Solução_Calculo_Stewart_6e

F.
270 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
11. F(x, y) =hy, xi corresponds to graph II. In the first quadrant all the vectors have positive x-andy-components, in the second
quadrant all vectors have positive x-components and negative y-components, in the third quadrant all vectors have negative x-
and y-components, and in the fourth quadrant all vectors have negative x-components and positive y-components. In addition,
the vectors get shorter as we approach the origin.
13. F(x, y) =hx − 2,x+1i corresponds to graph I since the vectors are independent of y (vectors along vertical lines are
identical) and, as we move to the right, both the x-andthey-components get larger.
15. F(x, y, z) =i +2j +3k corresponds to graph IV, since all vectors have identical length and direction.
17. F(x, y, z) =x i + y j +3k 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.
19.
The vector field seems to have very short vectors near the line y =2x.
For F(x, y) =h0, 0i we must have y 2 − 2xy =0and 3xy − 6x 2 =0.
The first equation holds if y =0or y =2x, and the second holds if
x =0or y =2x. So both equations hold [and thus F(x, y) =0]along
the line y =2x.
21. f(x, y) =xe xy ⇒
∇f(x, y) =f x (x, y) i + f y (x, y) j =(xe xy · y + e xy ) i +(xe xy · x) j =(xy +1)e xy i + x 2 e xy j
23. ∇f(x, y, z) =f x (x, y, z) i + f y (x, y, z) j + f z (x, y, z) k =
x
x2 + y 2 + z 2 i +
y
x2 + y 2 + z 2 j +
z
x2 + y 2 + z 2 k
25. f(x, y) =x 2 − y ⇒ ∇f(x, y) =2x i − j.
27. We graph ∇f along with a contour map of f.
The length of ∇f(x, y) is √ 4x 2 +1.Whenx 6= 0,the
vectors point away from the y-axis in a slightly downward
direction with length that increases as the distance from
the y-axis increases.
The graph shows that the gradient vectors are
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.