86 .FOET.TULATTON OF TWO OR, T}IN-EE

DIHCN�TONAL BoUNDAEY-VALU€ PBOBTEM S

xe! =

[::]

PROBL€M S}rAIN

BoUNDABY oT THg DoHAIN

PAgr oF TUC IOUNOAIV Wll€tE

EISQNAIAL GOTTNDAIV @Nolrloll lS

specrFrsD ( 6-

ryFE touNDABy coxD.)

Dlcr oF lllg buNDAr- trlllgRE NAIUIAL

BourDAEy coNornoN rt sfiFrFtgD ({-noe)

NG|fATION :

-sL

r

\

I

r t :

x =l I r? -- -?DINATE v€croR'

A?'

- (r.! : c'oosDlNATE vEcTroR'

R 3|:t l : uNrr NelrAL . lEcrce,'A tnu)

PSELIFITITRITS :

DNSR6IEI\|C€ TIICOEEM :

( $ . , d o - = ( f n i d F_ J

' t L - E - JJ L r

t .e . ?, r s r 5 g f d & = S f n d rJ L r

NITEEAATION BY PASTS

I ti g d& = - I+ g,iJ-s- + I+a ni dt'-O- .3L f

EE.

i . c .

ltyt)7d-4. = (c ,o,])ca + S+ardrO ) . ^ . t d

= q ' ' - - J r \ v ,

"

pnooF :

J {,.a drL = - j ?,rf csa + 5,*t),;crliL

_fL lL

( e:ooueT BuLE )

= - Ig;fe-a, +, I+a ni Jr3' rr

( Dl\er6€\lc€ ileeg! )

gl .

LIN€AE RSAT CONbUCTTON

NOTATIOI ;

tA : TEMFERATURE

9: r lfEA? FLut VGCIOR ( t )I t

9i r - *ij \j psurreei LAw oFIIEAT eoNOUC,lf 0N

Ii . C . I r - F ! U-

,3 : CoU! UC"IUITY f*lATE|l-

E ,E Syt rrST:rC

liosrLY wF nevF AN lsorEoptc llgDlutf

t^lT l r T N i $ e e I \ t D € N ? l t t i l A " r t r

cr A.i =, ^V PloNecrst D51"o

+ : HgAr SouGtF Pe3 uNfT VOtulrF

10.

DIFF€T-ENTIAL RUATIOTT'

ii,i B +

g ' q B +_

(-r i; %ilrr' = t

g.(-s 'gu) - f

]rEAr EquATf oN

( Polsso\rs tquA't lolf )

tN VEErol Fos|Yl

BourrDAsv c'oNDlTloNS B.c . )

t r= t

o N v TEi"|PEFATU:,S B.c.

( esseNrrAL B.c . )(onrrHt€T B.c . )( p-TYPg B.c . )

ft€AT Ftul B .c.

t xeru?AL 3.c. )( xculrAxN B.g. )( {- TtPg B.e. )

-1 in t- 9 ' 1 1

- l -

- l3 r ' )

9 tLINEAS TEAT CONDUCTION :

\\,EAK FORMUIATION I

l l , L - + = o

( qi,i -f ) dsr-l*$

REITOUAL.sLwEt6TTlN6 FUilC'TION

BY PASTS ..INTE63AT€

(- ! w i g i

SL .sL

+ !*f in idf 'Pt

oN frr

ot\f fr,

I'

!*tin;v(

) wl in;

It

E€M Eri3CQ. oL

FUltcTlON cr'.Als€'ll :

st!$l,At To l-D cAsg

ugA a u- t ox f 'qwe 1f =t w=% o* f.

Yw e\f

dsr- + t

w g;ni dr- I*f*=

o.

dr' - JT

THAT \At =

ginr =

f - ; - - - - r - - - r - . D - - - - - - - - - - - - - - . D - - - J

i - \ w , r j i d s L = ( * l J - s - + [ o [ d r ia J . ! r r

J J i. r L J L 1 . 1 IIL----- - --- ---- --- -- --- - ----- ---- - J

9 Z

THqBt FogC I

- 5.cL

\^r,iqi J.rL -- 5 w{ Ct' - Iwf dsa =e.

