Multivariable calculus, linear algebra, and differential equations

984 Pages·1986·61.17 MB·English

by  Stanley I Grossman

MULTIVARLIEA B
CALCULUS,
LINEAARL GBERA,
AND DIFEFRENTILA
EQUATIONS
SECOND EDITION
STANLEICY.R OSSMAN
UniveorMfso inttya na
(HarcBoruaJrcotev anoPvuibclhi,s hers)
AcadePmriecsI sn,c .
OrlanSdaoDn i egSoa Fnr ancNieswYc oor k
LondoTno ronMtoon trSeyadln Teoyk ySoa Poa ulo
TOA ARONE,R IKA,N DK ERSTIN
Copyr©i 1g9h8bt6yA cademic Press, Inc.
Alrli grhetsse rved.
Nop artt hpoiusfb limcaaybt eir oenp roodrut creadn simnia tntye d
foromrb ya nmye anesl,e ctorrmo encihca niinccallup,dh iontgo copy,
recorodrai nniygn ,f ormsattoiraoanngrd ee trsiyesvtaelm , without
permiisnws riiotnfi rnotgmh p eu blisher.
AcademicI ncPr.e ss,
OrlaFnldoor,3i 2d8a8 7
UnitKeidn gdEodmi tPiuobnl ibsyAh ceadd ePmrieIcsn s(c,L. o ndLotnd). ,
24/O2v8aR lo aLdo,n dNoWnl7 DX
ISB0N-2:1: 304380-8
LibroafCr oyn grCeastsaC laorNgdu mbe8r5:- 70253
Prinitnte hdUe n itSetdao tfAe mse rica
TOA ARONE,R IKA,N DK ERSTIN
Copyr©i 1g9h8bt6yA cademic Press, Inc.
Alrli grhetsse rved.
Nop artt hpoiusfb limcaaybt eir oenp roodrut creadn simnia tntye d
foromrb ya nmye anesl,e ctorrmo encihca niinccallup,dh iontgo copy,
recorodrai nniygn ,f ormsattoiraoanngrd ee trsiyesvtaelm , without
permiisnws riiotnfi rnotgmh p eu blisher.
AcademicI ncPr.e ss,
OrlaFnldoor,3i 2d8a8 7
UnitKeidn gdEodmi tPiuobnl ibsyAh ceadd ePmrieIcsn s(c,L. o ndLotnd). ,
24/O2v8aR lo aLdo,n dNoWnl7 DX
ISB0N-2:1: 304380-8
LibroafCr oyn grCeastsaC laorNgdu mbe8r5:- 70253
Prinitnte hdUe n itSetdao tfAe mse rica
Praecfe
In1 977f iterhdseit t oifmo ybn o oCka lcwualpsuu sb liIstlh,iet kdhe.se e coanndd
thierddi ttihofanotsl locwoendt,aa i nceodm preihnetnrsoidtvuotec h tceia olnc ulus
of aonnsdee vevraarli aMbalnieyns s.t rsuucgtgoetrshstsa eitdn ce nau mlboaefrr g e
studsetnotpsap fetsdet ru doynivena gr icaabllceau s lhuosrv,te errso ifmo ynt ext
shoublead v ailable.
Tom eetth naete Adc,a dePmriepcsu sb liCsahlecdPu alruIit sn 1, 8 91w,i th a
second ien1d 9i8tT6ih.oi ns onec-avlacrtuielacxubotsln ett ahifienr ssf to urteen
chapottfeh rmesa ibno oTkh.e orpilgwainatn sopa ulb lsiismhu,l taneously, a second
"shboorotck o"n tatihnleia snsegt v cehna potfte hrmesa itne xt.
