fea_lecture notes
DESCRIPTION
lecture notes from the Coursera FEA online courseTRANSCRIPT
SLAN Mfle 2oiS
i~t~: (1) iD
(2 ) iD
,~I .i L
f tnc1
-tl € 'X-=- 0 i S LA '- 0 ) -v----
Ll (o): l\ o
~ +kt (OY\Shruhv-E.- te1Rh0,1: ()= &L-\ ,-x.
-rlt\tS 1~ OUY
+h.tt.r -1 ~~ h ll ow1r:j
D (} 1 t - 0 In dx d ~ f K I '( t.! l t\ I <c~ : (' "J ll
1 1 r ' r • '(lt 1 ( 1 1' 'l'\ ''~ '"'11 C'11•101l\ (; I ~
. , Co) ., \A o f.:
t L\(L)=- u
5 -OJZ- cr(!-)=t
L OJ W ct~S ~0 \d~
\'vH'\ ~ (~ \Jc on{?. or ·\'At o\ h . .Q v-.
(j)
v(o)- ~o
* (ov b{ o thOV\ !?:>.c.) 1 t I~V.IY'O ,,, f.':oi»1CI(t, :~ (r"r~·J.:",'
> QV(.Jiied. to +~ . S.--Q(_I(LI df\\'J(I..h.IC, (1!-
, . f..~·~ r '.(1 (Y' ,._.; "\ e '('\
v. (o) ~ 1,\1)
1!--~--. -
~"'
~l V'J f. 'Y '> f\
DI~I(H •.f\ ~G
@, \.\ (o)
!J( G)
rr-(t..)-
I- ">7. . L
CI\N Be (trHCt-L
~·f:>'(\.'L[I ('l~ 1JCVtAANN E..c,.
WE Po N 0 I (ON S 1 DE~<.. 1\ ( {\ '.. r;.. \J J •- C f~ .__
;3ot1 0.G. f.\t<E NCur--'~Nt-J .' (~Qd·h.c..vcnJ)
r : ..;.. t ~n I • C>
)..-(,
r I
tl-C U, X: I •X, I
\..
. . ( . c I
n. t' C\ .
( q ll (l h c I) . ( ',''i(\(1 :w (
d~ t {- {\ dx \..'. (c
J
}x ( E 1A 1 -x) i (- - o ? _,_.l- r•(y_) +-
Cz -
-==:7 f1 \ rz_ \ n-t 1 r r 0 _r .': r1 /', t (' n n '"' · 1 .L V ""' "''=""' U v :::VILA ((A..tl 'v 1,:\ \..\..r•'QL\.( S:C'IL\.110r1:
==i NQumcunl B.G.<> Cdc~u con ~-t r:i~~cihcd r :~\(:,
[\ Jl.
0
\ \t'Y\.t -dept ndtld- ,Q) 0.$'1(,! h--J p) :' bLtl"11 (1'1"\j~ ic; t'lv,.pc~ i)()\ic., Poe r)
~~ 'ow n9\'~ ll(..vv1
~'.e o'e utll~ fiJns,·oJYif1J S\-{o/ys:~ok ~·!'), ( € t ic-,f-: c.. PDf s)
't./"" \: ( rr I(\ r 0'1' ~('{~ of e I OS htdlJ T
,. f
~ \), \..Q.. \ ~ ( )~J :
oo- r ~ t 1Y.; - o
d' ry , ...
L J d.u~':'h-JfllYlChofl
f (I{)~ r ·' 'f f....\ -I
t (t, L)
tr!..fYVo.\. (~v,,p\\oJ 1~"0..1 ~~\t
(\r,\Y'O 1n Lt.~-wt~t1
( d ·Kn d(nct-rd b~ £=- un)
x--o ~ x- L ) {~u ·~\·: I "'-.C. PV'.J po t·VS 0 lL ~ ~U.H'~C'' :c:..)
-11 r- ~. c.:s 1 ctkt c etr( ot ~)
I ~'( :.t. ~ \ t'1~ S '"V\.C
p~ot1LC 1'1 v\J ~ lA \d G~ ( J{ V C G) 1St Y(l iN d
·~ h< '{.:1) va' clc:('J.
' I ( {_
lvtt \D (SrO-'O.X) ''(\JC\X ~'f->tlG pv\JH-rm oJ~o tYic,atl~ ~s.o+- cr- ('.:·r-....(S\,,-e.,
IV CIS~ \-ycun~,f-crt. 1 , 1 ( n-t¥1 flC '( 0. tA I~Q_
I v 4 ;::\l"o L1\Cl£): (o>L) ~ ti ()_-xn l;\t> 0-.11d v3-vl<· 1
+- 0-X' ~ H·-.t l c Y' S~1~ Lt fl \,Q
rc I cchurl
~f.o,t- .-
J-=--ku,-x, ~-~£h t\'c.d'1lo
~luY ,'~~)'_ -f-(-x) l(\ '"' -)
'V\J ~ oJi \I'C t I - c I - · ' v I
c\ .Jt\ s-<-il'~' ~a y o\ \ll ' '',L} -
.ft vt 'X. Sov, ~DI-\ t< ~ ( Cl'J 1) \\-\ON~ :
-· 1 ( ~ lA \'\_, {t. YU~ p \ {. ': I IV) pi J
~(!..)- -~.
