NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH303

Transcription

1 NTION OPEN UNIVERSITY OF NIGERI SCHOO OF SCIENCE ND TECHNOOGY COURSE CODE: MTH COURSE TITE: VECTORS ND TENSORS NYSIS

2 MTH COURSE GUIDE COURSE GUIDE MTH VECTORS ND TENSORS NYSIS Course Tem Dr. nole ol (Developer/Edtor) - NOUN Dr. J. O. denrn (Wrter) - FUN NTION OPEN UNIVERSITY OF NIGERI

3 MTH COURSE GUIDE Ntonl Open Unversty of Nger Hedurters 4/6 hmdu ello Wy Vtor Islnd gos u Offe No. 5 Dr es Slm Street Off mnu Kno Cresent Wuse II, u e-ml: entrlnfo@nou.edu.ng UR: Pulshed y: Ntonl Open Unversty of Nger Frst Prnted ISN: ll Rghts Reserved

4 MTH COURSE GUIDE CONTENTS PGE Introduton. Wht you wll ern n Ths Course.. Course m.. Course Oetves Worng through Ths Course.... Presentton Shedule..... ssessment.. Tutor-Mred ssgnment..... Fnl Exmnton nd Grdng.. 4 Course Mrng Sheme Course Overvew 4 Flttors/Tutors nd Tutorls 4 Summry.. 5 v

5 Introduton Vetor nd Tensors (MTH), reures the nowledge ured n (MTH4-Vetor nd Geometry) whh you studed t the -evel. good mstery of the ourse ontent n Mthemtl Methods (MTH8) wll e helpful n lernng ths ourse suessfully. Ths s three-redt unt ourse. It s ompulsory ourse for ll students morng n mthemts t undergrdute level or.s. (Eduton) Mthemts. It s lso vlle to students offerng the helor of Sene (.S.) Computer nd Informton & Communton Tehnology. ny student wth suffent ground n mthemts n lso offer the ourse f he/she so wsh though t my not ount s redt towrds grduton f t s not reured ourse n hs/her feld of study. The ourse s dvded nto two modules s enumerted elow: Module Unt Unt Unt Elementry Vetor lger Vetor Dfferentton (Grdent, Dvergene nd Curl) The ne Integrl Module Unt Unt Green s Theorem, Dvergene Theorem nd Stoes s Theorem Tensor nlyss Wht You Wll ern n Ths Course Ths ourse gude tells you refly wht the ourse s out, wht ourse mterls you wll e usng nd how you n wor wth these mterls. In ddton, t dvotes some generl gudelnes for the mount of tme you re lely to spend on eh unt of the ourse n order to omplete t suessfully. It gves you gudne n respet of the Tutor-Mred ssgnment whh wll e mde vlle n the ssgnment fle. There wll e regulr tutorl lsses tht re relted to the ourse. It s dvsle tht you ttend these tutorl sessons. The ourse wll prepre you for the hllenges you wll meet n Vetors nd Tensors

6 MTH VECTORS ND TENSORS NYSIS Course m The mn m of the ourse s to provde you wth n understndng of Vetors nd Tensors. It lso ms to me ler dstntons etween the wys we hndle prolems n hgher dmensonl spes, nd provde solutons to some prolems tht my rse n engneerng, physs, nd other res of humn endevour, where the nowledge of dvne vetor s reured. Course Oetves To heve the ms set out, the ourse hs set of oetves. Eh unt hs spef oetves whh re nluded t the egnnng of the unt. You should red these oetves efore you study the unt. You my wsh to refer to them durng your study to he on your progress. You should lwys loo t the unt oetves t the ompleton of eh unt. y dong so, you would hve followed the nstrutons n the unt. elow re omprehensve oetves of the ourse s whole. y meetng these oetves, you should hve heved the ms of the ourse s whole. In ddton to the ms ove, ths ourse sets to heve some oetves. Thus, fter gong through the ourse, you should e le to: prove ontnuty nd estlsh lmt funtons n Vetors nd Tensors defne Dvergene, Grdent of Slr Funtons, nd Curl of Vetors estlsh Green s Theorem, Stoes s Theorems nd Dvergene Theorem wor out nswers to smple prolems n Tensor nlyss. Worng through Ths Course To omplete ths ourse you re reured to red eh study unt, red the textoos nd red other mterls whh my e provded y the Ntonl Open Unversty of Nger. Eh unt ontns self-ssessment exerses nd t ertn ponts n the ourse you would e reured to sumt ssgnments for ssessment purposes. t the end of the ourse there s fnl exmnton. The ourse should te you out totl of 6 wees to omplete. elow you wll fnd lsted ll the omponents of the ourse, wht you hve to do nd how you should llote your tme to eh unt n order to omplete the ourse n tme nd suessfully.