CJEAC3ANSI\Ig :

'rHrS lS TtlGi S|€AE "Oe,fYl Vrrelf

s\fBsnr\rnNe FoR, 9i= -r<ijuri

fq

+

f *,, rij urjd$,ilwfdsr

* 5*[cr.sL

VwelfWEAK FOR.M

ct3

NOTATTON :

a (w,u) = I*,

"i j urj Csr-JL

(w, +) = I wf d-sL.(L

(w, 1)., = I w[ dt-t l't

THctc An g ALL syHlt qg!_c . BILINqAR

FORlvls .

wtrlf Tltrs NSTATION lN FrtND, T]f€ w€AE

FOET.IULATION CAN B€ WRITTgN A3

FOLLOWS :

a(w ,u ) = (w , f ) + (w , l . ) rVwel/

t\, '?trlt FcSrrt, Tutg wgAE roarru\A1lo N

LAoES vgEy Slr{lLAP, TO T}fg TWO OnttR

wgAE s|trrs otreruED prcvtor,sty (ttr pp. lz al|rp)

j { .

LINSAR. HEAT CONDUSTIO N !

6ArgE,E,tN ?OErlUIfrtoN

INi.IEAD OF FUNGTTON €LASIS U AND A

w5 u3C FrNtTe DIHBIXSIONAt AFFTOXIT'IATIoNS

?th A*.D d

TIIAS ll t

h h hr F w e ? l l c \ t w l = O o N q

( o : * b i o ) o

rF tt. d ruGh ot = E

olr Pt

(os *t =a )

JulT L|EE l^|g DID tN Otl€-DlHErrSplrAL

C n l l , h

Assut{E rr{Ar 1f K :T*

AND ^ -h

L|Ft Vs colrE'r EucT JJ 8Y urlal6 v .

1 5 .

pg3!e A ruuernN 2'

(A 6lvGN Fulrcrrou)

sucu TIrAT t^

= ?

o\r fl

( o R 0 ' : A )

Fot' e,vt?Y tn- th

w3trg ,lh = art ,,t h, ?Now c,ttBE, TO SIFF ;F Uh BGtON6S'To -/5

h

o th. ' . d ' =

? O u | 6 , /

E '

Naw wQ,tt€ Tllg wEAs For,t l Fot

*te U'v h e U h

uh e, Ah

q 6 .

a(wru) = (w,f I + (w, 1)n VweV

w G - * h

u+ , *h

.n a(* ] , t ) = (*1f) + (* t r { )" vwLuh

uh r tt* A'

e a(w] .,'*7) = (*1 +l + ("1 {).. v*'euh

a(*1 vt)UNfril|l-f

= ( ",1 +) *(*l {),"-a(*] ilICNOt^f N

:+

( coFtgAlc te THq SALegtslu trc3rrr

ostArNGD Dtfr|teuttY ! nf aI) )

FINITS EL€F|CNT DTS3ESTI?ATION

DrSCI?Ertt€

E\.EtnENT

rooEi

Tuq DonAlN .St trfto

Doprarus -ge Q < e, S nel )

3? .

Drsccell2fTlohl

oF -St AND

r

ltAU€ A1 tg11gT Ttrg€g uoDEl.

rtoD€t5 .

4- NoDti RErr€rsn$ uortuv)

ELE$5NIS WILL

o E , 4 1 3 r , , , ,

( rlrc r^llLL ul3

Horf Do wE geuatg NoDE OBtrEErr6 TO

Ttrrg oCDGS|N6 Oc gquATtorr S 7

AP?Rqllt51q1

oF 3 t t rn

o N n B tt

Ltsgas

NTERFotATtoN

n-- 1 sgr oF oF- 'np J

, '6roBAL noo€S

hnD : NumtG?' oF

NODSS

LET tl =lt

v

I r r 2 r " ' t

g ?

l"

e1LET tla Bs TlfE srr oF NoDtEs AT wlftclf

h r l x* " = ? . t ' A - N o D € s ' .

rHE csretGrrcrr oF n(O rn nt ( i... T)rrr

gtr^Atlltr3 r.nuut ) t3 THE sFr oF

NoDGlS AT wtllclf Uh ls uuruotrtf '

wrlA't t3 Turg NumBGE oF gQvA'tloxr .?

