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VECTORANALYSIS
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Preface
Vectoranalysis,whichhaditsbeginningsinthemiddleofthe19thcentury,hasinrecent
yearsbecomeanessentialpartofthemathematicalbackgroundrequiredofengineers,phy-
sicists,mathematiciansandotherscientists.Thisrequirementisfarfromaccidental,fornot
onlydoesvectoranalysisprovideaconcisenotationforpresentingequationsarisingfrom
mathematicalformulationsofphysicalandgeometricalproblemsbutitisalsoanaturalaid
informingmentalpicturesofphysicalandgeometricalideas.Inshort,itmightverywellbe
consideredamostrewardinglanguageandmodeofthoughtforthephysicalsciences.
Thisbookisdesignedtobeusedeitherasatextbookforaformalcourseinvector
analysisorasaveryusefulsupplementtoallcurrentstandardtexts.Itshouldalsobeof
considerablevaluetothosetakingcoursesinphysics,mechanics,electromagnetictheory,
aerodynamicsoranyofthenumerousotherfieldsinwhichvectormethodsareemployed.
Eachchapterbeginswithaclearstatementofpertinentdefinitions,principlesand
theoremstogetherwithillustrativeandotherdescriptivematerial.Thisisfollowedby
gradedsetsofsolvedandsupplementaryproblems.Thesolvedproblemsservetoillustrate
andamplifythetheory,bringintosharpfocusthosefinepointswithoutwhichthestudent
continuallyfeelshimselfonunsafeground,andprovidetherepetitionofbasicprinciples
sovitaltoeffectiveteaching.Numerousproofsoftheoremsandderivationsofformulas
areincludedamongthesolvedproblems.Thelargenumberofsupplementaryproblems
withanswersserveasacompletereviewofthematerialofeachchapter.
Topicscoveredincludethealgebraandthedifferentialandintegralcalculusofvec-
tors,Stokes'theorem,thedivergencetheoremandotherintegraltheoremstogetherwith
manyapplicationsdrawnfromvariousfields.Addedfeaturesarethechaptersoncurvilin-
earcoordinatesandtensoranalysiswhichshouldproveextremelyusefulinthestudyof
advancedengineering,physicsandmathematics.
Considerablymorematerialhasbeenincludedherethancanbecoveredinmostfirst
courses.Thishasbeendonetomakethebookmoreflexible,toprovideamoreusefulbook
ofreference,andtostimulatefurtherinterestinthetopics.
TheauthorgratefullyacknowledgeshisindebtednesstoMr.HenryHaydenfortypo-
graphicallayoutandartworkforthefigures.Therealismofthesefiguresaddsgreatlyto
theeffectivenessofpresentationinasubjectwherespatialvisualizationsplaysuch anim-
portantrole.
M.R.SPiEGEL
RensselaerPolytechnicInstitute
June,
AVECTORisaquantityhavingbothmagiiitud anddirectionsuchasdisplacement,_velocity,force
and acceleration.
GraphicallyavectorisrepresentedbyanarrowOP(Fig.l)de-
finingthedirection,themagnitudeofthevectorbeingindicatedby
thelengthofthearrow.Thetailend0ofthearrowiscalledthe
originorinitialpointofthevector,andtheheadPiscalledthe
terminalpointorterminus.
Analyticallyavectorisrepresentedbyaletterwithanarrow
overit,asAinFig.1,anditsmagnitudeisdenotedby IAIorA.In
printedworks,boldfacedtype,suchasA,isusedtoindicatethe
vectorAwhileJAIorAindicatesitsmagnitude.Weshallusethis
boldfacednotationinthisbook.ThevectorOPisalsoindicatedas
OPorOP;insuchcaseweshalldenoteitsmagnitudebyOF,OPI,
orof.
Fig.
ASCALARisaquantityhavingmagnitudebut(n
direction,e.g.
m, a
ISh,tfe,temer
and
anyrealnumber.Scalarsareindicatedbylettersinordinarytypeasinelementaryalge-
bra.Operationswithscalarsfollowthesamerulesasinelementaryalgebra.
