltotheplaneofa,b,ando. 12.Whatisthecondition whichmustsubsistbetween the coefficients intheexpansion ofadyadicintononionformif LINEAR VECTOR FUNCTIONS 831 thedyadicbeself-eonjugate? What,ifthedyadicbeanti self-conjugate ? 13.Provethestatements madeinArt.116concerning the number ofwaysinwhichadyadicmaybereduced toits normalform. 14.Thenecessary andsufficient condition thatananti self-conjugate dyadic fPbezeroisthatthevectorofthe dyadicshallbezero. 15.Showthatif(/)beanydyadictheproduct (/).(/)cis self-eonjugate. 16.Showhowtomakeuseoftherelation fP~=0to demonstrate thattheantecedents andconsequents ofaself conjugate dyadicarethesame(Alt.116). 17.Showthat f/J'J~fP'J=(/)S2f/J and (f/J+7JI')'J=fP'J+(/)~'F+7JI''J' 18.ShowthatifthedoubledotproductfP:(/)ofadyadic byitselfvanishes, thedyadicvanishes. Henceobtainthe condition foralineardyadicintheformfP'J:fP'J=O. 19.Showthat«(/)+ef)s=fPs+e·(/)2·f. 20.Showthat«(/J+7JI')g=(/Js+(/)'J:7JI'+(/):7Jl'2+7Jl'a. 21.Showthatthescalarofaproduct ofdyadicsisun· changed bycyclicpermutation ofthedyadics. Thatis (f/J.IJl'.jJ) 8=(JJ•(/J.7JI')8=(7JI'•!J•fP)8. CHAPTER VI ROTATIONS ANDSTRAINS 123.]INtheforegoing chaptertheanalytical theoryof dyadics hasbeendealtwithandbrought toastateof completenesll whichisneadyfinalforpractical purposes. Thereare,however, anumberofnewquestions whichpresent themselves andsomeoldquestions whichpresentthemselves underanewformwhenthedyadicisappliedtophysics orgeometry. Moreover itwasforthesakeoftheapplica tionsofdyadicsthatthetheoryofthemwasdeveloped. Itis thentheobjectofthepresentchaptertosupplyanextended application ofdyadicstothetheoryofrotations andstrains andtodevelop, asfarasmayappearnecessary, thefurther analytical theoryofdyadics. Thatthedyadic(/)maybeusedtodenoteatransformation ofspacehasalreadybeenmentioned. Aknowledge ofthe precisenatureofthi8transformation, however, wasnotneede<l atthetime.Consider rasdrawnfromafixedorigin,andr' asdrawnfromthesameorigin. Letnow r'=(/).r. Thisequation therefore mayberegarded asdefining atrans formation ofthepointsPofspacesituatedattheterminus of rintothepointP',situatedattheterminus ofr'.Theorigin remainsfixed.Pointsinthefiniteregionsofspaceremainin thefiniteregiollsofspace.Anypointuponaline r=b+xa becomes apoint r'=(/).b+x(/)oa. ROTATIONS ANDSTRAINS 333 Hencestraight linesgooverintostraight linesandlines paralleltothesamelineagooverbythetransformation into linesparallel tothesamelinef/J.a.Inlikemannerplanes gooverintoplanesandthequalityofparallelism isinvariant. Suchatransformation isknownasahomogeneous strain. Homogeneous strainisoffrequent occurrence inphysics. For instance, thedeformation oftheinfinitesimal sphereinafluid (Art.76)isahomogeneous strain.Ingeometry thehomo geneous strainisgenerally knownbydifferent names. Itis calledanaffinecollineation withtheoriginfixed.Oritis knownasalinearhomogeneous transformation. Theequa. tionsofsuchatransformation are x'=anx+a12Y+a13z Y'=a~l1x+a2lY+a23% %'=a3lx+a32Y+a33%. 124.]TMorem:Ifthedyadic f/Jgivesthetransformation ofthepointsofspacewhichisduetoahomogeneous strain, f/J2'thesecondoff/J,givesthetransformation ofplaneareas whichisduetothatstrainandallvolumesaremagnified by thatstrainintheratiooff/J3'thethirdordeterminant off/J tounity. Let f/J=a1+bm+cn r'=f/J·r=al.r+bm.r+cn.r. Thevectors1',m',n'arechanged byf/Jintoa,b,o.Hence theplanesdetermined bym'andn',n'and1',l'andm'are transformed intotheplanesdetermined byband 0,canda, aandb.Thedyadicwhichaccomplishes thisresultis f/J2=bx c m x n+cXa n x1+a xb 1xm. HenceifIdenoteanyplaneareainspace,thetransformation duetof/Jreplaces IbytheareaI'suchthat II'=I/J'J••' 334 VECTOR ANALYSIS HenceItisimportant tonoticethatthevector Idenoting aplane areaisnottransformed intothesamevectorI'asitwould beifitdenotedaline.Thisisevidentfromthefactthatin thelattercase~actsonIwhereas intheformercase ~2acts uponI. Toshowthatvolumes aremagnified intheratioof~3to unitychooseanythreevectorsd,e,fwhichdetermine the volumeofaparallelopiped [defJ.Express ~withthevec torswhichformthereciprocal systemtod,e,fasconsequents. ~=ad'+be'+cf'. Thedyadic ~changes d,e,fintoa,b,c(whicharedifferent fromthea,b,caboveunlessd,e,fareequalto1',m',n'). Hencethevolume[def]ischanged intothevolume[abe]. ~3=[abc][d'e'f'] [d'e'f']-l =[def]. [aboJ=[def] ~3. Theratioofthevolume[abe]to[def]isas~3istounity. Butthevectorsd,e,fwereanythreevectorswhichdeter mineaparallelopiped. Henceallvolumes arechanged by theactionof~inthesameratioandthisratioisas~3isto1. Rotatim8 aboutaFixedPoint.Versors 125.]TMorem: Thenecessary andsufficient condition that adyadicrepresent arotationaboutsomeaxisisthatitbe reducible totheform ~=i'i+j'j+k'k (1) wherei',j',k'andi,j,karetworight-handed rectangular systemsofunitvectors. Let r=xi+yj+.k ~.r=x;,-+-l,j'+zk'. ROTATIONS ANDSTRAINS 335 HenceiffPisreducible tothegivenformthevectorsi,i,k arechanged intothevectorsi',i',k'andanyvectorl'is changed fromitspositionrelativetoi,i,kintothesameposi tionrelative toi',i',kI.Hencebythetransformation no changeofshapeisetlected. Thestrainreducestoarotation whichcarriesi,i,kintoi',i',kI •Conversely suppose the bodysutlersnochangeofshape-thatis,supposeitsubjected toarotation. Thevectorsi.i,kmustbecarriedintoanother right-handed rectangular systemofunitvectors. Letthese bei',i',k'.ThedyadicfPmaytherefore bereduced tothe form fP=i'i+i'i+k'k. Definition: Adyadicwhichisreducible totheform i'i+i'i+k'k andwhichconsequently represents arotationiscalled l'lo versO'r. Theorem: Theconjugate andreciprocal ofaversoI'are equal,andconversely iftheconjugate andreciprocal ofa dyadicareequalthedyadicreduces toaversoI'oraversoI' multiplied bythenegative sign. Let fP=i'i+i'j+k'k, fP0=ii'+ji'+kk'; fP.{/}0=i'i'+i'j'+k'k'=I {/}-1=fPC' Hencethefirstpartofthetheoremisproved. Toprovethe •secondpartlet If HencefP=ai+bj+ck, fP0= ia+jb+kc, fP.{/}0= aa+bb + cc. ~-1=fPc.{/}.{/}0=1. aa+bb+cc=1. 336 VECTOR ANALYSIS (1)'orHence(Art.108)theantecedents a,b,candtheconsequents a,b,cmustbereciprocal systems. Hence(page87)they must.beeitheraright-handed oraleft-handed rectangular systemofunitvectors. Theleft-handed systemmaybe changed toaright-handed onebyprefixing thenegative signtoeachvector. Then fP=i'i+j'j+k'k, iP= -(i'i+j'j+k'k). Thethirdordeterminant ofaversorisevidently equalto unity;thatoftheversorwithanegative sign,tominusone. Hencethecriterion foraversoI'maybestatedintheform fP0fPc=I,iPs=IfP1=1. (2) Orinasmuch asthedeterwinant ofiPisplusorminusone if(/JofPc=I, itisonlynecessary tostatethatif fP0fPc=I,iPs=I(iJI>0, (2)' fPisaversor. Therearetwogeometric interpretations ofthetransforma tionduetoadyadicfPsuchthat iP0(/Jc= IfPs=IfPI= -1 (3) iP=-(i'i+i'i+k'k). Thetransformation duetofPisoneofrotationcombined with reflection intheorigin.Thedyadici'i+j'j+k'kcausesa rotationaboutadefiniteaxis-itisaversor. Thenegative signthenreversesthedirection ofeveryvectorinspaceand replaceseachfigurebyafiguresymmetrical toitwithrespect totheorigin. Byreversing thedirections ofi'andj'the systemi',j',k'stillremains right-handed andrectangular, butthedyadictakestheform fP=i'i+jIj-k'k, or f/)=(i'i'+j'j'-k'k') 0(nofi'i+k'k). ROTATIONS ANDSTRAINS 337 Hencethetransformation dueto~isarotation dueto i'i+j ,j +k'kfollowed byareflection intheplaneofi'and i'.Forthedyadici'i'+j'j'-k'k' causessuchatransfor mationofspacethateachpointgoesoverintoapointsym metrically situated toitwithrespecttotheplaneofi'andj'. Eachfigureistherefore replaced byasymmetrical figure. Definition.: Atransformation thatreplaces eachfigureby asymmetrical figureiscalledaperversion. andthedyadic whichgivesthetransformation iscalledaperversor. Thecriterion foraperversor isthattheconjugate ofa dyadicshallbeequaltoitsreciprocal andthatthedetermi nantofthedyadicshallbeequaltominusone. 4'•(/Jc=I,I~I= -1. (3) Orinasmuch asif~.~c=I,thedeterminant mustbeplus orminusonethecriterion maytaketheform I(/JI<0, (3)' (/Jisaperversor. Itisevidentfromgeometrical considerations thattheprod uctoftwoversorsisaversor;oftwoperversol's, aversoI'; butofaversoI'andaperversol' takenineitherorder,a perversor. 126.]Iftheaxisofrotation bethei-axisandiftheangle ofrotation betheangleqmeasured positive inthepositive trigonometric direction, thenbytherotation thevectors i,j,karechanged intothevectorsi',j',k'suchthat i'=i' j'=jcosq+ ksinq, k'=-jsinq+ kcosq. Thedyadic ~=i'i+j'j+k'kwhichaccomplishes thisrota tionis 22 338 VECTOR ANALYSIS ,~=i i+cosq(jj+kk)+sinq(kj -jk).(4) jj+kk=l-ii, kj-jk=1xi Hence qJ=i i+cosq(I-it)+sinqI xi.(5) Ifmoregenerally inplaceofthei-axisanyaxisdenoted bytheunitvectorabetakenastheaxisofrotationandifas beforetheangleofrotationaboutthataxisbedenoted byq, thedyadic qJwhichaccomplishes therotationis ~=a a+cosq(I-aa)+sinq1Xa.(6) Toshowthatthisdyadicactually doesaccomplish the rotationapplyittoavectorr.Thedyadaaisanidemfactor forallvectorsparalleltoa;butlIDannihilator forvectors perpendicular toa.ThedyadicI -aaisanidemfactor forallvectorsintheplaneperpendicular toa;butan annihilator forallvectorsparalleltoa.Thedyadic1 x a isaquadrantal versor(Art.113)forvectorsperpendicular toa;butanannihilator forvectorsparalleltoa.Ifthen rbeparalleltoa ~• r=a a • r=r. Hence ~leavesunchanged allvectors(orcomponents of vectors)whichareparalleltoa.Ifrisperpendicular toa ~• r=cosqr+sinqa xr. Hencethevectorrhasbeenrotatedinitsplanethrough the angleq.Ifrwereanyvectorinspaceitscomponent parallel toasuffersnochange; butitscomponent perpendicular toa isrotatedaboutathroughanangleofqdegrees. Thewhole vectoristherefore rotatedaboutathroughthatangle. Letabegivenintermsofi,j,kas a=a,i+8-2j+ask, aa=a12ii+ala2ij+alaSik ROTATIONS ANDSTRAINS 339 Hence+a2atji+a22j j+a2asjk +aSatki+aSa2kj+as2kk, I=ii+jj+kk, IXa=0ii-asij+a2ik, +asji+Ojj-atik, -a2ki+atkj+0kk. ~={at2(1-COBq)+COBq}ii +lata2(1-cOBq)-a ssinq}ij +{atas(1-COBq)+a2sinqIi k +la2at(1-cosq)+assinq}ji +{a22(l-cOBq)+COBq}jj +{a2as(1-cosq)-atsinqlj k +{aSat(1-COBq)-a2sinq}ki +{asa2(1-cosq)+atsinq}kj +{aa2(1+COBq)+COBq}kk. (7) 127.]IffPbewrittenasinequation (4)thevectorof~ andthescalaroffPmaybefound. ~x=iXi +COBq(jXj + k Xk)+sinq(kXj - jXk) fPx=-2sinqi fP8=id+cosq(j•j + kk)+sinq(k.j -j •k), fP8= 1 + 2 COBq. Theaxisofrotation iisseentohavethedirection of-fPxt thenegative ofthevectoroffP.Thisistrueingeneral. Thedirection oftheaxisofrotation ofanyversoristhe negative ofthevectoroffP.Theproofofthisstatement depends 011theinvariant property offPx'Anyversor ~ maybereduced totheform(4)bytakingthedirection of4 340 VECTOR ANALYSIS coincident withthedirection oftheaxisofrotation. After thisreduction hasbeenmadethedirection ofthe'axisisseen tobethenegative ofrp)l.'Butrp)l.isnotalteredbythe reduction ofrptoanyparticular form-noristheaxisof rotationalteredbysuchareduction. Hencethedirection of theaxisofrotationisalwayscoincident with-rp)l.'thedirec tionofthenegative ofthevectorofrp. Thetangentofone-halftheangleofversionqis qsinqII/rp)l.•rp)l.tan-= ='}1+cosq 1+rp8(8) Thetangent ofone-half theangleofversionistherefore determined whenthevaluesofrp)l.andrp8areknown. The vectorfixandthescalarrplJ'whichareinvariants ofrp,deter minecompletely thevenwrrp.LetQbeavectordrawn inthedirection oftheaxitlofrotation. Letthemagnitude ofQbeequaltothetangent ofone-half theangleqof version. 1Q.Q=tan22q. ThevectorQdetermines theversoI'rpcompletely. Qwillbe calledthevectorsemi-tangent ofversion. By(6)aversoI'rpwasexpressed intermsofaunitvector paralleltotheaxisofrotation. rp= aa +cosq(I- aa)+sinq1Xa. HenceifQbethevectorsemi-tangent ofversion rp=QQ.~ +cosq(1-QQ.~)+sinq1Xv:.Q'(10) Thereisamorecompact expression foraversoI' (/Jinterms ofthevectorsemi-tangent ofversion. Letcbeanyvectorin space.Theversionrepresented byQcarries c -QXcintoc +Qxc. ROTATIONS ANDSTRAINS 841 Itwillbesufficient toshowthisincase0isperpendicular to Q.Forif0(oranycomponent ofit)wereparalleltoQthe resultofmultiplying byQxwouldbezeroandthestatement wouldbethat0iscarriedintoo.Inthefirstplacethemag nitudesofthetwovectorsareequal.For (0-Qx0)'(0-Qx0)=o·0+Qx(].QX 0 -20.QX 0 (0+QX0)'(0+QX0)=0·0+QXo'QX 0+20'QX 0 o•0+QXo.QX 0=0 • 0+Q•Qo.0 -Q•0Q•o. SinceQand0arebyhypothesis perpendicular 1o·0+QXo·QX 0=ell(1+tanll2q). Theterm0 •QX 0vanishes. Hencetheequality. Inthe secondplacetheangle between thetwovectorsisequaltoq. (0-QX0)'(0+QX0)o·0 -QXo.QX 0= ell(1+tanll~q) ell(1+tanll~q) 1e\l(1-tanll-q)2 e:l(1+tan\I~q)=cosq (o-Qxo)x (o+Qxo) 20X(QX0) = ell(1+tanll~q) ell(1+tanll~q) 2elltan~q ~------- =sinq. ell(1+tanll~q) Hencethecosineandsineoftheangle between 0 -QX 0 and0+QX 0areequalrespectively tothecosineandsineof theangleq:andconsequently theanglebetween thevectors mustequaltheangleq.Now 842 VECTOR ANALYSIS o -QX 0=(I- 1XQ)·0 and (0+QX0)=(I+1 xQ).0 (I+1xQ).(I- 1xQ)-l.(I- 1xQ)=I+IxQ. Multiply by0 (I+IxQ).(I- IxQ)-l.(0-Qx0)=0+Qxo. Hencethedyadic rP=(I+1xQ)•(I- 1xQ)-l (10)' carriesthevector 0 -Qx0intothevector 0+Qx cnomatter whatthevalueofo.Hepcethedyadic rPdetermines the versionduetothevectorsemi-tangent ofversion Q. Thedyadic1+1xQcarriesthevectorc-Qx cinto (I+Q.Q)o. (I+1XQ).(0-QXc)=c+Qxc-Qx0 -Qx(QX0) (I+1XQ).(0-Q+0)=c+Q.Qo=(1+Q.Q)o. Hencethedyadic 1+IxQ=(I_ 1XQ)-l 1+Q.Q carriesthevectorc-QX 0intothevector 0,if0beperpen diculartoQashasbeensupposed. Consequently thedyadic (I+1XQ)2 1+Q.Q produces arotationofallvectorsintheplaneperpendicular toQ.If,however, itbeappliedtoavectorxQparalleltoQ theresultisnotequaltoxQ. (I+IXQ)•(I+IXQ)xQ=x(I+JXQ)Q=xQ. l+Q.Q l+Q.Q l+Q.Q ROTATIONS ANDSTRAINS 343 Toobviatethisdifficulty thedyadQQ,whichisanannihilator forallvectorsperpendicular toQ,maybeaddedtothenu merator. Theversor f/Jmaythenbewritten f/J=QQ+(I+ 1 xQ)Z (10)" 1+Q.Q (I+ J xQ)•(I+ J xQ)=1+2 1xQ+(IxQ).(IxQ) (IxQ)•(IxQ)=(IxQ)xQ=I.QQ-Q.Q1. Hencesubstituting: f/J=(1-Q.Q)I+2QQ+2IxQ. (10)/1/ / l+Q.Q Thismaybeexpanded innonionform.Let Q=ai+bj+ek. {(I+aZ-bZ-eZ)ii +(2ab-2e)ij+(2ac+2b)ik} +(2ab+2e)ji+(l-aZ+1?-c2)jj+(2be-2a)jk(11) tP= +(2ac-2b)ki+(2bc+2a)kj+(l-az-1?+cZ)kk. 1+aZ+b'l.+cZ 128.]Ifaisaunitvectoradyadicoftheform (/)=2aa-1 (12) isabiquadrantal versor. Thatis,thedyadic (/)turnsthe pointsofspaceabouttheaxisathrough tworightangles. Thismaybeseenbysettingqequalto'IT'inthegeneral expression foraversor f/J= aa+C08q(I- aa)+sinq1 xa, oritmaybeseendirectly fromgeometrical considerations. Thedyadic (/)leavesavectorparalleltoaunchanged butre verseseveryvectorperpendicular toaindirection. Theorem: Theproduct oftwobiquadrantal versorsisa versortheaxisofwhichisperpendicular totheaxesofthe 344 VECTOR ANALYSIS biquadrantal versol'8andtheangleofwhichistwicethe anglefromtheaxilJofthesecondtotheaxisofthefirst. Letaandbbetheaxesoftwobiquadrantal versors. The product !J=(2bb -I)•(2a a -I) iscertainly aversor; fortheproduct ofanytwoversOr8 isaversor. Consider thecommon perpendicular toaandb. Thebiquadrantal venlOr2aa-1reverses thisperpendicular indirection. (2bb-1) againreversesitindirection andcon sequently bringsitbacktoitsoriginalposition. Hencethe product!J leavesthecommon perpendicular toaandbun changed. !Jistherefore arotationaboutthislineasaxis. !J.a=(2bb-I).