7 MTH VECTORS ND TENSORS NYSIS Ths ourse entls tht you spend lot of tme to red nd prte ll relted exerses. I wll dvse tht you vl yourself of the opportuntes of the tutorl lsses provded y the unversty. Presentton Shedule Your ourse mterls hve mportnt dtes for the erly nd tmely ompleton nd sumsson of your TMs nd ttendng tutorls. You should rememer tht you re reured to sumt ll your ssgnments y the stpulted tme nd dte. You should gurd gnst fllng ehnd n your wor. ssessment There re three spets to the ssessment of the ourse. The frst s mde up of self-ssessment exerses, seond onssts of the Tutor-Mred ssgnments nd thrd s the wrtten exmnton/end of ourse exmnton. You re dvsed to do the exerses. In tlng the ssgnments, you re expeted to pply nformton, nowledge nd tehnue you gthered durng the ourse. The ssgnments must e sumtted to your flttor for forml ssessment n ordne wth the dedlnes stted n the presentton shedule nd the ssgnment fle. The wor you sumt to your tutor for ssessment wll ount for % of your totl ourse wor. t the end of the ourse you wll need to st for fnl or end of ourse exmnton of out three hour durton. Ths exmnton wll ount for 7% of your totl ourse mr. Tutor-Mred ssgnment The TM s ontnuous ssessment omponent of your ourse. It ounts for % of the totl sore. You wll e gven 4 TMs to nswer. Three of these must e nswered efore you re llowed to st for the end of ourse exmnton. The TMs would e gven to you y your flttor nd returned fter you hve done the ssgnment. ssgnment uestons for the unts n ths ourse re ontned n the ssgnment fle. You wll e le to omplete your ssgnment from the nformton nd mterl ontned n your redng, referenes nd study unts. However, t s desrle n ll Degree level of eduton to demonstrte tht you hve red nd reserhed more nto your referenes, whh wll gve you wder vew pont nd my provde you wth deeper understndng of the suet. Me sure tht eh ssgnment rehes your flttor on or efore the dedlne gven n the presentton shedule nd ssgnment fle. If for

8 MTH VECTORS ND TENSORS NYSIS ny reson you nnot omplete your wor on tme, ontt your flttor efore the ssgnment s due to dsuss the posslty of n extenson. Extenson wll not e grnted fter the due dte. Fnl Exmnton nd Grdng The end of ourse exmnton for MTH (Vetors nd Tensors) wll e for out hours nd t hs vlue of 7% of the totl ourse wor. The exmnton wll onsst of uestons, whh wll reflet the type of self-testng, prte exerse nd tutor-mred ssgnment prolems you hve prevously enountered. ll res of the ourse wll e ssessed. Use the tme etween fnshng the lst unt nd sttng for the exmnton to revse the whole ourse. You mght fnd t useful to revew your self-test, TMs nd omments on them efore the exmnton. The end of ourse exmnton overs nformton from ll prts of the ourse. Course Mrng Sheme ssgnment ssgnments -4 End of ourse exmnton Totl Mrs Four ssgnments, est three mrs of the four ount t % eh -% of ourse mrs. 7% of overll ourse mrs. % of ourse mterls. Flttors/Tutors nd Tutorls There re 6 hours of tutorls provded n support of ths ourse. You wll e notfed of the dtes, tmes nd loton of these tutorls s well s the nme nd phone numer of your flttor, s soon s you re lloted tutorl group. Your flttor wll mr nd omment on your ssgnments, eep lose wth on your progress nd ny dffultes you mght fe nd provde ssstne to you durng the ourse. You re expeted to send your Tutor-Mred ssgnment to your flttor efore the shedule dte (t lest two worng dys re reured). They wll e mred y your tutor nd returned to you s soon s possle. Do not dely to ontt your flttor y telephone or e-ml f you need ssstne. v

9 MTH VECTORS ND TENSORS NYSIS The followng mght e rumstnes n whh you would fnd ssstne neessry, hene you would hve to ontt your flttor f: you do not understnd ny prt of the study or the ssgned redngs you hve dffulty wth the self-tests you hve ueston or prolem wth n ssgnment or wth the grdng of n ssgnment. You should endevour to ttend the tutorls. Ths s the only hne to hve fe to fe ontt wth your ourse flttor nd to s uestons whh re nswered nstntly. You n rse ny prolem enountered n the ourse of your study. To gn muh eneft from ourse tutorls, prepre ueston lst efore ttendng them. You wll lern lot from prtptng tvely n dsussons. Summry MTH (Vetors nd Tensors) s ourse tht ntends to provde solutons to prolems normlly enountered y engneers, physsts nd mthemtns n the ourse of dong ther norml os. It lso serves s tool whh often enles the mthemtns to wden the fronters of ther nlytl, onerns to ssues tht hve sgnfnt mthemtl mpltons. Nevertheless, do not forget to pply the prnples you hve lernt to your understndng of Vetors nd Tensors. I wsh you suess n the ourse nd I hope tht you wll fnd t omprehensve nd nterestng. v

10 MTH VECTORS ND TENSORS NYSIS Course Code Course Ttle MTH Vetors nd Tensors nlyss Course Tem Dr. nole ol (Developer/Edtor) - NOUN Dr. J. O. denrn (Wrter) - FUN NTION OPEN UNIVERSITY OF NIGERI v

11 MTH VECTORS ND TENSORS NYSIS Ntonl Open Unversty of Nger Hedurters 4/6 hmdu ello Wy Vtor Islnd gos u Offe No. 5 Dr es Slm Street Off mnu Kno Cresent Wuse II, u e-ml: entrlnfo@nou.edu.ng UR: Pulshed y: Ntonl Open Unversty of Nger Frst Prnted ISN: ll Rghts Reserved v

12 MTH VECTORS ND TENSORS NYSIS CONTENTS PGE Module... Unt Elementry Vetor lger... Unt Vetor Dfferentton (Grdent, Dvergene nd Curl) Unt The ne Integrl.. 4 Module. Unt Green s Theorem, Dvergene Theorem nd Stoes s Theorem.. Unt Tensor nlyss 9 v