VECTORALGEBRA.Theoperationsofaddition,subtractionandmultiplicationfamiliarinthealge-
braofnumbersorscalarsare,withsuitabledefinition,capableofextension
toanalgebraofvectors.Thefollowingdefinitionsarefundamental.
1.TwovectorsAandBareequaliftheyhavethesamemagnitudeanddirectionregardlessof
thepositionoftheirinitialpoints.ThusA=BinFig.2.
2.AvectorhavingdirectionoppositetothatofvectorAbuthavingthesamemagnitudeisde-
notedby-A(Fig.3).
Fig.2 Fig.
1
VECTORSandSCALARS
itsnamefromthefactthatarightthreadedscrewrotat-
edthrough900fromOxtoOywilladvanceinthepos-
itivezdirection,asinFig.5above.
In general,threevectorsA,BandCwhichhave
coincidentinitialpointsandarenotcoplanar,i.e.do
notlieinorarenotparalleltothesameplane,aresaid
toformaright-handedsystemordextralsystemifa
rightthreadedscrewrotatedthroughananglelessthan
180°fromAtoBwilladvanceinthedirectionCas
showninFig.6.
COMPONENTSOFAVECTOR.AnyvectorAin3di-
mensionscan arepre-
sentedwithinitialpointattheorigin0ofarecangular
coordinate system (Fig.7).Let(Al,A2,A3)
bethe
rectangularcoordinatesoftheterminalpointofvectorA
withinitialpointat0.ThevectorsAli,A2j,andA3k
arecalledtherectalarcomponentvectorsorsimply
componentvectorsofAinthex,yandzdirectionsre-
spectively.
A1,A2andA3arecalledtherectangular
componentsorsimplycomponentsofAinthex,yandz
directionsrespectively.
Thesumorresultantof Ali,A2j
vectorAsothatwecanwrite
andA3k isthe
A=A1i+A2I +Ak
ThemagnitudeofAis A=
IAI Al+A2+A
Fig.
Fig.
Inparticular,thepositionvectororradiusvectorrfrom0tothepoint(x,y,z)iswritten
r
=xi+yj+zk
andhasmagnitude r = IrI =
x2+y2+z2.
3
.y0,1to$
.
SCALARFIELD.Iftoeachpoint(x,y,z)ofaregionRinspacetherecorrespondsanumberorscalar
then iscalledascalarfunctionofposition
orscalarpointfunction
andwesaythatascalarfield0hasbeendefinedinR.
Examples.(1)Thetemperatureatanypointwithinorontheearth'ssurfaceatacertaintime
definesascalarfield.
(2)ct(x,y,z) =x3y-z2 definesascalarfield.
Ascalarfieldwhichisindependentoftimeiscalledastationaryorsteady-statescalarfield.
VECTORFIELD.Iftoeachpoint(x,y,z)ofaregionRinspacetherecorrespondsavectorV(x,y,z),
thenViscalledavectorfunctionofpositionorvectorpointfunctionandwesay
thatavectorfieldVhasbeendefinedinR.
Examples.(1)Ifthevelocityatanypoint(x,y,z)withinamovingfluidisknownatacertain
time,thenavectorfieldisdefined.
(2)V(x,y,z) =xy2i-2yz3j+x2zk definesavectorfield.
Avectorfieldwhichisindependentoftimeiscalledastationaryorsteady-statevectorfield.
VECTORSandSCALARS
SOLVEDPROBLEMS
1.Statewhichofthefollowingarescalarsandwhicharevectors.
(a)weight
(c)specificheat
(e)density
(g)volume (i)speed
(b)calorie (d)momentum (f)energy (h)distance (j)magneticfieldintensity
Ans.(a)vector (c)scalar (e)scalar (g)scalar (i)scalar
(b)scalar (d)vector (f)scalar (h)scalar (j)vector
2.Representgraphically(a)aforceof10lbinadirection30°northofeast
(b)aforceof15lbinadirection30°eastofnorth.