(2aa-I).a =(2bb-1).a=2bb.a-a. Thecosineoftheanglefromato!J.ais a.JJ•a= 2b.ab.a -a.a = 2(b.ayo!-1 =cos2(b,a). HencetheangleoftheversorJJisequaltotwicetheangle fromatob. Theorem: Conversely anygivenversormaybeexpressed astheproductoftwobiquadrantal versors,ofwhichtheaxes lieintheplaneperpendicular totheaxisofthegivenversor andincludebetween themanangleequaltoonehalfthe angleofthegivenversor. Forlet!Jbethegivenversor. Letaandbbeunitvectors perpendicular totheaxis-!J"of~hisversor. Furthermore lettheanglefromatobbeequaltoonehalftheangleof thi8versor. Thenbytheforegoing theorem !J=(2bb-1).(2aa-1). (14) Theresolution ofversorsintotheproduct oftwobiquad mntalversorsaffordsanimmediate andsimplemethodfor compounding twofiniterotations aboutafixedpoint.Let ~andlJfbetwogivenversors. Letbbeaunitvectorper· ROTATIONS ANDSTRAINS 345 pendicular totheaxelJoffPand1Jf.Letabeaunitvector perpendicular totheaxisoffPandsuchthattheanglefrom atobisequaltoonehalftheangleoff/J.Letcbeaunit vectorperpendicular totheaxisof1Jfandsuchthattheangle frombtocisequal to onehalftheangleof1Jf.Then fP=(2bb-I).(2all.-I) 1J!=(2cc -I)•(2bb -I) fj!'•fP=(2cc-I)•(2bb -I)2.(2aa-I). But(2bb-1)2isequal to theidemfactor, 8I:lmaybeseenfrom thefactthatitrepresents arotationthrough fourrightangles orfromtheexpansion (2bb -I).(2bb-I)=4b.b b b - 4 bb +I=I. Hence 1J!•f/J=(2cc-I)•(2aa-I). Theproductoffj!'intofPisaversortheaxisofwhichis perpendicular toaandcandtheangleofwhichisequal to onehalftheanglefromatoo. Iff/Jand1J!aretwoversorsofwhichthevectorsemi tangents ofversionarerespectively Q]and~,thevector semi-tangent ofversionClaoftheproduct 1Jf.f/Jis Q 3=Q]+Q2+Q2XQ] (15) 1 -Q]•Q2 Let f/J=(2bb -I)•(2aa -I) and lJT=(2cc -I)•(2bb -I). W.f/J=(2cc -I)•(2aa -I). -fP -1Jf -(1Jf.f/J)x Q1=1+~; Q2=1+~.' Q3=1+-(1Jf.f/J). f/J=4a•b ba - 2 a a - 2 b b +I, f/Jxa.::4A•b bXa, 346 HenceVECTOR ANALYSIS tfls=4(a0b)2-1, 1Jf=4cobcb - 2 bb - 2 ce +1, 1Jfx=4Cobc xb, Wa=4(c ob)2-1 1JI'0tfl= 4 c 0a ca - 2 cc - 2 aa + I, (1Jf0tfl)x=4c0a cXa, ('Jf0tfl)a=4(c0a)2-1. axb bxc axeQ1=--, ~=--, Qa=--aob boc aoc (bxc)X(ax b) [abe]b ~xQl=-· =- .aobboc aobboc But[abc]r=bxcaor+cxabor+axbeor, Hence [abc]b = b x c a 0b + c xabob + aXb e 0b. Qbxcaxbaxe Q2x1= ------+ .bocaobaobboe Hence Q_Q1 XQ2+Q2XQ1 a- a0c aobboc (axb) 0(bxc) aobbocaoebob Q10Q2=...a0bb 0c~-= a 0b b 0c-a0bboo Hence Hencea0c a-ob-b-o-c=1--~0Ql' Qa=Q1XQ2+Q2+ Q10 1 -Q10Q-; ROTATIONS ANDSTRAINS 347 Thisformula givesthecomposition oftwofiniterotations. Iftherotations beinfinitesimal Q1and~arebothinfinitesi mal.Neglecting infinitesimals ofthesecondorderthefor mulareduces to Qa=Q1+Q2' Theinfinitesimal rotations combine according tothelawof vectoraddition. Thisdemonstrates theparallelogram lawfor angular velocities. Thesubjectwastreatedfromdifferent standpoints inArts.51and60. Cyclics,RightTensors, Tonics,andeyclotonics 129.]Ifthedyadic f/Jbeaversoritmaybewritteninthe form(4) f/J=ii+cosq(jj+kk)+sinq(kj-jk). Theaxisofrotationisiandtheangleofrotationaboutthat axisisq.LetlJf'beanother versorwiththesameaxisand anangleofrotationequaltoq'. P"=ii+cosq'(jj+kk)+sinq'(kj- jk). MUltiplying: fP.'F='F•f/J= i i+cos(q+q')(jj+kk) +sin(q+q')(kj-jk).(16) Thisistheresultwhichwastobeexpected -theproductof twoversorsofwhichtheaxesarecoincident isaversoI'with thesameaxisandwithanangleequaltothesumofthe anglesofthetwogivenversors. IfaversoI'bemultiplied byitself,geometric andanalytic considel"dtions alikemakeitevidentthat f/J2=i i+cos2q(jj+kk)+sin2q(kj- jk), andf/J-=ii+cosnq(jj+kk)+sinnq(kj-jk). 848 VECTOR ANALYSIS Ontheotherhandlet(/)1equaljj+kkj aud(/)2equal kj- jk.Then (/)-=(ii+cosq(/)1+sinq(/)2)-' Theproductofiiintoeither (/)1or(/)2iszeroandintoitselfis ii.Hence (/)-=ii+(cosq(/)1+sinq(/)2)- (/)-=ii+cos-q(/)1-+ncos--1qsinq(/)1--1•(/)2+... Thedyadic (/)1railiedtoanypowerreproduces itself.(/)t=(/)1' Thedyadic (/)2railiedtothesecondpowergivesthenegative of(/)1;raisedtothethirdpower,thenegative of(/)2;raised tothefourthpower,til;raisedtothefifthpower, (/)2andso on(Art.114).Thedyadic (/)1multiplied by(/)2isequalto (/)2'Hence (/)-=i i+cos-q(/)1+ncos--1qIiinq(/)2 n(n-1) .-2!cos--2qsm2(/)2+... But (/)-=i i+cosnq(/)1+sinnq(/)2' Equating coefficients of(/)1and(/)2inthesetwoexpressions for,p- cosnq=cos-qn(n-1)2!cos--2qsin2q+... . .n(n-1)(n-2) .smnq=ncosll-1qsmq- 3! cos--SqS1D3q+.. Thustheordinary expansions forcosnqandsinnqare obtained inamannerverysimilartothema.nnerinwhich theya.regenerally obtained. Theexpression foraversoI'maybegeneralized asfollows. Leta,b,0beanythreenon-coplanar vectors; anda',b',0',the reciprocal system. Consider thedyadic (/)= aa'+cosq(bbI+0 C')+sinq(0bI -b0').(17) ROTATlONSANDSTRAINS 349 Thisdyadicleavesvectorsparalleltoaunchanged. Vectors intheplaneofband 0sufferachangesimilartorotation. Let r=cospb+sinp0, r'=~.r=cos(p+q)b+sin(p+q)c. Thistransformation maybegivenadefinitegeometrical interpretation asfollows. Thevectorr,whenpisregarded asavariable scalarparameter, describes anellipseofwhich band 0aretwoconjugate semi-diameters (page117).Let thisellipseberegarded astheparallel projection ofthe unitcircle r=cospi+sinqj. Thatis,theellipseandthecirclearecutfromthesame cylinder. Thetwosemi-diameters iandjofthecirclepro- .jectintotheconjugate semi-diameters aandboftheellipse. Theradiusvectorrintheellipseprojectsintotheradiusvector rintheunitcircle.Theradiusvectorr'intheellipsewhich isequalto~.r,projects intoaradiusvector1"inthecircle suchthat i'=cos(p+q)i+sin(p+q)j. Thusthevectorrintheellipseissochanged bytheapplica tionof~asaprefactor thatitsprojection tintheunitcircle isrotatedthrough anangleq. Thisstatement maybegivenaneaterformbymakinguse ofthefactthatinparallel projection areasarechanged ina definite constant ratio.Thevectorrintheunitcirclemay beregarded asdescribing asectorofwhichtheareaistothe areaofthewholecircleaRqisto27T'.Theradiusvectorr thendescribes asectoroftheellipse. Theareaofthissector istotheareaofthewholeellipseasqisto27T'.Hencethe dyadic ~appliedasaprefactor toaradiu.$vectorrinanellipse ofwhichband0aretwoconj?J,gate semi-diamcters advances thatvectorthrough aSl'ctortheareaofwhichistotheareaoj 850 VECTOR ANALYSIS eMwholeellipse tUqisto2?r.tSuchadisplacement ofthe radiusvectorrmaybecalledanellipticrotation through a sectorqfromitssimilarity toanordinary rotation ofwhich itistheprojection. Definition: AdyadicfPoftheform fP=aa'+cosq(bb'+00')+sinq(0b'- b0')(17) iscalledacyclicdyadic. Theversorisaspecialcaseofa cyclicdyadic. Itisevidentfromgeometric oranalytic considerations that thepowersofacyclicdyadicareformed,asthepowersofa versorwereformed,bymultiplying thescalarqbythepower towhichthedyadicistoberaised. fPa=aa'+cosnq(bb'+00')+sinnq(0b'- b0'). Ifthescalarqisanintegralsub-multiple of2?r,thatis,if 2?r-=m,q itispossible toraisethedyadicfPtosuchanintegralpower, namely, thepowerm,thatitbecomes theidemfactor fP"=I fPmaythenberegarded asthemthrootoftheidemfactor. Inlikemannerifqand2?rarecommensurable itispossible toraisefPtosuchapowerthatitbecomes equaltotheidem factorandevenifqand2?rareincommensurable apowerof fPmaybefoundwhichdiffersbyaslittleasonepleasesfrom theidemfactor. Henceanycyclicdyadicmayberegarded as arootoftheidemfactor. 1Itisevidentthatfixingtheresultoftheapplication of4>toallradii..actors inanellipsepractically fixeaitforallvectorsintheplaneofbandc.Forany ..actorinthatplanemayberegarded &8aecalarmultiple ofaradiUBvectorof theellipee. Definition: Thetransformation represented bythe 130.] dyadicROTATIONS ANDSTRAIN:S ~=aii+bjj+ckk351 (18) wherea,b,carepositivescalarsiscalledapurestrain. The dyadicitselfiscalledarighttensor. Arighttensormaybefactored intothreefactors fP=(aii+ j j +kk)•(ii+bj j+kk)•(ii+j j+ckk). Theorderinwhichthesefactorsoccurisimmaterial. The transformation r'=(ii+ii+ckk)•r issuchthattheiandicomponents ofavectorremainun alteredbutthek-component isalteredintheratioofcto1. Thetransformation maytherefore bedescribed asastretchor elongation alongthedirection k.Iftheconstant cisgreater thanunitytheelongation isatrueelongation: butifcisless thanunitytheelongation isreallyacompression, fortheratio ofelongation islessthanunity.Between thesetwocases comesthecaseinwhichtheconstant isunity.Thelengths ofthek-components arethennotaltered. Thetransformation duetothedyadic ~mayberegarded asthesuccessive orsimultaneous elongation ofthecom p<ments ofrparallel toi,i,andkrespectively intheratios ato1,bto1,cto1.Ifoneormoreoftheconstants a,b,c islessthanunitytheelongation inthatorthosedirections becomes acompression. Ifoneormoreoftheconstants is unity,components parallel tothatdirection arenotaltered. Thedirections i,i,karecalledtheprincipal axesofthestrain. Theirdirections arenotalteredbythestrainwhereas, ifthe constants a,b,cbedifferent, everyotherdirection isaltered. Thescalarsa,b,careknownastheprincipal ratwsof elongatiun. InArt.115itwasseenthatanycomplete dyadicwas reducible tothenormalform ~=±(ai'i+bj,i+ck'kJ 352 VECTOR ANALYSIS wherea,h,carepositive constants. Thisexpression maybe factored intotheproductoftwodyadics. (/)=±(ai'i'+hj'j'+ck'k') 0(i'i+j'j+k'k),(19) orf/J=±(i'i+j'j +k'k)0(aii+hj j +ckk). Thefactor i'i+j'j+k'k whichisthesameineithermethod offactoring isa.versor. Itturnsthevectorsi,j,kintothevectorsi',j',k'.This Versormayberepresented byitsvectorsemi-tangent of versionas .,..,. , i xiI+jXj'+kxk' 1 1+J J+ k k=1-+--·-·'-+--· .,+ kk"101JOJ 0 Theotherfactor ai'i'+hi'j'+ck'k', or aii+hjj+ckk isarighttensorandrepresents apurestrain.Inthefirst casethestrainhasthelinesi',j',k'forprincipal axes:in thesecond,i,j,k.Inbothcasestheratiosofelongation are thesame,-ato1,hto1,cto1.Ifthenegative signoccurs beforetheproduct theversionandpurestrainmusthave associated withthemareversal ofdirections ofallvectorsin 8pace-thatis,aperversion. Hence Theorem: Anydyadicisreducible totheproduct ofa versorandarighttensortakenineitherorderandapositive ornegative sign.Hencethemostgeneral transformation representable byadyadicconsists oftheproduct ofarota tionorversionaboutadefiniteaxisthrough adefiniteangle accompanied byapurestraineitherwithorwithout perver 8ion.Therotation andstrainmaybeperformed ineither order.Inthetwocasestherotationandtheratiosofelong& tionofthestrainarethesame;buttheprincipal axesofthe straindifferaccording asitisperformed beforeorafterthe • ROTATIONS ANDSTRAINS 353 rotation, eithersystemofaxesbeingderivable fromtheother bytheapplication oftheve~orasaprefactor orpostfactor respectively. Ifadyadic ~begiventheproductof.~anditsconjugate isarighttensorthe·ratiosofelongation ofwhicharethe squaresoftheratiosofelongation of~andtheaxesofwhich arerespectively theantecedents orconsequents of~accord ingas~0followsorprecedes ~intheproduct. ~= ±(ai'i+bj'j+ek'k), ~0=±(aii'+bjjI+ekk'), (/)•(/)0=alli'i'+01j'j'+ellk'k',(20) ~o•(/)=alli i+bllj j+ellkk. Thegeneralproblem offindingtheprincipal ratiosofelonga tion,theantecedents, andconsequents ofadyadicinits normalform,therefore reducestothesimplerproblemoffind ingtheprincipal ratiosofelongation andtheprincipal axes ofapurestrain. 131.]Thenaturalandimmediate generalization ofthe righttensor aii+bjj+ekk, isthedyadic ~=aaa'+bb b'+e0 0'(21) wherea,b,earepositiveornegative scalarsandwherea,b,0 anda',b',0'aretworeciprocal systems ofvectors. Neces sarilya,b,0anda',b',0'areeachthreenon-coplanar. Definition: Adyadicthatmaybereducedtotheform (/)=aaa'+bbb'+e0 0'(21) iscalledatonic. Theeffectofatonicistoleaveunchanged threenon .coplanar directions a,b,0inspace.Ifavectorberesolved intoitscomponents parallel toa,b,0respectively these 23 354 VECTOR ANALYSIS components arestretched intheratiosato1,hto1,cto1. Ifoneormoreoftheconstants a,h,carenegative thecom ponents parallel tothecorresponding vectora,b,0arere versedindirection aswellaschanged inmagnitude. The tonicmaybefactored intothreefactorsofwhicheach stretches thecomponents paralleltooneofthevectorsa,b,0 butleavesunchanged thecomponents parallel totheother two. (/)=(aaa'+bb'+00')•(aa'+hbb'+00')(aa'+bb'+coo'). Thevalueofatonicf/)isnotalteredifinplaceofa,b,0 anythreevectorsrespectively collinear withthembesub stituted, provided ofcoursethatthecorresponding changes whicharenecessary bemadeinthereciprocal systema',b',o'. Butwiththeexception ofthischange, adyadicwhichis expressible intheformofatonicissoexpressible inonly onewayiftheconstants a,h,caredifferent. Iftwoofthe constants sayhandcareequal,anytwovectorscoplanar withthecorresponding vectorsband 0maybesubstituted inplaceofbando.Ifalltheconstants areequalthetonic reducestoaconstant multiple oftheidemfactor. Anythree non-coplanar vectorsmaybetakenfora,b,o. Theproductoftwotonicsofwhichtheaxes",b,0arethe sameiscommutative andisatonicwiththeseaxesand withscalarcoefficients equalrespectively totheproducts of thecorresponding coefficients ofthetwodyadics. (/)=alaa'+hIbb'+cI00' 7J!'=a2aa'+h2bb'+c200' (/).7J!'=7J!'.(/)=ala2aa'+bIb2bb'+cIc200'.(22) Thegeneralization ofthecyclicdyadic aa'+cosq(bb'+00')+sinq(0b'-b0') is (/)=aaa'+h(bb'+00')+c(0b'- b0'),(28) ROTATIONSANDSTRAINS 355 wherea,b,0arethreenon-coplanar vectorsofwhicha',b',0' isthereciprocal systemandwherethequantities a,b,c,are positive ornegative scalars. Thisdyadicmaybechanged intoamoreconvenient formbydetermining thepositive scalarpandthepositive ornegative scalarq(whichmay alwaysbechosenbetween thelimits±'11)80that andb=pcosq c=psinq. (24) (24)'P=+y'b2+cl 1p-b tan2q=p+b·Thatis, and Then f/J=aaa'+pcosq(bb'+00')+Psinq(ob'-bo').(25) Thismaybefactored intotheproductofthre~dyadics (/)=(aaa'+bb'+00')•(aa'+pbb'+P00')• {aa'+cosq(bb'+00')+sinq(ob'-be')}. Theorderofthesefactorsisimmaterial. Thefirstisatonic whichleavesunchanged v.ectorsparalleltobandcbut stretches thoseparallel toaintheratioofato1.Ifais negative thestretching mustbeaccompanied byreversal indirection. Thesecondfactorisalsoatonic.Itleaves unchanged vectorsparalleltoabutstretches allvectorsin theplaneofbandeintheratiopto1.Thethirdisa cyclicfactor. Vectors paralleltoaremainunchanged; but radiivectorsintheellipseofwhichband 0are conjugate semi-dia.meters arerotatedthrough avectorsuchthatthe areaofthevectoristotheareaofthewholeellipseasqto 2'11".Othervectorsintheplaneofbandemayberegarded 88scalarmultiples oftheradiivectorsoftheellipse. 856 VECTOR ANALYSIS Definition: Adyadicwhichisreducible totheform fP=aaa'+Pcosq(bb'+00')+Psinq(0b'-b0'),(25) owingtothefactthatitcombines theproperties ofthe cyclicdyadicandthetoniciscalledacyclotonic. Theproductoftwocyclotonics whichhavethesamethree vectors, a,b,0asantecedents andthereciprocal system aI,b',0'forconsequents isathirdcyclotonic andiscom mutative. fP=a1••'+P1COBq1(bb'+00')+PIsinql(0b'- b0') W=a~aa'+PsCOBqs(bb'+co')+P~sinqs(eb'-be') fP.W=w.