## Cambridge Philosophical Society

- J. Harrison, Geometric Hodge Star Operator with Applications to the Theorems of Gauss, Green, Proceedings
- January
- 2006

1 Excerpt

- Published 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 “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 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. 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. Date: December 21,2005, edited May 12, 2006. 1 2 J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY 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 the “rigorization” period of calculus. Leibniz had searched for an ’algebra of geometry’, but this was not available until Grassmann realized the importance of k-vectors in his seminal 1844 paper. But his ideas were largely ignored until Gibbs and Clifford began to appreciate them in 1877. Cartan used the Grassmann algebra to develop the exterior calculus. Meanwhile, Hamilton and others were debating the rigorization of multivariable calculus. It was not even clear what was meant by “space” in the late 19th century. The notions of div, grad and curl of Gibbs and Heaviside, were finally settled upon by the bulk of the community, and there have been two separate, and often competing, approaches ever since. But the Cartan theory, rather than the coordinate theory we teach our freshmen, has given mathematicians and physicists a clearer vision to guide great leaps of thought, and this theory has led to much of the mathematics behind the prizewinning discoveries of mathematics and mathematical physics. The group of those who now understand the importance of the Cartan theory is growing, as evidenced by the many books and papers which now start with this theory as their basis. The reader might well ask, what are some of the problems of coordinate calculus? Does this Cartan theory have limitations? Why do we need something new? First of all, the coordinate theory requires multiple levels of limits upon limits to be able to understand interesting applications in manifolds or curved space that take years of training to understand. We become proud virtuosos of our coordinate techniques. But these methods are a barrier to others who do not have the time or patience to learn them. And they also form a barrier to those of us who rely on them for they may cloud our vision with their complexity. After we become experts in Euclidean space with our div, grad and curl operators, we then move into curved space. Everything works quite well in restricted settings of smooth Riemannian manifolds and submanifolds. We can manipulate the limits, know when to change their orders, and do term by term integration with Fourier series and wavelets, etc. But

@inproceedings{Harrison2006arXI,
title={ar X iv : m at h - ph / 0 60 10 15 v 4 7 J un 2 00 6 DIFFERENTIAL COMPLEXES AND EXTERIOR CALCULUS},
author={Jolie Harrison},
year={2006}
}