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65_20111201thu_calc3_94160_f11

65_20111201thu_calc3_94160_f11 - Thwmol til CALC E MAC...

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Unformatted text preview: Thwmol. til CALC E MAC 236-9460 START 6.3:Pau. de ‘ - . [331W E3 P1.— Pawlnaewaence (24:45) , , “u . Work In’teflrals ate often eau'u‘ to evalaqh, if the) are Path thdependent. ” E] Qef'); Lei E“ be a. few defined an an Open @an D 1% space. and Suppose that for any fans Pocifls A(1u3uza) and 5(12)32)E;> I." D B - ,., J; Fadr’ , as the same. for all pafis {3mm A448, Then S Ecol?“ is PAIN INDEEEN'DENT IN D and E (s I conserva-l-We on D. W —_—_..—""_‘”M ._______ - _______— (w...— [2] De“: .253 F n‘s afiéld on D and I? 3 :9 (scalar-vafu d, funcho'n on D) 5.1:. __s "'5': V}, the! y is called a ohnkial nc-hén r f? A“ —-—-—-——- Ar—____—__———— M————~A~ Assumg’uo'ns on C and D.‘ E] C is Eiecewise smooH,‘ 41: C = H CK ) CK smoom, (652,...” and Cn/s are Conned‘eci end-b-end.~ E1 D. f” ‘ \ x D I \‘ r P K \D [Q I i K Ln [2 '5 Deg“. \ \ /} // ) Cohhefi cl " OPP“ y _ ffi/ (\"\_//and [“4 I2 {5 couned’ed. 1‘ \ sm‘w'x 3| \ conned d o . \ / ts em 1 comedy}, _, _ \ ~a‘ r A "’ X D ( ( I: ..— e’ K \ } 0 EV! 37d \ \ -/ nof 5\mp\vj connecfea‘ Arm—1 M A» An [C1] The Fondame fol Theore LmeI rals. [. (He) _:r;__hvn1. [3 Let F: MU‘MIE)L +MN0113113J +P(7l)jjz>k be a, vac-[or grew 51 Nam! P cure3 cont-mums daft)" D in on an open connec {-ed Then 3 :F (f is difierenttdble) s.+. a -—-V' . 5 V? WHJUJLULLHLA EELIECEIILE <<<g<< 3333333 000000 nnnnn §%% a m w :79 gammy =_<£,s,+21>. *Nationa! DBland jg 33 a? the Path Jon'nifig A and B i-n-D; B“ How D’You. Know Hid: 3 53 7 fl COWPOMNF Tes’c for ConscNmi-ive Fields: (9. H48), TW =< M N P) I? conservahve ($37 I: 9...: GE 5 N_aP] 9’14 -—~- '5? OhMyGask —- How can I. remember THIS? “ // <M)N)P>X <%)%,%>= (0,010} don't”? I ‘ ‘ ‘~ ' MAcara '” ' ’ 9410 CALC f2. Izll/u 4 n w “c1 ‘2 11‘ 37:! 05 ‘ y‘éflfi'fig‘anSW—Qi-é—M—E §1G.3‘. .1152:#75,FM4 5: “VF-:75? (5’: 2x?+353‘+4z? S‘tt‘k BM— ~P_ , —. " a a] 3—27-63 €74) 3154—0 %:O)- fl:o a :0 A A .‘. F 1's canSWI/a‘I-nfe. 3 f s-t- {73-F=?. '7 We med ffiggjzgg zx+3y+4z ‘ . (a f(m3,z)=A%Z+_3_§;+z#+cS \/ VF— dZfi/t’i’fA—W 67C: (ZZJ33J4E> : E) W , —/ (5°70) ‘ . - (OM) 6) for exam?!“ if the Problem were 41> PM! I? (I?) (0/910) We social use tin-Fm: {a wrih- . (1,1,0 .(t,u) f F¢dF’=j 6;. air _- 300,11) Home) l°1°1°)(o,o,o) ((11 J)>=l+—3— :9 2+2 ,. (°)°)°7 Z ==W+ Wj-l-ZE qu—flk m \ ————1.. OWL—m] l So we Solved {he Problem. We found the pafenél‘al fund-(6n :9, (9qu 3+ seems like we Md 0(— lucued {Mo 'd'. Ma% [‘3 supposed tn be a 52+ More and £5 sysfemafic. 1.8:; (so {Waugh the (213+ par-1v op this Pmcess one more és'me and try +5 sysfimim flie pmcess. [(3 We have rigorously deII-CFMl'ned 153+ E7 is CoMServa4-10e . N0 19541? abou+ éha ‘ .‘. We know 3 § at. $5? = E .———- so LE 33C : E = <M) NJ P> Eg- we she know flxd‘ by (16;? fi¥=<¥z)£3)?z>. 5:9 fir/14) fizN) 5‘22? m a) m m (a m m (\l I: E 93 e (\u I: -E E i We 5e{' up a. Process {'n “bv‘dd” (or “c¢vss+ruc+") sp- “b’CDru'/ PM" 6? {he {,mblem. a, con5+an+ as far r9 Z. l 6‘“ @- {1: M = 21- . Ihfiflmle: f= Z2+ 3(3):) (*1; “emu“ This is a “Subhfial.” Pu+ fiis on 100‘ for the 1mm b81143 . b fir 3“”‘0‘3/ 'facl' af‘fllt @' 7C}, = N = By pfoblcm. (AME) 4———\ Bit also 9mm ean. (*9) qbove F}: 0 4' é}— 6H4!) 6‘? I. .3 = 3:, .'.! gzngz-f ACE) I (*D Now egg). (1?) 310:5 us more ufiformah‘m 44:4er (1“). So We re—wn‘l-e (4'): ‘ f=12+gja+PKZ> (*1?) Q. How do you See the process) Hie pafler'n ? 36%: 522: P: 4?. Algo %:;= 32(12+%32+fi®>: 5% 4“ ii: 43- .'- fiMZ): 222+C ...
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