Mosfitr st-cyaelcaurl ucs ovcseoirum rimslaeatsre rial.I s oHoonw efvoeurn,d
thtahtwi ass tnhoceta fsoetr hs ee condc-oyuerSasorems .ec hocoolvosen rlm yu lti­
variablei ntc hsaeel ccoyuneldauO rst. h eirnsc ludlei nsaeolamgree s bormdaei, f fer­
enteiqaula toirbo ontsMh,o. r eosvoemuren, i veirnsciltuiedased vmaoncrcaeel dc u­
lumsa teirnti hasele condt oypesiaucrcsa:h sT ayltohre'osir nev ma riaabnlde s
n
mapping�s"t of� mr.o m
Thuwse,m adteh dee citsowi roinat b eo ok thabteu swaobullwed i dien a
varioefct oyu rMsyeg so.ha alb se etnor ettahifenl aovfto hroe r igcianlacbluo louks
whimlaek ianl ga rnguem boeftr r adit"ipoonsatl-lcyta olpcaiucclcsue ss"tso i ble
sophomores.
Toa ccomtphliiIhss a,hv deo naen umboeftr h inMgossi.tm porIth aanvte,
contitnoiu necdl ude laorefgx ea mnpaulnmeedbxs ee rrscM iossmetas t.h ematicians
xii
PREFACE
agreteh atth eo nlwya yt ol earcna lcuwleulsil s b ys olvipnrgo blemasn,dt hibso ok
isd esignteode ncourasgteu denttosl earbny t hi7m3se t0hodM .u ltiiavbCalarel culus,
LinAelagrreaa b,nD dffi erteinEaqlu ations
contaionvse r examples-mamnoyr et han
arec ommonly fionut nedx attst hilse veEla.c he xampilnec ludaeltslh ea lgebrsatiecp s
needetdoc ompletthees olutiAosna .s tudenIwt a,s i nfuribayts etda temelnitk"sei t
nowe asifloyl lotwhsa .t. ".w heni tw asn ota ta lela sfyo mre .S tudenhtasv ea r ight
tos eet he" wholhea nd,s"o t os peaks,ot hatth eayl wayksn owh owt og etf rom" a"
to" b.I"n m anyi nstanecxepsl,a natiaornehs i ghligihntc eodl otrom akea s teepa sier
tof ollow. 4300
The texti ncludaepsp roximatelye xercises-incbloutdhid nrgi laln d
applied-ptryopbel emMso.r ed iffipcruolbtl eamrsem arkewdi tahn a ster(i*sa)kn da
(**).
fewe specially odniefsaf riecm ualrtk ewdi tha doublaes teriskT hee xercises
provitdheem osti mportlaenatrn itnogo ilna nyu ndergradmuaatthee mattiecxst book.
I stretsosm y studentthsa nto m attehro w weltlh eyt hintkh eyu nderstamnyd
lectuorret sh et extbotohke,dy o n ot reallyt hkemn aotwe ruinatl il thewyo rhkaevde
problemAs v.a sdti ffereenxciesb test weeunn derstansdoimnego neel ses'osl ution
ands olvian nge wp roblebmyy ourseLlefa.rni ngm athematwiictsh oduoti npgr ob­
lemsi sa bouats e asays l earnitnosg k wii thoguoti ntgo t hes lopes.
Ih avea lstor ietdoi ntrodduicfefi ccuolntc epbtysf irdsits cusssiinmgp lcea ses
anda pplicatiAosna sn.e xampleb,e gIit nh es ectioonnl inetarra nsforma(tSieocntiso n
9.w6i)t thw ov ersyi mplaep plicatTihoenrsae.r ep lenotyfd ifficult tphreoovreedm s
int hibso okb,u tI havet rietdo p utt hemo ffu ntitlh ec oncepatrses ufficiently
exemplified.