= uurotFE oF \lCDlEls I N Tl.lg

sq" \-rb

= ..1 .

Tt{tx, JFclt AtL Wh C, tf b

tr.|lF cA\l uretrei

wh( l t= Z NA(I , )crAcl"lt

trlt{nl,E NA1I) ts rrr€ sFAp€ ?tlucTrorf

AStOCIATIID WrTlt NODS A.

ts wh=o oN f-. ?tl-

wlfY ?

loct.

NODA L VALUS!

OF \A

SiHILARLY !

vhcr )= f NA( r )dA- AQlne t- uNENo.^,l,

dA : Tn€ uNrrbNN (ro Eq FrrEEJlrrao )A'r uoDt A .

i.c , TtF}rPglA'tUtE At NODE A.

lf,lTEEPoLATlorr oF ol t r ) oN lta s 3 , c

A'rt)= I NAt!) ?*Ael

w lrEp,rE fle

s A (L^)

sulrrtrrr€ lFr. *b, V

6ALERr3tlf FoR t'l I

f-"I S t / u r l

h hAHD

0

lo t

rN rnC

: a(NA,t lc) dE = (N ̂ ,+) t (Nert ) r8e1-1'

-f a( gArNs) ?sBelt

Va e?t n7rlr€t€ AEF h-- tFqgATlg;ll ttggg--l -

Fos tlal \l!EW

wtrCgt h..t lt TlrF NUFrBge- oF

NGDE: rN Tlre sqt 1tb.

To DgrgEHlUg Tt*lF STIFFN|F�SS FrA.tPlI,

AhlD rnc hP3g uRTo:,, EQUATIOU lJurlB€II

t.tulT BF ESTABLISIIED.

frDft g\CC,Y NoDC tN ,tna

AN FqUATION NUIYIBCE.

Ft rne NoDet rN ltg

NunnIEg lt AitlSuGt . (

sDe,rFlE AT r)fAT NoDg )

,

Dg$.TTNATION AEEAY ; Ib

lol,

urtF ASSfeN

NO QUfrTIOTTI

hEEC,AUIE U, IS

: NUMi€P Otr NODALDot lrTl

: N01{EE3 oF EqJATnrt

Asst6NrlENt OF

uoDG tS ACH

A R R A v ! D .

I D ( A ) =+I

-teLoBAU troOE

tSorrA'Ttorrl rt t'lGEPS To

tsfEr) 0) Dtsnpqr|crl

e f oLoBAL EluAllss .8.

i F ,F Ae rl-rfl

\ o r F A e " ( l )

NumgeR.

NGTT' THfiT

t < A ( h " P

l < P S 6 c l

nnp

h=t

lo3

wrlAT tt Tfr€ trlHEfi$ox eF rxE

:E,D ARRAY ? . hnp ,

MATR,IX SUATlorf I

K d = F- r - i ,

K"u = a (NarNg)

P = fD (A )q = I D ( B )

F p = ( N e , { ) t ( N e r { ) r

Z a(NArnr.) eeec1?

utuee*. neff cOttDUCnON :

ELEFT€NT SIIFFNESS MATRU A}ID FOEC€

vEcaoR

ELC,;feN" STIFF NCSS t'lATBlI

ELCF|CNT FGl, \,ggToR,

tfLCI.I€NT

iloD€ xunaB€ R

n

f,lutrrtB€B OF

EIEF,I€NT NODES

S a r b ne

L

k'= L klr]

f"= tf: l

*;i*u,j d&-tge'

o-l(N o r

k : = Nor.a ( No, Ltr)' -

4t r-t c r - ( No,f )e t

Tg N " ) * ( g N u ) d s u

nr

i),.-F: klufi

t06

NcFr€ TllAa :

- c - |t [ = t ( IL) te b rs A ?-tuoDc

e O l F b l t N o t A ! l - N s l C ,- o

A COiI\,GNIGT|T FORT'I OF WR,TTIilO B. :

. G c , Tn . u = I ( g N . ) x ( g N b ) J - s

ge

DCFlrrC t

D= * ( corlutrlultt mattlr)

E.= Y N"

E = [E ' Err . . . , E. . r r lTHCII :-

k iuE I B :pEbJ-o .gc

B" =a[ g'P E J-sr

lo6

coFrgrraATtoll oF HEAI FLUI U€C"OI,

LEr d-'= [11"1wH€rE d: 3 uh( I: )

rlr€N q = . D B d C- f l e n

3 , - ' P r g o d :i -

a 3 t

fo?LINEAL }I€AT C.oNDTTTIOil :

DAaA PcocCSslNG ABEATS r grI9, !!!