13 MTH VECTORS ND TENSORS NYSIS MODUE Unt Unt Unt Elementry Vetor lger Vetor Dfferentton (Grdent, Dvergene nd Curl) The ne Integrl UNIT EEMENTRY VECTOR GER CONTENTS. Introduton. Oetves. Mn Content. Revson of Elementry Vetor lger.. Slr Produt of Two Vetors.. w of Slr Produt..4 Trple Produts 4. Conluson 5. Summry 6. Tutor-Mred ssgnment 7. Referene/Further Redng. INTRODUCTION Ths ourse s n ntrodutory wor to Vetors nd Tensors. In ths unt, n ttempt wll e mde to revse some elements of vetor lger. You should note tht vetor untty s dstngushed from slr untty y the ft tht slr untty possesses only mgntude, wheres vetor untty possesses oth mgntude nd dretons. It s onvenent to represent vetor geometrlly s n rrow, pontng n the dreton ssoted wth the ones hvng length proportonl to the ssoted mgntude. These nd other propertes of vetor wll e explned n ths unt. You should red ths unt refully efore proeedng to other unt.. OJECTIVES t the end ths unt, you should e le to: desre vetors nd slr unttes lst slr produt of two vetors stte the w of slr produt expln vetor produts solve prolems on vetor lger.

14 MTH VECTORS ND TENSORS NYSIS. MIN CONTENT. Revson of Elementry Vetor lger Defnton: slr untty hs only mgntude whle vetor untty hs oth mgntude nd dreton. ngle etween two vetors. et nd e two vetors suh tht nd n e represented n the dgrm elow Then the ngle etween the two vetor n suh tht.. Slr Produt of Two Vetors et nd e ny two vetors (of the sme dmenson). The slr produton of nd denoted y s defned y Cos where nd re the mgntude of vetors nd respetvely nd s the ngle etween nd,, nd,, then The ngle etween nd s gven y Cos Provded nd re non-zero vetors. If,, re the usul unt vetors long the x, y, z xs respetvely then,,,, Where nd

15 MTH VECTORS ND TENSORS NYSIS.. ws of Slr Produts () () slr) ny m m m m m () f then nd re perpendulr... Vetor Produt et & e ny two vetors wth the sme dmenson nd let n e unt vetor perpendulr to oth &, the vetor produt of & wrtten s s defned y Sn n, n Where s the ngle etween &. If Then = =..4 Trple Produts The slr nd vetor produts of vetors, nd my hve menngful produts, nd Hene, the followng lws re vld. () () () (v) (v)

16 MTH VECTORS ND TENSORS NYSIS Remr. The produt s sometmes lled the slr trple produt or ox produt nd my e denoted y.. The produt s lled the vetor trple produt.. Reder re dvsed to provde proofs to to v ove y ssumng tht Remr () The untty s vetor untty () If then t lest one of or nd then y mplton the ngle etween them s zero. () If nd re lw then (v) where s ny slr. (v) The mgntude of nd s the re of the prllelogrm wth sdes nd (v) If nd nether nor then nd re prllel, 4. CONCUSION We hve lernt out vetor lger n ths unt. The mterls n ths unt re suffent enough ground to enle you understnd the next unt. 5. SUMMRY The followng fts re to e rememered: tht vetor hs mgntude nd dreton, unle slr untty whh hs only mgntude tht two vetors n e multpled n two wys. slr produts whh result n slr untty vetor produt whh result n vetor untty f nd, then s prllel to tht the mgntude of vetors whose produt s s the re of prllelogrm wth sdes nd. 4

17 MTH VECTORS ND TENSORS NYSIS 6. TUTOR-MRKED SSIGNMENT. Fnd the length nd dreton oms of theorem from the (, -, ) to the mdpont of the lne segment from orgn to the pont (6, -6, 4). Prove tht nd. If denotes the ngle etween the vetors nd use theorem n elementry geometry tht Cos 7. REFERENCE/FURTHER REDING Frns,. Hldernd (nd). dvned Clulus for pplton (nd ed.). 5

18 MTH VECTORS ND TENSORS NYSIS UNIT VECTOR DIFFERENTITION (GRDIENT, DIVERGENCE ND CUR) CONTENTS. Introduton. Oetves. Mn Content. Dfferentton of Vetor. Grdent, Dvergene nd Curl.. Dretonl Dervtves.. Dvergene of Vetor.. The Curl of Vetor Funton. Dfferentl Dervtves 4. Conluson 5. Summry 6. Tutor-Mred ssgnment 7. Referene/Further Redng. INTRODUCTION In unt, we lernt out vetor lger nd estlshed some propertes of vetors. In ths unt, we shll onsder vetor dfferentton nd derve some mportnt formul nd propertes of vetor dfferentton.. OJECTIVES t the end ths unt, you should e le to: dfferentte vetor unttes fnd the grdent of ny slr nd vetor feld dentfy the url nd dvergene of vetor feld.. MIN CONTENT Defnton The vetor funton gven u u u s sd to e ontnuous t pont u f ( u) ( u whenever, we n fnd some lm u u Ths s euvlent to u u 6

19 MTH VECTORS ND TENSORS NYSIS Defnton The dervtve of the vetor funton u lm t u u u d d d u u s gven s d provded tht. du du du du d d Hgher dervtves suh s, d u d u Remr e.t.. re smlrly defned If, nd C re dfferentle vetor funtons of slr u nd s dfferentle slr funtons then d du () d du () d du d du d du d du d du d du d du () d du (v) d du d d du du d du d du (v) C C C d du d du d du d du = C C d du d du d du = C C d du d du (v) C C C d du d du = C d du 7

20 MTH VECTORS ND TENSORS NYSIS. Dfferentton of Vetor. then d d d d () d d d () d d d (4) f n, y, z then d dn dy dz n y z Exmples Gven vetor - Sn t + Cos t + t dq d Q dq d Q Otn... d. dt dt dt dt () dq dt d dt d d (Sn) + (Cos t) + t dt dt () d Q dt d dt dq dt d dt (Cos t Sn t + ) = - Sn t Cos t () dq Cos t Sn t Sn t Cos t dt =.. Grdent, Dvergene nd Curl of Vetors Consder the vetor opertors lled del or nl defned y n y z If ( n, y, z hs ontnuous frst prtl dervtves n prtulr regon, we defne the grdent s: Grd n y z 8