if
Mathematibcesc om�s more intoenreek sntoiwnssg o methianbgo utth eh istor­
icadle velopmoefnt th es ubjeIct try.t oc onvinmcye s tudentthsa tc,o ntratrowy h at
themya yb eliemvaen,y g reat mathemaltiicvieiadnn tse resatnidon fgt ecno ntroversial
liveTsh.u st,o m aket hes ubjemcotr ei nteresatnidn,pg e rhapmso,r ef unI,h ave
includaen du mbeorf f ull-pbaigoeg raphsikceatlc hoefms a thematicwihaonh se lped
develotphe c alculIunst .h essek etchsetsu denwtisll le arfno,re xamplea,b outth e
dispute betweena nNdLe ewitbonnti hzegi, ftf olra nguagoefHs a miltoann,dt hel ove
lifoefL agrangIeti. sm y hopet hatth esneo tewsi lblr intgh es ubjetcolt i fe.
Thea nswertsom osto dd-numbeerxeedr cisaepsp eara tt heb acokf t heb ookI.n
additiaos nt,u denmta'nsu aclo ntaining dseotlauitliteooand ls ol d d-numbered prob­
lemsa nda ni nstrucmtaonru'ascl o ntaining dseotlauitliteoodna sle lv en-numbered
probleamrsea vailaTbhlees.we e rep reparbeydP rofesLseoornG erbeartS tJ.o hn's
UniversiinNt eyw YorkO ty.
Thitse xcto veras w ider angoef t opicbsu,tt heo nlpyr erequiissai ctoeu rsien
one-variable PcrailnccuilpuIaes lx.lp ye,tc hta ats tudeunsti ntgh ibso okw ilkln owt he
following:
howt oc omputlei mits
D
howt od ifferenatniyea lteem entfaurnyc tion
D
theb asitce chniqoufei sn tegraitnicolnu diensgp,e ciailnltye,g rabtyip oanr tasn d
D
integrabtyia o vna rieotfsy u bstitutions
theb asiacp plicatoifdo infsf erentainaditi notne graitinocnl udciunrgv sek etching,
D
computianrge aasn,d c omputinvgo lumes
Thes tudewnhto c omest ot hec ourse twhietshse k ilwlisld lo w ell.
PREFACE
xiii
Theb ooiksd ivided, 1irno05tfiuovg ep -h alr2yTt,hs e.fi rspta rcto nsiosfCt hsa pters
1-6T.he saers ei mitloCa hra ptersi nCa lculTuhsiE,r ddi tainodi nnc,l buadsei c
multivariablmea tcearTlihcaebul al.su disic f feirste hnacIite n trotdhusecp ea IRce"i n
Sect3i.oan7n dt heinn,l atseerc tiIog nesn,e ratloIRi "bz aes tiocp iicnIRs 2a ndIR3 •
Howevetrha,ep proiasgc rha duVaelc.t ionrt sh pel aanrede i scuisnsC ehda pt1e,r
witvhe ctfourn ctiinto hn2pesl ainnCe h apteCrh apt3ce orn taainen xst enosfti hoen
2.
materiinCa hla pt1ea rnsd t ot hraenedm ordei mensiCoonmsp.u tderra wgnr aphs
ofc ertqaui4and s ruircf aacrgeeis v iennS ect3i.o6n .
Chaptceorn taiinnst rodutcott hciealo cnu olufts w oa ndm orvea riaTbhlee s.
gradiiesin n5tt r odiuannS c eecdt 4i.oa5nst hnea tuerxatle nosfit ohoner dindaerryi va­
tivCeh.a ptperro viadnie nst rodutcomt uilotnii pnltee grwaittaihnoe nm phaosni s
applicaCthiaopnt6sce .or n taadin est ailed inttovr eocdtauoncrat liyinoscnil sud ai ng
discuswsiitophnr ,o oafnsda pplicaotfGi roenesn,S' tso,k easn'dts h dei vergence
theorems. 78,,
Thes ecopnadro tft hbeo oiksa ni ntrodutcolt iinoaenla gre ibnCr haa pters
and9 T.h imsa terreiaqlu inrome tisl tivariabelxec eacfap amtli cluilawuriisttv yhe ctors
inIR 2IR,3 a ndIR "T.h esceh aptcearnbs ec overaendyt imaef tCehra pt1e,Sr e ctions
3.1.-,5a3 ndS e7ct 3i.o7n.