ELEFTENT t|Ob€: ^*31" : lEN

A = X E N ( c . r e )t

' f-c.I.-EL€r'r€NT uurrB€R1 t-Loc.a1 (ererrGrr) rsOS {f

6tOBAL NOD€ {3

D€s'lrlNATloN lglfY I ID

P I rD ( t]- sroBAL NoD€ surrB€?f

euoStL Eq,r|rnON ll$}|t€R

LocAT\o$ gAfetl : LM

P = L M ( q r e )

t: Er€}tcr|r NrlFrBgrEL€FrErn rloDE Nr,P|B€P6r.o8AL €QITAT|oN NUF|S€:

S ,sNg

AND

6s"s uP I!:NTHCN OT}TAINS

l € .

ABI,ATt

?OLtotrll :

ArtD gD

LM AS

LM ( q , re ) = ID ( f EN (ore) )

E"IO AEQAV ; 6LoB,qL rloD€ t

I z a 4 s a 7 t t

rosExAlrtrL€ 1

E33t}JtTAL

B.c. DA?A

SPGCI?lco

( i .e.

€TCFI€NT NODCNul'rtCElNG

l = t 1'h= l' l

\-11-- I I

1'10 HAt NoDel

. ' . UNE,f\FWilt

EQVATIOU t

hntr=

7 -Nosct

lx .

l_xf

A :

2

2

P = E D ( A) 6LogAL SrATloll il .

q.l =

i loElAr'rgL€ 1

cS3tlrTtAL

B.c. bA?A

ggtFcl?lcD

( i .e.

. o. , ,2!

r o l

t t = [ t

t l l = l

, L r . . . o

t r { ,7 t

1-1t= [t't t 3'orl,rrtrlt

-Noscs

lx.

l_xl

ETCHCNT }TIDCNUTiSCEINcr

ID lrl,QAv r cuoBAL mo€ |l

S 6 . T t t l o l l l

ll-ln n.r E NoDel. a ,

.'. A uNE,Nowsl

A EeuATlot s

l.lntr= 12.

q . l = 8

E.A :

P = E D ( A) gLogAt SfATlOll * .

E(€HCN.r

t q 5

i l Ir€$ AER.AY :

LocagNODC

cr.|lutqttt

hen =

LM ACT.AY :

t\g,nE

\ruFl Et t

3

As fEn (cre )t atoBAL NooG

€LEFf ElfT Nut\. Cte

I Z t g B 6.csnL ( t

Irroot o I .

uu,rqtl l !I\.t

hel 3

usr.aB€R

h c l =

= ED(fer.rlerc))

Qvnnot.' uumB€R

P= Lt*1 (ore)

t'- Gtog4g

C\€t{CN.rT€N ASR,AY :

LocaluoDC

q.luuiet

h36 = '4

LM AC:,AY ..

hen= 4

As fEtr (qre )t eroBAL NgOG N$r,rt€lR.

Nut\i Ctt

hc l = 6

I ta

\Jut teC,l,

h e l = 6

= tb (f er.rlarc))

QVerrOu uUr.rB€R

r.ccAL ( tr.rooG " J

.tttl,ttCl

| !

\. '

3

6

P= LM (erc)

t'- €tog41

I L + 5 -l I2 3 5 6 g 35 6 E 7 f l l!

