Math 227 Problem Set VIII
Due Wednesday, March 23
1. Let F = (x − yz)ı̂ı + (y + xz)̂ + (z + 2xy)k̂ and let
S1 be the portion of the cylinder x2 + y 2 = 2 that lies inside the sphere x2 + y 2 + z 2 = 4
S2 be the portion of the sphere x2 +y 2 +z 2 = 4 that lies outside the cylinder x2 +y 2 = 2
V be the volume bounded by S1 and S2
Find
RR
with n̂ pointing inward
(a) S1 F · n̂ dS
RRR
(b) V ∇ · F dV
RR
with n̂ pointing outward.
(c) S2 F · n̂ dS
Use the divergence theorem to answer at least one of parts (a), (b) and (c).
2. Show that the centroid (x̄, ȳ, z̄) (where, for example, x̄ is the average value of x) of a solid V
with volume |V | is given by
(x̄, ȳ, z̄) =
1
2|V |
ZZ
(x2 + y 2 + z 2 ) n̂ dS
∂V
3. Let V be the solid in 3-space defined by
0≤z≤
9 − x2 − y 2
9 + x2 + y 2
Let S be the curved portion of the boundary of V oriented with outward normal and let
F = zy 3 ı̂ı + yx ̂ + (2z + y 2 )k̂
Assume that the volume of V is α and compute
outward pointing normal.
RR
S
F · n̂ dS in terms of α. Here n̂ is the
4. Find the flux of F = (y + xz)ı̂ı + (y + yz)̂ − (2x + z 2 )k̂ upward through the first octant part
of the sphere x2 + y 2 + z 2 = a2 .
5. Let E(r) be the electric field due to a charge configuration that has density ρ(r). Gauss’ law
states that, if V is any solid in IR3 with surface ∂V , then the electric flux
ZZ
E · n̂ dS = 4πQ
∂V
where
Q=
ZZZ
ρ dV
V
is the total charge in V . Here, as usual, n̂ is the outward pointing unit normal to ∂V . Show
that
∇ · E(r) = 4πρ(r)
for all r in IR3 . This is one of Maxwell’s equations. Assume that ∇ · E(r) and ρ(r) are
well–defined and continuous everywhere.
see over
r−a
6. Fix any point a ∈ IR3 and define the vector field E(r) = |r−a|
3.
(a) Compute ∇ · E(r).
RR
(b) Let Sǫ be the sphere of radius ǫ centered on a. Compute Sǫ E · n̂ dS where n̂ is the
outward pointing unit normal to Sǫ .
(c) Let V be a solid in IR3 with surface ∂V . Suppose that a ∈
/ V and a ∈
/ ∂V . Evaluate
RR
E
·
n̂
dS
where
n̂
is
the
outward
pointing
unit
normal
to
∂V
.
∂V
(d) Let V be a solid in IR3 with surface ∂V . Suppose that a ∈ V but a ∈
/ ∂V . Evaluate
RR
E · n̂ dS where n̂ is the outward pointing unit normal to ∂V .
∂V
7. Let V be a solid in IR3 with surface ∂V . Show that
ZZ
r · n̂ dS = 3 volume(V )
∂V
See if you can explain this result geometrically. Do not hand this part in.
Reminder: Midterm II is scheduled for Wednesday, March 16.
see over