Chaptearnsd 8 contaiinnt rodutcotm iaotnrsi dceetse,r minaanndtt hse,
Gauss-Jotredcahnn fioqsruoe l vsiynsgt eomfes q uatiCohnasp.t9 ei rn cludes more
advanmcaetde roinva elc tsopra caensdl inetarra nsformations.
Thet hipradro tft hbeo ocko nsiosftt hsse i ngClhea pt1e0Tr.h ec hapctoemr­
binteesc hnifqruocemas l cualnudls i naelagre abnrdca o ntadiinssc usosfsi oomnoesf
thmeo setl egant inr tehsceua lltcsu lus includingi nnTv aayrlioatrbh'lese st,h eorem
multivarmieaabnvl ael tuhee oraenmdt, h iem plfiucnictt tihoeno rNeomn.eo f t he
resuhletraser feo unidns tandcaarldc ulus texts.
Chapt1e1ra sn d1 2c omprtihsfeeo urptahro tft hbeo oakn dp roviado en e­
quarotres re mester inttoro ordduicndtaiirfoyfn e reeqnutaitailo ns.1 1iC sh apter
indepenodfCe hnatp t1e-r1sa0 n dc anb ec overaetad n yt imIetc. o ntadientsa iled
discusosfifi ornsst -oarnddle irn esaerc ond-eoqrudaetriA olnsison. c luadreoedp ­
tiondails cusosfie olnescci trrciucai ntdvs i bramtootriyo n.
Chapt1e2or,n s ysteomfds i ffereeqnutaitaibloe ngsiw,ni stt hh rseeec titohnast
requniorm ea trtihxe oTrhyer. e mainodfte hrce h apctoemrb imnaetsr tihxe oarnyd
linseyasrt eTmhsed. i agonaltiezcahtniiiosuqn su eeid nS ect1i2o.tn6o c ompute
thper incmiaptarslio xl utoifalo inn heoamro genseyosutsoe fdm i ffereeqnutaeiAti,1ao ln s.
Somoen e-varciaalbclucelo uusrc soevsie nrf isneirtiaeen sdo thedrons o tF.o r
tharte asIoh na vien cluidneC dh,a pt1e3ra, d iscusosfTi aoynl or's steh­eorem,
quencaensds, e riTehsim.sa tereixaclef,po Stre cti1o.3n1 o4np owesre rsioelsu tions
tod iffereeqnutaitaicloa nbnse c, o vearetad n yt imTeh.i isst hfief tpha rotft htee xt.
Ind iscusesiignegn vaanldue eisg enveicitts no ercse,s tsoak rnyo ws omething
aboucto mplneuxm beFrostr.h arte asIoh na vper oviadd eids cusosfci oomnp lex
numbeirnsA ppend3i.Ax l sion cludeAdp,p end5i,ix sa prooofft heP icard
in
theorreemg artdhieenx gi staennduc nei queonfaew sisd vea rioeffit rys t-doirfdfeer r­
enteiqaula tions.