1 5 7 I l0 ff

€LCr{E}I?

o I o 3 o 5

I z 3 I 5 g

3 I 5 b 'l Io 3 o , o 7

EXAMPLE 2

COtlSlDEn, THE

i'|ATRTI ANb

f

FOLLOWIilg E,LEI{ENT

FOnCe v€croR :

l t3

STIFFTIESS

{.'

r lq '\ l

k t=-

o

{.:?:i

k;

k;

6f

$.

iit)

f,

ki

t 5

f==k!,

@. , - ^ la r -

- - ' r

ik i j tkir l\- -- l \ oJ

9

ASSUr.r E

E LEFTE N"

TIIAT TIIS LFT

15 6l\tc,N At

ARTAY FoQ, Tlll t

L]*1 ( t, €)

L l r (2 , €)

LFr (9, e)

LH (rrc)

5oo9

l |4\ rnAT 13 Tll|F

ELCt.t€ilT TO

*tu

K t t

K g r

collrr.lBuTloN

K A N D F ?i r - - '

n E

Ktp

K r,,

Fs

Fl

OF THTS

+

@@@

@@

+

+

<-

(

f,e = - ti,u:-[&;+ ; G

, - --'o" - iuiint\- - -

-$itt

f t5

BILINEAR, quADf,lLATERr\L €LEM€ NT

(-tr-t) (tr- l)

ll ttPARENT DOMAIN

:;ll(r;, g: IFt

(x;f)

(xi, g: )Dor{A111

lfPAR.€t|T DOTqAN ,

nIIATUR,AL CtrORDINATT1S

!=[;] =[;Bl1 I

5= [ ; ]

!=[;] =t;l

(ri, ei)

r ) I 1r,r1 3

I 2q

rf tr tl6TH€

--NIVSTCAL ceo:,Dlurffis oF A DotlT

rN .$l.,e :

x -- I xl -- I x(lrl )lR' L :J tU( l , l ) lAe€ 6luQ,N tY THC FottrOuftilo MAPplt{Gl :

r (i,1) =4z

C,=l

eN" ( 1,1) X.

e (q,1) =+I(l--l

No(t,l) g:

u'(q,1) =1z

lNo(\rtl ) d:

oEr tN v€cToR. FoBH I

+I ( I) = I N.(!) I :

C=t

|$TECDOT.SnON G A Fur|cTrct| HAS TTg

sAtlE Foc,ttl :

(OORDINATT

TRANSFoRItATtoN

I}ITIRPOLATION

ll?SHAPE FONCTIOT'IS :

ONE_DIH€NSION : ( R€,r,flNDeR)

5,=- |€ . = + t

N , ( t ) B * t t - q )

Nr(h)= t ( t * i )

ruro- DtH€NSloNS :

N.(t , r t ) = *( , *h. tXt+r lo1)

t ,= - f " l l= - lN,(i,'t)=*(rtx,-l ) t= *, 1tr = _,Nr(i,r)=t(rq) 0-q) l'=

*' t3 = +rl q 3 - l r l . t = + l

ur(t'l)= +(q)(r+q)^l(i,l) = *(t-rx,+q)

N,(h) = * t t+ i . t ):t

TTrs ts nouf A snAtrE ruucTfoNI rg

LOors ur,C :

NA

t r lA T€NT .Ftort NoDe

rf ,l

€F Tlfg .r€rn Eulttlrgst, ll

Por€Tlfg

A

nlutueaR TRtAtrlguLAn €L€FI€NT :

ItAN EXANiPT€ O? Dg6gN€EATIOil

-t LIN€AG, TR|ANGUIAR ELeH€NT

<^AN BC D€BU€D FEOT}i

+ BIL|N€AR QUADR[ATEIAL gL€T'r€rtT

t.| c,oAtFscttlg soDES 3 Arro { .

{

r N K = t N q I : r . , s s e r I l = I ; .C--f

TllErt :

! = 1,, I:r N.X: +(Nr+Nr)I:

= ui li *NiX: {r N:, I:

X B-

3 r -rlD - -

) N - XI

- s + ( l

o= l

COORDINATS

TP,AilIfrBT{ATKIX

= N r = ( r - t ) ( r -q )

s N l B

= NrtNr =

( t+t)(r-q )

(l +'1 1

I1

!4

I

z

,

Nf

lNr

lN3

taowHcrr Tllg srnD€ FulfsTloNs AtE:

t TTl3 wAY ) htll€TlfgQ trr€ lfAU€ A

quADEl LAT€e.AL €tgr.f€llT OP A

TR|ANGULAR €t€Frl€Nr Ul€ CaN

u5€ TUE SAH€ PAPtrlT DO]"IAIN.

