1. Find the work done by the force F = (x2 + 2y, 1/ (y + 1)) acting on a body as it moves along the line segment from (1,0) to (2,2).
2. Let F = ( 2xy, x2 - 4y ) and let C be the path from (0,0) to (0,1) shown in the figure, consisting of a straight line segment and a parabolic arc generated by a parabola y =1 - x2 . Find the work
(Hint: Show that F is a gradient field)
3. Let F = (2x y2 - 3y, - y3 + 2 x2 y - 3x).
(a) Find a function f (x,y) such that grad f = F .
(b) Find C F dr where C is the oriented curve shown below.
4. A particle travels along the curve C given by r (t) = [cos t, sin t, t] and is subject to a forceF = (x , z, -2xy) . Find the total work done on the particle by this force for t between 0 and p .
5. Let F = (y - cos x + 4, - 2 x + y4 - 4 ey ). Find C F dr where C is the oriented curve bounding the region R whose area is 4. R and C are shown in the figure.
Use Green's Theorem.)
6. Find the flux of the constant vector field F = (2, 1, 4 ). through the triangle with vertices at (0,0,0),(2,0,0), and (0,4,0) oriented upward.
7. Let F =
(2xy , - y2 , 2z ).
(a) Find a div F.
(b) Find S
dA where S is the surface
of the sphere centered at the origin with radius 5,
(Hint: Use the Divergence Theorem.).
8. Let S be the part of the surface z = x2 + 3y lying above the rectangle in the xy - plane with vertices at (0,0), (2,0), (0,2), (2,2) and oriented upward. Let F = (y , z , x ). Find S F dA.
9. Let F = (y2 ,4). Let S be the portion of the (half) cylinder of radius 4 centered on the z axis, whose points (x,y,z) satisfy 0<=z <= 2 and y=> 0. Suppose S is oriented toward the z-axis. Find the flux integral S F dA.
10. Let F
= ( xz + 2 z2 , 2y +
1, 2z + x2 ).
(a) Find a div F at the point (1,1,-2).
the Divergence Theorem to calculate the flux of this vector field throgh
the unit cube.
(c) Find the curl F .
the work S
dA where S
is the upper hemisphere x2
+ y2 + (z-1)2 = 1
(Hint: Use Stokes' Theorem )