(/)=alasaa'+PIPsCOB(ql+qs)(bb'+00') +PIPssin(q1+qs)(0b'-bo~. (26) Reduction ofDyadica toCanonical Forms 132.]TlwYrem: Ingeneralanydyadic (/)maybereduced eithertoatonicortoacyclotonic. Thedyadicsforwhich thereduction isimpossible mayberegarded aslimiting cases whichmayberepresented toanydesireddegreeofapproxi mationbytonicsorcyclotonics. Fromthistheorem theimportance ofthetonicandcyclo tonic which havebeentreatedasnaturalgeneralizations of therighttensorandthecyclicdyadicmaybeseen.The proofofthetheorem, including adiscussion ofallthe specialcasesthatmayarise,islongandsomewhat tedious. Themethodofproving thetheorem ingeneralhowever is patent.Ifthreedirections a,b,0maybefoundwhichare leftunchanged bytheapplication of(/)then(/)mustbea tonic.Ifonlyonesuchdirection canbefound,thereexists aplaneinwhichthevectorssufferachangesuchasthatdue tothecyclotonic andthedyadicindeedprovestobesuch. ROTATIONS ANDSTRAINS 357 Thequestion istofindthedirections whichareunchanged bytheapplication ofthedyadic ip. Ifthedirection aisunchanged, then or (ip-aI).a=O.(27) Thedyadicip-aIistherefore planarsinceitreducesvectors inthedirection atozero.Inspecialcases,whichareset asideforthepresent,thedyadicmaybelinearorzero.In anycaseifthedyadic (/)-al reduoesvectorsoollinear withatozeroitpossesses atleast onedegreeofnullityandthethirdordetenninant of(/) vanishes.«/)-aI>a=o. (28) Now(page331)«/)+")s=iPs+(/)'J:fJ.T+(/):"'J+"s' Hence«/)-aI)s=(/)8-aiP'J:1+a2(/):~-a81I ~=1and18=1. But ip:1=(/)" iPs:I=(/)2'" Hencetheequation becomes as-a2ip8+a(/)28-(/)s=O. (29) Thevalueofawhiohsatisfiestheoondition that isa8olution ofaoubicequation. Letxreplacea.The cubioequation becomes 868 VECTOR ANALYSIS Anyvalueofzwhichsatisfies thisequation willbesuch that (28)' Thatistosay,thedyadicfP-zIisplanar. Avectorper pendicular toitscODdequents isreduced tozero.HencefP leavessuchadirection unchanged. Thefurtherdiscussion ofthereduction ofadyadictotheformofatonicoracyclo tonicdepends merelyuponwhether theoubicequation inz hasoneorthreerealroots. 133.]Theorem:Ifthecubicequation z&-z2f/J8+Zf/J28-f/J&=° (29)' hasthreerealrootsthedyadic f/Jmayingeneralbereduced toatonic. Forlet z=a,z=b,z=c bethethreerootsoftheequation. Thedyadics f/J-aI,f/J-bI,fP-cI (30)(f/J-aI)•a=0, (f/J-bI)•b=0, (f/J-cI)•c =O.areingeneral planar. Leta,b,cberespectively three vectorsdrawnperpendicular totheplanesoftheconsequents ofthesedyadics. Then f/J.a=aa, f/J.b=bb, f/J.c=cc.(30)' Iftherootsa,b,caredistinct thevectorsa,b,carenon coplanar. Forsuppose c=ma+nb (f/J-cI).(ma+nb)=0, But Hence and HenceROTATIONS ANDSTRAINS m(/)•a-mea+n(/)•b -ncb=O. (/)•a=ita,(/)•b=bb. m(a-c)a+n(b-c)b=0, m(a-c)=O, n(b-c)::zO. m= 0ora=c,n= 0orb=c.359 Consequently ifthevectorsa,b,carecoplanar, therootsare notdistinct; andthereforeiftherootsaredistinct, the vectorsa,b,carenecessarily non-coplanar. Incasetheroots arenotdistinctitisstillalwayspossible tochoosethree non-coplanar vectorsa,b,cinsuchamannerthattheequa,. tions(30)hold.Thisbeingso,thereexistsasystema',b',c' reciprocal toa,b,candthedyadicwhichcarriesa,b,cinto aa,bb,ccisthetonic f/J=aaa'+bbb'+ccc. Thwrem:Ifthecubicequation x8-x2(/)8+X(/)28-(/)3=0 (29)' hasonerealrootthedyadic (/)mayingeneralbereduced to acyclotonic. Thecubicequation hasonerealroot.Thismustbeposi tiveornegative according as(/)3ispositiveornegative. Let therootbea.Determine aperpendicular totheplaneof theconsequents of(/)-aI. «(/)-aI)•a=O. Determine a'alsosothat a'•«(/)-aI)=0 andletthelengthsofaanda'besoadjusted thata'.a=1. Thiscannotbeaccomplished inthespecialcaseinwhicha 860 VECTOR ANALYSIS anda'aremutually perpendicular. Letbbeanyvectorin theplaneperpendicular toa'. a'•(tP-aI)• b=o. Hence(tP-aI)•bisperpendicular toa'.HencetP.bis perpendicular toa'.Inasimilarmanner tPs•b,tPa•b,and tP-I•b,tP-JI.b,etc.,willallbeperpendicular toa'andliein oneplane.ThevectorstP.bandbcannotbeparallel orfP wouldhavethedirection baswellasaunchanged and thusthecubicwouldhavemorethanonerealroot. ThedyadicfPchanges a,tP.b,bintotP.a,tPs•b,tP.bre spectively. Thevolumeoftheparallelopiped But[tP.a tP2.btP.b]=tPa[a(/).bbJ.(31) tP.a=aa. Hence aa.«(/)2.b)x(tP.b)=(/)aa.(tP.b)Xb.(31)' Thevectors (/)2.b,(/).b,balllieinthesameplane.Their vectorproducts areparalleltoa'andtoeachother.Hence a(tP2•b)X(tP.b)=(/)atP.bXb.(31)" Inasmuch asaand(/)ahavethesamesign,let r=a-ItPa• (32) Letalso ba=P-ItP.b b2=p-2(/IJ.b,etc.(33) and b-1=p2(/)-I•b b -2=rtP·-2•b,etc. b2Xbl=blXb2• or Thevectorsb2+bandbIareparallel. Let b2+b=21'1.bl' Thenba+bl=21'1.b2bi+b2=21'1.ba bl+b_1=21'1.b b -I+b-'.l=21'1.b_1etc., etc.(84) (35) ROTATIONS ANDSTRAINS 861 Layofffromacommon originthevectors b,bI,bll,etc.,b_I,b-ll'etc. SincefPisnotatonic,thatis,sincethereisnodirection in theplaneperpendicular toa'whichisleftunchanged byfP thesevectorsb.passroundandroundtheoriginasmtakes onallpositive andnegative values. Thevalueofnmust therefore liebetween plusoneandminusone.Let Thenn=cosq. b_I+bI=2cosqb.(86) ThenDetermine cfromtheequation bI=cosqb+sinqo. b_I=cosqb -sinqc. Leta',b',0'bethereciprocal systemofa,b,c.Thisispos siblesincea'wassodetermined thata'• a=1andsince a,b,carenon-coplanar. Let 1jJ"=cosq(bb'+co')+sinq(cb'- bc'). Then 1jJ"•a=0, 1jJ"•b=bI, 1jJ"•b_1=b. Hence (aaa'+pIJT).a=aa=fP.a, (aaa'+pIJT)•b=Pbll=fP•b, (aaa'+pIJT)•b_I=Pb=f/J.b_1• Thedyadicaaa'+p1jJ"changesthevectorsa,bandb_1into thevectors fP•a,fP•b,an!!f/J•b-llrespectively. Hence f/J=(aaa'+pIJT)=aaa'+pcosq(bb'+CC') +Psinq(cb'-bo'). Thedyadic f/Jincasethecubicequation hasonlyonereal rootisreducible exceptinspecialcasestoacyclotonic. Thetheoremthatadyadicingeneralisreducible toatonic orcyclotonic hastherefore beendemonstrated. 862 VECTOR ANALYSIS 134.]Thereremaintwocases!inwhichthereduction illimpossible, 88canbeseenbylookingovertheproof.In thefirstplaceiftheconstant nusedinthereduction tocyclo tonicformbe±1thereduction fallsthrough. Inthesecond placeiftheplaneoftheantecedents of (f>-aI andtheplaneoftheconsequents areperpendicular the vectorsaanda'usedinthereduction tocyclotonic formare perpendicular anditisimpossible todetermine a'suchthat a.a'shallbeunity.Thereduction fallsthrough. If "=±1,b_1+b1=±2b. Let b-1+b1=2b. Choose 0=b1-b=b -b_1• Consider thedyadic IJ!=aaa'+p(bb'+0Of)+pCb' 7J!.a=aa=(f>.a, HenceIJ!.b=Pb+Pb1=fP.b, 7Jf•0=p0=pb1 -Pb=fP•e. tP=aaa'+p(bb'+00')+p0b1•(37) Thetransformation duetothisdyadicmaybeseenbestby factoring itintothreefactorswhichareindependent ofthe orderorarrangement tP=(aaa'+bb'+00').{aat+p(bb'+oo')} •(aa'+bb'+co'+ob'). 1Intheeeeasesitwillheseenthatther.ubicequation hasthreerealroots. Inoneeasetwoofthemareequalandintheothereasethreeofthem.Thus theeedyadiesmayberegurded 118limiting caseslyingbetweenthecyclotonie in whichtwooftherootsareimaginary andthetonicinwhichalltherootsarereal anddiatinct. Thelimitmayheregarded astaldngplaceeitherbythepure imaginary partofthetwoimaginary rootsofthecyclotonic becoming zeroorby twooftherootsofthetonicapproaching eachother. ROTATIONS ANDSTRAINS 863 Thefirstfactorrepresents anelongation inthedirection aina ratioato1.Theplaneofband 0isundisturbed. The secondfactorrepresents astretching oftheplaneofband 0in theratiopto1.Thelastfactortakestheform 1+ob'. (I+ob').xa=xa, (I+0b')•xb=xb+xc, (I+0b')•x0=xo. Adyadicoftheform1+ob'leavesvectorsparalleltoaand0 unaltered. Avectorxbparalleltobisincreased bythevec tor0multiplied bytheratioofthevectorxbtob.Inother wordsthetransformation ofpointsinspaceissuchthatthe planeofaand0remains fixedpointforpointbutthepoints inplanesparalleltothatplaneareshiftedinthedirection 0 byanamountproportional tothedistance oftheplanein whichtheyliefromtheplaneofaandc. Definition: Adyadicreducible totheform 1+cb' iscalleda6hearing dyadicorshearerandthegeometrical transformation whichitcausesiscalledashear.Themore generaldyadic ~=aall.'+p(bb'+00')+obi (87) willalsobecalledashearing dyadicorsMart'". Thetrans formation towhichitgivesriseisashearcombined with elongations inthedirection ofaandisintheplaneofbando. Ifn=-1insteadofn=+1,theresultismuchthesame. Thedyadicthenbecomes ~=aall.'-p(bb'+00')-ob' (37)' • ~=(aaa'+bb'+00')•{aa'-p(b0'+co')}•(I+cb'). 864 VECTOR ANALYSIS Thefactorsarethesameexceptthesecondwhichnowrepre sentsastretching oftheplaneofband 0combined witha reversal ofallthevectorsinthatplane.Theshearing dyadic ~thenrepresents anelongation inthedirection a,anelong& tioncombined withareversal ofdirection intheplaneof band0,andashear. Suppose thattheplaneoftheantecedents andtheplaneof theconsequents ofthedyadict/J-aIareperpendicular. Let theseplanesbetakenrespectively astheplaneofjandkand theplaneofiandjk.Thedya.dicthentakestheform ~-aI=Aji+Bjj+0kk+Dkj. Thecoefficient Bmustva.nish. Forotherwise thedyadic ~-aI -B1=(-Bi+Aj+0k)i+k(Di -Bk) isplanarandthescalara+Bisarootofthecubicequation. Withthisrootthereduction tothefonnofatonicmaybe carriedonasbefore. Nothing newarises.ButifBvanishes anewcaseoccurs. Let 1Jl'=~-aI=Aji+0k k+Dkj. Thismaybereduced asfollowstotheform ab'+bo' where a •b'=a •0'=b •0'=0andb •b'=1. Square 7J!' 7J!'II=ADki=a0'. ThenHenceamustbechosenparallel tokjand0',paralleltoi. Thedyadic 7J!'maythenbetransfonned into 1Jl'=ADk(0iA+:j)+Aji. ADkb'__0i+Di=, AD b=Aj 0'=i. ROTATIONSANDSTRAINS 365 Withthischoiceofa,b,b',0'thedyadic lJfreducestothe desiredformab'+be'andhencethedyadictPisreducedto tP=aI+ab'+bo' or tP=aaa'+abb'+aoo'+ab'+bo'.(38) Thismaybefactored intotheproduct oftwodyadicsthe orderofwhichisimmaterial. rp=aI.(I+ab'+b0'). ThefirstfactoraIrepresents astretching ofspaceinall directions intheratioato1.Thesecondfactor fJ=I+ab'+b0' represents whatmaybecalledacomplexshear.For r'=I.r+ab'.r + b0'.r=r+ab'.r + b0'.r. Ifrisparalleltoaitisleftunaltered bythedyadicfJ.If risparalleltobitischangedbytheaddition ofaterm whichisindirection equaltoaandinmagnitude propor tionaltothemagnitude ofthevectorr.Inlikemanner ifrisparalleltoeitischangedbytheaddition ofa~rm whichindirection isequaltobandwhichinmagnitude is proportional tothemagnitude ofthevectorr. fJ.xb=(I+ ab'+be').xb=xb+xa fJ•x0=(I+ab'+b0')•xe=x0+xb. J)efinitwn: Adyadicwhichmaybereducedtotheform tP=aI+ab'+b0' (38) iscalledacompk~sMarrt'. Thecomplex sheareraswellasthesimpleshearermen tionedbeforearelimitiLtg casesofthecyclotonic andtonic dyadics. 366 VECTOR ANALYSIS 135.]Amoresystematic treatment ofthevariouskinds ofdyadicswhichmayarisemaybegivenbymeansofthe Hamilton-Cayley equation lP3-tPalP2+tPutP-tP31=0 (39) andthecubicequation inx X3_tPax2+tP2ax-tP3=0. (29)' Ifa,b,caretherootsofthiscubictheHamilton-Ca.yley equation maybewrittenas (tP-aI).(tP-bI).(tP-cI)=O. (40) If,however, thecubichasonlyoneroottheHamilton-Cayley equation takestheform IngeneraltheHamilton-Cayley equation whichisanequa tionofthethirddegreeintPistheequation oflowestdegree whichissatisfied bytP.Ingeneraltherefore oneoftheabove equations andthecorresponding reductions tothetonicor cyclotonic formhold.Inspecialcases,however, thedyadic tPmaysatisfyanequation oflowerdegree. Thatequation oflowestdegreewhichmaybesatisfied byadyadiciscalled itscharacteristic equation. Thefollowing possibilities occur. I. (tP-aI)•(tP-bI)•(tP-cI)=O. II.(lP-aI).(lP2-2pcosqtP+p2I)=O. III. (tP-aI).(tP-bI)2=O. IV. (lP-aI)•(tP-bI)=O. V. (tP-aI)3=O VI. (lP-an2=""~ VII. (tP-aI)=O. ROTATlONSANDSTRAINS 367 Inthefirstcasethedyadicisatonicandmaybereduced totheform lP=aaa'+bbb'+coo'. Inthesecondcasethedyadicisacyclotonic andmaybe reducedtotheform lP=aaa'+pcosq(bb'+00')+Psinq(ob'-be'). Inthethirdcasethedyadicisasimpleshearerandmaybe reduced totheform lP=aaa'+b(bb'+oe')+eb'. Inthefourthcasethedyadicisagainatonic.Twoofthe ratiosofelongation arethesame.Thefollowing reduction maybeaccomplished inaninfinitenumberofways. lP=aaa'+b(bb'+oe'). Inthefifthcasethedya.dicisacomplex shearerandmaybe soexpressed that lP=aI+ab'+bo'. Inthesixthcasethedyadicisagainasimpleshearerwhich maybereduced totheform lP=a1+0b'=a(aa'+bb'+00')+ob'. Intheseventhcasethedyadicisagainatonicwhichmaybe reduced inadoublyinfinitenumberofwaystotheform lP=aI=a(aa'+bb'+0 0'). Thesesevenaretheonlyessentially different formswhicha dyadicmaytake.Therearethenonlysevenreallydifferent kindsofdyadics-threetonicsinwhichtheratiosofelonga tionarealldifferent, twoalike,orallequal,andthecyclo tonictogether withthreelimiting cases,thetwosimpleand theonecomplex shearer. 368 VECTOR ANALYSIS Summary ofChapuTVI Thetransformation duetoadyadicisalinearhomogeneous strain. Thedyadicitselfgivesthetransformation ofthe pointsinspace.Thesecondofthedyadicgivesthetrans formation ofplaneareas.Thethirdofthedyadicgivesthe ratioinwhichvolumes arechanged. r'=tP.r, .'=(/)2·" v'=tPa",· Thenecessary andsufficient condition thatadyadicrepre sentarotation aboutadefiniteaxisisthatitbereducible to theform orthat orthattP=i'i+j'j+k'k tP.(/)c=ItPa=+1 tP·(/)c=I (/)a>O(1) (2) Thenecessary andsufficient condition thatadyadicrepre sent1\rotation combined withatransformation ofreflection bywhicheachfigureisreplaced byonesymmetrical toitis that orthat orthattP= -(i'i+jfj+k'k) tP.tPc=I,tPa= -1 tP.tPc=I,tPa<O.(1)' (8) Adyadicoftheform(1)iscalledaversor;oneoftheform (1y,aperversor. Iftheaxisofrotation ofaverBorbechosenorthei-a.xis theversorreducesto tP=ii+cosq(jj+kk)+sinq(kj-jk) (4) or tP=ii+cosq(I-ii)+sinq1Xi.(5) Ifanyunitvectoraisdirected alongtheaxisofrotation tP= aa+cosq(I-all.)+sinqIXa(6) Theaxisoftheversorcoincides indirection with-tPx• ROTATIONSANDSTRAINS 369 Ifavectorbedrawnalongtheaxisandifthemagnitude of thevectorbetakenequa.ltothetangentofone-half theangle ofrotation, thevectordetermines therotation completely. 'Phisvectoriscalledthevectorsemi-tangent ofversion. Q= -f/J" 1+(j)/J(9) (10)'" orInterlIlBofQtheversorf/Jmaybeexpressed inanumberof ways. f/J=:.~+cosq(I-:.~)+sinqIxv':.Q(10) or f/J=(I+1xQ)•(I- IXQ)-1 (10)' or f/J=QQ±(I+1XQ)3 (1,0)" 1+QQ f/J=(1-Q•Q)1+2_QQ+2 IXQ .1+Q.Q Ifaisaunitvectoradyadicoftheform f/J=2aa-1 (11) isabiquadrantal versor. Anyversormayberesolved into theproduct oftwobiquadrantal versorsandbymeansof suchresolutions anytwoversorsmaybecombined into another. Thelawofcomposition forthevectorsemi-tangents ofversionis Q 3=Q1+Q2+Q2XQ1• 1 -Q1·Q2 Adyadicreducible totheform f/J=aa'+cosq(bb'+0Of)+sinq(0b'- b0')(17) iscalledacyclicdyadic.Itproduces ageneralization of simplerotation-anellipticrotation, sotospeak.Thepro 24 370 VECTOR ANALYSIS ductoftwocyclicdyadicswhichhavethesameantecedents a,b,0andconsequents a'b'0'isobtained byaddingtheir anglesq.Acyclicdyadicmayberegarded asarootofthe idemfactor.A dyadicreducible totheform ~=aii+bjj+ckk (18) wherea,b,carepositive scalarsiscalledarighttensor.It represents astretching alongtheprincipal axisi,j,kinthe ratioato1,bto1,cto1whicharecalledtheprincipal ratios ofelongation. Thistransformation isapurestrain. Anydyadicmaybeexpressed astheproduct ofaversor, arighttensor,andapositiveornegative sign. ~=±(ai'i'+bj'j'+ck'k')(i'i+j'j+k'k) or ~=±(i'i+j'j +k'k).(aii+bjj+ckk).(19) (21)Consequently anylinearhomogeneous strainmayberegarded asacombination ofarotationandapurestrainaccompanied orunaccompanied byaperversion. Theimmediate generalizations oftherighttensorandthe cyclicdyadicistothetonic ~=aaa.'