# Differential complexes and exterior calculus

@article{Harrison2006DifferentialCA, title={Differential complexes and exterior calculus}, author={Jenny Harrison}, journal={arXiv: Mathematical Physics}, year={2006} }

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…

## 10 Citations

### Topological Aspects of Differential Chains

- Mathematics
- 2011

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that…

### Towards Geometric Integration of Rough Differential Forms

- Mathematics, Computer Science
- 2020

We provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity,…

### Towards Geometric Integration of Rough Differential Forms

- MathematicsThe Journal of Geometric Analysis
- 2020

We provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity,…

### D ec 2 01 9 INTEGRATION OF NONSMOOTH 2-FORMS : FROM YOUNG TO ITÔ AND STRATONOVICH

- Mathematics
- 2019

Ω fdg∧dg can be defined over a two-dimensional domain Ω when the functions f , g, g : R → R are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω has sufficiently…

### CAN COMPATIBLE DISCRETIZATION , FINITE ELEMENT METHODS , AND DISCRETE CLIFFORD ANALYSIS BE FRUITFULLY COMBINED ?

- Mathematics
- 2011

This paper describes work in progress, towards the formulation, implementation and testing of compatible discretization of of differential equations, using a combination of Finite Element Exterior…

### Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to It\^{o} and Stratonovich

- Mathematics
- 2019

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…

### GEOMETRICAL METHODS IN COMPUTATIONAL ELECTROMAGNETISM

- Physics
- 2006

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,…

### Topological Aspects of Differential Chains

- Materials ScienceJournal 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}…

### Geometric Aspects of Discretized Classical Field Theories: Extensions to Finite Element Exterior Calculus, Noether Theorems, and the Geodesic Finite Element Method

- Mathematics
- 2016

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…

### CAN COMPATIBLE DISCRETIZATION, FINITE ELEMENT METHODS, AND DISCRETE CLIFFORD ANALYSIS BE FRUITFULLY COMBINED?

- Mathematics
- 2012

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…

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