MA 174  FINAL EXAM  5/7/99

D. Gottlieb Name We will use the following three vector fields in this exam. The position vector field:

r = xi + yj + zk.

The Coulomb vector field:

E = \frac{r}{∥r∥^{3}} = \frac{1}{(\sqrt{x^{2} + y^{2} + z^{2}})^{3}} (xi + yj + zk).

The rotation vector field:

R = −yi + xj + 0k.

(5 pts) a) For suitable F and region D, state the divergence theorem. (5 pts) b) For suitable F and surface S, state Stokes’ theorem.

ERRATA

2. Recall r is the position vector field. (5 pts) e) If the volume enclosed by a closed surface S equals 3π, what is the flux ∫∫_{S} r·ndσ

equal to?

(5 pts) f) Show there is no closed surface S whose normal vector field n is always perpen-

dicular to r. An alternative problem for 5c), worth five points: Find three real numbers whose sum is 9 and the sum of whose squares is as small as

possible.

3) Recall E: = \frac{r}{∥r∥^{3}} is the coulomb vector field. (10 pts) a) Show ∇ · E = 0. (10 pts) b) Show ∇(\frac{−1}{r}) = \frac{r}{r^{3}} = E where r = \sqrt{x^{2} + y^{2} + z^{2}} = ∥r∥. (5 pts) c) Find the flux ∫∫_{S} E · ndσ for a closed surface S not containing 0. (5 pts) d) Find the flux for S containing 0. (5 pts) e) Find ∫_{C} E · nds where C is a circle in the x–y plane about the origin with radius

1.

4. Suppose E_{1} = \frac{(r − a)}{∥r − a∥^{3}} where a = (1, 0, 0) and let E_{2} = \frac{r − b}{∥r − b∥^{3}} where b = (0, 2, 0).

(0 pts) a) True or False: The fact that the divergence of E is zero leads you to believe the same for E_{1} and the fact that the flux of E through a surface S depends only on whether 0 is enclosed in S or not enclosed in S leads you to believe the flux of E_{1} through S only depends on whether a is enclosed by S or not.

(0 pts) b) True or False: Your calculus skills are such that you could verify the mathematical statements in part a).

(5 pts) c) Evaluate ∇ · (E + E_{1} + E_{2}).

(5 pts) d) What is the flux of E + E_{1} + E_{2} through a sphere S of radius 1/2 centered at (0, 0, 1)?

(5 pts) a) Show ∇ · (aF_{1} + bF_{2}) = a∇ · F_{1} + b∇ · F_{2} where F_{1} and F_{2} are plane vector

fields and a and b are constants.

(5 pts) b) Show ∇(f · g) = g(∇f) + f(∇g) where f and g are differentiable functions of

two variables x and y.

(5 pts) c) Suppose F (x, y) = f(x, y)g(x, y) and suppose that F (x, y) has a maximum at (1, 2). If g(x, y) has a minimum at (1, 2) and if f has no critical point at (1, 2), what must g(1, 2) equal?

(5 pts) d) Evaluate ∇ · (r + 3E).

(10 pts) 6. Suppose the surface S is bounded by the curves C_{1}, C_{2} and C_{3}. Suppose ∇ × F = 0 on S and suppose the circulations ∫_{C_{1}} F · dr = 1

and ∫_{C_{2}} F · dr = −2 where the integrals are taken in the

direction of the arrows in the diagram.

What is ∫_{C_{3}} F · dr?

(10 pts) a) Find the work done by moving a particle of mass 1 around a circle of radius 1 around the origin of the x–y plane opposite to the direction of the force field R = −yi + xj.

(5 pts) b) Calculate the curl, ∇×R, of R and then use Stokes’ theorem to get a circulation integral around the boundary of a bounded region A in the x–y plane which gives the area of A.

8. ∫_{1}^{4} ∫_{0}^{\sqrt{x}}\frac{3}{2} e^{y/\sqrt{x}}dydx.

(10 pts) a) Sketch the region of integration. (10 pts) b) Reverse the order of integration.

(10 pts) a) What is the unit outward pointing normal vector at the (1, 1, 1) point on the

ellipsoid S given by x^{2} + 2y^{2} + 3z^{2} = 6?

(10 pts) b) What is the equation of the tangent plane to S at (1, 1, 1)? (10 pts) c) S intersects the sphere (x−1)^{2} +y^{2} +z^{2} = 2 (of radius \sqrt{2} and center at (1, 0, 0)),

in a curve C. Show (1, 1, 1) is on this curve C.

(10 pts) d) What is the unit tangent T to C at (1, 1, 1)?

(10 pts) a) Consider the set of all vectors x in space so that ∥x−a∥ = b where a is a constant vector and b is a positive number. What quadric surface does this set represent. Can you describe the roles of a and b?

OR

(alt. 10 pts) b) Where does the line t(1, −1, 1) + (1, 1, 1) intersect the ellipsoid 3x^{2} + 4y^{2} + 2z^{2} = 10?