2005/05/24 by Jenny Harrison, Harrison, Jenny
Mathematics · Physics and Astronomy · #58C99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:58C99
paper · pdf · doi:10.48550/arxiv.math-ph/0505063
153 page, 7 figures
arxiv created 2005/05/24 · arxiv updated 2009/12/01
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. They begin with a new theory of exterior calculus at a single point. This infinitesimal theory extends, by linearity, to a discrete exterior theory, based at finitely many points. A general theory of calculus culminates by taking limits in Banach spaces, and is valid for domains called ``chainlets'' which are defined to be elements of the Banach spaces. Chainlets include manifolds, rough domains (e.g., fractals), soap films, foliations, and Euclidean space. Most of the work is at the level of the infinitesimal calculus, at a single point. The number of limits needed to get to the full theory is minimal. Tangent spaces are not used in these notes, although they can be defined within the theory. This new approach is made possible by giving the Grassmann algebra more geometric structure. As a result, much of geometric measure theory is simplified. Geometry is restored and significant results from the classical theory are expanded. Applications include existence of solutions to a problem of Plateau, an optimal Gauss-Green theorem and new models for Maxwell's equations.