2005/01/01 by Jenny Harrison, Harrison, Jenny
Mathematics · Physics and Astronomy · #49Q15 #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Theoretical and Computational Physics #math-ph #math.CA #math.MP #msc:49Q15
paper · pdf · doi:10.48550/arxiv.math-ph/0501001
83 pages
openalex publication_date 2005/01/01 · arxiv created 2005/01/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 points of the development. For example, k-dimensional tangent spaces are replaced by representations of simple k-vectors supported in single points as limits of simplicial k-chains in a Banach space (much like Dirac monopoles). This subtle difference has powerful advantages that will be explored. Through these ``infinitesimals'', we obtain a coordinate free theory on manifolds that builds upon the Cartan exterior calculus. An infinite array of approximating theories to the calculus of Newton and Lebiniz becomes available and one can now revisit old philosophical questions such as which models are most natural for the continuum or for physics. Applications include three extensions of calculus: calculus on fractals, bilayer calculus (soap bubbles) and discrete calculus. This paper is a draft of the first half of the lectures.