FINAL EXAMINATION

MATH 2331

The point value of each problem is indicated in parentheses. There are 30 points. Show your work or otherwise explain what you are doing. Write in complete sentences. Begin each problem on a new page of your blue book in the order given. There are 12 problems.

1. Consider the vector field F(x, y) = ⟨1 + 4x^{3}y^{3}, 3x^{4}y^{2}⟩ on R^{2}.

(a) (1) Determine whether or not F is conservative.

(b) (2) Use the Fundamental Theorem of Line Integrals to find ∫_{C} F · dr, where C is the upper semicircle from P (−1, 0) to Q(1, 0), with center at the origin.

2. (2) Use Green’s Theorem to evaluate ∫_{C} y^{2} dx + 3xy dy, where C is the boundary of the semi-annular region D in the upper half plane between the circles x^{2} + y^{2} = 1 and x^{2} + y^{2} = 4, traversed in the counterclockwise direction when looked at from above.

3. Consider the vector field F(x, y, z) = ⟨xz, xyz, −y^{2}⟩ defined on R^{3}.

(a) (1) Find the curl of F.

(b) (1) Explain why there is, or is not, a function f(x, y, z) such that F = ∇f on R^{3}.

(c) (1) Find the divergence of the curl of F.

4. (2) Consider the vector field F(x, y, z) = e^{1−x^{2}−y^{2}−z^{2}}⟨x, y, z⟩ on R^{3}. Use the Divergence Theorem to find

∫∫∫ _{B} ∇ · F dV,

where B = {(x, y, z) : x^{2} + y^{2} + z^{2} ≤ 1} is the unit ball in R^{3}.

5. (2) Use Stokes’s Theorem to evaluate ∫_{C} F · dr, where F(x, y, z) = ⟨x^{2}y, \frac{1}{3}x^{3}, xy⟩ and C is the curve of intersection of the hyperbolic paraboloid z = y^{2} − x^{2} and the cylinder x^{2} + y^{2} = 1 oriented counterclockwise as viewed from above.

6. (2) True or False: If a, b and c are three vectors in a plane, then a · (b × c) = 0. (If true, explain why. If false, give a counterexample).

Date: Wednesday, May 5, 1999. Continued on back.

7. (2) Find symmetric equations for the line of intersection of the planes x + y + z = 1 and 2y − x + 3z = 1.

8. (3) Find the area enclosed by one loop of the four-leaved rose r = sin 2θ.

9. (3) Find an equation of the tangent plane to x^{2} + 3xyz − y^{2} + z^{3} − 4 = 0 at the point (1, 1, 1).

10. (3) Find the maximum of f(x, y) = xy on the ellipse 9x^{2} + y^{2} = 4. Indicate which method you are using.

11. (3) Find the volume of the solid region which lies inside the cylinder x^{2} + y^{2} = 1, below the plane z = x + 2y + 3 and above the xy-plane.

12. The attached contour map is of a function z = f(x, y) which possesses continuous partial derivatives.

(a) (1) Mark on the graph the points where you predict that the gradient of f is zero.

(b) (1) Indicate the points where you predict f has a saddle point.