2006/01/09 by Jenny Harrison, Harrison, Jenny
Mathematics · Physics and Astronomy · #15A75 #16E45 #49Q15 #58C35 #Black Holes and Theoretical Physics #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph #math.CA #math.DG #math.MP #msc:15A75 #msc:16E45 #msc:49Q15 #msc:58C35
paper · pdf · doi:10.48550/arxiv.math-ph/0601015
50 pages
openalex publication_date 2006/01/09 · arxiv created 2006/06/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.