Differential complexes and exterior calculus

TL;DR

This paper introduces a unified calculus framework over smooth manifolds that generalizes classical and exterior calculus, enabling analysis of complex, nonsmooth, and non-Euclidean domains through the chainlet complex and new prederivative concepts.

Contribution

It develops a broad, coordinate-free calculus theory on manifolds that unifies discrete, exterior, and continuum approaches, extending to nonsmooth and complex domains.

Findings

Unified calculus over manifolds via chainlet complex

Applicable to nonsmooth, non-Euclidean domains

Introduces prederivative and preintegral concepts

Abstract

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 ``pointed 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 antisymmetric 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.