Displacements

TL;DR

This paper introduces a novel calculus framework based on the concept of displacement, integrating topology, measure, differentiation, and integration to develop displacement equations and their relation to Stieltjes differential equations.

Contribution

It establishes a comprehensive theory of calculus centered on displacement, including new notions like displacement derivatives and path-based integrals, connecting them through a Fundamental Theorem of Calculus.

Findings

Defined displacement-based derivatives and integrals.

Linked displacement equations to Stieltjes differential equations.

Developed a foundational framework for displacement calculus.

Abstract

In this work we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation and integration. We find interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equations.