Homework 7 – Math 118C, Spring 2010
Due on Tuesday, June 1st, 2010
1. Verify Stokes’ theorem for
ω
=
x dx
+
xy dy
with
D
as the unit square
with opposite vertices at (0
,
0) and (1
,
1).
2. Evaluate the integral of
ω
= (
x
−
y
3
)
dx
+
x
3
dy
around the circle
x
2
+
y
2
= 1 using Stokes’ theorem.
3. Find a 1form
ω
for which
dω
= (
x
2
+
y
2
)
dx
∧
dy
, and use this to
evaluate
integraldisplay
D
(
x
2
+
z
2
)
dx dy,
where
D
is the region inside the square

x

+

y

= 4 and outside the
circle
x
2
+
y
2
= 1.
4. Show that the volume of a suitably wellbehaved region
R
in space is
given by the formula
v
(
R
) =
1
3
integraldisplay
∂R
(
x dy
∧
dz
+
y dz
∧
dx
+
z dx
∧
dy
)
.
5. Verify the invariance relation (
dω
)
T
=
d
(
ω
T
) when
ω
=
x dy
∧
dz
and
T
is the transformation
x
=
u
+
v
−
w
,
y
=
u
2
−
v
2
, and
z
=
v
+
w
2
.
6. Let
F
=
F
1
e
1
+
F
2
e
2
+
F
3
e
3
be a
C
1
mapping of an open set
E
⊂
R
3
into
R
3
. With this vector field we associate the 1form
λ
F
=
F
1
dx
+
F
2
dy
+
F
3
dz,
and the 2form
ω
F
=
F
1
dy
∧
dz
+
F
2
dz
∧
dx
+
F
1
dx
∧
dy.
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 Fall '09
 Garcia
 Math, Topology, Empty set, Open set, dy, Closed set, dω

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