A Topological View on Integration and Exterior Calculus

TL;DR

This paper develops a unified topological framework for integration and exterior calculus that generalizes classical integrals and derivatives, enabling applications across fractals, stochastic processes, and discrete systems.

Contribution

It introduces a novel construction of integration and exterior calculus based on unital magmas, extending classical calculus to arbitrary topological spaces and new mathematical contexts.

Findings

Unified framework for integration and exterior calculus.

Generalization of classical integrals like Riemann and Lebesgue.

Applications to fractals, stochastic analysis, and discrete systems.

Abstract

A construction of integration, function calculus, and exterior calculus is made, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann integral, geometric product integral, and Lebesgue integral are shown as special cases. Notions similar to chain complexes are developed to allow this form of integration to define notions of exterior derivative for differential forms, and of derivatives of functions as well. Resulting conclusions on integration, orientation, dimension, and differentiation are discussed. Applications include calculus on fractals, stochastic analysis, discrete analysis, and other novel forms of calculus.