M
427L
Differential
form
Practice
for
the
third
exam
6. Explain why a 2form on
R
n
is the same thing as a antisymmetric
n
×
n
matrix.
7. Let
F
= (
F
1
, F
2
, . . ., F
n
) (take the corresponding column vector) be a vector
field on
R
n
.
Let
ω
=
∑
F
i
dx
i
be the corresponding 1from.
Show that
d
(
ω
)
has the same data as
A
, the antisymmetric part of the matrix
D
=
D
(
F
), i.e.
A
= 1
/
2(
D

D
T
). Explain why for
n
= 3, exterior derivative of 1forms is the
same thing as curl of vector fields. Explain why exterior derivative of 2forms
(on
R
3
) is the same thing as divergence.
8. Let
f
:
R
2
→
R
2
be the map
f
(
r, θ
) = (
rcos
(
θ
)
, rsin
(
θ
)). Compute
f
*
(
dxdy
)
(here the first
R
2
has coordinates (
r, θ
) and the second has (
x, y
)).
9. Compute the exterior derivative of the 2 form
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 Spring '11
 staff
 Vector Space, Ω, Vector field, 427L, corresponding column vector

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