Calculus IV Exam 3 Fall 1999 12/2/99

 
 

Each problem is worth 10 points. Be sure to show all work for full credit. Please circle all answers and keep your work as legible as possible. Not liable for consequences of acts of God.
 

1. Compute  where the path C is a line segment from (1,3) to (2,0).
 
 

2. Compute  for F(x,y,z)=yi+(x-z)j+(2-y)k for any path C from (0,3,-2) to (2,0,1). How do you know the answer is independent of the particular path chosen?
 
 

3. Evaluate , where C is the triangular curve consisting of the line segments from (0,0) to (1,0), from (1,0) to (0,1), and from (0,1) to (0,0).
 
 

4. A really goofy cult leader has predicted that at midnight on New Year's Eve 1999, every point within and around the Earth will suddenly begin radiating jelly beans according to the vector field F(x,y,z)=x3i+y3j+z3k. Just in case this incredibly unlikely event should come to pass, compute the total number of jelly beans that would radiate outward through the surface of the Earth, i.e. the flux integral where S is a sphere of radius approximately 4000 miles.
 
 

5. Compute the surface integral  where F(x,y,z)=yi-xj-2k and S is the surface of the paraboloid z=x2+y2 (with upward orientation) within the cylinder x2+y2=4.
 
 

6. Use Stokes' Theorem to compute the surface integral  where F(x,y,z)=2yi+j+xyk and S is the surface of the paraboloid z=x2+y2 (with upward orientation) within the cylinder x2+y2=4.
 
 

7. Show that for any function f(x,y,z) with continuous second partials, curl(grad f) = 0. Make clear how you use the continuity condition.
 
 

8. Show that Green's Theorem is a special case of Stokes' Theorem (where the surface lies entirely in the xy-plane) by applying Stokes' Theorem to F(x,y,z)=P(x,y)i+Q(x,y)j+0k.
 
 

9. Buffy says "Like, Calculus is so unfair. My professor gave us this test, and we were supposed to show that for this line integral thingy it didn't matter what way you went, like the path or whatever. So I like, did it for two different parama-whatevers, and, like, I got the same thing both ways, which is like a miracle for me anyway, but my professor wrote all this bad stuff that I totally don't understand about how that wasn't the right thing to do. She gave me almost no credit at all! But, my God, how else could I do it? I mean, it's not like I have enough time to do every way of connecting those two points, cause there's, like, lots of them."

Explain (clearly enough for Buffy to understand) how such a thing can be done without actually trying an infinite number of paths, and tell her what's flawed about her approach.
 
 

10. Show that the surface area of a unit sphere (with parametrization x(u,v)=sin u cos v, y(u,v)=sin u sin v, z(u,v)=cos u) is 4.
 
 

Extra Credit (5 points possible):

You might think the surface area of the ellipsoid with parametrization x(u,v)=2 sin u cos v, y(u,v)=2 sin u sin v, z(u,v)=cos u would work out very much like that of the sphere in problem 10, but it turns out to be much harder. See what you can do with it (estimates might be good).