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of differentials. However, by having expressed everything carefully, the reader who
wants to use the notation in a serious way is less likely to make mistakes with it.
A limited and specialized development of differential forms for dimensions 1, 2,
and 3 can be found in various advanced calculus topics like [Buck78]. Such develop-
ments do not show the complete picture, however, and are more along the lines of
applications or examples. A thorough discussion of differential forms is based on the
“exterior algebra” of differential forms. Unfortunately, this involves a fair amount of
abstract algebra, especially if one starts with an axiomatic approach to tensor and
exterior algebras. We outline this approach in Appendix C. No matter how one devel-
ops the subject, the reader is forced to “suffer” through a large number of definitions
and theorems. Although most of the theorems follow trivially from the definitions, it
takes a while before one gets to the applications. The subject is actually a good
example where the right mathematical definitions lead in the end to significant con-
sequences, consequences that, because of the definitions, are trivial to prove. Many
topics on differentiable manifolds have a section on differential forms but most
readers would probably find their presentations hard reading. In the opinion of the
author, good references for differential forms are [Flan63], [Spiv65], and [Spiv70a].
Flanders does not prove everything but covers the subject and a great many applica-
tions well. He starts with an axiomatic formulation of the properties that one wants
the exterior algebra to posses. The advantage to his approach is that one quickly starts
using the usual notation associated with differential forms. Spivak has more limited
goals but
does
prove all his results and does a good job in presenting the subject in
a completely self-contained manner. He starts by defining the “tensor algebra” of mul-
tilinear maps and derives the exterior algebra from this. This more computational
approach avoids the existence proofs and requires less familiarity with advanced alge-
braic concepts, but still forces a reader to wade through quite a few definitions and
proofs of simple consequences before one gets to the actual differential forms them-
selves. By and large, our approach here will follow Spivak's, except that some of the
algebraic preliminaries have been off-loaded to Appendix C. Another topic worth
looking at is [GuiP74]. This topic also follows Spivak's approach. The reader who is
overwhelmed by the many definitions and abstract concepts should look ahead to the
end of this section and Section 8.12 to see that it will all be worth it.
Differential forms are derived from the exterior or Grassmann algebra E(
R
n
*) of
the dual space
R
n
*. This algebra can be identified with the algebra L(
R
n
) of exterior
forms on
R
n
. The theoretical basis of these algebras is developed in Section C.6
and will not be reproduced here, but we summarize the essential aspects of the latter.
The reader who is not interested in the mathematical details that justify our asser-
tions can simply take them as axioms that specify a language of forms and how the
symbols in that language are manipulated. All future properties will be deduced from
these.
Let
V
be an n-dimensional vector space over the reals and
V
* its dual.
Notation.
Let k ≥ 0. The vector space of alternating multilinear maps (also called
exterior k-forms of
V
)
k
w:
VVV
=¥¥ ¥Æ
12
...
VR
.
44
44
3
k
