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In-Depth Information
is denoted by L
k
(
V
). By convention,
V
0
=
0
and L
0
(
V
) =
R
. There is an
exterior
or
wedge
product
r
()
¥
s
()
Æ
r
+
s
()
Ÿ
: L
VV
L
L
V
that is distributive and associative. (The classical notation such as dxdy is really an
abbreviation of dxŸdy.) Define
n
Â
()
=
k
()
L
V
L
V
.
(4.25a)
k
=
0
Vector addition and the product Ÿ make L(
V
) into an algebra called the
algebra of
exterior forms
on
V
,
The defining properties of the algebra of exterior forms on V:
(1) If wŒL
r
(
V
) and hŒL
s
(
V
), then
1
rs
Ÿ=
()
Ÿ
wh
hw
.
(4.25b)
In fact, for any permutation s of {1,2, ...,k} and a
i
ŒL
1
(
V
), we have that
=
(
)
aa
Ÿ
Ÿ
...
Ÿ
a
sign
s aa
ŸŸŸ
...
a
.
(4.25c)
(
)
(
)
(
)
s
1
s
2
s
k
1
2
k
(2) Let a
1
, a
2
,..., anda
n
form a basis for
V
*. If 1 £ k £ n, then the set of all
aa
ŸŸŸ
...
a
,
1
£<<<£
i
i
...
i
n
,
i
i
i
12
k
1
2
k
n
k
Ê
Ë
ˆ
¯
is a basis for L
k
(
V
). It follows that L
k
(
V
) has dimension
.
(3) If a
i
Œ
V
*, then the element a
1
Ÿa
2
Ÿ ...Ÿa
k
ŒL
k
(
V
) satisfies
(
)(
)
=
(
()
)
aa
ŸŸŸ
...
a
vv
,
,...,
v
det
a
v
(4.25d)
1
2
k
1
2
k
i
j
for all
v
j
Œ
V
.
(4) If T :
V
Æ
W
is a linear transformation, then the induced map T* :
W
* Æ
V
*
on dual spaces defines an
induced map
k
()
Æ
k
()
T
*:
L
WV
,
(4.25e)
satisfying
( (
)
=
(
()( )
( )
)
k
()
Œ
T
*
a
vv
,
,...,
v
a
T
v
,
T
v
,...,
T
v
,
a
Œ
L
Wv V
,
(4.25f)
12
k
1
2
k
i
(
)
=
()
Ÿ
()
r
()
Œ
s
()
T
*
ab
Ÿ
T
*
a
T
*
ba
,
Œ
L
WW
,
b
.
(4.25g)
We summarize a few basic consequences of the above for emphasis.
