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Geometric Hodge star with applications to the theorems of Gauss and Green

JENNY HARRISON

Mathematical Proceedings of the Cambridge Philosophical Society / 140 / Issue 01 / January 2006, pp 135 ­ 155 DOI: 10.1017/S0305004105008716, Published online: 11 January 2006

Link to this article: http://journals.cambridge.org/abstract_S0305004105008716

How to cite this article: JENNY HARRISON (2006). Geometric with applications to the theorems of Gauss and Green. Mathematical Proceedings of the Cambridge Philosophical Society, 140, pp 135­155 doi:10.1017/S0305004105008716

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Geometric Hodge star operator with applications to the theorems of Gauss and Green

By JENNY HARRISON Department of , University of California, Berkeley, U.S.A.

(Received 22 June 1999; revised 23 December 2004)

Abstract The classical theorem for an n-dimensional domain A and a smooth vector field F in n-space   F · n = divF ∂A A requires that a vector field n(p) be defined a.e. p ∈ ∂A.Inthispaperwe give a new proof and extension of this theorem by replacing n with a ∂A of 1-dimensional polyhedral chains taken with respect to a . The operator  is a geometric dual to the Hodge star operator and is defined on a large of k-dimensional domains of integration A in n-space called chainlets. Chainlets include a broad range of domains, from smooth to soap bubbles and fractals. We prove as our main result the Star theorem   ω =(−1)k(n−k) ω A A where ω is a k-form and A is an (n − k)-chainlet. When combined with the general Stokes’ theorem ([H1, H2])   ω = dω ∂A A this result yields optimal and concise forms of Gauss’   ω =(−1)(k)(n−k) dω ∂A A and Green’s theorem   ω = dω. ∂A A

1. Introduction In this paper we develop a theory of on a large class of domains by taking limits of k-dimensional polyhedral chains in n-space with respect to a one parameter family of norms depending on a parameter r  0. We take r to be an N r integer; extensions to real r are possible ([H5]). Elements of the Banach spaces k obtained on completion are called k-chainlets of class N r . The norms are decreasing 136 Jenny Harrison N r N ∞ with r. The direct limit of the k is a normed linear space k . The parameter r reflects the roughness of the chainlets. Concepts such as smooth manifolds, fractals, vector fields, differential forms, foliations and measures have counterparts in chainlet . There is no geometric wedge for all pairs of chainlets as this would lead to of distributions. However, other product