- ~
\;\ (()) - '-''·o & 1l}(t.j-= V\j 1 -0~-
~0 tA-l'\ dcdt" .. ~ ~ ( L-} = - -~
I
~ r I 1 '\ 1• :,1(' 1 1
T -:.- ~..( ClJ \ ~ ~ l t \ 't aJ- 'Y - L .
\Lc~J ~~~ ';{'ron~ tl>fn'l C'~ a hNOr PDt;- of £•,pn·c- ~t}P'./ tn Oru dit'Y\.~.J'1S\ort.
F 1t1J ~ (-x) ~\v-cn V.~ 1
V j or<- t- > + Cx) o .. F'o t:t\.C Cons-.hhth•A:.,
V .C \ o..h D• \
\;:: r· U t:; >)(.
su <..Vl t-~cU-
.--t do- T F ~ 0 d"?
1\t\.\ S. ' s. c C\.1 1-e J I \ U. $TR D N C, -'~ ( 1)\'\c\.\ tiCn MV\ ~~ no' 6
I \}. ~L) ~ lAj ~
-0 R.()(L)-: t-
f 0 ~' " \ 0 ~ -t w c ct t' cd 1 r" . Cl\. t E v ( f2 'I po t n.-'"".
- I
--; I J { ~ ~ )' t· .1- =- 0 ! d~ d--x \
I
NOI~ ~ \l S\01.)\A\~ 'Q~ 0. h.H1diDr1 2>1A<.~ t'r"cd -\~0 JQ..V\Jeth-tfJ Qr{ pussi\-;Ul
(i.<. V\() ~~(O)'dilll.-llh.({).
\J\IQ rcqvt.u"t' Slf2.oNl\ CaNDtnDN~ o~ 1 Sn1cot~u\f!:.<' cr1 ~((x) 1C'(Ot'.st
~~\f 5.twYj ~ 'VYY' t1os ohNtJ_ Spo.h(l' aQV\vO-hvf~.
·J ~ llf.C\tiHCttt1~ 0-t:IV lf ed ~0 \ t.\..nO{l
f !i'' Y~ f,J,,) \IJ~:YL ~ be lc"@S +o [ 0 1t-l 0 v ~ t- LD \ L l
~ =) o- (j)- o-( o)- -- J+·Jx
)
0 0
~
J f d~. 0
~ d~ i d X I
0 ?.
j ~ ~~ \ 0 0
f diA I :r. Jy 0
ft,J OR_ --<:v rr ·r..Q J
\t?lj ~.G. @ l_
~I
lAb -
8·o~ Th-e. Weak FonYl t:.:.. u. \ ,'!(ir· ~ 1iph<.- voG- i~' Gt\Q di\Mr~lo-,1-
I f..{.~~ IS
l ~ 'N£-1\l'r 012. t-'1
\l.ol\ ___ _,
Fn''d tA C :i) E 3 co >po.c..c oC.. -h\vx.h~Yl:
s r), \. " ( ") ~ .,.1 ~ (1)"' tC\ I 1\ s Glt I V\ $.1.,.\ U/1 1.-!rOv-\- \J ( 0 )-::. !) 0 ,
~C\S\cC\lli- 1 ~Q \o(Jk_~ \n ~~ '.)irrc.~lct 6.C ..
S\AlVl -tno___+ {\)( w \ w b..c.\DV1t}\f\&\ -t·o
\} \N c 1'\j::
0
tJ.vJnr;l~,\ {hr n'vcvC C'tjvto..hcd 4-ltlvV\t\j~l lA-t P.,(Y-), n eo 1~ va\un~P 1\'"\H.=·s· ol~.
L.
1\ (X) Jx - ( \fl ~ 1\ ( ")d y \- 't l ( L) PI ( )<.) . t \ j \..._...-.~ - y- J
'\......--'"y---.../
\J () l 'v\ JYU. \: \ C rhQ 11\. +
0 \f0\\1\n"(
.Q \.QIIW: v\,-J.
n-tf FtNir£ GLc Mf.NI N\E-ntuD
11-\ IS, \N f-1\ t ro ~M .
~ovn~C\I'tj ~ (}({j_.
\\- \~ oJS.6 rN \?O.~f~ of oH\..CY VOYill~lut\(t"q i;(l.:.cJ 1
\1-..1 s~'l\~ f"ofm OX\d WQG,l r-ovrv) Ctrc r~ lttiv t\\--e~J- ~ -.J
4 (po.c¥1 tMp\~~ -lw oHur)
do- + ( - 0 ( 0 L) dx -t - In '
8.G. ~ · ~((o)-= \;\~>
" ( L) ~ t- w~cre ()- ( u,)( )
lnrroJIAo. 'II (-; 11= i w \ w(c), of
W; '>J\1Ql9~'-ht"~ ThVI<.hDYl
L L
tv\vJhyi~ Sh~ro. torn"~ lAf w a-hd \+e~(cd-c ovev lo, L)
Sw Atx) J.u d X ~ clX
i 'N (\ { ;<) + J y ::: ()
0 :------- ___.)
