• Corpus ID: 14063518

# Differential complexes and exterior calculus

@article{Harrison2006DifferentialCA,
title={Differential complexes and exterior calculus},
author={Jenny Harrison},
journal={arXiv: Mathematical Physics},
year={2006}
}
• J. Harrison
• Published 9 January 2006
• Mathematics
• arXiv: Mathematical Physics
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…
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We show that geometric integrals of the type $\int_\Omega f\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $\Omega$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to For almost one century (see [6]), it has been known that vector fields E, H, D, B, etc., in the Maxwell equations, are just "proxies" for more fundamental objects, the differential forms e, h, d, b, • Materials Science Journal of Geometric Analysis • 2011 In this paper we investigate the topological properties of the space of differential chains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} In this dissertation, I will discuss and explore the various theoretical pillars re- quired to investigate the world of discretized gauge theories in a purely classical setting, with the long-term This paper describes work in progress, towards the formulation, implementation and testing of compatible discretization of differential equations, using a combination of Finite Element Exterior ## References SHOWING 1-10 OF 26 REFERENCES In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. • Mathematics • 2003 This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney This paper proves an isomorphism theorem for cochains and differential forms, before passing to cohomology. De Rham’s theorem is a consequence. This leads to an extension of much of calculus and • J. Harrison • Mathematics Mathematical Proceedings of the Cambridge Philosophical Society • 2006 The classical divergence theorem for an$n$-dimensional domain$A$and a smooth vector field$F$in$n$-space $\int_{\partial A} F \cdot n = \int_A div F$ requires that a normal vector field$n(p)\$
We present here the fundamentals of a theory of domains that offers unifying techniques and terminology for a number of different fields. Using direct, geometric methods, we develop integration over
• Mathematics
• 1992
v is the 1-vectorfield “dual” to ω: if ω = ∑ (−1)fi dx1 ∧ · · · ∧ ∧ dxi ∧ · · · ∧ dxn, then v = (f1, . . . , fn).) There has been considerable effort in the literature (e.g. [JK], [M], [P]) to extend