Graphics Reference
In-Depth Information
C.6
The Tensor and Exterior Algebra
This section contains some rather technical mathematics. It is needed because without
a notion of tensor and exterior algebras one cannot discuss differential forms in a rig-
orous way. We could have simplified some of the discussion by restricting ourselves
to the algebras of multilinear and alternating multilinear maps as is done, for example,
by Spivak in [Spiv65] and [Spiv70a]. Those algebras will be highlighted in the dis-
cussion below, but they are special cases of a general construction and by consider-
ing only them the reader would have gotten an incomplete picture of the subject. For
that reason, we considered it worthwhile to outline the “correct” development of these
algebras. The reader still has the choice of skipping uninteresting material. Other ref-
erences for tensor and exterior algebras are [AusM63] and [KobN63].
In this section the field for a vector space will always be the reals
R
.
Notation.
Let
V
i
, 1 £ i £ k, and
W
be vector spaces. The set of multilinear maps
f
:
VV
¥
¥◊◊◊¥
V W
Æ
1
2
k
will be denoted by L
k
(
V
1
,
V
2
,...,
V
k
;
W
). If
V
=
V
1
=
V
2
= ...=
V
k
, then L
k
(
V
1
,
V
2
,
...,
V
k
;
W
) will be abbreviated to L
k
(
V
;
W
) and L
k
(
V
;
R
) is abbreviated to L
k
(
V
). It
follows that L
k
(
V
) denotes the multilinear maps
k
f
:
VVV
=¥¥◊ ◊ ◊ ¥Æ
12
VR
.
44
44
3
k
By setting
V
0
=
R
, the notation L
0
(
V
) makes sense and denotes the set of linear maps
R
Æ
R
.
We may always identify L
0
(
V
) with
R
using the natural map L
0
(
V
) Æ
R
that sends
a to a(1).
Definition.
A
tensor product
of vector spaces
V
i
, 1 £ i £ k, is a pair (
A
,a), where
A
is
a vector space and
a :
VV
¥
¥◊◊◊¥
V A
Æ
1
2
k
is a multilinear map that satisfy the following
universal factorization property
:
If f :
V
1
¥
V
2
¥ ...¥
V
k
Æ
W
is any multilinear map into a vector space
W
, then
there is a
unique
linear transformation g :
A
Æ
W
so that f = g
a. See Figure C.3.
a
...
V
1
¥ V
2
¥
¥ V
k
A
g
f
Figure C.3.
The universal factorization property of tensor
products.
W