21 MTH VECTORS ND TENSORS NYSIS = n y z Exmples () If n, y, z nyz fnd n y z nyz nyz n y z = nz nz nyz = nyz () f n, y, z n y y z fnd n y y z n y y z n y y z n y z = 6ny n y z y z () If n, y, z s slr funton P, we now tht Grd n y z nd dr dn dy dz Grd dr dn dy dz n y z = d Ths mples d grd dr dr... Dretonl Dervtves We rell d d r, dr ds dr d r dr hene ds d r ds d ds d Then ds d r ds grd grd s the unt vetor n the dreton of d r, f dr ds 9

22 MTH VECTORS ND TENSORS NYSIS NOTE: ds d s the propgton of grd on the unt vetor nd the dretonl dervtve of n the dreton of Remr () The unt Norml vetor N = () The grd of sum nd produts of slrs. z y n = z y n z y n = z y n = z z y y n n = z y n z y n = z y n z y n =.. Dvergene of Vetor The dvergene of vetor funton z y n,, s defned y dr = z y n = z y n

23 MTH VECTORS ND TENSORS NYSIS Exmples. Otn the dvergene of the vetor funton nz yz ny t the pont (, -, ). If n yz slr funton nd nz yz ny Fnd t the pont (, -, ). Determne the ontent so tht the vetor v n y y n n z s solenod Soluton. nz yz ny n y z nz yz ny n y z = z + z = z =. n yz n y z n y z = 6n yz n yz n y vetor v s solenod of ts dvergene zero V mples tht + + = = -.. The Curl of Vetor Funton The url of vetor funton defne denoted y url Exmples n y z =,, n y z y z z n n y. Fnd the url of f n y y z n z. If n y nd = n z y z n y Otn url

24 MTH VECTORS ND TENSORS NYSIS Soluton. Curl = yz, z, n n n y y yz z nz = yz z n. n yz n y z n y z n y n yz n y z z n y z = ny z n z, 4n yz n y z,ny z n z 4. CONCUSION Ths seton s very mportnt for the understndng of the remnng unts. You re to mster ths unt very well. 5. SUMMRY We hve estlshed formul to the followng: vetor dfferentton the grd of slr feld the url of vetor funton we show tht the unt norml vetor N = 6. TUTOR-MRKED SSIGNMENT. prtle moves long the urve n t, y t 4t, z t 5 where t s the tme. Fnd the omponent of ts veloty nd elerton t tme t =. If 5t t t nd Sn t Cos t

25 MTH VECTORS ND TENSORS NYSIS d dt d dt d dt Fnd () () (). If n, y, z ny z nd nz ny yz Fnd 4. et r x, y, z t pont (, -, ) nz e vetor. m Prove tht r m r 5. Fnd the dvergene of the vetor y xyz, xy x z,6z x yz 6. If x 8xy z,x y ny,x z m, show tht s not solenodl. 7. r Show tht the vetor r s rottonl where r = x + y + z 8. If x y y z z x show tht. 9. Gven tht E nd H re two vetors whh re ssumed to hve ontnuous prtl dervtves wth respet to poston nd tme. Furthermore, Suppose E, H H E E ; H t t Prove tht E & H stsfy the euton dt Use the ove reltons to show tht E H E H t 7. REFERENCE/FURTHER REDING Frns. Hldernd (nd). dvned Clulus for pplton (nd ed.).

26 MTH VECTORS ND TENSORS NYSIS UNIT THE INE INTEGR CONTENTS. Introduton. Oetves. Mn Content. The ne Integrl.. The Surfe Integrl.. The Volume Integrl 4. Conluson 5. Summry 6. Tutor-Mred ssgnment 7. Referenes/Further Redng. INTRODUCTION In lne ntegrl unle other ntegrls, we hve to onsder two or more funtons t tme, for ntegrton purposes. Suppose these two funtons re M x, ynd N x, y suh tht they re sngle vlued nd ontnuous t every pont of urve. Dvde the urve nto prts y mens of P x, y,,... y (,d) P n- (x n-, y n- ) P P (x,y ) (x,y,) (,) O x et x x x, y y y where x, xn, yn d et, n e defned y x x y y We form the produt nd then dd then to get n M, x N, y 4

27 MTH VECTORS ND TENSORS NYSIS mt of ths sum nd ll x, y smultneously s defned s lne ntegrl long the urve of two funtons M nd N smultneously. Thus, we wrte mt x y Curve n M, x N, = x, ydx Nx y M, y dy. OJECTIVES t the end ths unt, you should e le to: evlute the lne ntegrl of vetor funtons dentfy prolems reltng to lne ntegrls.. MIN CONTENT. The ne Integrl et x, y, z e vetor funton of poston defned nd ontnuous long urve C. The ntegrl of the tngentl omponent of long C s wrtten s: d r dn dy dz Where dr dx dy d z. For nstne n erodynm nd flud mehns ths lne ntegrl d r s lled the rulton of out Where represents the veloty of r or the veloty of the flud s the se my e. 5