N N
Unit=5lb
W
S
E
W
Fig.(a)
S
Fig.(b)
Choosingtheunitofmagnitudeshown,therequiredvectorsareasindicatedabove.
F
3.Anautomobiletravels3milesduenorth,then5milesnortheast.Representthesedisplacements
graphicallyanddeterminetheresultantdisplacement(a)graphically,(b)analytically.
VectorOPorArepresentsdisplacementof3miduenorth.
VectorPQorBrepresentsdisplacementof5minortheast.
VectorOQorCrepresentstheresultantdisplacementor
sumofvectorsAandB, i.e.C=A+B.This,isthetriangle
lawofvectoraddition.
TheresultantvectorOQcanalsobeobtainedbycon-
structingthediagonaloftheparallelogramOPQRhavingvectors
OP=AandOR(equaltovectorPQorB)assides.Thisisthe
parallelogramlawofvectoraddition.
(a)GraphicalDeterminationofResultant. Layoffthe1mile
unitonvectorOQtofindthemagnitude7.4mi(approximately).
Angle
EOQ=61.5°,usingaprotractor.ThenvectorOQhas
magnitude7.4mianddirection61.5°northofeast.
(b)AnalyticalDeterminationofResultant.FromtriangleOPQ,
denotingthemagnitudesof A,B.CbyA,B,C,wehaveby
thelawofcosines
C2=A2+B2-2ABcosLOPQ =32+
52 -2(3)(5)cos135° =34+15V2=55.
andC=7.43(approximately).
Bythelawofsines,
A C
Then
sinLOQP sinLOPQ
s
VECTORSandSCALARS
Sincetheorderofadditionofvectorsisimmaterial,wemaystartwithanyvector,sayFl.ToFladd
F2,thenF3,
etc.ThevectordrawnfromtheinitialpointofFltotheterminalpointofF6istheresultant
R,i.e.R=F1+F2+F3+Fμ+F5+F6.
TheforceneededtopreventPfrommovingis-RwhichisavectorequalinmagnitudetoRbutopposite
indirectionandsometimescalledtheequilibrant.
F
8.GivenvectorsA,BandC(Fig.1a),construct(a)A-B+2C(b)3C--z(2A-B).
(a)
Fig.1(a)
(b)
Fig.2(a)
Fig.1(b)
Fig.2(b)
VECTORSandSCALARS
9.Anairplanemovesinanorthwesterlydirectionat
125mi/hrrelativetotheground,duetothefact
thereisawesterlywindof50mi/hrrelativeto
theground.Howfastandinwhatdirectionwould
theplanehavetravelediftherewerenowind?
LetW =windvelocity
Va=velocityofplanewithwind
Vb=velocityofplanewithoutwind
-w
ThenVa = Vb+W or Vb = Va-W=
Va+(-W)
Vbhasmagnitude6.5units=163mi/hranddirection33°northofwest.
7
10.Giventwonon-collinearvectorsaandb,findanexpressionforanyvectorrlyingintheplanede-
terminedbyaandb.
Non-collinearvectorsarevectorswhicharenotparallelto
thesameline. Hencewhentheirinitialpointscoincide,they
determineaplane.Letrbeanyvectorlyingintheplaneofa
andbandhavingitsinitialpointcoincidentwiththeinitial
pointsofaandbatO.FromtheterminalpointRofrconstruct
linesparalleltothevectorsaandbandcompletetheparallel-
ogramODRCbyextensionofthelinesofactionofaandbif
necessary.Fromtheadjoiningfigure
OD=x(OA)=xa,wherexisascalar
OC=y(OB)=yb,whereyisascalar.
