A change of variables theorem for the multidimensional Riemann integral

zbMATH Open1224.26042arXiv0804.2333MaRDI QIDQ3074450

I. Nagy, Zoltan Molnar, Tivadar Szilagyi

Publication date: 8 February 2011

Abstract: The most general change of variables theorem for the Riemann integral of functions of a single variable has been published in 1961 (by Kestelman). In this theorem, the substitution is made by an `indefinite integral', that is, by a function of the form (tmapsto c+int_a^tg=:G(t)) where (g) is Riemann integrable on ([a,b]) and (c) is any constant. We prove a multidimensional generalization of this theorem for the case where (G) is injective -- using the fact that the Riemann primitives are the same as those Lipschitz functions which are almost everywhere strongly differentiable in ((a,b)). We prove a generalization of Sard's lemma for Lipschitz functions of several variables that are almost everywhere strongly differentiable, which enables us to keep all our proofs within the framework of the Riemannian theory which was our aim.

Full work available at URL: https://arxiv.org/abs/0804.2333

zbMATH Keywords

Lipschitz functionsRiemann integrabilitySard's lemmaChange of variables

Mathematics Subject Classification ID

Integration of real functions of several variables: length, area, volume (26B15) Implicit function theorems, Jacobians, transformations with several variables (26B10) Calculus of vector functions (26B12)

Related Items (2)

The change-of-variables theorem for the Lebesgue integral ⋮ Corrigendum to “A New Proof of the Change of Variable Theorem for the Riemann Integral”

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