Numberiinnt ghb eo oiksfa irsltya ndWairtdh.ei anc she cteixoanm plperso,b ­
lemtsh,e oreamnsde, q uataironenu sm berceodn secutsitvaerlwtyii,tn1 hg.R efer­
encteoa ne xampplreo,b ltehme,o roerme ,q uatoiuotns tihdseel:! ctiinow nhi ciht
appeairsrs e ferebnycc headp ter, ansden cutmiboeTnrh,.u sf,oe rx ampElxea,m ple
xiv
PREFACE
4i nS ect2i.oi3nsc allseidm,p Elxya,m p4l eint haste ctbiuootnu ,t stihdseee ctiiso n
refertroae sEd x amp2l.e3 T.h4e.m ordeif ficpurlotb laermmesa rke)d*o ro cca­
(
sion(a*l*al)ny,td h per oblwehmesrt ehu es oefa c alcuilasad tvoirs aarbmela er ked
Sectiownhsi cahrm eo rdei ffiacnudlc tab ne o mitwtietdh looustos fc ontinauriet y
labe"loepdt ioFnianla.tl"hl eeyn ,d osfp roooffts h eoraermmesa rkweidt ah • ·
Acknowledgments
Ia mg ratteof umlai nnyd iviwdhuoha ellsp ine tdh per eparoaftt hitioesnx Mta.n oyf
three vieowfeC rasl cuTlhuiEsrd,doi ntp,ir oviudseedfcr uilt ictihsaimtm protvheed
mateirniC ahla pt1e-r6s. rFeivviee pwaeirnss tawkionrgkleyd wtahyte hirro uthgeh
entitreexo tft hfier setd itiooftn h ibso oakn dp rovihduendd roefdd se taiilne­d,
sighstufgugle sIta imop nasr.t icgurlaatrteloPfy ru olf eGsesoorrCg aeio nft hGee orgia
InstiotfTu etceh noPlroogfye,As rsCtoo rp elaattn hdUe n iverosfNi etwHy a mpshire,
ProfeCsasrColor w eantP urdUunei verPsriotfye,Cs hsaorrDl eensl iantgM eirl lers­
vilSlteaC toel leagnedP, r ofeKsesioYtrah l aett hUen iverosfMi otnyt ana.
Inp repatrhiinssge coenddi tIiw oansf, o rtutnooa bttea miann syu ggestions
botfho irm provtihlneog g iocradloe frm aterainadfl oc ro rretchtkeii nngod fsp ed­
agogiecrarlot rhsac tr eienpt aon yf iresdti tOinoenm .a jocrh ansg4ueg ,ge sbtyae d
numbeorfr eviewwaesrt soi ncl�u"dm ea teriinCa hla pt3ea rnsd rathtehra n
postpioutnn et Cihla pt1e0r.
Iw isthot hatnhkf eoD ll.oiw nidnigv ifdoutrah leisin rv aluhaeblilpnpe r eparing
thsee coenddi tAilofnr:e Adn dreGwe,o rgIinas tiotfTu etceh nolJoagmeyMs;.
EdmondsSoann,tB aa rbara CoCmomlulneiNgtaeyt; h aniel GUrCoLsAs;m an,
DaniSe.Kl a hnN,o rthweUsntievrenr Ts.Ji .Rt ayn;s fUonridv,e rosfLi eteydJ so;h n
VenablLeask,e woCoodm muniCtoyl leWghei,tB ee aLra kea;n dP auYle arout,
BrighYaomu nUgn iversity.
ProfeLsesooGnre rbwehro,p repatrheSedt udeanntdI' nss trumcatnoura'lss ,
madmea nuys efsuulg gesftoitroh inems p roveomfte hnpetr oblseemtT sh.ib so oiks
ab ettteearc htionobgle caoufsh ei m.
Ia ma lsvoe rgyr atteoft uhWlea dsworth PCuobmlpiasnIhyni,cfn .ogp r e rmis­
siotnou smea terfiraomlmy b ooEkl emeLnniteaAarylr geSbercaEo,dn idt( i19o8ni4 n) ,
Chapt7e,8r, as n 9d, t ot hAed dison-PWuebslliesCyho imnpga nIynf,co .pr e rmission
tou smea terfiraomlmy b oo(kw iWtilhl iRa.mD erriEclke)m,e Dnitffaeryr Eeqnutai­al
tiownisAt php tliiocSnaesc,Eo dnidt( 1i9o8in1nC ) h,a pt1e1r1,s2 a ,n dA ppen5d,ai nxd
tot hSea undPeurbsl isChoimnpga nIyn,cf .op re rmistsoui somena terfiraotlmh eir
booAkn,I tnrotdiutoctonh Heit sooryMf a thecmsFa,i tftEihdo in(t 1i98b3yH) o,w arEdve s,
int hbei ograpshkiectoacfBlh e rnoounlp laig6 e4 9.
