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8/3/2019 Test 4A Solns

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3. Consider thevector f ie ldlrJr*a *2xev,xJz+4+x2zew,xy'

*x2yev' \ '\ ' 2Jz+4 |

; lv

r.{U

*Z-r

*ar"r^"rl^rd$ry.ar; (-, -4) L wt"4)vtt pV cla,/\prj<-\ oa\^d\tt' Nt't\t a,!z

a/-eM oA\A dS"Ah//\ (L,,u)=6, F r, c,b4€,qvoj-r).v-

= (uro,.:)

b. Find hepotentialunction.^

4,"$z r.=S{z14t

u

n - xt tf -tb

Zl -Zfq

J,f.Y#6t

I f1e1z

r fd*Jh(u=xg.lfi tr-.$'+$gO f=Uffi +f#+h(Ye)

' z ' Y zvqvt+ t \c

c. EvatuateI[tr:,2'r|,t,.+zxe.)ax(rJ, *a +x2zew)ar.h* r'v"wl,

14,1,5)€(3,z,o)

f=

' :3 t c 5 - 1 2 - o l

#.jn ." r(u,v) /2r"o.n,r, nv, 2)indhe quationf he angentlay! the urfacet(A"'8J,4\-,0

G= (Zc*r, srnv, ZtD

Fu= -2uanv,L^.6v,O)

vr--Z ')ctnv= 2G srny

C6vc{g V=

G, ,<)o)

n =(- 7-(3,-t,4) r U,tot - / - ; \

G(x-zfa),\-r) -r,- 1):- z t t - - o

rftfr ={; i;- t=l . ; G " l s,141 (",n, 2.,

lJ-gXZ

- i : +- - l

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S(&+2.\x{co'fu)d/H

IS3dA3G"*^%*;/")

6. Find hesurface reaof hepartof heparaboloid= x2+y2 with 2 < z <6.- =+- *

:

, . a t

d5='l(z*\'r(zrTI dA=rl4f t4tit r d,A

(+r+,)u'

S. UseGreen'sheoremoevaluateJf

.Arwhere =("t sin2y3x+2excrrs2y)f C s hepositivelyriented

c

boundaryf x2+y2=25.

-. [4u'11dA

6r a,lb

\\'[+r+l v^ ,'dAc

*''rft-

^{+.t r:lz al-z 54'-? t,1-C_1A-

J : *[,zs l=fl{rd-

3(r)(5Y

8/3/2019 Test 4A Solns

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7.

I : (,6rW)

ffi)&rd{I (+*

t | %dx=\:@q nr.r[o)r*Ift"u rr-Grldx

=( (u,fzuzJdx= Gt',r-trax[]

Findhe ux rFthroushrF= 0,"' ,0) ano,.,ili1',lriilJ"CXt Zwith1<y<band riented

towardshe ositive-axis.15-r

.; cls 6 = ( -Zy , t :ZzJ= ( - Z y , t ; 2 2 ) l

II rdAb

7'l{s^zrlll+[]\ :

\ \<",f ,O.?*,tF=) dAb

) r l+x'+--b

x'taz =57

f..3saott.*h ae

2

f zTf Gr

\. )ott"3e ' r d-,^dt

(* r d,'r

) D

d,fr @

1(-6{z rra)

1q-zlQ

- (rr.X?)

fd" c

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9. UseStoke's heoremoevaluatell"qf_yAlf .=!u,?!, x +z) whereS s hehemisphere2+y2+ 22='l ,

*,,nffintedupwardr*, \F.d; l.| c iv=.f'= l,z=o -t1-- cost,Jnt,o>+ 4)F'--(-srnt,-** ' , i l . t ' t )Azf Y

-ea\\t")= 3T

'10. EvaluatelJr.nos

when =(xy2,yz2,x2z)noS sthesurface f hesolid nclosedy hehemisphere

Pade

=JTL-cost,lJf(J]5:

zi f t ,1 [ '+ ]

lslpT5