-) rLElncr$r INT€RDorATtoN nA: Tlfg SAHE

tuElr AS COOEDINATE TEA[lSFoEttATlON. t.e.'

a^h = i Nld:Csl

N; d: + Nld;0 .e '

+ Nsd,

r! llOw STIAPC FUNCTION

lal

Loors 1rt€ :A

nri

ELEHTNT STAD€

FrrreTtOil

Pt?Atf lDII

EtoBAL $TAD€

FUTCTIOI'

t'i/ l

}f, 'eer-)

la2-

T}IR€,E. DITIEN SIONAL SIIAPIE FUNCTIOIIS "

TSILIN€A:, II€IA TITDPAL EL€}I€NT

tfH€lAlfEDSO N , BRI(,B

5= q ([)

TIf t tPAr'C i{r Do}lAlN

PT$|CAL Domerl

X = X ( I )r l

DArS\r D$rAtN , tlAlUSAL COOIbINATS: :

t q lq B I n l' - t l l

- t - o .

rz3rt tl

ru€ Pfl.rsKAL csoEDlhrA.ttl oF A DOINr

f N - Q e s

l r ? t r ( t r t r B ) Ix = { : t = 1 v ( \ , ' r , i ) t- r \ u ( l , I ) S ) I

Ar,t 6lulEN Br TH6' Fo,.tohfltf. '{APPIN6S :

x ( l " t r5 ) B6z

C3,

Irq 3 f

aI4 = l

No(q,l,q ) x:

9(t t,5) = ru.(t tI ,5) yi

z( \ r1 ,5) s No(\, t l r5) " :

a2 t frf vEsToS FO3M :

ax(1)= Z \ t :) [ l

e3l

(ou:ilNAr€

TRNININXOiI

Er€r.r€rlr nrERxttATlorfAS 'rhtc cooBDtNATE

12"4

llAS rt.;" SA]f€ Fo:t{

TEANSFoBt T ATION :

uh(trntr S ) =E

T No(\,t tr5) d:Cr3f

EtEruNT NTERTOIATNN

S}IAPE FUNCTIONS :

No(l ,I,9) = t(r+h.lxt *1"1 Xr+5"q)

v.lff gt€q, H. 1o 5oI - l - f - f

z + l - f - f

3 + l + f - f

q s f + l - f

5 - f - l + f

6 f - f - l + f'l r l + l + l

I - l { - , + t

t , l f

DE6ENEfI.ATION OF B?TCT. TOrr rl

weDCt,fz3

eoALESC'€ ItoD€5 AND +Ah|D

Axb E7

1s$ll:

I ( ! ) =

trf tfEF€

tNa ( : )

? ,+ Z 'N . .

creS

?.= t e2- Nq IoCl,3l

el .

Inutr)= l"'(:)t*+

(!l

, Q-lrlr5r6

tNr q) ,tNa(5) ,7

a=3

q = 7

jO O E O

- e - - o - -

n e

DGo€N€:ATlo$ oF t wEDo€tt To TgrEAn€DRort

cort.EseC ;1obEt 5 1 6 A N D 7

Tl€lf :

x( ! ) =

$HCR€ :

,,

N.(l) =

t ,

I:, + Ns(!li N:(r)a3l - x

t-

[* l ( 'D[ " ;

( : ) *

,

,

N. (I)

4 = l r a r 3

+ N'"(I) q,= 5

- r . G I - e ' - 1 - e O - - ' -

l2-?

lll6uER - oIDCR C\CHerlTS

IN ON€-DIHEN:ION "

Na

N , ( l )

N r ( l )

N3 ( t )

= * t ( t - r )

3 (r-q)= ( r+txr- t )

e.sralrl'E, :

h c n = 3

@

@

= tt ( f+ , )

$16$€l-oRD€l t\€MErlts lN

Trdo oR T$tsg Drn€NlloN:

C.Ar| BE OBTAINCD IvoCSUrsT BUU€

rrETlfls ts florr fi wotrs .'

i l(1+r;

@ ( l-rt.)

ETA'iPLS ..

@ ! I (1-')

* t (t-n) ( r-ht) t t(t+r)6) @

Ne(f,r ) = * t (l-t) (,-rl')

lz?