Butbytheparallelogramlawofvectoraddition
OR=OD+OC or r=xa+yb
whichistherequiredexpression.Thevectorsxaandybarecalledcomponentvectorsofrinthedirections
aandbrespectively.Thescalarsxandymaybepositiveornegativedependingontherelativeorientations
ofthevectors.Fromthemannerofconstructionitisclearthatxandyareuniqueforagivena,b,andr.
Thevectorsaandbarecalledbasevectorsinaplane.
11.Giventhreenon-coplanarvectorsa,b,andc,findanexpressionforanyvectorrinthreedimen-
sionalspace.
Non-coplanarvectorsarevectorswhicharenotparal-
leltothesameplane. Hencewhentheirinitialpointsco-
incidetheydonotlieinthesameplane.
Letrbeanyvectorinspacehavingitsinitialpointco-
incidentwiththeinitialpointsofa,bandcatO.Through
theterminalpointofrpassplanesparallelrespectively
totheplanesdeterminedbyaandb,bandc,andaandc;
andcompletetheparallelepipedPQRSTUVbyextensionof
thelinesofactionofa,bandcifnecessary.
Fromthe
adjoiningfigure,
OV =x(OA)=xa
wherexisascalar
OP=y(OB)=yb whereyisascalar
OT=z(OC)=zc wherezisascalar.
ButOR=OV+VQ+QR=OV+OP+OT
or r
=xa+yb+zc.
Fromthemannerofconstructionitisclearthatx,yandzareuniqueforagivena,b,candr.
VECTORSandSCALARS
g
18.LetP.,P
1
P3bepointsfixedrelativetoanorigin0andletr1,r2,r3bepositionvectorsfrom
0toeachpoint.Showthatifthevectorequationalrl+a2r2+a3r3=0 holdswithrespectto
origin0thenitwillholdwithrespecttoanyotherorigin0'ifandonlyif al+a2+a3=0.
Let r3bethepositionvectorsofPI,P2andP3withrespectto0'andletvbetheposition
vectorof0'withrespectto0.Weseekconditionsunderwhichtheequationa,r+ar'+ar` =0 will
holdinthenewreferencesystem.
FromFig.(b)below,itisclearthat r1=v+ri,r2=v+r2,r3=v+r
sothat a1r1+a2r2+a
3
r
3
=
becomes
alrl+a2r2+a3r3=a,(v+r')+a2(v+r2)+a3(v+r3)
_ (al+a2+a3)v+alr1+a2r2+a3r
0
Theresult alrj+a2r2+a3r3=0willholdifandonlyif
(al+a2+a3)v=0, i.e.
al+a2+a3 = 0.
Theresultcanbegeneralized.
O'
Fig.(a) Fig.(b)
19.FindtheequationofastraightlinewhichpassesthroughtwogivenpointsAandBhavingposi-
tionvectorsaandbwithrespecttoanorigin0.
LetrbethepositionvectorofanypointPontheline
throughAandB.
Fromtheadjoiningfigure,
OA+AP=OP
or a+AP=r, i.e.AP=r-a
andOA+AB=OB or a+AB=b, i.e. AB=b-a
SinceAPandABarecollinear, AP=tABor r-a=t(b--a).
Thentherequiredequationis
r =a+t(b-a) or r =(1-t)a+tb
Iftheequationiswritten (1-t)a+tb-r=0,thesum
ofthecoefficientsofa,bandris1-t+t-1=0.Henceby
Problem18itisseenthatthepointPisalwaysontheline
joiningAandBanddoesnotdependonthechoiceoforigin
0,whichisofcourseasitshouldbe.
AnotherMethod.SinceAPandPBarecollinear,wehaveforscalarsmandn:
Solving, r
ma+nb
m+n
mAP=nPB or m(r-a)=n(b-r)
whichiscalledthesymmetricform.
10
VECTORSandSCALARS
(a)Findthepositionvectorsr1andr2forthe
pointsP(2,4,3)andQ(1,-5,2)ofarectangular
coordinatesystemintermsoftheunitvectors
i,j,k. (b)Determinegraphicallyandanalyti-
callytheresultantofthesepositionvectors.
