Math 227 Problem Set VII
Due Wednesday, March 9
RR
F · n̂ dS where n̂ is the outward normal to
1. Evaluate, for each of the following, the flux
S
the surface S.
(a) F = (x2 + y 2 + z 2 )n (xı̂ı + y ̂ + z k̂) and the surface S is the sphere x2 + y 2 + z 2 = a2 .
(b) F = xı̂ı + y ̂ + z k̂ and S is the surface of the rectangular box 0 ≤ x ≤ a, 0 ≤ y ≤ b,
0 ≤ z ≤ c.
p
(c) F = y ı̂ı + z k̂ and S is the surface of the solid cone 0 ≤ z ≤ 1 − x2 + y 2 .
2. When a fluid of constant density flows through a circular pipe, its velocity is v = U (1−r 2 /a2 )k̂
where r and z are cylindrical coordinates, a is the radius of the pipe and U is the velocity at
the centre of the pipe. Find the rate at which mass is transported down the pipe. If the pipe
remains circular at each cross section, but the radius changes slowly along the pipe, show that
U a2 will remain constant.
a
z
3. Prove the mean value theorem for flux integrals: If F is a continuous vector field on a connected
surface S, then
ZZ
F · n̂ dS = [F(x0 ) · n̂(x0 )] A(S)
S
for some point x0 on S. Here A(S) is the surface area of S.
Hint. First prove that, for any continuous real valued function f and any positive function g,
ZZ
ZZ
f g dS = f (x0 )
g dS
S
S
for some point x0 on S.
4. (Hand in only parts (b) and (c) of this problem.)
Evaluate ∇ · F and ∇ × F for each of the following vector fields.
(a) F = xı̂ı + y ̂ + z k̂
(c) F = r̂ (the polar basis vector in 2d)
(b) F = xy 2ı̂ı − yz 2̂ + zx2 k̂
(d) F = θ̂θ (the polar basis vector in 2d)
5. Does ∇ × F have to be perpendicular to F?
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6. (Hand in only parts (a) and (c) of this problem.)
Verify the vector identities
(a) ∇ · (f F) = f∇
∇ · F + F · ∇f
(b) ∇ · (F × G) = G · (∇
∇ × F) − F · (∇
∇ × G)
(c) ∇ × (F × G) = F(∇
∇ · G) − (∇
∇ · F)G + (G · ∇ )F − (F · ∇ )G
(d) ∇ · f (∇
∇g × ∇ h) = ∇ f · (∇
∇g × ∇ h)
(e) ∇ · (∇
∇ × F) = 0
(f) ∇ × (∇
∇f ) = 0
(g) ∇ × (∇
∇ × F) = ∇(∇
∇ · F) − ∇2 F
7. Let r(t) be the position of a particle of mass m at time t, v(t) its velocity and a(t) its
acceleration. Suppose that F is a given vector field and ma(t) = F r(t) .
a) Show that
d
mr(t) × v(t) = r(t) × F r(t)
dt
(i.e. “rate of change of angular momentum=torque”).
b) In the case of planetary motion (about a sun at the origin) F r(t) is parallel to r(t).
What conclusion do you draw?
c) Show that a planet moving about a sun under the influence of gravity remains in a fixed
plane.
8. Let D be a subset of IR3 with the property that each point of D can be joined to the origin
by a straight line segment in D. Suppose that ∇ · F = 0 in D. Let r(t) = txı̂ı + ty̂ + tz k̂ be a
parametrization of the line segment from the origin to (x, y, z) in D. Define
G(x, y, z) =
Z
0
1
tF r(t) ×
dr
(t) dt
dt
Show that ∇ × G = F throughout D.
Reminder: Midterm II is scheduled for Wednesday, March 16.
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