FinaIlo lwye,a considedreabtbtolt eh eed itoarnidpa rlo ducsttiaoofnff s
AcadePmriecs wsh,op rovihdeelidpn a g renautm boefrw ayisn t hwer itianngd
producotfti hoibnso ok.
StanIl.Ge ryo ssman
January 1986
Vcetionr Pstl haen e
Witthh icsh apwteeb re gain ne ws ubjientc htse t udyc aolfc ulusst-utodhfye
vectaonrvdse ctfourn cstT.ih oins leaudsi ,nC hap4t,te ort hset uodffy u nctions
will
oft waon mdo rvea riaabs lheasdrp,e parftruoyrmoe u srt uodfoy n e-vacrailacsbul.le u
Them odesrnt uodfvy e ctboergsae ns senwtiitathlh lweyo rokft hger eIarti sh
mathemaStiriW ciilalnRi oawma Hna mil(t1o8n0 5-w1h8o6w 5o)r kweidt whh ahte
calqlueadt ernAifotHneasrm! i ltdoena'tshw ,o rhokinq s u aternwiaossn usp planted
byt hmeo raed abpltweao rokn v ectaonra lbyyst ihAsem erican matahnedm atician
physiJcoissiWtai hl lGairbdb (s1 839-a1n9d0t 3h)ge e netrraela tomfeo nrtd ered
n-tupblyt ehsGe e rmamna themaHtiecrimaanGn r ass(m1a8n0 9-1877).
ThrougHhaomuitl tloinaf'nesdf otrh ree mainodfte hrne i neteenth century,
thewraesc onsiddeerbaabotlveeet rh ues efuolfnq eusast erannivdoe ncst Aotrt sh.e
endo ft hcee ntutrhyge,r eBarti tpihsyhs iLciosrKtde lvwirno tteh aqtu aternions
". ..a lthobuegahu tiifnuglelnyih oauvbsee, e ann u nmixeevdti olt howsheo h ave
toucthheedim na nwya y. .ve.c t.o.rh.sa vnee vbeere onft hsel ighutsteeos a tn y
creature."
BuKte lvwianws r onTgo.d anye arallbylr ancohfce lsa sasnimdco adle prhny sics
arree presuesnittnehgdle a nguoafvg eec tVoercst.ao rraesl suos ewdi tihn creasing
frequiennt chsyeo cainabdli olosgciiecnacQleu sa.t ernitoonohs,a, vr ee cebnetelny
useidnp hysicpsa-ritnti hceloaern ydo thaerre as.
tseteh aec companbyiionggr apshkiectaclh .
1805S-ir RWoiwlHalanim aiml ton
1865
Borinn1 80i5nD ublwihne,r hees pemnots otfh ilsi fe,
William HRaomwialntw oansw ithout question Ire­
land's gmraetahteemsattH iacmiialnt.o n'(sa anft a­ther
torne)ay ndm othdeire wdh ehne w asas mabolyl. H is
unclael ,i ngutiosootkv, e trh beo ye'dsu catiBoyhn i.s
t fiftbhi rthday, cHoaumlridel atEdon ng liHsehb,r ew,
Latiann,dG reeBkyh. i1s3 tbhi rthhdeha aydm astered
nootn ltyh lea nguaogfce osn tinEeunrtoapbleu ,at l so
SanscCrhiitn,e Pseer,s iAarna,b Miacl,a y, Hindi, Ben­
galain,ds eveortahlea rsws e lHla.m illtioknte owd r ite
poetbroyt,ah sa c hialndda sa na dulatn,dh ifsr iends
inclutdheegdr eat EpnogeltSisas mhu Tealy lCoorl e­
ridagnedW illiam Wordswortpho.e twrHayas m ilton's
considseobr aeddh, o wevtehra,iti t s f ortutnhaahttee
develooptehdie nrt erests-eisnmp aetchieamlaltyi cs.
