Calculus IV Exam 3 Fall 1998 11/19/98

Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. Danger: Not responsible for losses due to unexpected thawing.
 
1. Compute  where C is a counterclockwise arc of the ellipse 4x2 + 9y2 = 36 beginning at (3,0) and ending at (0, -2).
 
 
 
2. Compute  if C is the portion of the parabola y = x2 from (0,0) to (2,4).
 
 
 
3. Show that the line integral in problem 2 depends on the path C taken from (0,0) to (2,4).
 
 
 
4. Calculate  where S is the surface of the rectangular box bounded by the planes x=0, x=3, y=0, y=2, z=0, and z=1.
 
 
 
5. When the ice sheet that forms over much of the arctic in winter breaks up during the spring thaw, occasionally a polar bear ends up floating off on a small iceberg which might be caught by ocean currents and swept out away from shore. Suppose this were to happen to a bear named Harvey, and he floated for weeks until the ice pretty much melted away to a piece just big enough for him to sit on. Then, unfortunately for Harvey, his ice raft gets caught in a rare ocean whirlpool effect like the vector field F = . Use a line integral to compute the amount of work (i.e., ) for Harvey to paddle his ice raft once around the whirlpool (counterclockwise of course) along a circle of radius 10 meters.
 
 
 
6. Explain how you can use the fact that curl(grad f) = 0 for any function f with continuous second partials to conclude that F = xyi - 2yzj + x3zk does not have a potential function.
 
 
 
7. Show that if F(x,y,z) is any constant vector field and G(x,y,z) = xi + yj + zk, then curl(FG) = 2F.
 
 
 
8. Show that the surface area of the helicoid (spiral ramp!) with parametrization x(u,v) = u cos v, y(u,v) = u sin v, and z = v, within the region S, is given by .
 
 


 
 

 
 
9. Biff says "So I was doin' this homework for calc, and there's somethin' I don't get. There's this problem about doin' the line integral of this function F =  on a circle with radius 2 goin' round the origin. The book says it's 2, but I used the Green's thing and got it to be zero. Is the book just wacked or what?"
 
 
 
10. Show that the flux of the vector field F(x,y,z) = xi + yj + zk outward through a smooth closed surface S is three times the volume V of the region E enclosed by the surface.
 
 
 
Extra Credit (5 points possible):
 
The book gave a couple alternate statements of Green's Theorem in the section on curl and divergence. In one of these the expression  is replaced with . Show that these are equivalent (Remember F = Pi + Qj + 0k).