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Calculus in the ring of Fermat reals. I: Integral calculus. (English) Zbl 1358.26023

Authors’ abstract: We develop 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.

MSC:

26E30 Non-Archimedean analysis
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26E35 Nonstandard analysis

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