(a) r1=OP=OC+CB+BP=2i+4j+3k
r2=OQ=OD+DE+EQ=
i-5j+2k
(b)Graphically,theresultantofr1andr2isobtained
asthediagonalORofParallelogramOPRQ.Ana-
lytically,theresultantofr1andr2isgivenby
r1+r2 = (2i+4j+3k)+(i-5j+2k) =
21.ProvethatthemagnitudeAofthevectorA=
A1i+A2j+A3kisA= A1+A2+A
.
BythePythagoreantheorem,
_
(OP)2= (OQ)2+(QP)
whereOPdenotesthemagnitudeofvectorOP,etc.
Similarly, (OQ)2=(OR)2+(RQ)2.
Then (5P)2=
(OR)2+(RQ)2+(QP)2or
A2=Ai+A2+A2, i.e.A=
Al+A2+A.
22.Given
r1=3i-2j+k, r2=2i-4j-3k,
r3=-i+2j+2k,
(a)r3, (b)r1+r2+r3,
(c)2r1-3r2--5r3.
(a) Ir3I = I-i+2j+2kI =V'(-1)2+(2)2+(2)
= 3.
findthemagnitudesof
(b)r1+r2+r3 = (3i-2j+k)+(2i-4;j-3k)+(-i+2j+2k) =4i-4j+Ok=
4i-4j
Then
Ir1+r2+r3I = 14i-4j+0k (4)2+(-4)2+(0)
=32=4/2.
(c) 2r1-3r2-5r3 = 2(3i-2j+k)--3(2i-4j-3k)-5(-i+2j+2k)
=6i-4j+2k-6i+12j+9k+5i-10j-10k=5i-2j+k.
Then
I2r1-3r2-5r
I =15i-2j+kI
=V'(5)2+(-2)2+(1)2 =
V130.
Y
23.If
r1=2i-j+k,
r2=i+3j-2k, r3=-21+j--3kandr4=3i+2j+5k,findscalarsa,b,csuch
thatr4=art+br2+cr3.
Werequire 3i+2j+5k =a(2i-j+k)+b(i+3j-2k)+c(-2i+j-3k)
_(2a+b-2c)i+(-a+3b+c)j+(a-2b-3c)k.
Since i,j,karenon-coplanarwehavebyProblem15,
2a+b-2c=3,
-a+3b+c=2, a-2b-3c=5.
Solving, a=-2,b=1, c=-3 and r4=-2r1+r2-3r3.
Thevectorr4issaidtobelinearlydependentonr1,r2,andr3;inotherwordsr1,r2,r3andr4constitutea
linearlydependentsetofvectors.Ontheotherhandanythree(orfewer)ofthesevectorsarelinearlyin-
dependent.
Ingeneralthevectors A,B,C,... arecalledlinearlydependentifwecanfindasetofscalars,
a,b,c,..., notallzero,sothataA+bB+cC+...=0.otherwisetheyarelinearlyindependent.
VECTORSandSCALARS
Letr1andr2bethepositionvectorsofPandQrespec-
tively,andrthepositionvectorofanypointRontheline
joiningPandQ.
r1+PR=
r or
PR=
r-r
r1+PQ= r2 or PQ= r2-r
But PR=tPQwheretisascalar. Thenr-r1=
t(r2-r1) istherequiredvectorequationofthestraightline
(comparewithProblem19).
Inrectangularcoordinateswehave,since r =xi+yj+zk,
(xi+yj+zk)-(x1i+y1)+z1k)
= t[(x2i+y2j+z2k)-(x1i+y1j+z1k)]
or
(x-x1)i+(y-y1)j+(z-z1)k =
t[(x2-x1)i+(y2-y1)j+(z2-z1)k]
Sincei,j,karenon-coplanarvectorswehavebyProblem15,
x-x1= t(x2-x1), y-y1= t(y2-y1), z-z1 = t(z2-z1)
astheparametricequationsoftheline,tbeingtheparameter.Eliminatingt,theequationsbecome
X-x
x2-x
Y-Y1 z-z
Y2 ` Y1 z2 - z
29.Giventhescalarfielddefinedby (x,y,z) = 3x22-xy3+5, find atthepoints
(a)
(0,0,0),
(b)(1,-2,2) (c)(-1,-2,-3).
