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 responsible for damage due to clogs.

1. Find , where C is a line segment starting at (0,1) and ending at (,-1).
 
 

2. Compute  along the counterclockwise quarter circle from (1,0) to (0,1).
 
 

3. Compute  for the path C consisting of the first-quadrant portion of a circle (centered at the origin) of radius 3 traversed counterclockwise, along with the line segments from (0,3) to (0,0) and from (0,0) to (3,0).
 
 

4. The Earth constantly radiates heat into space, partly due to internal cooling and partly due to reflected energy from the Sun. Suppose this heat is radiated according to the vector field F(x,y,z) = 10xi +10yj + zk. What is the flux of this vector field through the surface of the Earth's upper atmosphere, which forms a sphere about 7300 kilometers in radius?
 
 

5. When water drains out of a bathtub and swirls down a pipe, the vector field representing the velocity of the water might be modeled by F(x,y,z) = -yi + xj - 5k. Suppose the pipe is a cylinder of radius of 2 cm (centered around the z axis) and has a filter screen shaped like the plane 2x + 3y + z + 6 = 0 in it. Compute the flux of this vector field through this surface (assume the surface is oriented positively).
 
 

6. Show that for any vector field F(x,y,z) = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k, so long as P, Q, and R have continuous second-order partial derivatives, div(curl F) = 0. How is the requirement that the partials be continuous necessary?
 
 

7. Buffy is having trouble with conservative vector fields. "Oh my God, I just, like, don't get it. I mean, like, we looked at this one, you know? And I, like, told this guy Biff that I was studying with that it was, like, conserva-whatever 'cause when you, like, draw a half circle going, like, either way around from up at the top to down at the bottom, it's like, totally across the arrow thingys so it's like zero either way, y'know? So I think it's conserva-whatever, but this Biff guy says it's not, but, y'know, he's not very smart, I think maybe, so I dunno, y'know?"

Who's right, Biff or Buffy, and what do you think of her reasoning?

 
 
 
 

8. Show that in any vector field F(x,y) = P(x,y)i + 0j, where P has continuous partial derivatives, the line integral through F along any vertical line segment C will be zero (this is essentially the portion of the proof of Green's Theorem that I hand-waved in class). [Hint: If you have trouble figuring how to do it in general, warm up with your favorite particular vertical line segment and a nice vector field like F(x,y) = xyi + 0j, then see if you can generalize.]
 
 

9. The parametric equations x = cosh u cos v, y = cosh u sin v, z = sinh u, represent a hyperboloid of one sheet. Show that the surface area of such a hyperboloid between the planes z = a and z = b is given by the integral . [Hint: Recall that (sinh x)' = cosh x and (cosh x)' = sinh x.]
 
 

10. Stewart gives a simpler formula for flux integrals through a vector field F = Pi + Qj +Rk when the surface involved can be expressed in the form z = g(x,y). Show that in this case the general version we used reduces to .
 
 

Extra Credit (5 points possible): How does the flux through the drain pipe in problem 5 depend on the slope of the plane and the radius of the pipe? Justify your answers.