NU'TIERICAL INTI6SATION

TO CAl,cuLArg EL$Fr€NT STIFFN€33iiATntI' AND Fotc€ U€CTOR rrr€

N€€D TO , tor|gHohr, t\lT€5EA'rE

TCRltrl Lftg Tllls ..

,flf t1)ds -- ?

AAr{OL€ : FoR' tlgAf CONDUGTIOr|

?rlr3 COUID B€ AS ?ot1'ouf I :

ft l)= (gNjs (gNu)rscN, Tlr|E t\lTg6BATtoN

6tugs 'rrrE a- b cor'reoilg}lr

oF TtrE el€rrertT S{IFFNESI

l.rATEl |, .

PoR Flsd=l f j ( lnco;r^ i l )

| - / - . - t t

_\ f t*rdx = \ t t r ( I ) ) *r \ ( t ) d\d J - , J

130

Fot' n3g =L

^"lf(.,g)dgJr_J dfl?i'fh(rr\,rr),e(\r)) j(1'r )dt rl

O | - f t

det

t--fr..:

JA(oE|AN , j =

Jlar

H

f

(A) = detgan3-r?1

Foe nsd = 3

t

Jf(\sp)j*- =+ drdgde

+l r l r l

.!.[F(xm,5 ), 9(r,T,5 ) ra (t,?,b ) ) j(t,?,5)- t r r r f

{ q { d l

j ,B dct

?.I"5u-"5??

-2"3

?r?t?.:l2 t

a!"q

?r"teq

J '7f

??fra-a1

13ltN CAss Cel€ , TH€

TAis ts :

INT€6:ATION

-l ts(h) rt = ,

e o g t \ - t = L-lE

l : f i

]tlt ?(t,r) d1\ = ?FoR nSd =, $

= ?

,WC D6Tr WANT 10 TPY TO TTIVC

TH€sI INTSBATTONS €IAgTIY.

Tlfcc€eonc, w€ AgPeorlMATE ?lltM

BY rlut{tF9rtAL rNTE6lATto}l

-5 fr(r ,1t5 ) J1 J1 J5+ l

wk

tr int

t(E)dl * Z ?(qr.)k= t

lltf3z

NrrtrrB€9 oF lNle6larloN 0otUtSn irtt^,

H c

wL

kt*

at

ao

1 CoOBDTNATE

INTIEGIA-IION

WCTI6HT OF

Forrrr

OF TIIE

DOTTIT

rHE TNT€6}ATIOI

w, wa

Acuplr6

OATTOMIATS CTACTIY

Tllk

btAr{DL€ : TEAP€?OIDAI, IUU€

nir* =1av

l , = - fq " a * lw r 3 I

w t = I 2*D oRDGt-

loogt€g

: f s , + lr-1-rrlta/\-r,

trrT€crATgs

e,IAr"rpt€ : SfltpSoXi R$LGra3

W3

nint^,q ,^,q.d

I g

|NTE6e56=

EI lc tLT. rT

Ws = t l l

wr - 4lI

w l = t l t

RD3f 3

= 3e - l

= O

t r + fq

wa hft

ggDE:- DolvrroirllA Ls

4t* oEbER. AGcuGATE .

RCl.fAB:3

l .

z .

FEh|€3

cstT

IH€P,€

ACr|lEri

FCI^I€C,

INTE6:ATION polrlrs ]rgltN tgSS

ARe trnggHTlox ILULE! wntctl

Tlr€ sAlrc AsB$tec v wlTtl

r1n.r€ql5tt6N fotrrl

t34

6AUSSTAN QuADrAtul€ RULCt :

u TH€r ae€ oFllifAt lN ortc DtFrENltor

ii) \rrrlr ( nint) FotNTt/ TnGt Aclflcug

(z n int) oe,D€" oF AccutAcv AND

nn€orfiG (Z nirrt- l) orDEB FotYxoFrlntt

EXACILY.

l . hint = |^,

{ , = O , \ r J f = Z

Z*o OEbtS acaUpATG

I ET oc.Dtl DotYltoFulrr lllr€lb?ATgD et5tY

Z , n i n t - 2 -

aE|

l , = - t / { E , W . = lx ^ - -f - = 1 t l { 3 tr J

4"* oet€e eceu:nrG

3LD oRDeC. FOLrNoFilAtr n$Q3qaau EtrAsTtY

W . * f

133

3 . n i n t = 3rylq , = - \ / F , w f = s / 1-

T.= o , \= a/ l

i =*{1F ' W3 = 5/?

