Graphics Reference
In-Depth Information
C.6.9. Theorem.
Let T : T
k
V
Æ
W
be a linear transformation. The map T is alter-
nating if and only if T(ker Alt) =
0
.
Proof.
See [AusM63].
Definition.
Let
V
be a vector space. Define the
k-fold exterior product
of
V
, denoted
by E
k
V
, to be the subspace of alternating tensors in T
k
V
, that is, using Theorem
C.6.8(2),
0
0
ET
E
VVR
V
=
=
,
(
)
Õ
k
k
k
=
Alt T
V
T
V
,
k
≥
1
,
Clearly, E
1
V
=
V
.
Next, we would like to define a product for alternating tensors that maps alter-
nating tensors to alternating tensors. The tensor product of two alternating tensors is
unfortunately not always an alternating tensor, but all we have to do is project back
into the set of alternating tensors using the Alt map.
Definition.
Define a map
r
()
¥
s
()
Æ
r
+
s
()
Ÿ
:E
VV
E
E
V
by
(
)
Ÿ=
+
rs
rs
!
!!
(
)
wh
Alt
w h
ƒ
.
(C.14)
The map Ÿ is called the
exterior
or
wedge product
.
The ugly factorials are added here in order to avoid them in other places, such as
in the definition of the volume element for differential forms.
C.6.10. Theorem.
Let
V
be an n-dimensional vector space.
(1) The exterior product for
V
is bilinear, that is,
(
)
Ÿ=
ww hw hw h
whh whwh
whw h
+
Ÿ+
Ÿ
1
2
1
2
(
)
=Ÿ +Ÿ
Ÿ= Ÿ =
ŸŸ
1
2
1
2
(
)
a
a
a
wh
Ÿ
for w
i
, wŒE
r
(
V
), h
i
, hŒE
s
(
V
), and a Œ
R
.
(2) If wŒE
r
(
V
), hŒE
s
(
V
), and qŒE
t
(
V
), then
(
)
rst
rst
++
!
(
)
Ÿ= Ÿ Ÿ
(
)
=
(
)
wh qw hq
Ÿ
Alt
w hq
ƒƒ
.
!!!
In particular, the exterior product is associative and
