18.02A PROBLEM SET 12 Due in recitation Tuesday, March 9

1. (10 points) This problem is about the unit ball B in ℝ^{3}:

B = {(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1}.

The ball is divided into a top half and a bottom half:

B^{+} = {(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1, z ≥ 0}, B^{−} = {(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1, z ≤ 0}.

The two half balls are separated by a disk E in the x-y plane. (The E stands for “equator.”) The top surface of the ball is

S^{+} = {(x, y, z) | x^{2} + y^{2} + z^{2} = 1, z ≥ 0},

and the bottom surface is

S^{−} = {(x, y, z) | x^{2} + y^{2} + z^{2} = 1, z ≤ 0}.

Orient each of the three surfaces S^{−}, T, and S^{+} with a generally upward normal (positive z coordinate). Consider the vector field F = (z + 1)k.

a) Calculate the flux of F (upward) through each of the three surfaces S^{−}, T, and S^{+}.

b) Calculate the two integrals

∫ ∫ ∫_{B^{+}} div F dV, ∫ ∫ ∫_{B^{−}} div F dV.

c) Explain how to use the divergence theorem as a check on your answers to (a) and (b).

2. (15 points) Suppose u(x, y, z) is a function defined everywhere in space, and having at least two continuous derivatives. The Laplacian of u is by definition

L(u) = \frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} + \frac{∂^{2}u}{∂z^{2}}.

a) Suppose that D is a region in space bounded by a closed surface S. Show that the flux of the gradient of u across S is equal to the integral of the Laplacian of u across D: that is,

∫ ∫_{S} ∇u·dS = ∫ ∫ ∫_{D} L(u)dV.

b) Suppose now that we have found a function u with the property that the gradient of u on the unit sphere is equal to the outward unit normal to the sphere. Calculate the flux of ∇u outward through the unit sphere.

c) If u is a function with the property specified in (b), and B is the unit ball, show that

∫ ∫ ∫_{B} L(u)dV = 4π.

Explain why it follows that L(u) must be at least as big as 3 somewhere in B. d) Find a specific function u with the property in (b), and find a specific place in B where L(u)

is at least 3. (Hint: part of what you want is that the gradient of u should be perpendicular to the surface x^{2} + y^{2} + z^{2} = 1. What is the gradient of a function perpendicular to?)

3. (10 points) A roller coaster is designed with the parametric equations

x(t) = cos t, y(t) = sin t, z(t) = 1 − cos(3t),

The force of gravity on a roller coaster car is

F(x, y, z) = −mgk,

with m the mass of the car and g the acceleration of gravity.

a) Let C_{1} be the curve followed by the roller coaster from t = 0 to t = π. Calculate the work

∫_{C_{1}} F·dr

done by gravity on the car in traversing this path.

b) Same question, but now for the path C_{2} followed from t = π to t = 2π.

c) Now we’ll move the roller coaster to a tiny asteroid, where the force of gravity is

E(x, y, z) = −\frac{mg}{(z + 1)^{2}} k.

Calculate the work

∫_{C_{1}} E·dr.

(Hint: if you write out the line integral and compute, this is a little bit unpleasant. There’s an easier way.)

4. (10 points).

a) Suppose f is a differentiable function of one variable, and F(x, y, z) = f(z)k. Show that F must be conservative, and explain how to use f to find a potential function for F.

b) Suppose E is the vector field

E(x, y, z) = \frac{1}{ρ}(xi + yj + zk).

(Here ρ(x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}.) The vector field is not defined at the origin. Show that curl E = 0, and find a potential function for E.