Exam III  MA 174  4/21/00  D. Gottlieb

1a. Suppose it is 2 miles from town A to town B along a certain road. If you drive from A to B without stopping or turning around, what is the difference on your odometer between the mileage at town A and the mileage at town B?

b. The above simple question illustrates what equation expressing length as a line integral of a curve in terms of a parameter t?

2. Write down the vector field F⃗ for the vector field such that F⃗ (P) points towards the origin and is of unit length for any point P in ℝ^{3}. Use the i⃗, j⃗, k⃗ vectors.

3. Find a potential function for the vector field F⃗ (x, y, z) = zk⃗.

4. Find the circulation around the ellipse x^{2} + \frac{z^{2}}{2} = 3 of the vector field F⃗ in Problem 3.

5. Suppose an umbrella’s shadow is the unit circle x^{2} + y^{2} = 1 when the sun is directly overhead on the z-axis. The umbrella can be described by the equation f(x, y, z) = 0 for some f whose gradient is given by xi⃗ + y^{2}j⃗ + z^{3}k⃗. Set up an integral giving the surface area of the umbrella. (You may leave z in your answer, since the information given does not completely describe z as a function of x and y.)

6. State Stokes’s theorem with all its hypotheses.

7. Find the curl of the vector field in Problem 3.

8. Is a sphere (x^{2} + y^{2} + z^{2} = r^{2}) simply connected?

9. Let

F⃗ = \frac{−y}{x^{2} + y^{2}} i⃗ + \frac{x}{x^{2} + y^{2}} j⃗ + z k⃗

Now the curl of F⃗ is zero, and the circulation of F⃗ is not zero around the unit circle in the xy plane. Why can’t you apply Stokes’s Theorem to calculate the circulation on the unit circle? (If you are unsure of the reason, then show the curl of F⃗ is in fact zero and calculate the circulation of F⃗ around the unit circle in the xy plane.)

10. You want to use Stokes’s theorem to calculate the circulation on the boundary of the surface S shown below. Can you choose a unit normal n⃗ to use in the surface integral of Stokes’s theorem integrated over S? If so, what is it?