\\1 v ~ (,_) (i)( + .., >y 0
... .. - - -
L L
) w,," ~~~)dx..:. f \N ~~~)f~'l 1 0 0
L ~
j '~'~,") o- 1\!v.l Jv ~ J w ~ l~l f Av .... 0 0
l t \T(L) 7 t-
J 'N [\(~} ~ dx +- 'N(L) t- t.~l-) i)
~y---.-'
ttOrll OCJfH.Q QlA)
D1r 1 ck~tL-
Find ~\ f- S =- {' u \ ~lfo)- 1.\o} svrh 1\•(1.1-
~ '1\1 f '-'\) = ~ \JIJ i 'N :o)~ t' 1 L l ( ~ \f\J l '/. cr- b. l'f) o X -=- ' ·' A ( 'j;J .( r\ 'I -\- \\J ( L) ( l-) r\· ( L) 0
- v--- _/
~ ln¥~r o.l c b~ pcu h
L
f ~ -; \'! {, C• X { 1/\J CJ A \ L - _\IN A (x) J )t } 'N (G) cr ( L) A ( t..) () 0 0
L
do- r ' -- 'N t ~' c: I
d~ J 0
L
IJ(~..-)0' I 0°u\ fJ.r,;
J lfJ A (X) f d 'I ~ \' J (:...) ( t ) {; ( L) u
L
J do- 1
d~ \f' 1 A ( x) cl )( + W ( L) A ( ~( L) - -t ) ~ D
j \J'IA')() [- f- ~i j ;X + W(L)A (o-(t..)-t) ~ D 0
1h 1 <;. lnDl ds \f W f: \T = -i W / W { o) o ~ 4 ~\so ~ o \ c\ <:. fur V 1 (x) -- cP { x) (- ~ x- t /
~--t- /1! vv V\.9 i"( C1 { t) ) o -hv -- f- f::- ( o 1 L-) o
0 ()<.) =- c o ... t-- X-;:- t n, L l ®
.! 't ' I rr ' . .
-=-)
N c J r (_)
L-r ( , , A 1 - [) ) cp (-x:) ( _ (fn. . • ) . o-;,- - 1 elY
l)
TYUS i~ /o
(Of f:,.,. ~
, ( L . I
- o- - . I A ( f ) 0 X .- 0 ) l:
'-----v----------
\1'~ 1..( ~~ bv >o s1r-t9. l l. IS ()..
•VI Levie< OlJW'O 1(1 S ~ tA (). r.e. .
0
\} V\J f -v:-0 \\ { L) A ( L) ( G"" ( L) -- {: ) - (_)
I I ./
\J( c) - t- ~I E u "-~ A f\J N g c. ! / (~ii\(L '-JI(L) & (\(1...)/b)
\-.l Oft-·. l~c 'NCC\k:. f~'{TI' (l1Yr~r1 / ~-(\~ ~~ Plf\Cl1\-Ct i?oLu1Jo1~
(.O{Idt~Dfl \Jo~J in·.
(
\.
(2.o.LJ 1hf. Str:i · 'l ~ 'Noes~
v=- r ·J ' I X
l OlT f- l" \(i ( ' - .l.
dx
tJio)~ Vo
') (I -' • J +-
- f\ rf'{t~Yi (Y' C• h :-'(!~ fJ~
~\!\-{> ~ v 1 (, d s
.....
--)
J J.
--1...-
\\Jf f.\~ r r r-- r-A : -
r, (\ J V\ & s -:: r u l u ( [") =- \1 01 St' ( ltl + \ '\Ct t-
(' ' .., ;.J W E '-'l- ~ JL Vv • iV l (;) -- 0 J
I
r- ;-J w,-x CJA dx = )"'1-f-Acix. + 'NfL)tl
0 0
N i(\Q, ~ h i'Y'.).. 'vi ~ 00 . ..J
ConStntc:t- Opp(l)'XfrYlCL~Cn.:. •r -G'r · ~c-ckN." f.··c~'o. i fv .. /\c hoY' S:.plU~ . ~-~: ~-x,)) Y1~o)l
~S)-y\(,1-- ih.C Soll,.JJtwl SpaLl ~ 'NO.•j~·}-l('J -~v Y'C~10tl 3DaJC
. ( sv,b3JJ-
fln6 L(~) E; s~ c. g <kn cro ~c ('
; )'I 0 ??il)X( m o,Ji Cf\
of LA [ u ''s~.q::J' k)
J' lc ==- f v."" e y, ( o, 1...) \ ,/' ( o J = v "1 \ I \f\1~ L 'A-;-\<~ C vu
.S\AU..1 -H, od- -r c:; u
'\) 'o ~ i \\J h (:; ~' ( o, L) \ '1\J v. [o) = 0 j
1 1 r • I I>
L
L
( ·~ r , \ VI· 1 ~o'Y .!-
)
0
\_
I
f \V\\k- D\fY\t 1\)iona \ - ---\N.QC\ l ror(•(1.
,, (1akr~in
' I Nra ~
rrviY''
- --
) w,., IT A J X "
0
f lJ\1 ~ ~. J~ L 'N ( L H h-
(
I J ,.. I ! \ I I r I I ·- ) r V,ll'l( 'f't ' ~ 1
'r S - MJ ~t f( t· <;; n fi_S.. 1 : co~( f 1 tr 'cr ( hr•(' ~~" ... I
I ((Cll\ v'' f. J£,_.:- [ u'· E= 1-f (r,L) I u"'!c) = ll, r t
C'DV'I. iJ.t r () (v,Y•tf"ICn I : (o1 L) ~ [/(._
lli.:-ir\C H\f 1u,V1c.hDt1 \j +r \:21:: CUl L2-·tLt~l(~l6r\. P-- i...
J Z I V OX- < o..._
0 ~i.e. H'\.t +~,~..~1chon ,~ boV\.t-1~(J ov" H\..0 tVlttvvCLO
4 i.-e. ihl hAY~tiio" v is L'- il i~ iS sqv<Ar~ 111 k' g v cdJ LQ . .
