Calculus in the ring of Fermat reals Part I: Integral calculus

TL;DR

This paper develops integral calculus for quasi-standard smooth functions on Fermat reals, establishing existence, uniqueness, and properties of primitives, and extending to infinite dimensional and multiple integrals within a Cartesian closed framework.

Contribution

It introduces a novel integral calculus framework on Fermat reals, including primitive existence, classical formulas, and infinite dimensional integral operators.

Findings

Existence and uniqueness of primitives for Fermat reals

Classical integral formulas extended to Fermat spaces

Framework supports infinite dimensional and multiple integrals

Abstract

We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.