MTH 243

University of Rhode Island

Practice problems for Exam #3
                                                     Kingston 4/16/99




1.  Find the work done by the force F = (x + 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, x - 4y )  and let 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

C F dr

(Hint: Show that  F  is a gradient field)

3.   Let  F = (2x y - 3y, - y + 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 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 + y - 4 e ). 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.

(Hint: 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 , -  y , 2z  ).
 
(a)   Find a div F.

(b)   Find S F dA  where S  is the surface of the sphere centered at the origin with radius 5, oriented outward.
(Hint: Use the Divergence Theorem.).

8.   Let S be the part of the surface z = x+ 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 = (y ,4). Let  S  be the portion of the (half) cylinder of radius  4  centered on the  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 z , 2y + 1, 2z +   x ).
 
(a)   Find a div F at the point  (1,1,-2).

(b)   Use the Divergence Theorem to calculate the flux of this vector field throgh the unit cube.
 
(c)   Find the  curl F .

(d)   Compute the work  S F dA  where  S  is the upper hemisphere  x+ y+ (z-1)= 1  
 
(Hint: Use Stokes' Theorem )