\J(y) =:.
V(v..}==
\J(y)::
\1 (y) =- hx--x:.) f:.- L.. ( o, ~__) 1
b.( I tC\ ttA~" di [H I
L2-
(
l
Or1 c etn \ n O:..c,HC 'f oJ rk ~ ~tt '-'
~ow a f.. :,a.+ cr.,,lrc~ cv"'f ~\\( du'ivcLhv€~ o~ \J? ( R(jVviClv'+j) I v ('I) e H- ( o, L; ~~ .
j[\J'-+L,k,, '(\ Jx < C<J
0 ~ -to .er1sure cov~c.J. LHliJ$.
R ei\11 a~ k:: ~ has bu Vi \(I rrv J{ I c.c 6. for d q'Y\.() n $. l CY\ oJ pt q fO~(' !... '/J
l n 50' tv o..~ 'J1 ~. \'Y\. ( C' 1 L) = L ) J . S f:)~th o.J cl\ l'n.Q vl)fo
} I d.:: \ fa-r I d\lh(l
1\t\.f n.s t \ re o { H\C C1cm Ct HL
L
\1 f K'(o,L) if J rv, • (M(o,LJ)'- v,.._2.},Jx. <. 00
C> \ __..,.~-- -, r- --____,
\JW o,\"E c t·VlfYt \It 0 _9 \1 ~ \f~ f \ r~.: h r c:1t V\ \J e,~._-nv-e .
Nt w: \Ah c J .:_ 1 u ~ {:; ~ I ( 0 I L) \ \A~ ( «>) ,-- iJ' ' ~
u e \At.., I~ b 0 t.;- h de. J 0, n d ~ o I S. U ~ x. .
"' s.q~oJ-e.. 1nk2ro.hlc..
-8 ·~. \·f ( O, LJ ~V\ ChOY\S
\J ( "?) -= w Yl s h:u tl-n \.::..
' I (1-.):::: 2 0 <.. 'Y \(:o v l I \•rJ_o ... :r---_
f '
\
\ -\--)X 7--!;) L
I·" r 'ctJ \: ~ lv. ~ oJex~n ( F\(\\k- 'D1 n\Cll;\o raJ/~ ,jJQt\ l t:"onvl.
fi(\6 11 11. f s"- = t ~"-E- ~I (o,c-) \ \,\(O)o V.o 1 J I , 1.~ \ c_ I. '
o.! 1-{... (oAtoo 1Yl0 I!AnC.rlCl'
'>Me-\., +~o.cL+- \j W" t: 'iJ-k = { \N '- & HI ( o, L) \ ·-N\ o) ~ D 1 L
_ ( 'N \,. ~ j.. 0 x ~ v·}''( LJ t t··. )
0 0
!:' -\JIU~ IS ~\v£.Vl I $:0 / (
Vk = ~ lf..~x. ) f lX) \'\o ~W-d tu 0-ffrt'<lrt"\(t)t. ( 1..JO,I {i tlt~'t( ~ e:siYi'\f.f,SICll
1 (\.' ~ olso ~\c \A> f ~ -t > buJ O\" · ~ i ·r- I D caso..
H-ow do \Ne_ obt-cun "~"' , 'J\l h ? C\ H-e vr"o..-t c (cf, f:;.. ;A;- h. ? =l t='a.xh~oVl Co,_) itV-D ''f\Yl(tf e).(..r-,vu4s'' Wh-ich. o~
d1·S k-o i h-t S'.Jvbc~ovv1 cc i 1 '<; D~ ( o l L).
------r -- ~ -__ l
_n = ( {) !..._) ..... _ \ I
'1' 9en.e ~ cd JoMcnn
D I (I L"' I ( e. J1~ I I \01 :.; HoY\.Qo .J _ :'f'v, u Sv odoYnCL·u· ~ - L ) (S O.VI ~ <;!At:: ootncti
JL s +~ c)c~l!t"e- ot _f)_ _ ./ tJo~A.noo.f~ pot f\1 s c-~ Q.
_n_- J1U JJl
\.
( ' Q, - ( e.. e ~ l) _J ,_ - X. 'X.
\ - - ..- .,
dor:. net- r·i1•!1tt1 \·<; tVdcd poti'\~S. ---
- ( t-- - ~ ' ht <; ( on\ 'll'l \. )( 1
1\1'< t> 'X' 1 j
t). -~ ---- --, - _____ J
-t tl- -~~_/(~._~ _ 7N ~ ()€ -+- -y c ~L .v
L
N '/'\·I 1 "':=:.. f ' ..,._ Y·O~C'J Qf.
----...,_, - r-----
N I L
\:j.Q~H\Q. \oto...( bos:\s {\A.YKhOr'.S. OY1 J'Le.. ___ -(\(\·\~ h·Arnbtr o~ v o S.t~ ttAn ch mts:' - o '~'<? (' - (25- rt nd -\ ~ re for€ 0 '/'e r J2 .
r
\we ba~\~ ~t,lr\(~lOAS 0\JC?..r C ... "·.. fc\'-11\C>''~''cSt) CC'rr,rAcfc.. \· (is;(
d~ : Q
1\ f \ .)... \""f..,} -=
e
Q __ ( L
f.\- I L. )
rx.(L
N A ( -x.) : h 0 d oJ ba Sol ~ f I,U\ c h' 0 fl !;
d~ · t'IC)dCL~ d( 9\'€--f>S o{ f~~d CIYI..