+bbb'+ccc' andcyclotonic (j)=aaa'+b(bb'+co')+c(ob'-bo) (23) (24)' whereorq;=aaa'+pcosq(bb'+co')+psinq(ob'- b0')(25) __ 1p-b P=+v'b2+c2andtan~q=p+b' Anydyadicingeneralmaybereducedeithertotheform (21),andistherefore atonic,ortotheform(25),andis therefore acyclotonic. Thecondition thatadyadicbea. tonicisthatthecubicequation x3-~sx2+~2Sx-(/)8=0 (29)' ROTATIONS A.NDSTRAINS 371 shallhavethreerealroots.Specialcasesmwhichthe reduction maybeaccompilllhed inmorewaysthanonearise whentheequation hasequalroots.Thecondition thata. dyadicbeacyclotome isthatthiscubicequation shallhave onlyonerealroot.Thereoccurtwolimiting casesinwhich thedyadiccannotbereduced tocyclotonic form.Inthese casesitmaybewrittenas f/J=aaa'+p(bb'+ee')+obi (37) andisasimpleshearer,orittakestheform f/J=aI+ab'+b0' (88) andisacomplex shearer. Dyadics maybeclassified accord ingtotheircharacteristic equations tonic cyclotonic simpleshearer specialtonic complex shearer specialsimpleshearer specialtonic.(f/J-aI)•(f/J-bI)•(f/J-cI)=0 UP-aI)•(r,(J2-2pcosqf/J+p2I)=0 (f/J-aI)•(f/J-b1)2=0 (f/J-al).(f/J-bl) =0 (f/J-aI)8=0 (f/J-a1)2=0 (f/J-aI) =0 CHAPTER VII JrnlCELLANEOUS APPLICATIONS Quadric Surfaces 136.]IflPbeanyconstant dyadictheequation r •lP•r=const. (1) isquadratic inf.Theconstant, incaseitbenotzero,may bedividedintothedyadic lPandhencetheequation takes theform orf·lP•r=1, f.lP•f=O.(1)' (2) (3)Thedyadic lPmaybeassumed ·tobeself-conjugate. Forif 'J!isananti~elf-eonjugate dyadic,theproduct f •"•rM identically zeroforallvaluesoff.Theproofofthisstate mentisleftasanexercise. ByArt.116anyself-eonjugate dyadicisreducible totheform iijjkk lP=±t±b2±-I•a c If f::...xi+yj+zk, .xlylZl f •lP•f=±"2±bl±,.a c(4) Hencetheequation represents aquadricsurfacerealorimaginary. Thedifferent caseswhicharisearefourinnumber.Ifthe signsareallpositive, thequadric isarealellipsoid.Ifone signisnegative itisanhyperboloid ofonesheet;iftwoare QUADRIC SURPACES 373 r0fP0r=const.negative,ahyperboloid oftwosheets.Ifthethreesigll8are allnegative thequadric isimaginary. Inlikemannerthe equation isseentorepresent aconewhichmaybeeitherrealor imaginary according asthesignsaredifferent orallalike. Thustheequation represents acentralquadricsurface. Thesurfacereducesto aconeincasethecOll8tant iszero.Conversely anycentral quadricsurfacemayberepresented byasuitably chosen8el£ conjugate dyadicfPintheform r0fP0r=const. Thisisevidentfromtheequations ofthecentralquadric surfaces whenreduced tothenonnalform.Theyare Thecorresponding dyadicfPisiijjkkfP=± - ± - ±-.a2b'lc2 Themostgeneralscalarexpression whichisquadratic in thevectorrandwhichconsequently whensetequaltoacon stantrepresents aquadricsurface,contaill8 termslike ror,(roa)(bor), roc,doe, wherea,b,c,d,eareconstant vectors. Thefirsttwoterms areofthesecondorderinr;thethird,ofthefirstorder;and thelast,independent ofr.Moreover, itisevidentthatthese foursortsoftermsaretheonlyoneswhichcanoccurina scalarexpression whichisquadratic inr. But andr0r=r0lor, (r0a)(b0r)=r0abor. 374 VECTOR ANALYSIS Hencethemostgeneralquadratic expression maybereduced to r •(j)•r+r •A+a=0, where (j)isaconstant dyadic,Aaconstant vector,anda aconstant scalar.Thedyadicmayberegarded asself conjug'd.Wifdesired. Toberidofthelineartermr •A,makeachangeoforigin byreplacing rbyr'-t. (r'-t)•(j)•(r'-t)+(r'-t)•A+0=0 r'•(j)•r'-t •(j)•r'-r'•(j)•t+t •(j)•t +r'•A-t •A+a=O. Since(j)isself-conjugate thesecondandthirdtermsare equal.Hence tr'•(j)•r'+2r'•(~A-(j)•t)+a'=O. Ifnow(j)iscomplete thevectortmaybechosensothat t 1 _2A=(j)•tort=2(j)1.A. Hencethequadricisreducible tothecentralform r'•(j)•r=const. Incase(j)isincomplete itisuniplanar orumlinear because (j)isself-conjugate. IfAliesintheplaneof(j)orintheline of(j)asthecasemaybetheequation tijA=(j)d issolublefortandthereduction tocentralformisstillp0s sible.ButunlessAissosituated thereduction isimpossible. Thequadricsurfaceisnotacentralsurface. Thediscussion andclassification ofthevariousnon-central quadrics isaninteresting exercise.Itwillnotbetakenup here.Thepresentobjectistodevelopsomuchofthetheory QUADRIC SURFACES 375 ofquadricsurfaces aswillbeusefulinapplications tomathe maticalphysicswithespecial reference tonon-isotropic media. Hereafter therefore thecentralquadrics andinparticular the ellipsoid willbediscUS'Bed. 137.JThetangentplanemaybefoundbydifferentiation. r0(/)0r=1. dr0(/)0r+r0(j)0dr =O. Since(j)isself-conjugate thesetwotermsareequaland dr0(/)0r=O. (5) Theincrement drisperpendicular to(j)0r.Hence (j)0ris normaltothesurfaceattheextremity ofthevectorr.Let thisnormalbedenoted by11andlettheunitnormalben. 1I=(j)or (j)or (j)orn= ~...... ..~..-v«(/) 0r) 0«(/) 0r)vr 0(/)2 0r(6) Letpbethevectordrawnfromtheoriginperpendicular to thetangent plane.pisparallel ton.Theperpendicular distance fromtheorigintothetangent planeisthesquare rootofpop.Itisalsoequaltothesquarerootofr0p. r0p =l'cos(r,p)p=p2. Hence Or Butrop=pop. pop p-=ro--=1.pop pop r0(j)0r=roll=1. Henceinasmuch aspand11areparallel,they areequal. p (/)0r=1I=-.pop(7) 876 VECTOR ANALYSIS Onpage108itwasseenthatthevectorwhichhasthedirec tionofthenormaltoaplaneandwhichisinmagnitude equal tothereciprocal ofthedistance fromtheorigintotheplane maybetakenasthevectorcoordinate ofthatplane.Hence theaboveequation showsthatfP.risnotmerelynormalto thetangentplane,butisalsothecoordinate oftheplane. Thatis,thelengthoffP.risthereciprocal ofthedistance fromtheorigintotheplanetangenttotheellipsoid at theextremity ofthevectorr. Theequation oftheellipsoid inplanecoordinates maybe foundbyeliminating rfromthetwoequations. 5r •fP•r=1, ifP.r=Ii. r=fP-1•Ii=Ii.fP-l. Hencer.fP•r=Ii•fP-1 •fP•fP-1 •Ii=11•fP-1•11. Hencethedesiredequation is Ii·fP-1·1I=1. (8) If Let andfP-1=a2i i+b2jj+c2kk. r=J!i+yj+zk, lI=ui+vj +wk, whereu,v,warethereciprocals oftheintercepts ofthe plane11upontheaxesi,j,k.Thentheellipsoid maybe writtenineitherofthetwoformsfamiliarinCartesian geometry. (9) QUADRIC SURFACES 377 138.]Thelocusofthemiddlepointsofasystemof parallel chordsinanellipsoid isaplane.Thisplaneis calledthediametral planeconjugate withthesystemof chords. Itisparallel totheplanedrawntangent tothe ellipsoid attheextremity ofthatoneofthechordswhich passesthroughthecenter. Letrbeanyradiusvectorintheellipsoid. LetIbethe vectordrawntothemiddlepointofachordparalleltoa. Let r=l+xa. b •(/)•11.=0.IIrisaradiusvectoroftheellipsoid r •(/)• r=(I+xa)•(/)•(I+xa)=1. HenceI.(/)• I+2xI •(/)•a+:z.2a •(/)•11.=1. Inasmuch asthe vector Ibisectsthechordparalleltoathe twosolutions ofxgivenbythisequation areequalinmag nitudeandopposite insign.Hencethecoefficient ofthe lineartermxvanishes.I •(/)•a=O. Consequently thevector Iisperpendicular to(/).a.The locusoftheterminus ofIistherefore aplanepassedthrough thecenteroftheellipsoid, perpendicular to(/).a,andpanillel tothetangentplaneattheextremity ofa. Ifbisanyradiusvectorinthediametral planeconjugate with&, Thesymmetry ofthisequation showsthataisaradius vectorintheplaneconjugate withb.Letcbe1\thirdradius "Vectorintheellipsoid andletitbechosenasthelineof intersection ofthediametral planesconjugate respectively withaandb.Then a •(/)•b=0, b •(/)•c=0, ,.(/).11.=0.(11) 378 VECTOR ANALYSIS Thevectors a.,b,0arechanged into~•a.,~•b,~•cby thedyadicfP.Let a'=fP•a.,b'=fP•b,0'=fP•o. Thevectorsa',b',0'formthesystemreciprocal toa,b,c. Fora •a'=a •rp•a=1,b •b'=b •rp•b=1, o •0'=0 •fP•C=1, and a •b'=a •fP•b=0,b.0'=b •fP•0=0, o •a'=0 •fP•a=O. Thedyadicrpmaybetherefore expressed intheforms "andrp=a'a'+b'b'+0'0', ~- 1=a a+b b+0o.(12) Ifforconvenience thethreedirections a.,b,0,becalleda systemofth"eeconjugate radiivectors,andifinasimilar mannerthethreetangentplanesattheirextremities becalled 'asystemofthreeconjugate tangent planes,anumber of geometric theorems maybeobtained frominterpreting the invariants offP.Asystemofthreeconjugate radiivectors maybeobtained inadoublyinfinitenumberofways. .Thevolumeofaparallelopiped ofwhichthreeconcurrent edgesconstitute asystemofthreeconjugate radiivectorsis constant andequalinmagnitude totherectangular parallelo pipedconstructed uponthethreesemi-axes oftheellipsoid. Forleta.,b,0beanysystemofthreeconjugate axes. fP-1=a a+bb+0o. Thedetenninant orthirdoffP-1isaninvariant andinde pendent oftheforminwhichfPisexpressed. Butif HenceQUADRIC SURFACES lP-1=a2ii+b2j j+c2kk, lPa-1=a2b2c2• [abc]=abc.879 HenceThisdemonstrates thetheorem. Inlikemannerbyinter preting lPs'lP8-1,andlP8itispossible toshowthat: Thesumofthesquaresoftheradiivectorsdrawntoan ellipsoid inasystemofthreeconjugate directions isconstant andequaltothesumofthesquaresofthesemi-axes. Thevolumeoftheparallelopiped, whosethreeconcurrent edgesareinthedirections oftheperpendiculars uponasystem ofthreeconjugate tangentplanesandinmagnitude equalto thereciprocals ofthedistances ofthoseplanesfromthe centeroftheellipsoid, isconstant andequaltothereciprocal oftheparallelopiped constructed uponthesemi-axes ofthe ellipsoid. Thesumofthesquaresofthereciprocals ofthethreeper pendiculars dropped fromtheoriginuponasystemofthree conjugate tangentplanesillconstant andequaltothesumof thesquaresofthereciprocals ofthesemi-axes. Ifi,j,kbethreemutually perpendicUlar unitvectors lP8=i.lP.i+j.lP.j+k.lP.k, lP8-1=i.~1.i+j •lP-1•j+k•lP-1•k. Leta,b,cbethreeradiivectors intheellipsoid drawn respectively paralleltoi,j,k. a.lP.a=b.lP.b=c.lP·o=l. i.lP.i j.lP.jk.lP.k lP~=+ +---...a.lP.a b.lP.b o.lP.c Butthethreetermsinthisexpression arethesquaresofthe reciprocals oftheradiivectorsdrawnrespectively inthei,i, kdirections. Hence: 380 VECTOR ANALYSIS Thesumofthesquaresofthereciprocals ofthreemutually perpendicular radiivectorsinanellipsoid isconstant. And inasimilarmanner: thesumofthesquaresoftheperpen dicularsdropped fromtheoriginuponthree mut~ally perpen diculartangentplanesisconstant. 139.]Theequation ofthepolarplaneofthepointdeter minedbythevectorais1 I·~.a=1. (13) Forlet1bethevectorofapointinthepolarplane.The vectorofanypointuponthelinewhichjoinstheterminus of 1andtheterminus ofais , yl+a:a a:+y Ifthispointlie8uponthesurface yl+a:a ~.YI+a:a=l a:+y x+y y2 2a:y x2-----:-I·~.1+ I.~.a +----- a•~•a =1.(a:+y)2 (x+y)2 (a:+y)2 Iftheterminusof.liesinthepolarplaneofathetwovalues oftheratiox:ydetermined bythisequation mustbeequal inmagnitude andopposite insign.Hencetheterminxy vanishes. Hence I·~·a=1 isthedesiredequation ofthepolarplaneoftheterminus ofL Letabereplaced byza.Thepolarplanebecomes I·~•za=1, or 1Iti.evidently immaterial whether thecentralquadricdetermined by•be realorimaginary, ellipeoid orhyperboloid. QUADRIC SURFACES 381 Whenzincreases thepolarplaneoftheterminus ofza approaches theorigin.Inthelimitwhenzbecomes infinite ~hepolarplanebecomes I'(/)•a=O. Hencethepolarplaneofthepointatinfinityinthedirection aisthesameasthediametral planeconjugate witha.This statement isfrequently takenasthedefinition ofthediame tralplaneconjugate witha.Incasethevectoraisaradius vectorofthesurfacethepolarplanebecomes identical with thetangentplaneattheterminus ofa.Theequation I •(/)•a=1orI •Ii=1 therefore represents thetangentplane. Thepolarplanemaybeobtained fromanother standpoint whichisimportant. IfaquadricQandaplaneParegiven, Q=r.(/).r-l=O and theequationP=r.C -c=0, (r.(/)• r -1)+k(r•C -C'!=0 represents aquadric surfacewhichpassesthrough thecurve ofintersection ofQandPandistangent toQalongthat curve.InlikemanneriftwoquadricsQandQ'aregiven, Q=r.(/).r-l=0 Q'=r •(/)'• r -1=0, theequation (r.(/)• r -1)+k(r•(/)'• r -1)=0 represents aquadricsurfacewhichpassesthrough thecurves ofintersection ofQandQ'andwhichcutsQandQ'atno otherpoints.Incasethisequation isfactorable intotwo equations whicharelinearinr,andwhichconsequently rep resenttwoplanes,thecurvesofintersection ofQand(j becomeplaneandlieinthosetwoplanes. 882 VECTOR ANALYSIS If...4.isanypointoutaideofthequadricandifallthetangent planeswhichpassthrough...4. aredrawn,theseplanesenvelop acone.Thisconetouchesthequadricalongaplanecurve theplaneofthecurvebeingthepolarplaneofthepointA. Forletabethevectordrawntothepoint...4.. Theequation ofanytangentplanetothequadricis I'fP.r=1. Ifthisplanecontains ...4.,itaequation issa.ti.qfied bya.Hence theconditions whichmustbesa.tisfied byrifitstangent planepassesthrough ...4.are a.fP.r=1, r.fP.r=1. Thepointsrtherefore lieinaplaner.(fP•a)=1which oncomparison with(13)isseentobethepolarplaneofA. Thequadricwhichpassesthrough thecurveofintersection ofthispolarplanewiththegivenquadricandwhichtouches thequadricalongthatcurveis (r.fP.r -1)+k(a.fP.r-1)1=O. Ifthispa.B8esthrough thepointA, (a.fP.a-I)+k(a.fP.a -1)2=O. Hence(r.fP•r -1)(a.fP•a-I)-(a•fP•r -1)2=O. Bytransforming theorigintothepointAthisiseasilyseen tobeaconewhosevertexisatthatpoint. 140.]LetfPbeanyself-conjugate dyadic.Itisexpres sibleintheform fP=Aii+Bjj+Ckk ...4.<B<C fP-BI=(C-B)kk-(B-A)ii.where ...4.,B,Carepositive ornegative scalars. moreletFurther- 383 QUADRIC SURFACES Let v'C-Bk=candv'B-Ai=a. Then(/)-BI=cc-aa=~ ~(c+a)(c-a)+(c-a)(c+a) I· Let Thenc+a=pandc - a=q. 1(/)=BI+2(pq+qp). (14) Thedyadic (/)hasbeenexpressed asthesumofaconstant multiple oftheidemfactor andonehalfthesum pq+qp. Thereduction hasassumed tacitlythattheconstants A,B,C aredifferent fromeachotherandfromzero. Thisexpression for(/)iscloselyrelatedtothecircular sectionsofthequadricsurface r·(/).r=l. Substituting thevalueof(/),rv(/)•r=1becomes Br.r+r.p q.r=l. Let r'p=n q •r(r•p -n)=O.beanyplaneperpendicular top.Bysubstitution Br.r+nq.r -1=O. Thisisaspherebecause thetermsofthesecondorderall havethesamecoefficient B.Iftheequation ofthissphere besubtracted fromthatofthegivenquadric,theresulting equation isthatofaquadricwhichpassesthrough theinter sectionofthesphereandthegivenquadric. Thedifference is Hencethesphereandthequadric intersect intwoplane curveslyingintheplanes q •r=0andr·p=n. 384 VECTOR ANALYSIS Inasmuch asthesecurveslieuponaspheretheyarecircles. Henceplanesperpendicular topcutthequadricincircles. Inlikemanneritmaybeshownthatplanesperpendicular to qcutthequadricincircles. Theproofmaybeconducted as follows:Br.r+r.pq.r=l. Ifrisaradiusvectorintheplanepassedthrough thecenter ofthequlldric perpendicular toporq,thetermr.pq.rvan ishes.Hencethevectorrinthisplanesatisfiestheequation Br.r=l andisofconstant length. Thesectionistherefore acircular section. Theradiusofthesectionisequalinlengthtothe meansemi;u:is ofthequadric. Forconvenience letthequadricbeanellipsoid. Thecon stantsA,B,Carethenpositive. Thereciprocal dyadic(/)-1 maybereducedinasimilarmanner. "'-1iijjkt V'=-+-+ABC Let Then Letf=I~_1tandd=11_1i. BCAB (/)-1_~I=ff-dd=!j(f+d)(f-d)B 2! +(f-d)(f+d)I. f+d=uandf -d=v. Then1fP-1=BI+~(uv+vu). (15) QUADRIC SURFACES 385 ForThevectorsuandvareconnected intimately withthecir cularcylinders whichenveloptheellipsoid r •qJ•r=1orIf.qJ-l•If=1. 