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.
Highlights
IntroductionDerivatives are, in the classical sense of Newton [1], infinitesimal rates of change of one (dependent) variable with respect to another (independent) variable
Derivatives are, in the classical sense of Newton [1], infinitesimal rates of change of one variable with respect to another variable
We introduce the concept of displacement derivative of a function defined over a compact interval endowed with a displacement structure
Summary
Derivatives are, in the classical sense of Newton [1], infinitesimal rates of change of one (dependent) variable with respect to another (independent) variable. This result conveys the true meaning of the absolute derivative—it is the absolute value of the derivative—and it extends the notion of derivative to the broader setting of metric spaces. Even so, this definition may seem somewhat unfulfilling as a generalization. The definition of ∆x does not have to depend on a rescaling, but its absolute value definitely has to suggest, in a broad sense, the notion, if not of distance, of being far apart or close as well as the direction—change of sign. The last section is devoted to the conclusions of this work and the open problems lying ahead
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