Calculus in the ring of Fermat reals Part I: Integral calculus
arXiv:1503.05965
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.
arXiv commentv2: simplify the thickening for the proof of the Fermat-Reyes theorem in the previous version