1-If.If+If.uv.If=l.B IfnowIfbeperpendicular touorvthesecondterm,namely, If• uv•If,vanishes andhencethe'equationbecomes If·If=B. Thatis,thevectorIfisofconstant length. Buttheequation If.u=O istheequation ofacylinder ofwhichtheelements andtan gentplanesareparalleltou.IfthenIf.Ifisconstautthe cylinder isacircular cylinder enveloping theellipsoid. The radiusofthecylinder isequalinlengthtothemeansemi-axis oftheellipsoid. Thereareconsequently twoplanespassing through the originandcuttingoutcirclesfromtheellipsoid. Thenormals totheseplanesarepandq.Thecirclespassthrough the extremities ofthemeanaxisoftheellipsoid. Therearealso twocircularcylinders enveloping theellipsoid. Thedirection' . oftheaxesofthesecylinders areuandv.Twoelements of thesecylinders passthrough theextremities ofthemeanaxis oftheellipsoid. Theseresultscanbeseengeometrically asfollows. Pass aplanethrough themeanaxisandrotateitaboutthat axisfromthemajortotheminoraxis.Thesectionisan ellipse. Oneaxisofthisellipseisthemeanaxisofthe ellipsoid. Thisremains constant duringtherotation. The otheraxisoftheellipsevariesinlengthfromthemajortothe minoraxisoftheellipsoid andhenceatsomestagemustpasi through alengtheqnaltothemeanaxis.Atthisstageof 25 386 VECTOR ANALYSIS therotation thesectionisacircle.Inlikemannerconsider theprojection orshadowoftheellipsoid castuponaplane parallel tothemeanaxisbyapointataninfinitedistance fromthatplaneandinadirection perpendicular toit.Asthe ellipsoid isrotatedaboutitsmeanaxis,fromthepositionin whichthemajoraxisisperpendicular totheplaneofprojec tiontotheposition inwhichtheminoraxisisperpendicular tothatplane,theshadowandtheprojecting cylinder havethe meanaxisoftheellipsoid asoneaxis.Theotheraxischan~es fromtheminoraxisoftheellipsoid tothemajorandhenceat somestageoftherotationitpassesthroughavalueequalto themeanaxis.Atthisstagetheshadowandprojecting cylinder arecircular. Thenecessary andsufficient condition thatrbethemajor orminorsemi-axis ofthesectionoftheellipsoidr.(/J•r=1 byaplanepassingthrough thecenterandperpendicular toa isthata,r,and(/J.rbecoplanar. Let and Differentiate : Furthermorer·(/J.r=1 r.a=O. dr.(/J.r=0, dr.a=O. dr·r=0, ifristobeamajororminoraxisofthesection; forrisa maximum oramininum andhenceisperpendicular todr. Thesethreeequations showthata,r,and(/J•rareallortho gonaltothesamevectordr.Hencetheyarecoplanar. Conversely if[ar(/J.r]=O. [ar(/J.r]=0,(16) dr.r=O.drmaybechosenperpendicular totheircommon plane. Then QUADRIC SURFACES 387 Hencerisamaximum oramininum andisoneoftheprin cipalsemi-axes ofthesectionperpendicular toa. 141.]Itisfrequently anadvantage towritetheequation ofanellipsoidintheform insteadofr.1JT2.r=1, r.fP.r=l.(17) TIllsmaybedone;becauseif iijjtt¢=2+ b2+-2'a c '"iijjtt '1'=-+-+abc(18) itladyadicsuchthatYJisequaltofP.1JTmayberegarded as asquarerootoffPandwrittenasfPl.Butitmustbere membered thatthereareothersquarerootsoffP-for example,iijjtt-+---abc and Forthisreasonitisnecessary tobearinmindthatthesquare rootwhichismeantbyfPlisthatparticular onewhichhas beendenoted by1JT. Theequation oftheellipsoid maybewrittenintheform r.1JT.1JT.r=1, or (1JT.r)•(1JT.r)=1. Letr'betheradiusvectorofaunitsphere. Theequation of thesphereis 388 VECTOR ANALYSIS Ifr'=1Jf.ritbecomes evidentthatanellipsoid maybe transformed intoaunitspherebyapplying theoperator " toeachradiusvectorr,andviceversa,theunitspheremay betransformed intoanellipsoid byapplying theinverseoper ator'F-1toeachradiusvectorrI.Furthermore ifa,b,0are asystemofthreeconjugate radiivectorsinanellipsoid a.1j!'i.a=b.1j!''J..b=c.~.0=1, a.'F'J..b=b •1Ft.c=o.7p"'J..a=O. Ifforthemoment a',b',0'denoterespectively 1j!'.a,'F.b, F.o, a'•a'=b'•b'=0'•c'=1, a'•b'=b'•0'=0'•a'=O. Henoethethreeradiivectorsa',b',0'oftheunitsphereinto whichthreeconjugate radiivectorsintheellipsoid aretrans formedbytheoperator 1J!-1aremutually orthogonal. They formaright-handed orleft-handed systemofthreemutually perpendicular unitvectors. T/teQrem: Anyellipsoid maybetransformed intoanyother ellipsoid bymeansofahomogeneous strain. Lettheequations oftheellipsoids be r •(fl•r=1, and r.'F.r=1. Bymeansofthestrain(flltheradiivectorsrofthefirst ellipsoid arechanged intotheradiivectorsr'ofaunitsphere r'=(fll.r,r'•r'=1. Bymeansofthestrain'F-ltheradiivectorsr'ofthisunit spherearetransformed inlikemannerintotheradiivectorsr ofthesecondellipsoid. Hencebytheproductrischanged intor.r='F-l•(fll•r. (19) QUADRIC SURFACES 389 HenceThetransformation maybeaccomplished inmoreways thanone.TheradiivectorsrIoftheunitspheremaybe transformed amongthemselves bymeansofarotationwithor without aperversion. Anythreemutually orthogonal unit vectorsinthespheremaybechanged intoanythreeothers. Hencethesemi-axes ofthefirstellipsoid maybecarriedover byasuitable strainintothesemi-axes ofthesecond. The strainisthencompletely determined andthetransformation canbeperformed inonlyoneway. 142.]Theequation ofafamilyofconfocal quadricsur facesis x2 y2 z2--+--+ --=1. (20)a2-nb2-n c~-n Ifr·rp•r = 1andr.fJf.r = 1aretwosurfaces ofthe family, rp-l=(a2-n1)ii+(b2-n1)jj+(c2-n1)kk, fJf-1=(a2-n2)ii+(b2-n2)jj+(c2-n2)kk. rp-1_fJf-1=(n2-n1)(ii+jj+kk) =(n2-n1)I. Thenecessary andsufficient condition thatthetwoquadrics andr.rp.r=l r.fJf.r=l beconfocal, isthatthereciprocals ofrpandfJfdifferbya multiple oftheidemfactor rp-l_fJf-1=xI. (21) 890 VECTOR ANALYSIS Iftwoconfocalquadrics intersect, theydo80atrightangles. Letthequadrics ber •fP•r=1, and Letr·1j!.r=1. • =fP•rand.'=1j!.r, r=fP-1••andr=1j!-1••'. Thenthequadrics maybewrittenintermsof.and.'as and••fP-1••=1, .'.1j!-1••'=1, wherebytheconfocal property, fP-1_1j!-1=xI. Ifthequadrics intersectatrthecondition forperpendicularity isthatthenormals fP•rand1j!•rbeperpendicular. Thatis, •••'=O. Butr=1j!-1••'=fP-1•I=(1j!-1+xI)•• =1j!-1••+x.. x•••'=.'.1j!-1••'-••1j!-1••'=1 _•.1j!-1••'. Inlikemanner r=r/J-l•I=1j!-1••'=(r/J-l-xI)•I'=fP-1••'-x.'. x•••'=•.tP-1••'-••r/J-l••=•.fP-1•• '-1. Add: Hence2x••I'=I'(fP-1-1j!-1)•I'=XI••'. I ••'=0, andthetheoremisproved. Iftheparameter nbeallowedtovaryfrom-octo+octhe resulting confocal quadrics willconsistofthreefamiliesof whichoneisellipsoids; another,hyperboloids ofonesheet; andthethird,hyperboloids oftwosheets. Bytheforegoing QUADRIC SURFACES 391 7jT=(p+Idn.theorem eachsurfaceofanyone familycutseverysurface oftheothertwoorthogonally. Thesurfaces formatriply orthogonal system. Thelinesofintersection oftwofamilies (saythefamilyofone-sheeted andthefamilyoftw~heeted hyperboloids) cutorthogonally theotherfamily-thefamily ofellipsoids. Thepointsinwhichtwoellipsoids arecutby theselinesarecalled.corresponding pointsuponthetwoellip soids.Itmaybeshownthatthemtiosofthecomponents of theradiusvectorofapointtotheaxesoftheellipsoid through thatpointarethesameforanytwocorresponding points. Forletanyellipsoid begivenbythedyadic iijjkk (p=:I+b2+-2'a c Theneighboring ellipsoid inthefamilyisrepresented bythe dyadic '"IIjjkk :t'=+ + ,a2-dnb2-elnc2-dn 7jT-I=(p-l+Idn. Inasmuch as(pand7jTarehomologous (seeEx.8,p.330) dyadics theymaybetreatedasordinary scalarsinalgebm. Therefore iftermsoforderhigherthanthefirstindnbe omitted, Thetwo neighboring ellipsoids arethen r •fP•r=1, and r.(fP+Idn)•r=1. By(19) r=(fP+Idn)-i•fPi•r, r=(I+(pdn)-i.r, _ 1 dn r -(I-2(pdn)•r=r - -2-fP•r. 892 VECTOR ANALYSIS Theratioofthesecomponents isThevectorsrandrdifferbyamultiple offP.rwhichis perpendicular totheellipsoid fP.Hencetheterminiofiand rarecorresponding points,fortheylieupononeofthelines whichcutthefamilyofellipsoids orthogonally. Thecom ponentsofrandrinthedirection iarer •i=xand dn. dnxioi=:c=r •i --.-1 •fP•r=x----.2 2as ~=l_dn. x 2as Theaxesoftheellipsoids inthedirection iare"-las-dnand a.Theirratiois "-laS-dn a1dnd-a--- 1 11. X2a=---=_.---- 2a'xa Inlikemanner"-Ib2 -dn=~and"-Icll -dn=~. b '!I cz Hencetheratiosofthecomponents ofthevectorsfandr drawntocorresponding pointsupontwoneighboring ellip soidsonlydifferatmostbytermsofthesecondorderindn fromtheratiosoftheaxesofthoseellipsoids. Itfollows immediately thattheratiosofthecomponents ofthevectors drawntocorresponding pointsupon any twoellipsoids, sepa ratedbyafinitevariation intheparameter n,onlydifferat mostbytermsofthefirstorderindnfromtheratiosofthe axesoftheellipsoids andhencemustbeidentical withthem. Thincompletes thedemonstration. ThePropagation ofLightinOrystals 1 143.]Theelectromagnetic equations oftheetherorofany infiniteisotropic mediumwhichistransparent toelectromag neticwavesmaybewrittenintheform 1Thefollowing diI!C113sion mustberegarded asmathematical notphysical. Totrea.tthe.nbjectfromthestandpoint ofphysicswouldbeoutofplacehere. (1) \1·D=OTHEPROPAGATION OFLIGHTINCRYSTALS 393 d~D Potdt~+ED+\1V=0, whereDistheelectricdisplacement satisfying thehydrody namicequation \1•D=0,Eaconstant ofthedielectric meas uredinelectromagnetic units,and\1Vtheelectrostatic force duetothefunction V.Incasethemediumisnotisotropic the constantEbecomes alinearvectorfunction fP.Thisfunction isself-conjugate asisevidentfromphysical considerations. Forconvenience it·willbetakenas4'1rfP.Theequations thenbecome d2DPotdt~+4'1rfP•D+\1V=0,\1•D=O.(2) Operate by\lx\lX. d2D\1x\lxPotdt2+4'1r\lx\1xfP•D=O.(3) Thelasttermdisappears owingtothefactthatthecurlof thederivative \1Vvanishes (page167).Theequation may alsobewrittenas Butd2DPot\lx\lxdt2+4'1r\1X\1xfP•D=O.(3)' \lX\lX=\l\l• -\l.\l. (4) \l•D=O.Remembering that\l.Dandconsequently \l.ddDand ~ t \l•d~vanishandthatPot\l•\lisequalto- 4'1rthedt· equation reducesatonceto d2D- =\l•\lfP•D-\l\l•(/).D,dt2 Suppose thatthevibration Disharmonic. Letrbethe vectordrawnfromafixedorigintoanypointofspace. 894 ThenVECTOR ANALYSIS D=Acos(m• r -nt) whereAandmareconstant vectorsandnaconstant scalar represents atrainofwaves. Thevibrations takeplacein thedirection A.Thatis,thewaveisplanepolarized. The waveadvances inthedirection m.Thevelocity '/)ofthatad vanceisthequotient ofnbym,themagnitude ofthevector m.Ifthiswaveisanelectromagnetic waveinthemedium considered itmustsatisfythetwoequations ofthatmedium. Substitute thevalueofDinthoseequations. Thevalueof'1.D,'1.'1fP.D,and'1'1•fP.Dmaybe obtained mosteasilybyassuming thedirection itobecoinci dentwithm.m·rthenreducestomi.rwhichisequal.to mx.Thevariables yandznolongeroccurinD.Hence D=ACOB(mx-nt) '1•D=i .dD= -i .Amsin(mx-nt) ~x '1.'1fP•D= -m2fP•Acos(mx-nt) '1'1•fP.D= -m2ii.fP.Acos(mx-nt). Hence Moreover"V•D=-m •Asin(m• r -nt) 'l."VfP.D=-m.m fP.D "V"V•(/)• D = - m m • (/)•D. Henceiftheharmonic vibration Distosatisfytheequa tions(4)ofthemedium andn2D=m • mfP.D - mm • fP•D m·A=O.(5) (6) TIlEPROPAGATION OFLIGHTINCRYSTALS 395 Thelatterequation statesatoncethattkevibrations must betransverse tothedirection mofpropagation ofthewaves. Theformerequation maybeputintheform Introducem.m mmD=-- (J)•D- - • (J)•D. (5)'nZ nZ m8=-•n Thevector Iisinthedirection ofadvance m.Themagnitude ofIisthequotient ofmbyn.Thisisthereciprocal ofthe velocityofthewave.Thevector Imaytherefore becalled thewave-slowness. D =I • I(J).D-I I •(J)•D. Thismayalsobewrittenas D= - (IX I X (J)•D)=I X«J)•D)X L Dividing bythescalarfactorcos(mx-nt), A=I X«J)•A)X I=I • I(J).A-II•(J)•A.(7) Itisevidentthatthewaveslowness Idepends notatall uponthephraseofthevibration butonlyuponitsdirection. Themotionofawavenotplanepolarized maybediscussed by decomposing thewaveintowaveswhichareplanepolarized. 144.]Letabeavectordrawninthedirection Aofthe displacement andletthemag'nitude ofabesodetermined that a •(J)•a=1. (8) Theequation (7)thenbecomes reducedtotheform a=I X«J)•a)X I=I • I(J).a=II.(J)•a(9) a •(J)•a=1. (8) ThesearetheequatioIlB bywhichthediscussion ofthevelocity orrathertheslowness ofpropagation ofawaveindifferent directions inanon-isotropic mediummaybecarriedon. a • a=I • Ia·(J)•a=I ••• (10) 396 VECTOR ANALYSIS Hencethewaveslowness sdue toadisplacement inthe direction aisequalinmagnitude (butnotindirection) tothe radiusvectordrawnintheellipsoida.fP.a=1inthat direction. axa=O=s.s axfP.a-axs s.(/).a 0=8. S(ax(/)oa)•fP.a=axs.fP.a8.(/).L Butthefirsttermcontains (/)•atwiceandvanishes. Hence ax8·(/)•a=[a8fP.a]=O.(11) Thewave-slowness Itherefore liesinaplanewiththe direction aofdisplacement andthenormal(/).adrawntothe ellipsoida.(/).a=1attheterminus ofa.Sincesisperpen diculartoaandequalinmagnitude toaitisevidently com pletelydetermined exceptasregardssignwhenthedirection aisknown. Giventhedirection ofdisplacement thelineof advanceofthewavecompatible withthedisplacement iscom pletelydetermined, thevelocity oftheadvance islikewise known. Thewavehowever mayadvance ineitherdirection alongthatline.Byreference topage386,equation (11)isseen tobethecondition thatashallbeoneoftheprincipal axesof theellipsoid formedbypassingaplanethrough theellipsoid perpendicular tos.Henceforanygivendirection ofadvance therearetwopossible linesofdisplacement. Thesearethe principal axesoftheellipsecutfromtheellipsoida.fP.a=1 byaplanepassedthrough thecenterperpendicular tothe lineofadvance. Tothesestatements concerning thedeter minateness ofswhenaisgivenandofawhen8isgivenjust sllchexceptions occurasareobviousgeometrically. Ifaand (/).aareparallel 8mayhaveanydirection perpendicular toa. Thishappens whenaisdirected alongoneoftheprincipal axesoftheellipsoid.Ifsisperpendicular tooneofthe circularsectionsoftheellipsoid amayhaveanydirection inthe planeofthesection. THEPRQPAGATION OFLIGHTINCRYSTALS 397 Whenthedirection ofdisplacement isallowedtovarythe slowness. varies.Toobtainthelocusoftheterminus of.,a mustbeeliminated fromtheequation or (I- •••rp+•••fP)•a=O. (12) Thedyadicintheparenthesis isplanarbecauseitannihilates vectorsparalleltoa.Thethirdordeterminant iszero.This givesimmediately (I- • • •rp+••.rp)8=0, or (rp-l-•••I+18h=O. (13) (14) HenceThisisascalarequationinthevector..Itisthelocusof theextremity of•whenaisgivenallpossible directions. A numberoftransformations maybemade.ByEx.19,p.331, (rp+ef>a=rp8+e. rp2·f=fP8+e. rpc-1•ffPa" Hence (fP-1-•••I)8+•.(rp-l_•••I)0-1••((fr-l_•••I)8=O. Dividing outthecommon factorandremembering thatfPis self-conjugate. 1+••(rp-l_•••I)-l••=O. rp or 1+•. .• =0,1-•.•rp.01.. rp---+.. ..=0•••I-•••rp ••(I+1~••::rp)••=O. I• • .8=O.1-•.•rp LetiijjkkfP=-+--+--a2b2c2 398 VECTOR ANALYSIS I(1)..(1)..(1)t IfP=--2 11+--z JJ+--2 t.-a·a l-~ 1-~ 1-~ a2bZC2 Let a=xi+Yj+zkand82=x2+y2+Z2. Thentheequation ofthesurfaceinCartesian coordinates is x2 yZ z2---2+---Z+---2=o. 1-!...- 1-~ 1-!...-aZbZ cZ(14)' (fP-1-••a1+88)3=O.Theequation inCartesian coordinates maybeobtained directly from Thedeterminant ofthisdyadicis a2_82+x2xy xz xy bZ_82+y2yz =0. xz yz c2_82+z2(13), Bymeansoftherelation 82=xZ+yZ+z2thisassumesthe forms or oraZxZbZyZ c2z2 8z_a2+Zb2+~2--:A=0,S- S-C Thisequation appearstobeofthesixthdegree. Itishow everofonlythefourth. Thetennsofthesixthordercancel out. Thevector 8represents thewave-slowness. Supposethata planewavepolarized inthedirection apassestheoriginata THEPROPAGATION OFLIGHTINCRYSTALS 399 certaininstantoftimewiththisslowness. Attheendofa unitoftimeitwillhavetravelled inthedirection ..adistance equaltothereciprocal ofthemagnitude of..Theplanewill beinthispositionrepresented bythevector 1(page108). If 8=ui+vj+wk theplaneattheexpiration oftheunittimecutsoffintercepts upontheaxesequaltothereciprocals ofu,v,w.These quantities aretherefore theplanecoordinates oftheplane. Theyareconnected withthecoordinates ofthepointsinthe planebytherelation ux+vY+wz=1. Ifdifferent planewavespolarized inallpossible different directions abesupposed topassthrough theoriginatthe sameinstanttheywillenvelopasurfaceattheendofaunit oftime.Thissurfaceisknownasthewave-surface. The perpendicular uponatangentplaneofthewave-surface isthe reciprocal oftheslowness andgivesthevelocity withwhich thewavetravelsinthatdirection. Theequation ofthewave surfaceinplanecoordinates u,v,wisidentical withtheequa tionforthelocusoftheterminus oftheslowness vectorI. Theequation is u2v2w2 --S-2+--8-2+--8-2=0 (15)1-- 1-- 1--a2 b2 c2 where 82=u2+v2+w2•Thismaybewritteninanyofthe formsgivenpreviously. ThesurfaceisknownasFresnel's Wave-Surface. Theequations invectorformaregivenon page397ifthevariablevector 1beregarded asdetermining a planeinsteadofapoint. 