Althohuege hn jomyaetdh emataiy cosu nboagys ,
SiWri llRioawma Hna miltonH amiltionnt'eswr eassgt r eaetnlhya nbceyad c hance
GranCgoelrl ectionm eetianttg h aeg oef 1 5w itZhe raCho lbutrhnAe,m er­
The
icalni ghtcnailncgu lSahtoorrat.fl tye rwHaarmdisl,t on
begatnor eaidm pormtaatnhte mabtoiockoasftl h tei me.
In1 82a3t,t haeg eo f1 8h,e d iscovaenre erdri onrS imoLna plaMceec'sa ncieqluaenesd t e
wrotei mapnr espsaipveoernt hseu bjAe cyte.al ra theeer n tered TriniinDt uyb lCionl.l ege
Hamiltuonni'vse rcsairtewyea rsa stonisAhtti hnaegg .oe f 2 1w,h isltiel anlu nder­
graduahtehe a,ds oi mprestshefeadc ultthya htew asa ppoinRtoeydaA ls tronoomfe r
IrelaanndPd r ofeosfsA osrt ronaottm hyUe n iverSshiotrytt.hl eyr eahfewt reortw,eh aits
nowc onsidaec rleads swiocraokln o ptiUcssi.no gn lmya thematthiecoahrley p ,r edicted
conirceaflr aicntc ieorntt ayipnoe fsc rystLaalttseh.rit sh eowrya sc onfirbmype hdy sicists.
Largbeelcya uosfte h iwso rkH,a milwtaoskn n ighitn1e 8d3 5.
Hamiltfiornsg'trs e pautr emlayt hemaptaipceaarpl p eairne1 d8 3I3n.t hiwso rhke
descrainba eldg ebwraayti ocm anipuplaaitorefrs e anlu mbeTrhsi.ws o rgki vreusl tehsa t
arues etdo dtaoya dds,u btrmaucltt,i apnlddy i,v icdoem plneuxm beArtsf .i rhsotw,e ver,
Hamilwtaosun n abtlode e viasm eu ltiplfioctrrai tpilooernns - tupolfne usm befrosnr > 2.
Fo1r0 y eahresp ondetrhepidrs o blaenmdi,i t ss a itdh ahtes olvieitdn a ni nspirwahtiiloen
walkionntg h Ber ougham iBnDr uibdlgiien1n 8 4T3h.e k eyw ast od isctahrfedam iliar
commutaptriovpeeo rfmt uyl tiplicTahteni eowon b.j ehcect rse atedc awlelqreueda ternions,
whicwhe rteh per ecurosfwo hraswt e n owc avlelc tors.
Fotrh ree sotfh ilsi fHea,m ilstpoenmn ots otf h itsi mdee velotphiaenl gg eobfr a
quaterniHoefn estl.ht at th ewyo ulhda vree volutsiiognnairfiyic nma antchee mapthiycsa­l
icHsi.s monumweonrtaokln t hissu bjTecrte,ao tnQi usaet erwnaisp oubnlsi,si hn1e 8d5 3.
Thereahfetw eorr,k oenda ne nlarwgoerdkE ,l emoefQn uteasrt niAolntsho.Hu agmhi lton
dieidn1 86b5e fohriEes lm eenwtassc ompletthewedo ,r wka sp ublibsyhh eidss o in n1 86.6
Studeonfmt ast hemaatincdps h yskincosHw a militnaov na rioefot tyh ceorn teIxnt s.
mathemaptihcyaslifo cres x,a mpolneee,n counttheHera sm iltofnuinacntw ihoinco,hf ten
represtehnteto ste anle rignay s ystaenmdt hHea milton-dJiafcfoebreieq nutaitaoilfo ns
dynamiIcnms a.t rtihxe otrhyCe,a yley-Hatmhielotrsoetnma ttheaset v emrayt rsiaxt isfies
itosw nc haracteeqruiasttiiocn .