28 MTH VECTORS ND TENSORS NYSIS ne Integrl out loses plne N C M Evlute the ntegrl I = xydy x dx of trngle motor vertes,,,, &, SEF-SSESSMENT EXERCISE where C mde up of the sdes ns =. Evlute the lne ntegrl I ydx from to where C s the rle x y. Fnd the wor done n movng prtle one round rle C n:. the xy-plne f the rle C hs entre t the orgn nd rdus. If the fore feld F x y z x y z x y 4z s s. gven.. If F x y y x nd C s the losed urve n the xyplne, n= ost y = snt, t= to t=. Evlute Fdr. 4. Fnd the totl wor done f prtle s moved n fore feld y F xy y long the urve y x n the xy -plne from (, ) to (, ). 5. If 5 x 4y 4yz xz. Evlute d r from (,, ) to (,, ) long the followng pths. ) t, y t, z t ) The strght lne from (,, ) to (,, ) then from (,, ) to (,, ) nd then to (,, ) 6. Clulte v dr where V= x nd s gven y 7. The strght lne onng (, ) to (, ) 8. The r of rle wth entre t theorgn nd rdus unts. 6

29 MTH VECTORS ND TENSORS NYSIS.. The Surfe Integrl X N P SSS Y R Z et represent n element of R nd S the orrespondng of re of S t the pont P ( x, y, z) on S. et lso x, y, z e funton of poston on S nd let Y denote the ngle etween two outwrd norml PN to the surfe P nd the postve Z xs. S Cos Y S nd Se Y x, y, zs s the totl vlue of x, y, z n ten over the surfe. Ths sum eomes the ntegrl x, y, zds x, y z I, Se y = R z n zy Ths mples tht Se y dxdy I s ds zn s R z y dxdy 7

30 MTH VECTORS ND TENSORS NYSIS et S e sded surfe nd let one sde e onsdered rtrry s the postve sde ssoted wth the dfferentl of the surfe re ds. vetor d s whose mgntude s ds nd dreton s tht of n. then ds n ds where n s the lmt vetor norml to ny pont of the postve sde of S. the ntegrl d s n ds s n exmple of surfe lled the flux of over C S S. others re:. ds.. nds d s we hve d ds Se Y. where f z f n, y mplyng tht z f n, y n f f z n, y n y z f n, y f Ths mples tht z f n, y f f n ds n y y If the ngle Y etween the z-xs nd n s ute the postve sgn s dopt nd f the ngle s otuse we dopt the negtve sgn. Hene d ndy n ds n dependng on the proeton n The proeton 8

31 MTH VECTORS ND TENSORS NYSIS Exmple n ds where 8 y nd S s tht prt of the plne, n y 6z whh s n the ny plne ( ln udrte) dndy n ds. n n n y 6z n 6 6 n 7 n n,, , 7, 6 7,, 7 8z,,y,, 6 6z 6 8y = 7 7 n y Now n y 6z gves z 6 6 n y 6 8y n 7 7 n 8y 6 8y 6 n = 7 7 Ths mples tht 7 7 nds = 6 ndndy dndy n n 6 n 7ndy 7 6 To otn the lmt of ntegrton of n we set y = z =, nd for y we set z = Hene we hve n 6 6 n y dndy 4 9

32 MTH VECTORS ND TENSORS NYSIS SEF-SSESSMENT EXERCISE. Evlute the lne ntegrl I ydx from to where C s the rle x y. Fnd the wor done n movng prtle one round rle C n:. the xy-plne f the rle C hs entre t the orgn nd rdus. If the fore feld F x y z x y z x y 4z s s. gven.. If F x y y x nd C s the losed urve n the xyplne, n= ost y = snt, t= to t=. Evlute Fdr. 4. Fnd the totl wor done f prtle s moved n fore feld y F xy y long the urve y x n the xy -plne from (, ) to (, ). 5. The strght lne onng (, ) to (, ). The r of rle wth en tre t the orgn nd rdus unts.. The Volume Integrl The volume or spe ntegrl s gven y dr or dr for Closed surfe Exmple If F n z yz. Evlute F. ds = s S F dr where V s the volume enlosed y the ue gven y n, y nd z. Soluton F ds s Where F dr dr dndydz nd F n y n y F dr dndydz

33 MTH VECTORS ND TENSORS NYSIS SEF- SSESSMENT EXERCISE. Evlute n ds where z n y z nd S s the surfe of the ylnder n y 6 nluded n the st udrnt etween z = nd z = 5. Evlute nds d n the st udr nt etween z = nd z = 5. F. If F y n nz ny. Evlute n ds surfe of the sphere 4. CONCUSION n y z from the ny. where S s the plne. The mterls n ths unt re very mportnt for the understndng of suseuent unts. You must understnd ths wor thoroughly efore movng to the next seton. 5. SUMMRY In ths unt, we estlshed the reltonshp etween lne ntegrl of slr feld nd vetor funtons. Thus, we hve used the lne ntegrl n fndng Surfe ntegrl Volume ntegrl or spe ntegrl whh s gven s: dr or dr, for losed surfe. 6. TUTOR-MRKED SSIGNMENT. et F nx n y. Evlute F dv where V s the regon ounded y the surfe n, y, z n, z 4, x, y 6. If F ny yz zn. Evlute F ds over the sphere gven y n y z where n z nz y y z nd S s the surfe of the sphere hvng entre t (, -, ) nd rdus. F n z, 4ny, 4n. Evlute F dr where r s the. Evlute ds 4. If losed regon ounded y the plne n, y, z nd n y z 4

34 MTH VECTORS ND TENSORS NYSIS 7. REFERENCES/FURTHER REDING Hldernd, F.. (nd). dvned Clulus for ppltons. Verm, P.D.S. (nd). Engneerng Mthemts.