145.]Inanisotropic medium thedirection ofarayof lightisperpendicular tothewave-front. Itisthesameas thedirection ofthewave'sadvance. Thevelocityoftheray 400 VECTOR ANALYSIS isequaltothevelocity ofthewave.Inanon-isotropic mediumthisisnolongertrue.Theraydoesnottravelper pendicular tothewave-front -thatis,inthedirection ofthe wave'sadvance. Andthevelocitywithwhichtheraytravels isgreaterthanthevelocityofthewave.Infact,whereasthe wave-front travelsoffalwaystangenttothewave-surface, the raytravelsalongtheradiusvectordrawntothepointoftan gencyofthewave-plane. Thewave-planes' envelop the wave-surface; theterminioftheraysaresituated uponit. Thusinthewave-surface theradiusvectorrepresents inmag nitudeanddirection thevelocity ofarayandtheperpen dicularuponthetangent planerepresents inmagnitude and direction thevelocityofthewave.Ifinsteadofthewave surfaoethesurfacewhichisthelocusoftheextremity ofthe waveslowness beconsidered itisseenthattheradiusvector represents theslowness ofthewave;andtheperpendicular uponthetangentplane,theslowness oftheray. Letv'bethevelocityoftheray.Then••y'=1because theextremity ofy'liesintheplanedenotedby.,Moreove:r thecondition thaty'bethepointoftangency givesdy'pel pendicular to..Inlikemannerif.'betheslowness ofthe rayandythevelocityofthewave,.'•y=1andthecondition oftangency givesd.'perpendicular toY.Hence ••y'=1and.'•y=1, (16) and••dy'=O,y.d.'=O, y'.dl=O, .'.dy=O, y'maybeexpressed intermsofa,I,andtPasfollows. da=2.·d. tP.a-l. tP.adl+I.1 tP.da - 1d.•tP•a=••.tP•da. Multiply byaandtakeaccountoftherelations a••=0and a~tP•da=0anda • a=•.•.Then orTHEPROPAGATION OFLIGHTINCRYSTALS 401 I •dI - a •dI I •rP•a=0, dI •(I- a I • rP•a)=O. Butsincev'.dl=0,TandI -aI.rP.ahavethesame direction. v'=x(I-aI •rP•a), I •T=x(I• I -I •aI •rP•a)=xI •I. HenceI-al.rP.av'=-----, I • I I·rP.a-a. rP.al.rP.av'•rP•a= =o. I • I(17) HencetherayvelocityTisperpendicular torP.a,thatis,the rayvelocityliesinthetangentplanetotheellipsoid atthe extremity oftheradiusvectoradrawninthedirection ofthe displacement. E"quation (17)showsthatv'iscoplanar with aandI.Thevectorsa,I,rP.a,andv'therefore lieinone plane.InthatplaneIisperpendicular toa;andT,torP.a. TheanglefromItoTisequaltotheanglefromatorP.a. Making useoftherelations alreadyfound(8) (9)(11) (16)(17),itiseasytoshowthatthetwosystemsofvectors a,v',aXv'andrP.a,I,«(/).a)x8 ar:ereciprocal systems.IfrP•abereplaced bya'theequa tionstakeonthesymmetrical form v'·a'=O v'•v'=a'•a'a •a'=1, I •v'=1, a=I Xa'xI I=axv'xaa'=v'x a xv' v'=a'x8xa' a'•(/}-1•a'=1.(18) Thusadualrelationexistsbetween thedirection ofdisplace ment,theray-velocity, andtheellipsoid rPontheonehandj 26 402 VECTOR ANALYSIS andthenormaltotheellipsoid, thewave-slowness, andthe ellipsoid rp-1ontheother. 146.]ItwasseenthatifIwasnormaltooneofthecir cularsectionsofrpthedisplacement acouldtakeplaceinany direction intheplaneofthatsection. Foralldirections in thisplanethewave-slowness hadthesamedirection andthe samemagnitude. Hencethewave-surface hasasingular planeperpendicular to..Thisplaneistangenttothesurface alongacurveinsteadofatasinglepoint.Henceifawave travelsinthedirection Itheraytravelsalongtheelements of theconedrawnfromthecenterofthewave-surface tothis curveinwhichthesingular planetouchestheaurface. The twodirectiollB Iwhicharenonna!tothecircularsectionsoffP arecalledtheprimary optica.Te8.Thesearetheaxesofequal wavevelocities butunequal rayvelocities. Inlikemannerv'beingcoplanar withaandrp•a [(/)•aTa]=[a'v'rp-1•a']=O. Thelastequation states t~tifaplanebepassedthrough thecenteroftheellipsoid (/)-1perpendicular toT,thena' whichisequaltorp.awillbedirected alongoneoftheprin ~ipalaxesofthesection. Henceifarayistotakeadefinite direction a'mayhaveoneoftwodirections. Itismorecon venienthowever toregardv'asavectordetermining aplane. Thefirstequation [(/)• av'a]=0 statesthataistheradiusvectordrawnintheellipsoid (/Jto thepointoftangency ofoneoftheprincipal elements ofthe cylinder circumscribed aboutrpparalleltov':ifbyaprincipa) elementismeantanelement passingthroughtheextremities ofthemajororminoraxesoforthogonal planesections ofthatcylinder. Hencegiventhedirection v'oftheray,the twopossibledirections ofdisplacement arethoseradiivectors VARIABLE DYADICS 403 oftheellipsoid whichlieintheprincipal planesofthecylin dercircumscribed abouttheellipsoid paralleltov'. Ifthecylinderisoneofthetwocircularcylinders which maybecircumscribed about(jJthedirection ofdisplacement maybeanydirection intheplanepassedthrough thecenter oftheellipsoid andcontaining thecommon curveoftangency ofthecylinder withtheellipsoid. Theray-velocity forall thesedirections ofdisplacement hasthesamedirection and thesamemagnitude. Itistherefore alinedrawntoone ofthesingular pointsofthewave-surface. Atthissingular pointthereareaninfinitenumberoftangentplanesenvelop ingacone.Thewave-velocity maybeequalinmagnitude anddirection totheperpendicular drawnfromtheoriginto anyoftheseplanes. Thedirections oftheaxesofthetwo circular cylinders circumscriptible abouttheellipsoid (jJare thedirections ofequalray-velocity butunequalwave-velocity. Theyaretheradiidrawntothesingular pointsofthewave surfaceandarecalledthesecondary opticaxes.Ifaray travelsalongoneofthesecondary opticaxesthewaveplanes travelalongtheelements ofacone. Variable D!Jadics. TheDifferential andI1tugral Calculus 147.]Hitherto thedyadicsconsidered havebeenconstant. Thevectorswhichenteredintotheirmakeupandthescalar coefficients whichoccurred intheexpansion innonionform havebeenconstants. Fortheelements ofthetheoryandfor elementary applications theseconstant dyadicssuffice. The introduction ofvariabledyadics,however, leadstoasimplifica tionandunification ofthedifferential andintegralcalculus of vectors,andfurthermore variable dyadicsbecomeanecessity inthemoreadvanced applications -forinstance, inthetheory ofthecurvature ofsurfaces andinthedynamics ofarigid bodyonepointofwhichisfixed. 404 VECTOR A.NALYSIS HenceLetWbeavectorfunction ofposition inspace.Letrbe thevectordrawnfromafixedorigintoanypointinspace. r=xi+yj+zk, dr=dxi+dyj+dzk, ~W ~W ~WdW=dx ~x+dy ~y+dz~z' ~~W.~W ~Wl dW=dr.~i ~x+J~y+k~f Theexpression enclosed inthebracesisadyadic.Itthus appearsthatthedifferential ofWisalinearfunction ofdr, thedifferential changeofposition. Theantecedents arei,j,t, andtheconsequents thefirstpartialderivatives ofWwithre spect ofx,y,z.Theexpression isfoundinamannerprecisely analogous todelandwillinfactbedenoted by\1W. .~W.~W ~W \1W=1~+J}Jj+k~z' (1) Then dW=dr·\1W. (2) Thisequation isliketheoneforthedifferential ofascalar function V.dV=dr.\1V. Itmayberegarded asdefining \1W.Ifexpanded into nonionform'VWbecomes 't"'7W..ClX"~Y.k~Zv=11-+1]-+1.-Clx ~x ~x if..~X..~Y.~Z+J1 -+JJ-+1k - (3)Cly Cly ~y .~X .~Y ClZ +k1~+kJ~+kk~' W=Xi+Yj+Zk. VARIABLE DYADICS 405 Theoperators 'V.and'VXwhichwereappliedtoavector function nowbecome superfluous fromapurelyanalytic standpoint. Fortheyarenothing morenorlessthanthe scalarandthevectorofthedyadic'VW. divW='V.W=('VW)8' curlW='VxW=('VW)x'(4) (5) Theanalyticadvantages oftheintroduction ofthevariable dyadic'VWaretherefore these.Inthefirstplacetheoper ator'Vmaybeappliedtoavectorfunction justastoascalar function. Inthesecondplacethetwooperators 'V•and,'Vx arereducedtopositions asfunctions ofthedyadic'VW.On theotherhandfromthestandpoint ofphysicsnothing isto begainedandindeedmuchmaybelostiftheimportant in terpretations of'V.Wand'VxWasthedivergence andcurl ofWbeforgotten andtheirplacestakenbytheanalytic idea ofthescalarandvectorof'VW. Ifthevectorfunction Wbethederivative ofascalar function V; wheredW=d'VV=dr.'V'VV, ~2V ~2V ~2V 'V'VV=i i~x~+ij~.x~y+ikdx~Z. d2V ~2V ~2V +j i~y~x+jjdy2+jkdy~z'(6) .a2V.a2V ~2V +kJaz~.x+kJ~z~!I+kk~Z2· Theresultofapplying 'Vtwicetoascalarfunction isseento beadyadic. Thisdyadicisself~onjugate. Itsvector\7x'VV iszero;itsscalar'V.'VVisevidently ~2V~2V ~2V '1.'1V=('V'VV)s=~+~+,,~.C'x C'y C'Z 406 VECTOR ANALYSIS Ifanattempt weremadetoapplytheoperator Vsymboli callytoascalarfunction Vthreetimes,theresultwouldbea sumoftwenty-seven termslike •••~8V..~~8V 1 1 1~X8'1JA~X~Y~z'etc. Thisisatriadic. Threevectorsareplacedinjuxtaposition without anysignofmultiplication. Suchexpressions will notbediscussed here.Inasimilarmanneriftheoperator'V beapplied tu-'icetoavectorfunction, oron~toadyadicfunc tionofposition inspace,theresult·willbeatriadicandhence outsidethelimitssettothediscussion here.Theoperators VxandV.mayhowever beappliedtoadyadic rjJtoyield respectively adyadicandavector. If•~rjJ•~fP ~fP VxrjJ=1 X~;+Jx~y+kx~'(7) ~fP ~rjJ dfP V•fP=i •~x+i .~y+k •~z. (8) fP=ui+vi+wk, whereU,v,warevectorfunctions ofposition inspace, VxrjJ=Vxui+Vxvj+Vxwk,(7)' and V•fP=V•ui+V•vj+V•wk.(8)' Orif rjJ= iu+jv+kw, VxfP=i(dw_~V)+j(~u_~W)+k(~v_~u)(7)" dy ~z dZ~x ~x ~y ~u~v~wand V•rjJ==-+- +-. (8)" ~x~y~z Inasimilarmannerthescalaroperators(a.V)and_(V."V) maybeappliedtofP.Theresultisineachcaseadyadic, VARIABLE DYADICS ~~ ~f/J ~f/J (a•\7)f/J=al~x+all~y+as~' ~2(j) ~2(j) ~II(j) (\7•\7)(j)=~xII+(iy2+~z2•407 (9) (10) Theoperators a •\7and\7•\7asappliedtovectorfunc tionsarenolongernecessarily toberegarded assingleoper ators.Theindividual stepsmaybecarriedoutbymeansof thedyadic\7W. (a:\7)W=a.(\7W)=a·\1W, (\7•\7)W=\7•(\1W)=\1•\1W. Butwhenappliedtoadyadictheoperators cannotbeinter pretedasmadeupoftwosuccessive stepswithoutmakinguse ofthetriadic\7(j).Theparentheses however mayberemoved without dangerofconfusion justastheywereremoved in caseofavectorfunction beforetheintroduction ofthedyadic. Formuloo similartothoseuponpage176maybegivenfor differentiating products inthecasethatthedifferentiation leadtodyadics. \7(uv)=\7uv-+-u\7v, \7(vxw)=\7v xw-\7wxv, V'x(vxw)=w•\7v -\7•vw- v •\1w+\1•wv, \7(v•w)=\7v •w+\7w•v, \7•(vw)=\7•vw+v •\7w. \7x(vw)=\7x"w-v x\1w, \7•(uf/J)=\7u•f/J+u\1•f/J, \7x\7x(j)=\7\7•f/J-\7•\1f/J,eU:. Theprinciple intheseandallsimilarcasesisthatenun ciatedbefore,namely: Theoperator \7maybetreatedsym- 408 VECTOR ANALYSIS andbolically asavector. Thedifferentiations whichitimplies mustbecarriedoutinturnuponeachfactorofaproduct towhichitisapplied;. Thus \1x(vw)=[\1x(vw)].+[\1x(vw)].., ['\7x(vw)]..=\1xvw, ['\7x(vw)]. ~-[vx\1w].= -v x\1w. Hence \1x(vw)=\1xvw-v x\1w. Again \1(vxw)=[\1(vxw)].+[\1(vxw)].., [\1(vxw)]..='\7v xw, [\1(vxw)].=[-\1(wxv)].= -\1wXv. Hence \1(vxw)=\1v x w - \1wxv. 148.]Itwasseen(Art.79)thatif0denoteanarcofa curve of whichtheinitialpointis1'0andthefinalpointisr thelineintegralofthederivative ofascalarfunction taken alongthecurveisequaltothedifference between thevalues ofthatfunctionatrand1'0' fdr.'\7V=V(r)-V(ro)· c InlikemannerJ:l'•\1W=W(1')-W(1'0)' J:dl'•V'W=o. Itmaybewelltonotethattheintegrals Jdr.\1WandJ\1W •dl' arebynomeansthesamething.'\7Wisadyadic.The vectordl'cannotbeplacedarbitrarily uponeithersideofit. VARIABLE DYADICS 409 Owingtothefundamental equation (2)thedifferential dr necessarily precedes "VW.Thedifferentials mustbewritten beforetheintegrands inmostcases.:Forthesakeofuni formity theyalwayswillbesoplaced. Passingtosurfaceintegmls, thefollowing formulre, some ofwhichhavebeengivenbeforeandsomeofwhicharenew, maybementioned. Ifdax"VV=IdrV fJdax"Vw=Jdrw Ifdao"VxW=Jdrow IJ da0"VxfP=Jdr0fP. Thelineintegrals aretakenoverthecomplete bounding curve ofthesurfaceoverwhichthesurfaceintegrals aretaken.In likemannerthefollowing relations existbetween volumeand surfaceintegrals. JfJdv"VV=JIdaV JffdV "Vw=JfdB W Iffdv"V0W=ffda.W JJIdv "VxW=Ifdaxw ffIdv"V.fP=Jfda.fP JIId 11"VxrP=IJda xrP. 410 VECTOR ANALYSIS Thesurfaceintegrals aretakenoverthecomplete bounding surfaceoftheregionthroughout whichthevolumeintegrals aretaken. Numerous formulre ofintegration bypartslikethoseupon page250mightbeadded.Thereaderwillfindnodifficulty in obtaining themforhimself. Theintegrating operators may alsobeextended toothercases.Tothepotentials ofscalar andvectorfunctions thepotential, PotfI,ofadyadicmaybe added.TheNewtonian ofavectorfunction andtheLapla. cianandMaxwellian ofdyadicsmaybedefined. Pot(/)=-JJJ(/)(x,~~";)dv,. NewW=JJJ raW(x"y"%,)dv,r3, t 12 Theanalytic theoryoftheseintegrals maybedeveloped as before. Themostnaturalwayinwhichthedemonstrations maybegivenisbyconsidering thevectorfunction Wasthe sumofitscomponents, W=Xi+Yj+Zt andthedyadic(/)asexpressed withtheconstant consequents i,j,kandvariableantecedents u,v,w,orviceversa, (/)=ui+vj+wk. Thesematterswillbeleftatthispoint.Theobjectofen teringuponthematallwastoindicatethenaturalextensions whichoccurwhenvariabledyadicsareconsidered. Theseex tensions differsoslightlyfromthesimplecaseswhichhave THECURVATURE OFSURFACES 411 gonebeforethatitisfarbettertoleavethedetailstobeworked outorassumed fromanalogywhenever theymaybeneeded ratherthantoattempttodevelop theminadvance.Itissuffi cientmerelytomention whattheextensions areandhowthey maybetreated. TheCurvature ofSurfaces 1 149.]Therearetwodifferent methods oftreating thecur vatureofsurfaces. Inonethesurfaceisexpressed inpara. meticformbythreeequations x=f1(u,t'),1=f2Cu,v),Z=f8(u,v), or r=f(u,v). Thisisanalogous tothemethodfollowed (Art.57)indealing withcurvature andtorsionofcurvesanditisthemethod employed byFehrinthebooktowhichreference wasmade. Inthesecondmethodthesurfaceisexpressed byasingle equatiou connecting thevariables x,71,z-thus F(x,71,z)=O. Thelattermethodoftreatments affordsasimpleapplication of thedifferential calculus ofvariable dyadics. Moreover, the dyadicsleadnaturally tothemostimportant resultsconnected withtheelementary theoryofsurfaces. Letrbearadiusvectordrawnfromanarbitrary fixed origintoavariable pointofthesurface. Theincrement dr liesinthesurfaceorinthetangentplanedrawntothesurface attheterminus ofr. d F=dr.\lF=O. Hencethederivative \lFiscollinear withthenormaltothe surface. Moreover, inasmuch asFandthenegative ofFwhen 1Mnchofwhatfollowsispractically freefromthenileofdyadies. Thillill especially trueofthetreatment ofgeodeties, Arts.155-157. 