ar X iv : m at h - ph / 0 60 10 15 v 4 7 J un 2 00 6 DIFFERENTIAL COMPLEXES AND EXTERIOR CALCULUS

In this paper we present a new theory of calculus over k-dimensional domains in a smooth n-manifold, unifying the discrete , exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of " monopolar chains, " culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior calculus become special cases. The chainlet operators , products and integrals apply to both symmetric and antisym-metric tensor cochains. As corollaries, we obtain the full calculus on Euclidean space, cell complexes, bilayer structures (e.g., soap films) and nonsmooth domains, with equal ease. The power comes from the recently discovered prederivative and preintegral that are antecedent to the Newtonian theory. These lead to new models for the continuum of space and time, and permit analysis of domains that may not be locally Euclidean, or locally connected, or with locally finite mass. Preface We put forward a novel meaning of the real continuum which is found by first developing a full theory of calculus at a single point – the origin, say, of a vector space – then carrying it over to domains supported in finitely many points in an affine space, and finally extending it to the class of " chainlets " found by taking limits of the discrete theory with respect to a norm. Local Euclidean structure is not necessary for the calculus to hold. The calculus extends to k-dimensional domains in n-manifolds. We do not rely on any results or definitions of classical calculus to develop our theory. In the appendix we show how to derive the standard results of single and multivariable calculus in Euclidean space as direct corollaries. This preprint is in draft form, sometimes rough. There are some details which are still linked to earlier versions of the theory. It is being expanded into a text which includes new applications, numerous examples, figures, exercises, and necessary background beyond basic linear algebra, none of which are included below. Problems of the classical approach. As much as we all love the calculus, there have been limits to our applications in both pure and applied mathematics coming from the definitions which arose during …