Coordinate free integrals in Geometric Calculus

TL;DR

This paper presents a coordinate-free method for evaluating integrals in geometric calculus by repeatedly applying the fundamental theorem of calculus, which could have practical applications and reveal new mathematical connections.

Contribution

It introduces a novel coordinate-free integration technique in geometric calculus based on manifold boundary manipulation and antiderivatives.

Findings

Provides a generalization of real-variable integration methods

Potential applications in various mathematical fields

Facilitates coordinate-free integral evaluation

Abstract

We introduce a method for evaluating integrals in geometric calculus without introducing coordinates, based on using the fundamental theorem of calculus repeatedly and cutting the resulting manifolds so as to create a boundary and allow for the existence of an antiderivative at each step. The method is a direct generalization of the usual method of integration on function of a real variable. It may lead to both practical applications and help unveil new connections to various fields of mathematics.