35 MTH VECTORS ND TENSORS NYSIS MODUE Unt Unt Green s Theorem, Dvergene Theorem nd Stoes s Theorem Tensor nlyss UNIT GREEN S THEOREM, DIVERGENCE THEOREM ND STOKE S THEOREM CONTENTS. Introduton. Oetves. Mn Content. Green s Theorem. Dvergene Theorem. Stoe s Theorem 4. Conluson 5. Summry 6. Tutor-Mred ssgnment 7. Referenes/Further Redng. INTRODUCTION In ths unt, we wll dsuss some mthemtl theorems suh s Green s theorem, dvergene theorem nd stoe s theorem. These theorems re useful n hndlng eutons of mthemtl physs, prtulrly n the re of veloty of flud n three dmensons. We now on pplton of these theorems prtulrly the dvergene theorem tht the veloty of n nompressle flud hs zero dvergene. Some other ppltons wll e onsdered n ths seton. The referenes t the of ths unt throw more lght on the ppltons of these theorems; you my wsh to ontt them.

36 MTH VECTORS ND TENSORS NYSIS. OJECTIVES t the end ths unt, you should e le to: pply the Green s theorem pply dvergene theorem nd stoe s theorem to solvng prolems rsng from mthemtl physs.. MIN CONTENT. Green s Theorem If R s losed regon on the ny -plne ounded y smple losed urve C nd f M nd N re sngle vlued funtons whh re ontnuous n n nd y hvng ontnuous dervtves n R, then N M Mdn Ndy dndy n y where the urve C s trversed ounter R lowse Exmple Verfy the Green s theorem n the plne for ny y dn n dy s lose urve of the regon ounded y Soluton The pont of nterseton of M ny y nd N x M N x y nd n y x y n nd ny y dn n dy n x y.h S I s ny y d n n dy R long dndy n n n dn n n long y n, dy dn I n n n dn n dn y n nd y n where C y n s (, ) nd (, ), y n, dy ndx I 9 I 4

37 MTH VECTORS ND TENSORS NYSIS Now R.H.S. n n ydndy n y n R 4 = ny y dn n n n dydn dn SEF- SSESSMENT EXERCISE Evlute hvng the Green s theorem the ntegrl where C s ounded y (, ), ),, (, ( y Sn n) dn Cos ndy. Dvergene Theorem If V s the volume ounded y losed surfe nd s vetor funton of poston wth ontnuous dervtves then v dr n ds or dv Fdv Fdv F d s F ds Exmple s v v s Verfy the dvergene theorem for 4x y z ten over the regon ounded y n y 4, z nd z.h. S = dv 4n y z 4 4y zdr 4n v n y 4 4y zdzdydx 4 4n R.H.S. The surfe S of the ylnder onssts of se S ( z ) nd the top S z nd the onvex poston S n y 4. Then the surfe s gven y z R n s n ds On z s nds s n ds s n ds S n, = 4x y n On z S n = 4, 4x y 9 so tht 5

38 MTH VECTORS ND TENSORS NYSIS S nds On S x y 4 x y 4 9ds 9 ds x y x y n x y n 4x y z = x y et x Cos, y Sn ds ddz Ths mples tht n ds S 8 Cos 8 Sn dzd 6 Cos Sn dzd 48 Cos Sn d 48 Cos Sn. dzd Hene the totl surfe S SEF -SSESSMENT EXERCISE. Evlute F n ds where F 4 xz y yz nd S s the S surfe of the ue ounded y n, x, y z, z. Evlute r n ds where S s losed surfe.. Stoe s Theorem If S s n open two-sded surfe ounded y losed non-ntersetng urve C (Smple losed urve) then f hs ontnuous dervtves. d r s n ds d s Where C s trversed ounter-lo wsely Exmple: Use stoe s theorem to determne n ds s where y z yz 4 xz 6

39 MTH VECTORS ND TENSORS NYSIS nd S s the surfe of the losed ue n, y, z, x, y, z the xy plne. Soluton: y stoe s theorem I = n ds dr s 4 = y z dx yz dy xzdz Sne t s ove the I y dx 4dy xy plne then z y pplyng the Green s theorem M N M y, N 4 y x I dydx dxdy 4. SEF- SSESSMENT EXERCISE Use the stoe s theorem to determne z z x ydx y dy y dz where S s the upper hlf surfe of the s sphere x y z s the xy plne. 4. CONCUSION The mterls n ths unt should e well understood efore movng to tensor nlyss whh mde use of ll the mterls we hve developed n the preedng hpters. 5. SUMMRY We hve estlshed the followng formul: N M Mdx Ndy dxdy x y R dr n ds. S dr n ds d s C. 7

40 MTH VECTORS ND TENSORS NYSIS Study the mterl presented n the preedng setons euse the whole des wll e used n the suseuent setons on vetor nd tensors. 6. TUTOR-MRKED SSIGNMENT. Evlute y usng Green s theorem x y dx x y dy Where C s losed urve formed y. C y x nd x y etween (, ) nd (, ). Use stoes theorem to trnsform ydx xdz zdy to surfe ntegrl. 4. Prove tht Grd grd grd 5. Prove tht dv dvu u. grd 7. REFERENCES/FURTHER REDING Hldernd, F.. (nd). dvned Clulus for pplton. Verm, P.D.S. (nd). Engneerng Mthemts. 8

