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

304 0 CHAPTER 16 VECTOR CALCULUS
11. F(x, y) ·= {x, -y) corresponds to graph IV. In the first quadrant all the vectors have positive x-components and negative
y-components, in the second quadrant all vectors have negative x- and y-components, in the third quadrant all vectors have
negative x -components and positive y-components, and in the fourth quadrant all vectors have ~os itive x - and y-compdnents.
In addition, the vectors get shorter as we approach the origin.
13. F (x, y) = (y, y + 2) corresponds to graph I. As in Exercise 12, aJI vectors in quadrants I and II have positive x~components
while all vectors in quadrants III and IV have negative x-components.Vectors along the line y = -2 are horizontal, and the
vectors are independent of x (vectors along horizontal lines are identical).
15. F(x, y, z) = i + 2j + 3 k corresponds to graph IV, since all vectors have identical length and direction.'
17. F(x, y, z)' =·xi+ y j + 3 k corresponds to graph III; the projection of each vector onto the xy-plane is x i + y j , which points
away from the origin, and the vectors point generally upward because their z-components are all 3.
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The vector field seems to have very short vectors near the line y = 2x.
For F(x, y) = (0, o'} we must have y 2 - 2xy = 0 and 3xy- 6x 2 = 0.
The first equation holds if y = 0 or y = 2x, and the second holds if
x = 0 or y = 2x. So both equations hold [and thus F ( x, y) = 0] along
the line y = 2x.
21. f(x, y) = xe"'Y =>
'i7 f(x, y) = f:,(x, y) i + jy (x, y) j = (xe'"Y · y + e'"Y) i + (xe"'Y. x) j = (xy + l )e"'Y i + x 2 e'"!l j
23. 'ilf(x,y, z) =f:z:(x,y,z)i+fy(x,y,z)j +f::(x,y, z) k =..) 2
x i +.J
2 2
y j +..)
2 2 2 2
z 2
~ k
x +y +z x +y +z x +y +z
25. f(x,y) = x 2 - y => 'ilf(x,y) = 2xi - j .
The length ofV' f(x, y) is ..)4x 2 + 1. When x =I= 0, the vectors point away
from they-axis in a slightly downward direction with length that increases
as the distance from the y-axis increases.
2x
4y
27.Wegraph\lf(x, y)=
2 22 i + 1
2
l +x+ y +x+ jalongwith 22 y
a contour map off.
The graph shows that the gradient vectors arc perpendicular to the
level curves. Also, the gradient vectors point in the direction in
which f is increasing and are longer where the level curves are closer
together.
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