SifY\·, \ ec r l'j -For w~ ( ;t} !
Nn <:.
w~ (-x)=- L N"Cx) c~ /', \........---r- .J / - '
n~l · I VJO<).- iS ~.( "'< -( r ~ .( ('(--f' C'l ()I'Y\. 'k IS. dt( ["Ct-evt+·.
\ Bltbnov- GoJ{J k.\n ~~\l \ ~~:dny~cz_
•-''\ ~
( ..... l '.;..)
0 r-....... .... -I- . \ ...... -~ ~-.)
V) .......
~ 8_-~ ~ 0 ~ '- ...::5
·--------{'I I
'X
dv..c lu t V..t I\"'-C1 f·P'-n ~ _.. )..,-... "'
N' ( >'--) = t\j C --x- c ~ '1 J :::= I
~\2tx) ~ I \2-{'X( ~))-==-
6\- u. n,J- d o VYlCA-i n
(a ko IV1Cd l--\.Q rr leU ' r cJ'1
J o t'Y) c· \ , ' )
N 1
( ~ ) ==- \ - f "'\ 2 Cf ) --- ( ) { 1-t-'F ~-\
A
I~
C-1
2..
~::I r.'l..
I
( \ i~ A- B s A~ :::. t O 1 f f\ 1 P.~ \./'0"--
k rt• .tc kt r ch lt f\
HI(~ ) +- N2 (~) "" 1-,_~ +
f'j' (-I) = l
[\J 1
[ I)= 0
(\f2 ( -l) ": 0
N 2 (r)~ I
~mu.r~: l1u \0(01 <kht1lhon o~ bo~tS ful'lctioY1S. {foJs
~ 0 ~~ 0 t;c.d \;u h) ;l<t he honL a~o cia k d w 1Hc. .o och n oJs, y ~ n P ··I 1 _()_ €..
With COIYtpa.d St.-t.pf>o(+ '" _\ L:- Ct..no
L e
---·
(
'f;- I (
()
~ 1- ~
_Q~ 't; ::- I
(
I
)
~'"A_.(: I I
-+-------
. l I r \ t" "-r + )'I
~ ~e (\
d~ ~ '( "
rx~- 'X-~
~z.."
'){' f-1
o------•
-· -~(. : {. c lf\ \.(1\:f- £.c I'~~ l L
1 \1 ( r.. 1\ L \ '0\tV ~ \.,{ S TD H J\ \) E
___ _..
~NO
I
' tJ " l'?
I"J I
t?·O I 1
... I '--
... , " I ... ~ ..,. .
\ "12. , I ('.
.c I
,o r n · · ·r f ~·- cJ , .. , i ~ ..,_ I
• ,..'r j_ nQ. ,.,~, •/,fr·l-~l- .
r ..-;:::; k.J3 r;. 2 \'f. _ ( ~ ' )~' ) h~ a l-
rrv c I I -+ k~l"( \(- 1\0 '2- .' n\ 0\f(' '( A. \1\J ~ ( (J)-= {J -:::-) V-J ~. I:_ t 12 G~ . (iZ)
(
\
' \ \
\
A
R-e C oJ(: N'(~}:: i- 'f. _,
2.
l ('('
I
=-? \ ,\ .r.; -:
) 2'1
< \ 1-- Ge_Gel -
- <"' 'r C,., ~ r I ' . I
r ')
\ - -:2
2_({\ -\'\~ ~I
~ C.l\ r--- 2 h-Q -~
~
,...,
1 1 (1 }11' k ~ .'
' \2 - ~ ) f. 2. .
\ . I • ' ' I
_''<JJ r 011 4- ., ' \:
1 (! · J 11 'r "4 ('! l 0
f! J
11-------I
2.
' I
J ; ~ ' I ,,,.... () . A CIX J'X
-;- J V'lh -t h d X, =
...n_e
r ,, \ e c~~-1\·i
2
;:>i ()'\.( Vl s \ 1)1, Cl (
_@
I QV\\tA CQrrQ<. v.f ~'.J.C'Ct !c. r t'fYVl -a-
' oJ H\.( '"''J· @ -clcl\L(J·t-
...--- ; . " . L I 'j( - ,~ 0 ")(
e. J h~"",. A J' c ·+ V · 1 ._ , -::: n S' r c_Q ~<'-L
~ ( ~ "' p\ Y' S" lL ....