412 VECTOR ANALYSIS equated tozerogivethesamegeometric surface,VFmaybe considered asthenormll,luponeithersideofthesurface. In casethesurfacebelongstothefamilydefinedby F(x,y,z)=const. thenormalVFliesuponthatsideuponwhichtheconstant increases. LetVFberepresented byBthemagnitude of whichmaybedenoted byN,andletnbeaunitnormaldrawn inthedirection ofB.Then "B='1F, B•B=N2=VF•VF, 1n=-VF.N(1) IfIisthevectordrawntoanypointinthetangentplaneat theterminus ofr,I-randnareperpendicular. Consequently theequation ofthetangentplaneis (I-r)•VF=O. andinlikemannertheeqnationofthenormallineis (I-r)xVF=0, or l=r+k'1F wherekisavariable parameter. Theseequations maybe translated intoCartesian fonnandgivethefamiliarresults. 150.]Thevariation dnoftheunitnormaltoasmface playsanimportant partinthetheoryofcurvature. dnis perpendicular tonbecaUI:lenisaunitvector. 1n=--VFN dN 1dn=-N2'1F+Nd'1F, THECURVATURE OFSURFACES 418 1 1dD= -dr.VVF- -dr.VNVF.N JV2 Thedyadic1-Dnisanidemfactor forallvectorsperpen diculartonandanannihilator forvectors parallel ton. Hence and Hence But Hence Let Thendn •(I-nD)=dn, VF.(I-DD)=0, 1dn=Ndr.VVF.(I-nD). dr=dr.(I-nD). dn=dr.(I-DD)•VVF.(I-Dn)• N (I-Dn)•VVF·(I-DD)fP= .N dD=dr.fP.(2) (8) (4) Inthevicinityofanypointuponasurfacethevariation dDof theunitnormalisalinearfunction ofthevariation ofthe radiusvectorr. ThedyadictPisself-conjugate. For NtPc=(I- nn)c•(VVF)c·(I- nn)c' Evidently(1-nn)c=(I-DD)andby(6)Art.147VVF isself-conjugate. HencefPcisequaltof/J.Whenappliedto avectorparallelton,thedyadic f/Jproduces zero.Itisthere foreplanarandinfactuniplanar becauseself-conjugate. The antecedents andtheconsequents lieinthetangent planeto 414 VECTOR ANALYSIS thesurface. Itispossible (Art.116)toreduce tPtothe form tP=ai'i'+bj'j' (5) wherei'andj'aretwoperpendicular unitvectorslyinginthe tangentplaneandaandbarepositiveornegative scalars. dD=dr·(ai'i'+bj'j'). Thevectorsi',l'andthescalarsa,bvaryfrompointtopoint ofthesurface. ThedyadictPisvariable. 151.]Theconicr •tP.r=1iscalledtheindicatrix ofthe surfaceatthepointinquestion. Ifthisconicisanellipse, thatis,ifaandbhavethesamesign,thesurfaceisconvexat thepoint;butiftheconicisanhyperbola, thatis,ifaandb haveopposite signsthesurfaceisconcavCKlonvex. Thecurve r·tP·r=1mayberegarded asapproximately equaltothe intersection ofthesurfacewithaplanedrawnparalleltothe tangentplaneandneartoit.Ifr.tP•rbesetequaltozero theresultisapairofstraight lines.Thesearetheasymp totesoftheconic.Iftheyarerealtheconicisanhyperbola ; ifimaginary, anellipse. Twodirections onthesurfacewhich areparalleltoconjugate diameters oftheconicarecalledcon jugatedirections. Thedirections onthesurfacewhichcoin cidewiththedirections oftheprincipal axesi',j'ofthe indicatrix areknownastheprincipal directions. Theyarea specialcaseofconjugate directions. Thedirections uponthe surfacewhichcoincide withthedirections oftheasymptotes oftheindicatrix areknownasasymptotic directions. Incase thesurfaceisconvex, theindicatrix isanellipseandthe asymptotic directions areimaginary. InspecialcasesthedyadictPmaybesuchthatthecoeffi cientsaandbareequal. tPmaythenbereduced tothe form fP=a(i'i'+j'j') (5)' THECURVATURE OFSURFACES 415 (5)" f/)=ai'i'.inaninfinitenumberofways.Thedirections i'andj'maybe anytwoperpendicular directions. Theindicatrix becomes a circle.Anypairofperpendicular diameters ofthiscircle giveprincipal directions uponthesurface. Suchapointis calledanumbilic. Thesurfaceintheneighborhood ofan umbilicisconvex. Theasymptotic directions areimaginary. Inanotherspecialcasethedyadic f/)becomes linearandredu cibletotheform Theindicatrix consistsofapairofparallellinesperpendicular toi'.Suchapointiscalledaparabolic pointofthesurface. Thefurtherdiscussion oftheseandotherspecialcaseswillbe omitted. Thequadricsurfaces affordexamples ofthe'variouskinds ofpoints. Theellipsoid andthehyperboloid oftwosheets areconvex. Theindicatrix ofpointsuponthemisanellipse. Thehyperboloid ofonesheetisconcavo-convex. Thein dicatrix ofpointsuponitisanhyperbola. Theindicatrix ofanypointuponasphereisacircle.Thepointsareall umbilics. Theindicatrix ofanypointuponaconeorcylinder isapairofparallellines.Thepointsareparabolic. Asur faceingeneralmayhaveuponitpointsofalltypes-elliptic, hyperbolic, parabolic, andumbilical. 152.]Alineofprincipal curvature uponasurfaceisa curvewhichhasateachpointthedirection ofoneoftheprin cipalaxesoftheindicatrix. Thedirection ofthecurveata pointisalwaysoneoftheprincipal directions onthesurfaceat thatpoint.Through anygivenpointuponasurfacetwoper pendicular linesofprincipal curvature pass.Thusthelines ofcurvature dividethesurfaceintoasystemofinfinitesi malrectangles. Anasymptotic lineuponasurfaceisacurve whichhasateachpointthedirection oftheasymptotes ofthe indicatrix. Thedirection ofthecurveatapointisalways oneoftheasymptotic directions upon the surface. Through 416 VECTOR AJ,ALYSIS anygivenpointofasurfacetwoasymptotic linespass.These linesareimaginary ifthesurfaceisconvex. Evenwhenreal theydonotingeneralintersect atrightangles. Theangle between thetwoasymptotic linesatanypointisbisectedby thelinesofcurvature whichpassthroughthatpoint. Thenecessary andsufficient condition thatacurveupona surfacebealineofprincipal curvature isthatasoneadvances alongthatcurve,theincrement ofdn,theunitnormaltothe surfaceisparalleltothelineofadvance. For dn=tP•dr=(ai'i'+bj'j')•dr dr=xi'+yj'. Thenevidently dnanddrareparallelwhenandonlywhen drisparalleltoi'orj'.Thestatement istherefore proved. Itisfrequently takenasthedefinition oflinesofcurvature. Thedifferential equation ofalineofcurvature is dnXdr=O. (6) Another methodofstatement isthatthenonnaltothesurface. theincrement dnofthenormal,andtheelement drofthe surfacelieinoneplanewhenandonlywhentheelement dr isanelementofalineofprincipal curvature. Thedifferential equation thenbecomes [ndndrJ=O. (7) Thenecessary andsufficient condition thatacurveupona. surfacebeanasymptotic line,isthatasoneadvances along thatcurvetheincrement oftheunitnormaltothesurfaceis perpendicular tothelineofadvance. For dn=dr.tP dn •dr=dr.rp•dr. Ifthendn •driszerodr.(/J•driszero.Hencedrisan asymptotic direction. Thestatement istherefore proved.It THECURVATURE OFSURFACES 417 isfrequently takenasthedefinition ofasymptotic lines.The differential equation ofanasymptotic lineis dD •dr=O. (8) 153.]LetPbeagivenpointuponasurfaceandnthe normaltothesurfaceatP.Passaplanepthroughn.This planepisnormaltothesurfaceandcutsoutaplanesection. Consider thecurvature ofthisplanesectionatthepointP. Letn'benormaltotheplanesectionintheplaneofthe section. n'coincides withnatthepointP.Butunlessthe planepcutsthesurfaceeverywhere orthogonally, thenormal n'totheplanesectionandthenormalntothesurfacewillnot coincide. dnanddn'willalsobedifferent. Thecurvature oftheplanesectionlyinginpis(Art.57). c=dt= d2r. d8d82 Asfarasnumerical valueisconcerned theincrement ofthe unittangent tandtheincrement oftheunitnormaln'are equal.Moreover, thequotient ofdrbyd8isaunitvector inthedirection ofdn'.Consequently thescalarvalueofCis C=dn'.dr=dn'.dr d8d8d82 Byhypothesis n =n'atPandn·dr =n'•dr =0, d(n•dr)=dn •dr+n.d2r =0, d(n'•dr)=dn'•dr+n'•d2r =O. Hence dn •dr+n •d2r =dn'•dr+n'•d2r. Sincenandn'areequalatP, dn •dr =dn'•dr. Hencedn.dr dr.rj).dr dr.·rj).dr C=.d82=--d-8-2--- dr.dr 27(9) 418 Hence orVECTOR ANALYSIS (i'•dr)2(j'•dr)2O=a-_.+ b .dr.dr dr.dr 0=aC082(i',dr)+bC082(j',dr), O=acos2(i',dr)+b sin2(i',dr). (10) Theinterpretation ofthisformulaforthecurvature ofa normalsectionisasfollows: Whentheplanepturnsabout thenormaltothesurfacefromi'toj',thecurvature 0ofthe planesectionvariesfromthevalueawhentheplanepasses through theprincipal direction i',tothevaluebwhenit passesthrough theotherprincipal direction j'.Thevalues ofthecurvature havealgt:.braically amaximum andminimum inthedirections oftheprincipal linesofcurvature. Ifaand bhaveunlikesigns,thatis,ifthesurfaceisconcavo-convex atP,thereexisttwodirections forwhichthecurvature ofa normalsectionvanishes. Thesearetheasymptotic directions. 154.]Thesumofthecurvatures intwonormalsections atrightanglestooneanotherisconstant andindependent of theactualposition ofthosesections. Forthecurvature in onesectionis 01=acos2(i',dr)+bsin2(i',dr), andinthesectionatrightanglestothis HenceO2=asin2(i',dr)+bCO~2(i',dr). 01+O2=a+b='fl8 (11) whichprovesthestatement. Itiseasytoshowthattheinvariant 'fl28isequaltothepro ductofthecurvatures aandbofthelinesofprincipal curv ature. 'flu=ab Hencetheequation x2-'fl8X+'fl28=0 (12) THECURVATURE OFSURFACES 419 isthequadratic equation whichdetermines theprincipal curv aturesaandbatanypointofthesurface. Bymeansofthis equation thescalarquantities aandbmaybefoundinterms ofF(x,y,z). (I-nn).\1\1F·(I-nn)rfl=. N tP=_\1_\1_F_-_2_n_n_._\1_\1~F_+_n_n_._\1_\1_F_._n_n .N (nn.\1\1F·nn)s=(nn.nn·\1\1F)s=(nn.\1\1Fh Hencerfl_(\1\1F)s_(nn •\1\1F)ss-N N (\1\1F)s=\1•\1F, (nn •\1\1F)8=nn :\1\1F=n •\1\1F•n. Hence or\1.\1F rfls=N\1F\1F:\1\1F ]{3 \1F·\1\1F.\1F N3(13) (13)' Theseexpressions maybewrittenoutinCartesian coordinates, buttheyareextremely long.TheCartesian expressions for fP28areevenlonger. Thevectorexpression maybeobtained asfollows: ,Ii_(I- nn)2•(\1\1F)2•(I- nn)2 '1'8- N2 (I-nn)2=nn. 155.]GivenanycurveuponaIlurface. Lettbeaunit tangent tothecurve,n aunitnormaltothesurfaceandm a 420 VECTOR ANALYSIS vectordefinedasn xt.Thethreevectorsn,t,mconstitute ani,j,ksystem. Thevectortisparalleltotheelementdr. Hencethecondition foralineofcurvature becomes Hence Hence Moreover Hence ortxdn=O. d(m.n)=0=m.dn+n.dm. n·dm=O. m·dm=O. txdm=0, dmxdn=O.(15) (16) (16)' Theincrements ofmandofnandofrareallparallelincaseof alineofprincipal curvature. Ageodetic lineuponasurfaceisacurvewhoseosculating planeateachpointisperpendicular tothesurface. Thatthe geodetic lineistheshortestlinewhichcanbedrawnbetween twopointsuponasurfacemaybeseenfromthefollowing considerations ofmechanics. Letthesurfacebesmoothand letasmoothelasticstringwhichisconstrained tolieinthe surfacebestretched between anytwopointsofit.Thestring actingunderitsowntensions willtakeaposition ofequili briumalongtheshortestcurvewhichcanbedrawnuponthe surfacebetween thetwogivenpoints. Inasmuch asthe stringisatrestuponthesurfacethenormalreactions ofthe surfacemustlieintheosculating planeofthecurve.Hence thatplaneisnormaltothesurfaceateverypointofthecurve andthecurve itself isageodetic line. Thevectorstanddtlieintheosculating planeanddeter minethatplane.Incasethecurveisageodetic, thenormal totheosculating planeliesinthesurfaceandconsequently is perpendicular tothenormaln.Hence (17) orTHECURVATURE OFSURFACES 421 D't xdt=0, Dxt.dt=O m.dt=O. Thedifferential equation ofageodetic lineistherefore [Ddrd2r]=O. (18) HenceUnlikethedifferential equations ofthelinesofcurvature andtheasymptotic line,thisequation isofthesecondorder. Thesurfaceistherefore covered overwithadoublyinfinite systemofgeodetics. Through anytwopointsofthesurface onegeodetic maybedrawn. Asoneadvances alonganycurveuponasurfacethereis necessarily someturningupanddown,thatis,aroundthe axism,duetothefactthatthesurfaceiscurved. Theremay ormaynotbeanyturningtotherightorleft.Ifoneadvances alongacurvesuchthatthereisnoturningtotherightor left,butonly the unavoidable turningupanddown,itistobe expected thattheadvance isalongtheshortestpossible route -thatis,alongageodetic. Suchisinfactthecase.The •totalamountofdeviation fromastraightlineisdt.SinceD, 1,mformani,i,ksystem I=tt+D D+mm. dt= tt.dt+DD.dt+mm.dt. SincetisIiunitvectorthefirsttermvanishes. Thesecond termrepresents theamount ofturning upanddown;the thirdterm,theamounttotherightorleft.Hencem•dtis thepropermeasure ofthispartofthedeviation froma straightest line.Incasethe.curveisageodetic thisterm vanishes aswasexpected. 156.]Acurveorsurfacemaybemapped uponaunit spherebythemethodofparallelnormals. Afixedoriginis assumed, fromwhichtheunitnormal DatthepointPofa 422 VECTOR A.NALYSIS givensurfaceislaidoff.Theterminus P'ofthisnormallies uponthesurfaceofasphere.Ifthenonnalstoasurfaceatall pointsPofacurvearethusconstructed fromthesameorigin, thepointsP'willtraceacurveuponthesurfaceofaunit sphere. Thiscurveiscalledthespherical imageofthegiven curve.InlikemannerawholeregionTofthesurfacemay bemapped uponaregionT'thesphere. TheregionT'upon thespherehasbeencalledthehodogramoftheregionTupon thesurface.Ifdrbeanelementofarcuponthesurfacethe corresponding element upontheunitsphereis dD=ifJ•dr. (19). HenceIfdabeanelement ofareauponthesurface, thecorre sponding elementuponthesphereisda'where(Art.124). da'=tP2•da. ifJ=ai'i'+bj'j' tP2=abi'Xj'i'Xj'=abDD. da'=abnn •da. TherntioofanelementofsurfaceatapointPtotheareaof itshodogram isequaltotheproductoftheprincipal radiiof curvature atPortothereciprocal oftheproductoftheprin cipalcurvatures atP. Itwasseenthatthemeasure ofturningtotherightorleft ism.dt.Ifthenaisanycurvedrawnuponasurfacethe tot..'tlamountofturninginadvancing alongthecurveisthe integral. (20) Foranyclosedcurvethisintegral maybeevaluated ina manneranalogous tothatemployed (page190)intheproof ofStokes's theorem. Consider twocurvesaandA'near THECURVATURE OFSURFACES 423 together. Thevariation whichtheintegral undergoes when thecurveofintegration ischanged from0to0'is oJm0dt. ofmodt =J0Cm 0dt)=Jom 0dt+Jmoodt dCm 00t)=dmoot+mod0t oJm0dt=fom 0dt-Jdm 0ot+fdCm 0ot). Theintegral oftheperfectdifferential tlCm 00t)vanishes whentakenaroundaclosedcurve.Hence Theidemfactor is1=tt+nn+mm, omodt=omolodt=omonnodt, fort0dtand0momvanish. Asimilartransformation may beeffecteduponthetermdmoot.Then ofmodt= fcomon nodt-dmon noot). Bydifferentiating therelations m0n =0andnot=0itis seenthat omon=-moon noot=-onot dmon=-modn nodt =-dnot. HenceoJmodt=fcmoon todn-modn toon) ofm0clt =fCmxtoon xdn)= -In 00n xdno 424 VECTOR ANALYSIS Thedifferential 0n xdnrepresents theelement ofareain thehodogram upontheunitsphere. Theintegral Jn·onxdn=Jn·da' represents thetotalareaofthehodogmm ofthestripof surfacewhichliesbetween thecurvesaanda'.Letthe curveastartatapointuponthesurfaceandspreadoutto anydesiredsize.Thetotalamountofturning whichisre quiredinmakinganinfinitesimal circuitaboutthepointis 27r.Thetotalvariation intheintegralis J0Jm •dt=Jm •dt -27r. JJn.da'~H, ButifHdenotethetotalareaofthehodogram.(21) HenceJm.dt=27r-H, or H=27r-Jm.dt, (22) orH+Jm·dt=27r. Theareaofthehodogram oftheregionenclosed byany closedcurveplusthetotalamountofturningalongthatcurve isequalto27r.Ifthesurfaceinquestion isconvexthearea uponthespherewillappearpositivewhenthecurveuponthe surfaceissodescribed thattheenclosed areaappearspositive. If,however, thesurfaceisconcavo-convex theareauponthe spherewillappearnegative. Thismatterofthesignofthe hodogram mustbetakenintoaccountinthestatement made above. THECC.:RVATURE OFSURFACES 425 157.]Iftheclosedcurveisapolygon whosesidesare geodetic linestheamountofturningalongeachsideiszero. Thetotalturningistherefore equaltothesumoftheexterior anglesofthepolygon. Thestatement becomes: thesumof theexterior anglesofageodetic po}ygon andoftheareaof thehodogram ofthatpolygon (takingaccount ofsign)is equalto27T'.Suppose thatthepolygon reducestoatriangle. Ifthesurfaceisconvextheareaofthehodogram ispositive andthesumoftheexterioranglesofthetriangleislessthan 27T'.Thesumoftheinterioranglesistherefore greaterthan '1r.Thesphereorellipsoid isanexample ofsuchasurface. Ifthesurfaceisconcavo-convex theareaofthehodogram is negative. Thesumoftheinterioranglesofatriangle isin thiscaselessthan7T'.Suchasurfaceisthehyperboloid ofone sheetorthepseudosphere. Thereisanintermediate casein whichthehodogram ofanygeodetic triangleistracedtwicein opposite directions andhencethetotalareaiszero.Thesum oftheinterioranglesofatriangleuponsuchasurfaceisequal to7T'.Examples ofthissurfaceareafforded bythecylinder, cone,andplane. Asurfaceis'saidtobedeveloped whenitissodeformed that linesuponthesurfaceretaintheirlength. Geodetics remain geodetics. Onesurfaceissaidtobedevelopable orapplicable uponanotherwhenitcanbesodeformed astocoincide with theotherwithout altering thelengthsoflines.Geodetics upononesurfacearechanged intogeodeticR upontheother. Thesumoftheanglesofanygeodetic triangle remainun changed bytheprocessofdeveloping. Fromthisitfollows thatthetotalanlOuntofturningalonganycurveorthearea ofthehodogram ofanyportionofasurfacearealsoinvariant oftheprocessofdeveloping. 