(a)0(0,0,0) = 3(0)2(0)-(0)(0)3+5 = 0-0+5 = 5
(b)00,-2,2) =
3(1)2(2)-(1)(-2)3+5 =
(c) )(-1,-2,-3) = 3(-1)2(-3)-(-1)(-2)3+5 = -9-8+5 -
30.Graphthevectorfieldsdefinedby:
(a)V(x,y)=xi+yj,
(b)V(x,y)_-xi-yj, (c)V(x,y,z)
=xi+yj+A.
(a)Ateachpoint(x,y),except(0,0),ofthexyplanethereisdefinedauniquevectorxi+yjofmagnitude
havingdirectionpassingthroughtheoriginandoutwardfromit.Tosimplifygraphingproce-
dures,notethatallvectorsassociatedwithpointsonthecirclesx2+y2=a2a>0havemagnitude
a.ThefieldthereforeappearsasinFigure(a)whereanappropriatescaleisused.
Y
Fig.(a) Fig.(b)
VECTORSandSCALARS
(b)Hereeachvectorisequaltobutoppositeindirectiontothecorrespondingonein(a).Thefieldthere-
foreappearsasinFig.(b).
InFig.(a)thefieldhastheappearanceofafluidemergingfromapointsource0andflowinginthe
directionsindicated.Forthisreasonthefieldiscalledasourcefieldand0isasource.
InFig.(b)thefieldseemstobeflowingtoward0,andthefieldisthereforecalledasinkfieldand
isasink.
Inthreedimensionsthecorrespondinginterpretationisthatafluidisemergingradiallyfrom(orpro-
ceedingradiallytoward)alinesource(orlinesink).
Thevectorfieldiscalledtwodimensionalsinceitisindependentofz.
(c)Sincethemagnitudeofeachvectoris x2+y2+z2, allpointsonthespherex2+y2+z2=a2, a>
havevectorsofmagnitudeaassociatedwiththem.Thefieldthereforetakesontheappearanceofthat
ofafluidemergingfromsource0andproceedinginalldirectionsinspace.Thisisathreedimension-
alsourcefield.
SUPPLEMENTARYPROBLEMS
31.Whichofthefollowingarescalarsandwhicharevectors?(a)Kineticenergy,(b)electricfieldintensity,
(c)entropy,(d)work,(e)centrifugalforce,(f)temperature,(g)gravitationalpotential,(h)charge,(i)shear-
ingstress,(j)frequency.
Ans.(a)scalar,(b)vector,(c)scalar,(d)scalar,(e)vector,(f)scalar,(g)scalar,(h)scalar,(i)vector
(j)scalar
32.Anairplanetravels200milesduewestandthen150miles600northofwest.Determinetheresultantdis-
placement(a)graphically,(b)analytically.
Ans.magnitude304.1mi(50Y'3-7),direction25°17'northofeast(arcsin3/74)
33.Findtheresultantofthefollowingdisplacements:A,20miles30°southofeast;
B,50milesduewest;
C,40milesnortheast;D,30miles60°southofwest.
Ans. magnitude20.9mi,direction21°39'southofwest
34.Showgraphicallythat-(A-B)_-A+B
.
35.AnobjectPisacteduponbythreecoplanarforcesasshowninFig.(a)below.Determinetheforceneeded
topreventPfrommoving. Ans.323lbdirectlyopposite150lbforce
36.GivenvectorsA,B,CandD(Fig.(b)below).Construct(a)3A-2B-(C-D) (b)
2
C+I(A-B+2D).
Fig.(a)
Fig.(b)
