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

308 0 CHAPTER 16 VECTOR CALCULUS
(b) r (O) = 0, F(r(O)) = (e- 1 , 0); 2.1
1 ) _ (l 1 ) F( ( 1 )) _ / -1/2 1 )·
r ( 72 - 2' 272' , .r 72 - \ e ' 4,72 '
F(r(ll)/
r(1) = (1, 1), F (r(1)) = (1, 1).
In order to generate the graph with Maple, we usc the 1 ine command in
the plot t ools package to define each of the vectors. For example,
vl :=line ( [0, 0),. [exp(-1), 0 ] ):
0 ~--....:.....:....c!..._ _ __ +i 2. 1
-0.2
generates the vector from the vector field at the point (0, 0) (but without an arrowhead) and gives it the name vl. To show
everything on the same screen, we use the display command. In Mathematica, we use List Plot (with the
Plot Joined - > True option) to generate the vectors, and then Show to show everything on the same screen.
31. x = e-t cos4t, y = e-t sin4t, z = e-t, 0 ~ t :::; 2rr .
Then:= e- 1 (-sin4t}(4) - e-tcos4t = - e- 1 (4sin4t+cos4t),
dy = e- 1 (cos4t)(4)- e-t sin4t = -e- t( - 4cos4t + sin4t), and dd.z = - e- L, so
dt
t
(
~~ y + ( ~; y + ( ~: y
= v(-e- 1 )2((4sin 4t + cos4t)2 + ( - 4cos4t + sin4t)2 + 1]
= e-t J16(sin 2 4t + cos 2 4t) + sin 2 4_t + cos 2 4t + 1 = 3 ..f2 e-t
Therefore
J~ x 3 1/ z ds = J 0
211" (e-t cos4t) 3 (e-t sin 4t) 2 (e- 1 ) (3 ..f2 e-t) dt
- r 2 -.r3.,J2e- 7tcos 3 4tsin 2 4tdt- 172·704 ..f2(1 - e- 14 -.r)
- Jo - 5,632,706
33. We use the parametrization x = 2 cost, y = 2 sin t, - ~ :::; t :::; 'i. Then
ds = (',~n 2 + (~~) 2 dt = v(-2sint) 2 + (2cost) 2 dt = 2dt, som = J~ kds = 2kJ::~~ 2 dt ~ 2k(7r),
Hence (x,·il) = · (~, 0).
35. (a) x = ..!._ { xp(x, y , z) ds, y = ..!._ ; · yp(x, y, z) ds, z = ..!._ { zp(x, y, z) ds where m = ./~ p(x , y, z) ds.
m}c m c m }c
(b) m = fc kds = k J~ -.r V4sin 2 t + 4cos 2 t + 9 dt = k vTIJ;"" dt = 2rrkJI3,
1 1 211" i 1 2-.r
x = JI3 2kVlJ sintdt = 0, y = M 2k v'i3 costdt = 0,
27rk 13 0 27rk 13 0
z =
1 M { 2 -.r (k Vi3) (3t) dt = 2
3 (2rr
2 ) = 3rr. Hence (x, y, z) = (0, 0, 37r).
27rk 13 /o 7r
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