A change of variables theorem for the multidimensional Riemann integral

TL;DR

This paper extends the classical change of variables theorem for Riemann integrals to multiple dimensions, utilizing Lipschitz functions and a generalized Sard's lemma to maintain a Riemannian framework.

Contribution

It provides a multidimensional change of variables theorem for Riemann integrals using injective functions and a generalized Sard's lemma for Lipschitz functions.

Findings

Established a multidimensional change of variables theorem for Riemann integrals.

Proved a generalization of Sard's lemma for Lipschitz functions in multiple variables.

Maintained proofs within the Riemannian framework using strong differentiability.

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 (t\mapsto 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.