,~
cz I
Jl"-~
I v I -(
(~ (-\ " o, , I I 1 I z ( {
hi 0 I J ,..-
__ _....
({)~tr ~~Ldiml frt'\Y' f ~.: n \.C \ ~ + \
- -
'~\'"'e ( ·1) - Z I \ =-, -~) c -~ e
I ' j ( I 1 ~ (' (• t' 1 '
- ~ I ' ( f' (' () 1 !\ I }
CD -
~-I~:"~ t= I I () ~( } f
-- - ___; \
( t>Vlh \ \;t,t h () Yl ·h fi~Y' t hr t c}·· t+ ~ \, c ( \ \\ nr n1 s 1
f
..t , n C. · "L • r, VltZ.t
-I 0
'€ 'I Q)
NOIT:
- - I \__-~
~AI '2 "" "'( 2) .1- ">\ ·1--=-L ! J ~ o Q :: L 0 "-.a P
A I A ,.-,L
Oe_,
S\ fY\ \\ a y\v : -'
kvor1rr ~9 v- ~~ lic1 pro ?<--Y ~ o~ f r I (
V\O~t{.' d.uy-ees o- .... r.?eoom 01 1 • · L r· •
1 ~ S..o :.{. noYl + ~z \d ,
I c _, 'tO l 'II.(, ox t~ Y1 tl drS:. ~ NOT t4ENE ~L\.. '{ ~}'! __ \/: : . 'oLd 1-Y~AQ. iw
4 . Q.Cfl '·'~ ~·(, 'l~rf-ors . , lv ~ -
tocoJ
S\iY\1 I o..r I~:
' \
-=- 0 01 I+ I- I
J"2 :::=. d \~'2-1 I
I -IJ I.!. "('0:)- l orv -
=
\VI (?, !1-. 1)'\~ ... 1 \ f- ~ \
d, -=- c., ~~-
dr
0.2-
= 0,('~
-12 c., ::. (".)r?1.'(' I -0 r-~Q ~ ~- 1'\-" 0 i" I
c~ = e
c Q.+A-1
'
. ~1 0-:iQ.. ~).llr'.(.<Q(' (}(
,J
\I
02. I'
12 .. z Dr,v o, 02
~- 6 -. I I \
d:L 03 oh~Q
J2 II
ch·c
(
-==- C1 + ~.\,'\ 1-,-
r,n i ~ ~ 1-(>r(\e\'J
~ ~$.QX'r ~ lj :
I --
2..
\ I
< C (\ ' C C ··· "' "" I 2 '~ "'« ··ettl hi - ' J---
1,.-._1 V\~ 1
I -1
tf!:-h2.
- '
' -I
r '. t (: ( 1 ~ 2' i r- (' -- j_
C.(/ f:"[.S'; \I " ( I- ..J
~
)
c (\fJ_ l.. ')
I I
I
\
<cz. C:h · .. CnQ.Q c\'\.(.Qi-1 I ')
()
()
• 1
' <c2
C3 ... Cn~ Cn~+ 1 ;f~
<cL c ...... c 1 ~~
[~.0~ lhi
Re0u.QJ:
__,_ _
~I
I"'
_I !_L h1 ' h"l.
_.L h'l.
I h~
+ <c c ,,. c c 2 3 n~ n.e.1 ~ , I
c n ~. '> r !-\ r.~- l_l~ -' ~ ;. , .Ql •• h fq ' h-z. Vh.
. \ i . -.t-- vh ~3 ~"L
b 0 0
_ _L
VI~
2lJ
, '
(_(l
I I
r, ... J?
-tA
7 - \
- \ 2
' ' f I ! , I l' , -• ( i .I(' ".U
r L
-· - --~ --
- I
l
'
-I 2... --\
- \ i '/r--/ ' ,, \lr.r11 \ ..Q ~'\ 9 t v' ~\. 1
(] ,..
0
tA
(
,..
{~hQ (
2
<c2 1'1 { l,~ .• , c~" I L '. .Q.I \ '\
c.-r
) , ~
b I (' \ . . I [A ,}.o i I ..... . ~
.... . '(
t - 0 ) • V\~
) l~AJ 0
0
..... I -\
Q I ~ 9. ::. Q ~ r ..{- N rAJYi 'I - ~HI (;'{ \N (i C\ k fo-nYI I ,- /.. ~
toco~: ·;l·o- £ f l Cc
' s -:.
~- " - I -r ~ - c - . -
(~)
~H'{ ~ _-:::=-
~ ...
f ( ...._
- ~ ~ h~ I 7 - , f I , . lr ,
2 . . . \ ~ j
\~
L
I I \ 1 · t?_ :- r
d~ I -I ~ - ( ~ I J
I ~ (. t - J, - L - ;' (- ~{..
A
~ k ..
' \<;, Sv J";\rt ..Qh-,G \C !Ylo::l, ,.._ L_
\
·- \.:- c
_, . ")
C-r '/ ' ~- (')
L.. t" 1(.
~ d E; \i(_YI - )
'\ \ o \Lo·-o -
r ( \ ~--
-
\ l J d ' 0 ~ - ~ 0.
r ,_ J·.f-o .
J·o -·
'' V' c tG\tlu:J' &r\ /(Jt' · oo.~
::::::. 0 In t W.. LCV'S\ oc Y1'0
Cfl ~0 •
---I
.. ~ I
-- & - a e "'(' ~~
_Q' ~~
w~=' ~ N2 (~) c~~! ' r J2.
~ SL1
'J ( . .--.. :_ ') ') ' ) r f\ l -~' J ) v,
"2. c, ~ /~-1 . e., " ·r ...
' I \/1:) ") U(LJ = ')~ ~
.J )
n!2.9-' ) < I 7- ) r h'l ( J L c<1 c~ A 2. l 1 I + n-,., \;:--
1-\ i 9 r~ < - O·u~R r f b \ '1 Y\ o vvwJ '6a <;, i c, { \il V\ 0-1 on S
r - • • 0
- .
• • • ____ ... --- .