426 VECTOR ANALYSIS Harmonic Vibrations andBivutor8 158.]Thedifferential equation ofrectilinear harmonic motionis Theintegral ofthisequation maybereduced byasuitable choiceoftheconstants totheform x=Asinnt. Thisrepresents avibration backandforthalongtheX-axis aboutthepointx=O.Letthedisplacement bedenotedby Dinplaceofx.Theequation maybewritten D= iAsinnt. Consider D=iAsinntcosmx. Thisisadisplacement notmerelynearthepointx=0 butalongtheentireaxisofx.Atpointsx=2k'7r,wherem lcisapositive ornegative inte~er,thedisplacement isata.ll timesequaltozero.Theequation represents astationary wavewithnodesatthesepoints.Atpointsmidwaybetween thesethewavehaspointsofmaximum vibration. Ifthe equation beregarded asinthreevariables x,y,zitrepre sentsapill-newavetheplaneofwhichisperpendicular to theaxisofthevariable x. Thedisplacement givenbytheequation D1=iA1cos(mx-nt) islikewise aplanewaveperpendicular totheaxisofxbut notstationary. Thevibration isharmonic andadvances alongthedirection iwithavelocity equaltothequotient of HARMONIC VIBRATIONS ANDBIVECTORS 427 Ifvbethevelocity; ptheperiod;andlthewave nbym. length, nv=-,m27rp=-,n27rl=-,ml 'V- -(2)p Thedisplacement D2=jA2cos(mx-nt) differsfromD1intheparticular thatthedisplacement takes placeinthedirection j,notinthedirection i.Thewaveas beforeproceeds inthedirection ofxwiththesamevelocity. Thisvibration istransverse insteadoflongitudinal. Bya simple extension itisseenthat D=Acos(mx-nt) isadisplacement inthedirection A.Thewaveadvances alongthedirection ofx.Hencethevibration isobliqueto thewave-front. Astillmoregeneralformmaybeobtained bysub8tituting m•rformx.Then D=Acos(m.r -nt). (3) Thisisadisplacement inthedirection A.Themaximum amountofthatdisplacement isthemagnitude ofA.The waveadvances inthedirection mobliquetothedisplace ment;thevelocity, period,andwave-length areasbefore. Somuchforrectilinear harmonic motion. Elliptic har- monicmotionmaybedefinedbytheequation (p.117). d2r-=- n2r.dt2 Thegeneralintegralisobtained as r=Acosnt+Bsinnt. Thediscussion ofwavesmaybecarriedthrough aspre viously. Thegeneral waveofellipticharmonic motion advancing inthedirection misseentobe 428 VECTOR ANALYSiS D=AC08(m• r -nt)-Bsin(m• r-nt).(4) dDI. Ide=nIAsm(m.r -nt)+Bcos(m.r -nt)I(5) isthevelocity ofthedisplaced pointatanymoment inthe ellipseinwhichitvibrates. Thisisofcourseentirelydiffer entfromthevelocityofthewave. Aninteresting resultisobtained byaddingupthedis placement andthevelocity multiplied bytheimaginary unitV-1anddividedbyn. v-IdD .D+---=Acos(m•r -nt)-Bsm(m• r -nt) ndt (6) +V-11Asin(m• r -nt)+Bcos(m• r -nt)}. D+V-1dD=(A+V_1B)V-lfa.r--q nd t e (6)' Theexpression hereobtained, asfarasitsformisconcerned, isanimaginary vector.Itisthesumoftworealvectorsof whichonehasbeenmultiplied bytheimaginary scalarV-1. Suchavectoriscalledabivectororimaginary vector. The ordinary imaginary scalarsmaybecalledbucalars. Theuse ofbivectors isfoundveryconvenient inthediscussion of ellipticharmonic motion. Indeedanyundamped elliptichar monicplanewavemayberepresented asabovebythepro ductofabivector andanexponential factor. Therealpart oftheproduct givesthedisplacement ofanypointandthe pureimaginary partgivesthevelocity ofdisplacement dividedbyn. 159.]Theanalytic theoryofbivectors differsfromthatof realvectorsverymuchastheanalytic theoryofbiscalars differsfromthatofrealscalars.Itisunnecessary tohave anydistinguishing character forbivectors justasitisneed- HARMONIC VIBRATIONS .ANDBIVECTORS 429 lesstohaveadistinguishing notation forbiscalars. Thebi vectormayberegarded asanaturalandinevitable extension oftherealvector.Itistheformalsumoftworealvectors ofwhichonehasbeenmultiplied bytheimaginary unitv-=-f. Theusualsymboliwillbemaintained forV-1.Thereis notmuchlikelihood ofconfusion withthevectoriforthe reasonthatthetwocouldhardlybeusedinthesameplace andforthefurtherreasonthattheItaliciandtheClarendon idifferconsiderably inappearance. Whenever itbecomes especially convenient tohaveaseparate alphabet forbivec torsthesmallGreekorGerman letteI'IJmaybecalledupon. Abivector maybeexpressed intermsofi,j,kwithcom plexcoefficients. If and orr.=r1+ir2 r1=Xli+Yd+ZIk, r2=x2i+Y2j+Z2k, r =(Xt+ix2)i +(Y1+iY2)j +(Zt+iZ2)k, r=xi+yj+zk. Twobivectors areequalwhentheirrealandtheirimaginary partsareequal.Twobivectors areparallelwhenoneisthe product oftheotherbyascalar(realorimaginary). If a.bivector isparalleltol\realvectoritissaidtohaveareal direction. Inothercase8ithasacomplex orimaginary direction. Thevalueofthesum,difference, direct,skew, a.ndindeterminate products oftwobivectors isobviouswith outspecialdefinition. Thesestatements maybeputinto analytic formasfollows. Let Thenif ifr = r1+ ir2andI=12+i12, r=l, r1=11andr2=112 430 VECTOR ANALYSIS r+1=(rl+11)+i(r,+I,). r·1=(rl•11-r,•I,)+i(rl•I,+r,•8t), rX1=(rtXIt-r,XI,)+i(rtXI,+rsXIt) rs=(rllt+r,I,)+i(rtl,+rslt). Twobivectors orbiscalars aresaidtobeconjugate when theirrealpartaareequalandtheirpureimaginary parts differonlyinsign.Theconjugate ofarealscalarorvector isequaltothescalarorvectoritself.Theconjugate ofany sortofproductofbivectors andbiscalars isequaltothepro ductoftheconjugates takeninthesameorder.Asimilar statement maybemadeconcerning sumsanddifferences. (rl+ir,)•(rt-ir2)=rt• rl+r,• r2• (rl+ir2) X(rl-ir,)=2ir2Xrt, (rt+ir,)(rl-ir,)=(rlrl+r,r2)+i(rtrl- rlr,). Ifthebivectorr=rt+ir,bemultiplied byarootofunity orcyclicfactorasitisfrequently called,thatis,byanimagi naryscalaroftheform wherecosq+isinq=a+ib, a2+h'=1,(7) theconjugate ismultiplied bya-ih,andhencethefour producta areunaltered bymultiplying thebivector rbysuchafactor. Thusif I I . I ( . h)(.) r=rt+tr,=a+trl+'tr,, HARJ,!Ol';[C VIBRATIONS ANDBIVECTORS 431 160.JAcloserexamination oftheeffectofmultiplying a bivector byacyclicfactoryieldsinteresting andimportant geometric results. Let f1'+i1'2'=(cosq+isinq)(rt+ir2).(8) Then f1'=f1cosq-f2sinq, f2'=r2cosq+f1sinq. Byreference toArt.129itwillbeseenthatthechangepro ducedintherealandimagina.ry vectorpartsofabivector by multiplication withacyclicfactor,isprecisely thesameas wouldbeproduced uponthosevectorsbyacyclicdyadic rp=aa'+cosq(bb'+cc')-sinq(cb'- bc') usedasaprefactor.bandcaresupposed tobetwovectors collinear respectively withrtandr2•aisanyvectornotin theirplane.Consider theellipseofwhichrtand1'2area pairofconjugate semi-diameters. Itthenappearsthat1'1' and1'2'arealsoapairofcs>njugate semi-diameters ofthat ellipse. Theyarerotatedintheellipsefromf2toward l'l'by asectorofwhichtheareaistotheareaofthewholeellipse asqisto27T".Suchachangeofposition hasbeencalledan ellipticrotation through thesectorq. Theellipseofwhich l'1and1'2areapairofconjugate semi diameters iscalledthedirectional ellipseofthebivector l'. Whenthebivector hasarealdirection thedirectional ellipse reducestoarightlineinthatdirection. Whenthebivector hasacomplex direction theellipseisatrueellipse. The angulardirection fromtherealpartl'1tothecomplex partl'2 isconsidered asthepositive direction inthedirectional ellipse,andmustalwaysbeknown.Iftherealandimagi narypartsofabivector turninthepositivedirection inthe ellipsetheyaresaidtobeadvanced; ifinthenegative direr tiontheyaresaidtoberetarded. Hencemultiplication ofa 432 VECTOR ANALYSIS bivectorbyacyclicfactorretardsitinitsdirectional ellip"by asectorequaltotheangleojthecyclicfactor. Itisalway!!pollBibletomultiply abivector bysuchacyclic factorthattherealandimaginary partsbecomecoincident ,withtheaxesoftheellipseandareperpendicular. r=(cosq+isinq)(a+ib)wherea •b=O. Toaccomplish thereduction proceedasfollows: Form r • r =(cos2q+isin2q)(a+ib)•(a+ib). Ifa'b=0, r • r=(cos2q+isin2q)(a• a -b •b). Let andr·r=a+ib, btan2q=-.a Withthisvalueofqtheaxesofthedirectional ellipseare givenbytheequation a+ib=(cosq-isinq)r. Incasetherealandimaginary partsaandbofabivector areequalinmagnitude andperpendicular indirection botha andbintheexpression forr • rvanish. Hencetheangle qisindeterminate. Thedirectional ellipseisacircle.A bivector whosedirectional ellipseisacircleiscalledacircu larbivector. Thenecessary andsufficient condition thata. non-vanishing bivector rbecircularis r • r=0,rcircular. If r=xi+yj+zk, r • r=x2+y2+z2=O. Thecondition r • r =0,whichforrealvectorsimpliesr=0, isnotsufficient toensurethevanishing ofabivector. The HARMONIC VIBRA.TIONSANDBIVECTORS 438 bivector iscircular, notnecessarily zero.Thecondition that abivector vanishisthatthedirectproductofitbyitscon jugatevanishes. (rl+ir~)•(rl-ir~)=rl•rl+r~• r2=0, then rl=r~=0andr=O. Incasethebivectorhasarealdirection itbecomes equalto itsconjugate andtheirproductbecomes equaltor •r. 161.]Thecondition thattwobivectors beparallelisthat oneistheproduct oftheotherbyascalarfactor.Anybi scalarfactormaybeexpressed astheproduct ofacyclic factorandapositivescalar,themodulus ofthebiscalar.If twobivectors differbyonlyacyclicfactortheirdirectional ellipsesarethesame.Hencetwoparallelvectorshavetheir directional ellipsesimilarandsimilarly placed-theratioof similitude beingthemodulus ofthebiscalar.Itisevident thatanytwocircular bivectors whoseplanescoincide are parallel. Acircularvectorandanon-circular vectorcannot beparallel. Thecondition thattwobivectors beperpendicular is orr·1=0, Consider firstthecaseinwhichtheplanesofthebivectors coincide. Let r=a(rl+irz),1=b(Ill-+:i~). Thescalarsaandbarebiscalars. rlmaybechosenperpen diculartor2,andIImaybetakeninthedirection ofr~.The condition r •1=0thengives r~•~=0andrl•I~+r~•II=O. 28 434 VECTOR ANALYSIS Thefirstequation showsthatr2aod12areperpendicular and hence 111aod~areperpendicular. Moreover, thesecond showsthattheangulardirections fromr1tor2andfrom11to 12arethesame,andthattheaxesofthedirectio~al ellipses ofrandIareproportional. Hencetheconditions forperpendicularity oftwohivectors whoseplanescoincide arethattheirdirectional ellipsesare similar,theangular direction inbothisthesame,andthe majoraxesoftheellipsesareperpendicular.1Ifbothvectors haverealdirections theconditions degenerate intotheper pendicularity ofthosedirections. Theconditions therefore holdforrealaswellasforimaginary vectors. LetrandIbetwoperpendicular hivectorstheplanesof whichdonotcoincide. Resolver1andr2eachintotwocom ponentsrespectively parallelandperpendicular totheplane of8.Thecomponents perpendicular tothatplanecontribute nothing tothevalueofr •8.Hencethecomponents ofr1 andr2paralleltotheplaneofIformabivector r'whichis perpendicular to8.Tothisbivector andIItheconditions statedaboveapply.Thedirectional ellipseofthebivector 1" isevidently theprojection ofthedirectional ellipseofrupon theplaneofI. Hence,iftwobivectors areperpendicular thedirectional ellipseofeitherbivector andthedirectional ellipseofthe otherprojected upontheplaneofthatonearesimilar,have thesameangular direction, andhavetheirmajoraxesper pendicular. 162.]Consider ahivectorofthetype D=Ae,IIm·r-..,~ (9) whereAandmarebivectors andnisabiscalar. risthe positionvectorofapointinspace.Itistherefore tobecon- 1Itshouldbenotedthatthecondition ofperpendicnlaritJ ofmajoraxesisnot thesame38thecondition ofperpendicularity ofrealpartsandimaginary parts. HARMONIC VIERATlONSANDBIVECTORS 435 sideredasreal.tis beconsidered asreal.thescalarvariable timeandisalsoto Let A=Al+i~, m=ml+im2, n=nl+in2• D=(AI+i~)e-lIIt•reot'eil.,.r-ol'~ (10) Ashasbeenseenbefore,thefactor(AI+i~)e'llII,·r-o,'1 represents atrainofplanewavesofellipticharmonic vibra tions.Thevibrations takeplaceintheplaneofAlandA2' inanellipseofwhichAland~areconjugate semi-diam eters.Thedisplacement ofthevibrating pointfromthe centeroftheellipseisgivenbytherealpartofthefactor. Thevelocity ofthepointafterithasbeendivided bynl isgivenbythepureimaginary part.Thewaveadvances inthedirection mI'Theotherfactorsintheexpres sionaredampers. Thefactore-....·risadamper inthe direction m2•Asthewaveproceeds inthedirection ~it lliesaway.Thefactore""isadamperintime.If112is negative thewavediesawayastimegoeson.If112isposi tivethewaveincreases inenergyastimeincreases. The presence (forunlimited time)ofanysuchfactorinanex pression whichrepresents anactualvibration isclearlyinad missible.Itcontradicts thelawofconservation ofenergy. Inanyphysical vibration ofaconservative systemn2isne cessarily negative orzero. Thegeneralexpression (9)therefore represents atrainof planewavesofellipticharmonic vibrations damped ina definitedirection andintime.Twosuchwavesmaybecom pounded byaddingthebivectors whichrepresent them.If theexponent m • r - 11tisthesameforboththeresulting trainofwavesad"ances inthesamedirection andhasthe 436 VECTOR ANALYSIS theresultant issameperiodandwave-length astheindividual waves. The vibrations, however, takeplaceinadifferent ellipse.Ifthe wavesare (A+B)elia•r-IIt). Bycombining twotrainsofwaveswhichadvance inopposite directions butwhichareinotherrespectsequalasystemof stationary wavesisobtained. Thetheoryofbivectors andtheirapplications willnotbe carriedfurther. Theobject in entering atalluponthisvery shortandcondensed discussion ofbivectors wasfirsttoshow thereaderhowthesimpleideaofadirection hastogiveway tothemorecomplicated butnolessusefulideaofadirectional ellipsewhenthegeneralization fromrealtoimaginary vectors ismade,andsecondtosetforththemannerinwhichasingle bivector Dmaybeemployed torepresent atrainofplane wavesofellipticharmonic vibrations. Thisapplication ofbi vectorsmaybeusedtogivetheTheoryofLightawonderfully simpleandeleganttreatment.l 1Snchnseofbivectors IsmadebyProfessor Gibbsinhiscourseoflectureson ..TheElectroma.'lnetic TheoryofLight,"delivered biannually atYaleUnh·ersity. Dil'ectors werenotnsedinthesecondpartofthischapter, becanseintheopinion ofthepresentauthortheyp088essnoessential advantage overrealvectorsuntil themoreadvanced partsofthetheory,rotation oftheplaneofpolarization by magnets andcrystals, totalandmetallic reflection, etc.,arereached. PRINTED INTHEUKITED STATES 01'AlLEllICA 1031 fa 1 1 J...,.. '"c.1 :);.2f)1 .Gl+• AOOOOOb2523bb SUBJECT DATEDUETORECAI r 1. O° HNNm1 AgoooobeSe3bb