MAT 362
Answers to selected exercises, week of Oct. 3
Graded problems
1. The divergence theorem applies to flux of a smooth vector field across a surface that encloses
a
volume
. The fallacy in the argument lies in its use of Stokes’ theorem, which cannot be
applied here. Stokes’ theorem applies to a
capping surface
. A sphere is not a capping
surface: it encloses a volume (hence is not a “cap”) and has no unique boundary curve.
2. If
u
and
v
are smooth functions (which you may assume), then
u
∇
v
is a smooth vector field,
so the divergence theorem applies. All that is needed here is to show that
∇ ·
(
u
∇
v
)
=
u
∇
2
v
+
(
∇
u
)
·
(
∇
v
)
,
which you can do by writing out the individual terms:
∇ ·
(
u
∇
v
)
=
∇ ·
(
uv
x
,
uv
y
,
uv
z
)
=
(
u
x
v
x
+
uv
xx
,
u
y
v
y
+
uv
yy
,
u
z
v
z
+
uv
zz
)
=
(
∇
u
)
·
(
∇
v
)
+
u
∇
2
v
.
The second step follows from the product rule. (To simplify the notation, I’ve used
v
x
=
∂
v
/∂
x
and
v
xx
=
∂
2
v
/∂
x
2
, etc.)
3. This statement follows immediately from Exercise 2: just substitute
v
=
u
in Exercise 2
and use the hypothesis that
∇
2
u
=
0. The dot product
x
·
x
=
k
x
k
2
for any vector
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 Fall '06
 CALLAHAN
 Advanced Math, Vector Calculus, Surface integral, 1 1 w

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