41 MTH VECTORS ND TENSORS NYSIS UNIT TENSOR NYSIS CONTENTS. Introduton. Oetves 4. Mn Content. Trnsformton of Co-oordntes. Crtesn Tensor. Summton Convetons.4 Contrvrnt nd Covrnt Vetor.5 Contrvrnt, Covrnt nd Mxed Vetor.6 Fundmentl Opertons wth Tensor.7 Outer Multplton.8 Multplton of Tensor.9 Doule Produt of Tensors 4. Conluson 5. Summry 6. Tutor-Mred ssgnment 7. Referenes/Further Redng. INTRODUCTION Ths seton ntrodues you to Tensor nlyss. The wor wll me use of the des developed n the prevous unts.. OJECTIVES t the end ths unt, you should e le to: defne tensors perform vrous opertons on tensors solve orretly ny exerse on tensor nlyss.. MIN CONTENT pont n the N-dmensonl spe s set of N numers denoted y N X, X,..., X where,,.,n re ten not s exponents ut s supersrpts. The ft tht we nnot vsulse ponts n spes of dmenson hgher thn hs of ourse nothng to do wth the exstene of suh spe. 9

42 MTH VECTORS ND TENSORS NYSIS. Trnsformton of Co-oordntes n et X, X, X,..., X nd X, X, X,... X e o-ordntes of pont n two dfferent frmes of referenes. Suppose there exst ndependent reltons etween the o-ordnte of the two systems hvng the form X X N N X X X N X X X, X, X, X,..., X,..., X,... X N N N () R R N Whh n e ndted refly s X X X, X,... X n where K =,, N. It s ssumed tht the funtons nvolved re sngled vlued, ontnuous nd hve ontnuous dervtves. The N onverse to eh set of oordntes X, X,..., X the wll orrespond N K N unue set X, X,..., X gven y X X X, X, X -_ (v) The relton gven y euton () nd () defnes trnsformton of o-ordnte from one frme of referene to nother frme of referene.. Crtesn Tensor Defnton: When the trnsformtons re from one retngulr oordnte system to nother the tensors re lled Crtesn tensors.. Summton Conveton Note tht N x N N x x... N x ut shorter notton s smply to wrte s x where the onventon tht whenever the ndex (super or susrpt) s repeted n gven term we re to sum from the ndex from to N unless otherwse spefed. Ths s lled the summton onventon nsted of movng the ndex we ould us p nother letter sy P nd the sum ould e wrtten s. ny ndex whh s repeted n gven term so tht the summton onveton ppled s lled dummy ndex. s ndex ourrng only one s lled free ndex nd n stnd for ny of the numer,,,n suh s s euton () nd () p X

43 MTH VECTORS ND TENSORS NYSIS Exmples. If X, X,..., X then the dfferentl of dx dx... dx N x X X d N dx X dx X N. d X t X t X X X X M X t X t X X M X t X... X N N t N. If S gx g X g X S g X.4 Contrvrnt nd Covrnt Vetors If N unttes,,... relted to N other unttes,,..., system X, X,..., X N N s n o-ordnte system X, X,..., X. y the trnsformton eutons N N re. In nother o-ordnte P N p P X X They re lled omponent of Contrvrnt vetor or Contrvrnt tensor of the frst rn or frst order. If N unttes,..., relted to N, other unttes system X, X,..., X N, N n o-ordnte system X, X,..., X re,,..., N n nother o-ordnte N. y the trnsformton eultes N x p, P,,..., N p x x y onveton p they re lled omponent of ovrnt p x vetor or tensor of the frst rn or frst order. Note: Tht supersrpt s used to ndte ontrvrnt omponent nd susrpt s used to ndte ovrnt omponent.

44 MTH VECTORS ND TENSORS NYSIS Exmples on Contrvrnt nd Covrnt Tensor. Wrte the lw of trnsformton for: x x x x x x. Illustrte: s n d for rememerng the trnsformton, note tht the reltve poston of ndes p r On the left sde of the trnsformton re the sme s those on the rght hnd sde of the euton. Some these ndexes re ssoted wth x o-ordnte nd sne ndes,, re esly wrtten s t ws done () () mn Consder m C Consder p rst p p x C m x p x x x x x,,,, m n r s x x x x x C.5 Contrvrnt, Covrnt nd Mxed Tensor If to N unttes N other unttes y trnsformton eutons m s N n o-ordnte system X, X,..., X re relted pr N n nother o-ordnte system X, X,..., X p r N N pr x x s r N s x x p,,,.., p r pr x x s t s lled ontrvrnt omponent of tensor of s x x the seond rn. The N unttes of tensor of the seond rn s p pr t mn s re lled ovrnt omponent s x x p r s e.g. of mxed tensor x x Slr or nvrnt: slr or nvrnt s lled tensor of rn zero. Symmetr nd Sew Symmetr Tensor tensor s lled symmetr w.r.t. of ndes f ts omponent remns unltered upon. Interhnge of the ndes e.g. f = the tensor s symmetr n p nd m. tensor s lled sew symmetr wth.respet.to. ontrvrnt or ovrnt ndes of ts omponent hnge sng upon nterhnge of the mpr pmr n m& p ndes e.g. s mpt s pmr s

45 MTH VECTORS ND TENSORS NYSIS Tensor of rn greter thn st l re omponent of mxed tensor of rn 5. (Contrvrnt of order 5 nd ovrnt of order ). The roneer Delt Ths wrtten s defned s rn. Exmples. Evlute () p Soluton If s mxed tensor of nd If r () p s r Sne p p r s f p & r s f p p x. Show tht x Soluton p If p =, then If p = then x x p p x nd = sne p X X x p x re ndependent then x x x p p.6 Fundmentl Operton wth Tensor ddton nd Sutrton: The sum of or more tensor of the sme rn nd type.e. sme numer of Contrvrnt ndes nd sme numer of ovrnt ndes s lso tensor of the sme rn nd type thus f mp nd re the tensor then mp C mp of vetor s Enumertve nd ssotve. The dfferene of tensors of the sme rn nd type s lso tensor of the sme rn nd type..e. mp mp mp D mp mp

