Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

330 0 CHAPTER 16 VECTOR CALCULUS
25. F(x, y, z) =xi - z j + y k, z = g(x, y) = .j4- x 2 - y 2 and Dis the quarter disk
{ (x, y) I 0 ~ x ~ 2, 0 ~ y ~ v'4 - x 2 }. 8 has downward orientation, so by Fonnula 10,
!1(
x2
= - - v' 4 - x 2 - y 2 y )
· + y dA
D v' 4 - x2 - y2 . v' 4 - x2 - y2
= - IIv x 2 (4- (x 2 +y 2 ))- 1 1 2 dA = - I 0
..,. 12 I 0 2 (r cos B?(4- r 2 )- 1 1 2 rdr dB
·= - .[ 0
..,.1 2 cos 2 B d8 I~ r 3 (4- r·2)- 1 1 2 dr [let u = 4- r 2 =? r 2 = 4 - u and-~ du = rdr]
= - I 0
..,. 12 (~ + t cos 28) d8 I~ -~(4 - u)(u)- 1 1 2 du
= - [~8 + t sin28]~ 12 (-~) [8v'U - iu 3 1 2 ]: = -t(-~)( -16 + ¥) = -~1!'
27. Let 8 1 be the paraboloid y = x 2 + z 2 , 0 ~ y ~ 1 and 82 the disk x 2 + z 2 ~ 1, y = 1. Since 8 is a closed
surface, we use the outward orientation.
On S 1 : F (r (x, z)) = (x 2 + z 2 ) j - z k and r , x r = = 2x i - j + 2z k (si~ce the j-component must be negative on S1). Then
fis
1
F · dS =
II [-(x 2 .+ z 2 )- 2z 2 ] dA = -I~..,. I;(r 2 + 2r 2 sin 2 8) r dr d8
o;2 +=2 $1
= -I~..,. I; r 3 (1 + 2sin 2 8) dr d() = - ]~..,. (1 + 1-cos 28) dB-I 0
1
r 3 dr
= - [28- t sin 28]~..,. [~r·4 ]~ = - 41!' . ~ = -1r
OnS2: F(r(x, z))= j -zkandr.., x r , =j. Thenifs F ·dS= JI (1)dA=1r.
Hence'Jis F · dS = - 1r + 1r = 0.
2 .,2 + ;:2 $ 1
29. Here 8 consists of the six faces of the cube as labeled in the figure. On 81:
F = i + 2yj + 3z k, rv x rz = i and IIs 1
F · dS = I~1 }~ 1 dydz = 4;
82: F =xi + 2j + 3z k, r= x r, = j and IIs 2
F · dS = I~ 1 I~ 1
2dxdz = 8;
8s: F = x i +2yj + 3k, r , x r v·= kand fis 3
F · dS = I~ 1 I~ 1 3dxdy = 12;
84 : F = -i +2yj +3zk,r: x r v = -i and IIs 4
F · dS = 4;
85: F = x i- 2j+3zk, r., x r: ~ - j andiJ~ 5
F · dS = 8;
86: F =xi+ 2yj- 3k, r 11 x r., = -k and J/~ 0 F · dS = J~ 1 j~ 1 3dxdy = 12.
6
Hence ffs F · dS = I: Jfs. F · dS = 48.
i=l 1.
31. HereS consists of four surfaces: S1 , the top surface (a portion ofth~ circular cylinder y 2 + z 2 = 1); S2, the bottom surface
(a portion of the xy-plane); Ss, the front half-disk in the plane X= 2, and s 4 . the back half-disk in the plane X = 0.
ll.