18.02A PROBLEM SET 13 Due in recitation Tuesday, March 16

1. (10 points) This problem concerns an analogue of the theory of conservative fields using surface integrals in place of line integrals.

a) Suppose that

F = M(x, y, z)i + N(x, y, z)j + P (x, y, z)k

is a vector field in space, and that each of the functions M, N and P has continuous second partial derivatives. Show that div curl F = 0.

b) You might say (although no one does!) that a vector field G in space is superficially conservative if the flux of G through a surface S depends only on the boundary curve C of S. That is, if S and S′ are two surfaces having the same boundary curve C, then

∫ ∫_{S} G·dS = ∫ ∫_{S′} G·dS.

Show that if G = curl F, then G is superficially conservative. (Hint: use Stokes’ theorem.)

c) Show that if G is superficially conservative, and S_{0} is a closed surface, then

∫ ∫_{S_{0}} G·dS = 0.

d) Show that if G is superficially conservative, and the first partial derivatives of its coordinates are continuous, then div G = 0.

e) Suppose

G = (y − z)i + (z − x)j + (x − y)k.

Find a vector field F with the property that G = curl F.

2. (15 points) Let S be the cylinder

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

oriented with an outward-pointing normal. Let C_{1} be the top circle

C_{1} = {(x, y, 1) | x^{2} + y^{2} = 1},

oriented counterclockwise, and C_{0} the bottom circle

C_{0} = {(x, y, 0) | x^{2} + y^{2} = 1},

also oriented counterclockwise. Finally (to make a tin can) look at the top and bottom discs

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

Orient T_{1} with the upward normal k, and T_{0} with the downward normal −k. Finally, let D be the inside of the can

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

Consider the two vector fields

F = xi + yj + zk,
G = (y − z)i + (z − x)j + (x − y)k.

a) Calculate the curl and the divergence of F and G.

b) Calculate the flux of F, G, curl F, and curl G through each of the three surfaces S, T_{0}, and T_{1}.

c) Calculate the integrals of F, G, curl F, and curl G around each of the closed curves C_{1} and C_{2}.

d) Calculate the integrals of div F and div G over the region D.

e) Stokes’ theorem and the divergence theorems provide a lot of relationships among your various answers to parts (b), (c), and (d). Write down as many of them as you can think of, in the forms that would apply to any vector field F. (This means I am looking for formulas like

∫ ∫_{T_{0}} curl F·dS = − ∮_{C_{0}} F·dr

instead of formulas with your particular functions written out.)

3. (5 points). For the vector fields F and G of problem 2,

a) Find a function f with grad f = F. Is there another function f′ with grad f’ = G?

b) Find a vector field H with curl H = G. Is there another vector field H′ with curl H′ = F?