46 MTH VECTORS ND TENSORS NYSIS Exmple p If r nd tensor. p r re tensor. Prove tht ther sum nd dfferene re Soluton x x x,, p x x x r p r r x x x p.. p r x x x x x x.. p x x x nd x x x.. p x x x r r p r p r p r p r.7 Outer Multplton The produt of tensors s tensor whose rn s of the gven tensor. Ths produt whh nvolves ordnry multplton of the omponent of the tensor s lled the outer produt e.g. pr C s the outer produt of m s prm s pr nd m s Contrton: If one ontrvrnt nd one ovrnt ndex of tensor re set eul, the result ndtes tht, summton over the eul. Indes re to e tng ordng to the summton onveton. Ths resultng sum s tensor of rn lesser thn the orgnl tensor. The proess s lled ontrton. SEF- SSESSMENT EXERCISE If tensor of rn 5 Tensor of Rn mpr s.e. set r = s to otn mpr s mp Exerse y settng the ove mp where p = D mp m 4

47 MTH VECTORS ND TENSORS NYSIS Internl Multplton y the proess of outer multplton of tensor followed y ontrton, we otn new tensor lled n nner produt of gven tensor, the proess s lled nner multplton. e.g. f nd mp r mp nd r st the outer produt mp r st, lettng = r, we otn r st nd ths s the nner produt, moreover, f p = s then nother nner produt s otn mp nd r pt Tensor form of grdent, dvergent nd url () Grdent: If s slr of nvrnt, the grdent of s defned y = grd, P where, P s the ovrnt p x p dervtve of w. r. t x p () Dvergent: The dvergent of s the ontrton of ts ovrnt dervtve w. r. t x p e the ontrton of Mthemtlly, p p p Dr g gx X () Curl: The url of p s p,, p X p X p (v) Tensor of rn, the url s lso defne s pr p, pln: The pln of s the dvergene of grd nd,. g X x = dr P g g In se g, g must e reple y g oth ses g & g n e nluded the wrtten g n ple g. lterntng Tensor,, Ths s defne y,, f f f,, re n yle,, re nt yle,, on. for 5

48 MTH VECTORS ND TENSORS NYSIS. sum O.K where s the ngle etween & r, Sum K K Usng lterntng tensor K = ( + + )K = K Notton et us denote the unt tensor y, s mtrx. = (,, ) = I unt vetor long rms = (,, ) = unt vetor long y rms = (,, ) = unt vetor long z rms = = unt x mtrx Use lterntng tensor to estlsh the result of vetors. y vetor produt u v where u v re u v u v u v u v v u v u v u v u v u v u y lterntng tensor u v U U V U V U V V 6

49 MTH VECTORS ND TENSORS NYSIS U V U V UV U V UV U V U V U V U V U V U V U V U V U V U V U V UV U V U V U V UV U V UV U V U V K U V UV K U V UV U V U V U V U V U V U V U V Defned proved. The vetor U U, U, U, (,,), (,,), (,,) e.t.. s lled frst order tensor. et e seond order tensor & I x.8 Multplton of Tensors We hve vrous multpltons Dynod produt et U U, U, U, V V V V e vetors.e. frst order tensor, Defnton: we defned the dynod produts of u & v wrtten s U V U V UV U V U V U V U U V V U V U U V U Hene & 7

50 MTH VECTORS ND TENSORS NYSIS et N et s e slr nd S S S N S e seond order tensor then Sngle Dot, Produt of Tensors Defnton: et Wrtten, e tensors. We defned the sngle dot produt of nd so tht the I omponent of l, S S S S S S where S l S S S = Dot Produt of Tensor nd Vetor et e tensor, let v e vetor. The dot produt of wrtten,v s defned y V V so tht th omponent of V V Exmple If & V V, V, V nd v 8

51 MTH VECTORS ND TENSORS NYSIS The frst omponent s V V V V The seond omponent s V V V V The thrd omponent s V V V V So [ V ]= ( S, S S ) et onsder v V or V The th omponent = V Frst omponent = Seond omponent = Thrd omponent = V V V V V V V V V V V V Note s symmetr f Otherwse s not symmetr hene V V Exmple Prove tht () v v v () u v w uv w.9 Doule Produt of Tensors et, e tensor Defnton: we defne the doule produt of s : l nd wrtten : 9

52 MTH VECTORS ND TENSORS NYSIS In the sme wy uw U W Smlrly () ( vw : xy) V W () :, f X Y Prove = =.e. f & uf 4. CONCUSION The tensor nlyss dsussed n ths unt s to fmlrse you wth hgh lnguge eng used n hgher mthemts. You need to study ths unt properly so tht you n pply t to lot of future mthemts ourses. 5. SUMMRY In ths unt we hve lernt out the followng: summton onventon n tensor nlyss produt of tensors ovrnt tensors, nd ontr vrnt tensors the grdent,dvergent nd url of tensors tensors of vrous rns. You re reured to mster them properly so tht you wll le to do vrous exerses nd Tutor- Mred ssgnments n ths unt. 4

53 MTH VECTORS ND TENSORS NYSIS 6. TUTOR-MRKED SSIGNMENT. Wrte eh of the followng usng the summton onventon. d dx x pr. If nd tensor m n dx x x x x.. p r x x x x x. m x x v... dx N x m s re tensor. Prove tht s r C r m s pr x x p x x r x x N. x x m C s lso prm s x x s r pr prm s m s dx dt x dx xdt x x d 7. REFERENCES/FURTHER REDING Hldernd, F.. (nd). dvned Clulus for pplton. Verm, P.D.S.(nd). Engneerng Mthemts. 4