6T'r o3!GB fcuBAT€

STtf o:,o€t Do\yr.oFtfAts r$rc6PATtD tl GTty

Nulrc3lcal |rlTElSt A.Trc|| t\l Tw0-Dtwlsltlout ..

nn1 :I6(i,fr) wrL= l

rrr€clArrorr PorNrt ({.rtr) ArrD

TlEll:, (OSg|ESDONDInG l^rGlCTT$ wk CAu

8e oBTAINgD Florr oNQdt|rNsloNlt

6hfssr0rr q\ADcAa\nE l,ur,5 Bv PSotucteuLc

B6

Htlrg ls ttow lr woPtss '.

Ef lraD\t !-a-

2.x 2- r\rT€6rNrlox :

(2)| w ,n*i l l turlil I

^r(3) .

tr e - tllj*fit I

t:-r{Ibrl? |

1

t

(r)g = {

,.ut+ t/(ir j -

"l{

t

FrtIIItI- - -.rn- -a;-i t(d,{.): wltvt'i

:

t t '. - -€€.- t - - -

ti.ltl?iru!'r^rG ) t' ' r ' --?

:I

i(,il i:)l/ "Tw,"'a \

- - . j f O o - o . . .

i

t3?

fieSumnrlAtv oF 2;tL c,uLc:

Ir int = 4

(q, 1"., ) = (-i r-ft )(d,{.)(i,{ ')(s* ,i.)

NOTS THAT :

hi.t

l-

L=l

= (-t )-t)(*fr 2*+)t t , * t )

, wf = I

w a =

w r 3

w r c

( ArGr oF Tlf€

DAg€rIr Dometrr)t\OtE: :

stlrltAgtY , oNg cAN APDtV TlfE PlODrr?

o$\€ To TltrcE'Dlnslltp$AL rllFBEPffilorf.

|3A

C-ALCULAIIGN OF ICBIUATIUEI OF

'tlt€ 6lrAP€ FUNCTrcNI :

wE N€€:D -ro gALGU1AT€ trrc DEPTVATTIF'

oF Tlr€ sl|A9{g Fn4alort wfTtf BBDrEtr.TC GiloBAL CoogDrrr ATE!5.

Wg HAVE Ttig IIflVATt\llHl W'9;r' \ocAL

Cq;g;ROtXAlF.

h s d = I

Nor, = Norl l, * ( eest.l;l-.r-../ t

V

w€ neuc 2-TPt r he

hS{ = 2-

Norl = Nort €rr + Norl ltrr

ntorb s Nari qr? {- No,l nr7

r31

llr Frtrttll FonFl :

Ir,*( *o r t t Nor ; ) = ( r t . r t rN . [ )

|

H Ln,,t 'u]

na J'TtrGtE

-

?a

[ ; ; ]

xrrt

?nH-l; ]

, lqn= f IFU,r

= 3 37x

_

f z x f l- . f -- l -

\ ?1 )

- l

h s d - 3

T(esf )'-

Jcof

4-:-

J

140 .

?N.-r

?t-

?N.-- =

? x-,

a+.€-

? x-

?x.t" t

-

?N.E

-

-- ?Nr (Ef t-EE , l

coF rr =

cdrL =

Cof rl =

gofzr =

Cofet =,

cofa3 =

9rtl zr5 -

9r\ zrt

9r t? , I D

Z\ t r t i r '

?rs x4 '�-'

Zrt Xrtl rD

gr5 a,?

9r; ?r5

grl arq

3,i x,l

7t\ Xrs

zq x' l

14 l

d., = Xrl 9r5 - ag 9,rl

t o { r . 3 t S 9 q - x r + 9 r s

tof sr a Xrl gl '�- *rT 9t+

j = Y,i 4o$,r r xr1 cofru + Xf GFlr

\, = ( Nort 6f,, * Nc,, c4re +Nqri C"f ,r)/ j

"o,b B ({rt @far * uo{ Cofar + Nory coFzr)/i

Nqr? a (uo4 Cd' + N.rt (ofrr + Uerl Cfl )/i

r - - : l -