(r..) Febcurr y e r 4h,rv* Le..t u r e .5 o-f - - PDF document

4h

e3E

4h,rv* e r Febcurr y

Le..t u r e

Llo ltt +a, l' + 4,, Y e t thar*c tettt ftc poly uortt'a-( wcth cawtplcx rools

Two !e o.r'ti Ja fJ

• E

{uv Jo- '- avr f c( solatrcu! Fr c*' p /os .

P/6u..

F9qe,aorlL

Pfecs c

(r..)

.5

• -f

Ro"-' flgt A",t"wy tl*t(

7\,r F;n'61q.y

• .wo/

3 Jar Fn 'l*y 5 . \erf Flo'/rY. ar-nr- ]

'ed f a

• ,/owr

Yor 3cc h'e,u

rW

?-hnr,'

: 4't!\a fesuft

(3r co-t plet 6.'veb *' , a" d lR rcols

the aoE

(t)

tf tt e

P(rl -

chnro< te r * lt

ll t4rf +ac

P"/, t ovt"(

hr I e.w^?f"x ;ool5

f

• n. *+r'n/ , lF.. r-,r'lz

1.,

wrth t, F c lR)

• fter. arch taf

lr(t) , e

|( /r)

• . fu*dc-r" a
• r. f,'1
• f ai.

(r).

(r+,1)r

a't s?t

{

{ l(, ll,X I ,ir o- {"."!"we.fut sg!

• F at. (t)

cff '1, , '1, o-r. 1l', ,.t{ *r,y. t o

* Rrc,r(r r tt I X , y. ) r'! o. { u,arrfo rra 'o f,*/

sel "f

• 4. (l) ,

,hPh tt^ P (uwefftur

f(t\. c, 1,ft) + ct l. al ,

w,'fln tr t Ct a.tltlltey cousttt lt, tt co.tlcaf thc ? e be_ry'l toft.ttch

• t eg

. /r).

iro-u p /e

a"{ .'

F= at

rr-j

tnlu low

• P1

J (r)

Fi|.J the ote\P.ol

Y"

• 2 Y' + r l

lo"[t"t (o. so /.. f.'c'r s wc

• bf.r'-

th} ch,!. f,arfr

r +a = O

v;7 _2

2 r 7l-7

f tts =

t y'-r D-l

c'

e?.

2

z

2

l,i=,.rvE'l

Fqn o{evr

• ar. fo /

So/q t.o\t i

Tho i e he,-r/

s. /.

.'t

cl

y (t) s tcz

• tr-t

(l)t

• a-tV7)t

L(r\. e

2"E,e

e

(.+.VT)t

(e)

.<

* Re\.dc h ! 0) The so/ulrovr t" (2) tuzlqle rec{- uo- /u ty' ,rr. y' cou^ plcr- t4(4eol :c1q f,'cvt.

(1I Re!/- v./e"./ 'olq f,'ov.l ".re {rflic,att

fc t\ Lc cP*r-, fto*' .ou'tPlar. uvlbe.l to/u *r.ov r c\r (7).

6) S,.ce e|.4) l"r rf*( coefltcrerft) tt ts qro{u(

• .pp

/ta4 t'.ors t"

• b

fo.r vr the r,"ost i cbcfal rP*l- v*tQa.l So/efrov.

{.

Ar"-o/a ,

sg.

Cow.Pa*-

Y, atl -

e.t+.'Vi') t

Ff*o/ the rt!,/ - r,,,r.lue{ 9e}t.0,( se/c f,'o '^ rf

Y"-al'+tl'o.

J .,,

f ,r ru/o- r,..er' f,u ( r"/c f.'rr ct'? l

, Il',, , "('-''nd)' I

A"y /.,,r ecr cot.6zv|.'f.or So/q t r "-

• l

(I ). f ", pcrf, c,r /e e,

\ (t\

• L

L /,ttt

r's ,*ilt c

, ittt )

Li,

tt) ln tt)

). /tt --

• L.

zt Recatartt I h,r/

U at+ v? ,') a Uo-V?,) 1

?

e'

rt6't

• , v7t

<'

I{r\:

I t ./i'f

Le e

Y,/t\.

( u"t71

• ,vFr

e

• r'y's-,

J

e'

2

{.(e

LI

_.'/E t \

• e

)

' vll

frr) s

4 /,r.r'ur r

?hata fuc f uu c l, ot"s

?arl- ve,,l tl-p a( .

Rec*l1 ; Zhe Eu (p r (c.wq l*

rrlE t

c"s((i t)

+ i Stv, (g!7)

ilr

covplox- co- ju;cfe

'''vFt : co\ (,/Vt) - ,'sre (vq-t)

e

{

Thevr i

.'VE t

c+

r'/8"

• itl? d
• N?1

a

cos(\/€at)

zi S,:n ( Y7 71

ThtreSof'.