I /
.,...,
------~-------~ ~ ~='
hln.(J~ 3
L
' r A-~~ C, T IQJ-
2
) ~/'Q_ -=- L N B ( 'X ( ~ )J c ~
t J '(~)~ -~(\-~) 2
N~(~-~:: 1-~2
IJO!~) = ·~ (i ~ ~)
B:::-t
1
(~
I
I ~-= - \
k 'f D Yl~ c ~ J;_( tc{ prop.e (~
. -- ·-~-~-=----- N (IQ.,
f'=--1 ~-o ,..___. "( ~-=\
(I I 'Y, -;--I
l r ,~_ B)
N ( ~ ) :: 1T ( f, -- ~ b'-·
1. r c t fT (1i"1J
g- '· .~~I
,
::(c_) -1-1
~l '5) v ~ '~~
(
---NV\~ l\ ( 0 ~ -~6) ~.,.\
e.-.~~
N(l~
IT (~A- ~B) \?>=I BtA
~ (f- f')r: P., I
'0-1 p.,
NV'~ (;<'"'A c 3.) Tf cl - cr 1.;- I BtP.
L 1 'f\ ( ~) -: \ ~
I
W 1s. -:>w s. or ~~~Js ~rrn
~(f)'f\.i!ck., ck \tc{
it/ ?YO VC 'f ~'{ ~
TVll~ +Crty) I~ 0 for e,-:: (.., ~ <:\()(,( \f I~(\ \)YVC\L-tC 4 .
-ihl V\J 1110 l~ pV\)d•J d- i~ 0 -
I -+- \f./' r L-) -t- A ~
\)l ( ,(}1\.L{-- Ne~Amo.hn
Pr.J -o 1.£.rvt .
(
J 0
-'L-R9 coJ!-C ·.
-i +~
N ~ ~ ~ dJ'i ( ~ \ ~ t I)~ ) " ~ " ~ t
[:1.o ~ ID com pvc '-t ~ 1 ~ - v.>.c + ~ 1 nvtr h 6 i lttvJ o ~ 1 ~-x mo.pp 1 "3 ~
( >:: ' ""') N .. (-e'\ A 'te i) =- L 4 J Y (J iSDpo.Y' O.r\'"\o+n c. IYl O-?f' (lf) .
f\ ::: 1 "' v
noooJ coordn,aJ-e~ 1r. pV1~s i(ru domCiiP
f
-x, ~ : Y\fi.id s-\~ ~od-L
:::::::; 'Y~ = r ~ - --vie.
2_
~-o
1 an3~ ~\if- o~ H tt g.c OYY .. dY'tc JV1Ct pp, n3 1 ~ ( DVlS i-cuJ-.
- A(htl.L fY'clpp;n3
) f IAJ ~'l [ /). U ~ x J 'X -=:: 00z N A ( 2 ) - C J ) ( A ( "') /V ;; ( 2. ) . J1' i ) ( A J~ he- e 'l L
8 )s: 1f e o X
r~O ..Q 1 __j,!... J2
{
_, ' I y>
J N '~ N J~Ol:J -, ( '2. . '.0
. \ • I oC. ) l~ . }~ f - .
l r
. -I
'
' ~ '
l ~ ' \D l.'}o.CJ • l't· I ) 'I
-I
j N·.~ ·~~ ~ ~ - ~J' r 2~ -1)\19 == ~-:4~ '-- -\;+'./a'>>{~ ~>, "- 1] -1 -1 -1
1 _, =-~ 'I) t- I - ( -~- I~ S N;C: N~~ n~ =- S M2~ - 1)h'1; )JJ; = i~ ~ ~
0 : _,
-I _,
·3 Oe_
_,
f .. I
- ' g - --- -~2 - lo ~ 3 J
k ~ -P.,.V.JY\.C td -e.. Sl-'l(h'-r~c:
L ~-01 ) NQ1~. (Cfll i d.t(' ~ w' ~ t\ h - jc~ N~~i~);i Jx "-' 2c.t l'h ~t~Y~ _fl..: e A ' 's; JL _\) 1
lGY1\~d-tY t A 0)~ Lt ~'\,\ hY 1)\. 0~ _5l.~ ( I
6 1
t
1 J 2 ~(~-I) d~ - L_~·~A~~ I ~~~d~
h.:.t l.... "' _, l;\~(1~ Mo.;tY\x v~ cl-c< Not-a.h Ort:
"' <. 0 ~ C 'e C: ) ~ (l. 1A" ~~ f\1: J ~ 2 J N2J~
il
-I
I
J I(~\ f \ ) ..o z '1) t ~ rt J ""'~ I
~s~(l_mbl\,} of ~\oloo__Q rt\C• tr\·'( -\JfcJx (iiV'l}loY1s
L_ J 'N\o-~ ~d"J(,.. Z J w11hh -\- '"""'qt A ~ ~ e n~
SL
V__oc~IL(: f{){ D\dc\~~t- Ntt.tn'\Ct..Vln t>ft:liolo.nr \A'v.lo)== lAd= 0
) \f\l"(o)=- o,
r.d <c). c2 /2(/\ L4''" ~13 -413
+ Z/c2~-~ en c2Qo~'/2(A. ~I /~o -4/3 7/lo J
d~ e:.:z... he-
d2Q-I -4 l· ---...
......____-- u----- 1/lo - I~ h~
7~j \S>l e t.P'MJ.•J- 0 - 4 .3 2f3 -v, j ~~!1
I ' - Lj I -,
<r}- o- I iflf ~ ~~(~ 'l b '3 . b
- G0
/ ' 1\'h 3 j I~ 2Q-' h 2~1 - ' - \1 +- c. c (., ) -2.. ·~
c- - _ _____; e=2. f) :1 • I I
~ ""J \'QSI .
. H, ~<tf 13 ~ eV'·~ q . -c (\. - 4/j ' --2.. 1/- (% ...