Desptihtgeer ewaotr hke w asd oinHga,m iltfoinny'aesla wresr aet ormetnoht i m.
Hiwsi fwea sas emi-inavnadhl ewi adsp lagubeyad l cohoIlitis ts hme.r egfroartei tfoy ing
poionuttt h at durilnagsy tet ahrteshs,nee e wlfoyr meAdm eriNcaatni oAncaald eomfy
ScienecleescS tieWrdi llRiaomw aHna milttoob nei tfsi rfsotr eaisgsno ciate.
VECTORASN DV ECTOORP ERATION3S
1.1
Int hicsh apwteew ri lelx plporroep eorftv ieecsti ontr hspe l anWeh.e ng oing
throutghhim sa tertiharele ,a dsehro uklede ipnm intdh alti,km eo sitm portant
discovveercitehosar,vbs ee eans ourocfge r ecaotn trovecrosnyt-raot vheawrtas sy
norte soluvnetwdie lli ln ttoht ew entieth century.+
1.V1E CTORSA ND VECTOR OAPTEIRONS
Inm anya pplicoaftmi aotnhse mattoti hcpesh ysiacnadbl i olosgciiecnacalen sd
enginesecriienngat,ri ceso tnsc ewrnietdeh n titthiahetas v beo tmha gnit(uldeen gth)
andd irecEtxiaomnp.l es tihnneco ltiuodonefsf orcvee,l ocaictcye,l eraantdi on,
moment.Iu itms f rqeuenutsleyft uoel x prtehsesqs uea ntgietoimeest rically.
LePta nQd b et wdoi ffeproeinnittnt s hp el anTeh.et nh dei relcitsneeedg ment
froPmt oQ ,d enoPQte,id s t hset raigshetg-mletinhntaee t x tefnrdosPm t oQ (see
Figulra)eN. o tteh atthd ei relcitnseeed g mePQn tasn d ardei ffesrientnchtee y
poiinnot p posdiitree c(tFiiognlusb.r ) e
QP
v
v
Q Q
/ ,/
,
0 (a) x 0 (b) x
FIG1U RE
Thep oiPni tn t hdei rectleindse e gmePQn its c alltehiden iptoiiaonlftt h e
segmeanntdt hpeo iQn itsc aleldt htee irnmpaoli Tnhtet. w oi mporptraonpte rties
ofad irelcitsneeedg meanritet m sa gnit(uldeena gntidht )s diIrtfew codt iiroenc.t ed
linseeg mePQn tasn d havteh sea mmea gnitaunddde i recwteis oanty,h atth ey
areeq uivnaolm eantttwe hre rteh eayr leo cawtietdrh e spteoct thoe r iingT.h e
direlctiensdee gments 2ia nr aeFl eilqg uuirvea lent.
RS
Deifnit1 iGoEnO METRDIECF INITOIFAO VNE CTORT hsee otfa ldli relcitneed
segments etqoau g iivvadeliner netlc itsneeedg meinsct a lalv eedc tAnoyrd i.r ected
lisneeg meinntt h astei tsc alalr eed preseonftt hvaee tcitoonr .
REAMRKT.h ed irelcitsneeedg meinnFt isg uarreae lr le presenotfta htseia omnes
2
vector.
tFoirnt erdeisstciunsgos fti hodeen vse lopomfme ondte vme catnoarl ysis, byM .J C.r owe,
book
AH istoofVre yc Atnoarl (yUsniiosvfN .o trDea mPer eNsost,Dr aem e1,9 6co7)or,Mn soutrlhrKteil si neexc'ellsen tb ook
MathematicfraolmA nTchioteuMongo thd teT rni m(eOsx fUonridPv r.e sNse,wY ork, 32.
1972),
Chapter