Y, (t\

t

cot (v7 t)

Yr (rI

e'

5i r.r ( Vs-

r)

e

3e s.1- urlrrr/

So/tr f,',rv3

f14ey r..- e -ol p ropor i'oro1 lo a..tr oft"e 7- lor tit,

so ths y

• ro r {,av/o.t-ar'l*/

sef for ef , /tJ .

Sol.

yft)

fo"

For

ct,

c.)

rt

rerl-v*lqeol

c"wplat

• V*

lucy'

c, e lR.

c, e e.

?e hc. a(

Yrt)

• [c,

coJ(/B-t) + cr S,h /VE-r)je

a:

• (.

Thf proo f rf Thrw. (clsc t") (o!lcw! exc-ctly ftr? so'r, d r'o{e 4'J *.'ve\ r!'

tha . exc-'F/e .:.bcvt rtpldccvrl the

roofi

t* r/Ea,

• r>
• ( +(1 r'

1- y'it,

• {:

,( - frl'

Thp leql- vo.lqe

• l

f,u.d^*c"f*/ sol .tte

i Rer^at|\ i

ct

cos (F t)

Iff\

• ). 1r)
• .(t

s... /p fJ

(

@r3,

Frrro/ mcl - y,r fu../ sclrr f.'or r "f

Y"+ e Yt +t =

i""a"*a" r"r /

, lo')

=O

• r r V-

eTt

• zr 2v-F

E!'

F:

P(r\.

• z

rz +tr +6

! V.t - 21

A.

(*<o)

Sc/e l','or.l

z

z

ln =

• ,

*rVr I

v7

t

• (:
• l

I Real- rz.r {trPc/

fu- "f+*' e'r f,* /

rr

c"s('E t)

r,ft) =

• t

.(

Row,o+ h.

riu /.'he

re d. ttcl f pel dc/tr.

Y, ,nl

r,\:'.-

(5) ./os zr,'lcp phystct(

fo dow;,O/ d.t llc ttorl wttr^ {.,'c f,'otr.)

procert t,

age3

So( r

• (so

/ =o

1 e hat-4/

• F. r,rol

+5

r.(. "f i

(1) )

't'

P(r) = rr+5

• 1.,-

vri I

r.-l

| ., - -Ve' , (

• G. VF.

y/r) '

c, coS (Vg- f )

i ca (gru*r lq

((7 t)

.-=-Jtnl

I

E+ r /r h.e

Pfoa ? St e:

t{)

• {cscr

wtth cut

t lce Thysc c

• (

dt s! t7 * troa

(f)

• ).

e"s (v7 t)

yzt(\ =

C,, CL e lR

(*t

e

r Aoc f,'co"fiovr r

Thp RLc .lrcu.'t.

Coutrefer

• '\.

l?e!t!f,!-<e !\,Jtrc l'o-"' a e

€(eztrt.

• etrcq. t

,r./,'f h

R , c*g* 4' l!7 C , c.v4

L . ?' v'r^ b,

l"+"\

\Lto,/

T.t)

Thr e

• lo. fed.

at'rctt.t l+ frt

c(t. t e c*erov | ,

€ lowiw1 t'w Sv,e h

a 9'rf ea, f

l.'e: r

A trtt) + R tr/t\ +

'.().li

=O

Den'y*f, e

both

I J./Al

e?,.h.vF

lgr'+Rfl

+r-I

_-o

It

l_l

cJt

.(

/ur'r.l

• le

f ,'r.,'Cro- r

phys,cs. )

f rl + L c r'

.l+

+-ttL

=O

(5J

:e!'

F-

so(t.

ra+

• eot

!

t rFl,F

d:

lrtt

• Vr.l,ae
• /

{q-d,r,-.4"'f*/

Ep , tS ). .\ €,'tc! t, .

be'o!t' '

+urtr = o

.l 4r - 4 El

I

L.

a_

2L

l-ec'

fo

a0(

Z+ie I i

I?

thew ura

R--o

It. $

C te

asc tl&Lta t

Th?rd FoFe I

fz/fl = s''vl

<f-.D

{

c.-l e l. ,'

, t heu

R1

lLL

*nz

rf

R.e <

(=1

G>

4L

a

lta

tttL,

• t. R

ZL

u=L

L'

I

I

Tha Fe srJ tav rt de,,?t tha t'a?r+- asct lb h, e,a s .

Thofa la F a.

I, lt) :

• |(t

cc5 ('

(

q.r,l

• cr

t)

t)

Ir /rl

• {i

c

,,,.1

• Ct'