-- 4f?
IJb
0
~,,_
) W1~{. I
c
~-
- 4/3
(.
-1l~ (-rfto _,. lb)
- 4t:;
yb
= < (} C 1 c4 C s .. rf"'-• C?ru.!l CLn,i I > )
. -QY 0 J_ /') 2 Yl.QJ.
<( .. '3 Czv'ltlrt, ... - G .. /
·,~. }1 .cJ
4:(?:>
9[~
-4:/3
( f~~cj
0 u 0
YW? . ' . -4/ 3··.
7(~r ...
---+71, --'\{?
Yro
4/3 V3+ '!3
4:{?, '( -QTF•·
-~'-;
- 4(3
/ro
-v~ ~b 't/s ·13
-4;, ~
IJ 0 0
-\-
-41 I 1
f{c
Bf?> -4/2>
-4i· 7/~ ~ -
·:z>W-1 0
d 2t1~
12,-J' 0
d' (
d 2~'o/ I
~ <c? c?) ,, . c?·,(\ ~ 1;
J2- \ J1 I
~ c'
- < c "1 c ':) / .. CLt'l.(.(\ I I
)
0 \Ao
D ..., -'+13
1- 0 - 2Qci' 'h, h / 0
0
y3 tA 0
(~
F -5o, c ~T ~ ~__: ~I : J
Q.Q fY\tt t ~~. 0)- t:- 1"1\o.J { i 'f (OvY1fUK•dS <:~)~ c\ 1\ kt< I 'C\- .f'IV r(\ \iN' (l Y (1\~C .
(?-\ \?oY'Jw ,J \-\.-, or 'f ;yul'r•X \f: lo V1t'f ( G; do -\u ~ 0~ ~ua-J V tA--h G QO £-\'. ~ V ~"G f-; OY1S.
~· tJ.,d-~~ n~M s V\o.Vf I' ill_o/" co,-.-tv\l::u;ho-11 ~ _f 1ec~r dv-•- \:<1> -\ 1M u.\:..Q. o~ Q.uod ,-cc-tic.. 'Bal/IS. flA"'C. h flY'S.
(
Con<tld.tY +~ \)<Ao~ D\nckltf frvio\..QJ'v\.
( y.d \ 2 -\)
fin~ \Ah f. S h C J) J h_ { u h t- H' (S2) I ~l"(o )· ll.y"CL)' v
3\.1~~-~ j. '••ctt \} w" E- u--1- 'V, u\ i w IE::~ '(.52.) I W0/o)' OJ w'Y~}l
J /-
®
c.' c~~- ~ )-o )
-ov--'lVi -1
:,j c l I; - -'-' r AI\( l, VI+ l Pure) 'f'ru bk 11")
-:) \L- J - r -=- D -- -~~ :: F +- S.ftA.V'dcor d ~ r11 f-o c hf\C cy- p ~D ~J\c n/1 s
d c 'f::.- I ~ ( dd)ll I j-\' Sot u_h llY> 0 f i" oddl f' t'b blt f1-, _ J -
(\ iot
wQ: L w~ = 2-Q:. I
'c c, Beco.~tSR. ~~ a (~)-: c ot'St. t J 1 J g ( ~ )d tj ~ 'J ( 1 od+.)
_ (on~+-_, ;:
= f j (;!j,) '~•_t ~ 2 ( (Onlt:)
, .. W.c?. (' {.l Yt I rd(' f) y-cJ C.. ( 0 t, r t(t.rr/ S
-" x:c~d l tJ._ -
\.
\'") . }:: \ ~ - D w,- 2 \i\ \ ( I )
n \1\,\ =- 2, i : _,/ l' ~)
\AJ :: l l R\A.l(\ from
f( 2. :- ·; ~ { 3 ) \N z. -= \ ~nu )~ia'()
·<t -= -J3rs- \tJ = ~/1 Qv 0 c! v C\.-ft\ r'{) ,
V\ \(I~ -- 3) l . I
~1-: D w"l-: Efq
~ '1 : J:)/s- VJ - - ?j~ -?
~ClUSSICl \1 Qvu.(_\ \ a:h,d'(' ~<u1t \.,\,\~~ \Atf~y((J-fS fb\t.JltiOVV1ietVi Ot Ofcln··
\ if\.Oa. f s
())..0! ( s
----- v-~nt 1 c S , <
M C i t1\ ~- d i f'r'\.C.'<' ~ i on cd .!-y\ cJl 1-v\Q' r\nl~ (' ." INI'-f 5c(t-\.hofl ·.
I~ 0
s Le__
W :,o G(tA.f' ~ Q Fo L~ no rn.1 oJ1 B>o~ \ ~ r:::.,I\/1\...DD'YIS 'r.ovt \oc:..~h c~nJm;c~-cJ ~ I () ( ~ ~ DV\ui- e, S2.) . rol-
J
c"'(n) hr Y\>O -
'v- .;:--- ,, I "
I} o. = 2 r \ o e- -flo
). 1~ - (OV''-1(1.AvU~ J 'X- i '0 .52-0..
co(SL~) 11:-
to Y'l ~ r VI o l) s. !
Y G0 (SL) v-v----- .
I'Y' (}.) ~ \ Q VV\ C\.. t-1 c.. \}"JO..( to 50-4 M Y\C \- -
·,s, COV'·h .. .AOvS
.- r tIN clOI'r'C\ f1 j
