A working knowledge of differential forms so strongly illuminates
the calculus and its developments that it ought not be too long
delayed in the curriculum. On the other hand, the systematic
treatment of differential forms requires an apparatus of topology
and algebra which is heavy for beginning undergraduates. Several
texts on advanced calculus using differential forms have appeared
in recent years. We may cite as representative of the variety of
approaches the books of Fleming [2], (1) Nickerson-Spencer-Steenrod
[3], and Spivak [6]. . Despite their accommodation to the innocence
of their readers, these texts cannot lighten the burden of
apparatus exactly because they offer a more or less full measure of
the truth at some level of generality in a formally precise
exposition. There. is consequently a gap between texts of this type
and the traditional advanced calculus. Recently, on the occasion of
offering a beginning course of advanced calculus, we undertook the
expe- ment of attempting to present the technique of differential
forms with minimal apparatus and very few prerequisites. These
notes are the result of that experiment. Our exposition is intended
to be heuristic and concrete. Roughly speaking, we take a
differential form to be a multi-dimensional integrand, such a thing
being subject to rules making change-of-variable calculations
automatic. The domains of integration (manifolds) are explicitly
given "surfaces" in Euclidean space. The differentiation of forms
(exterior (1) Numbers in brackets refer to the Bibliography at the
end.
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