The most general change of variables theorem for the Riemann integral offunctions of a single variable has been published in 1961 by H. Kestelman. In this theorem,the substitution is made by an 'indefinite integral', that is, by a function of the form t H c + .t +∫_a~tg =: G(t)wheregis Riemann integrable on[a, b]andcis any constant. We provea multidimensional generalization of this theorem for the case where G is injective — usingthe fact that the Riemann primitives are the same as those Lipschitz functions which arealmost everywhere strongly differentiable in (a, b). We prove a generalization of Sard's lemmafor 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 whichwas our aim.
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机译：H. Kestelman于1961年发表了有关单个变量的Riemann积分函数的变量定理最普遍的变化。在该定理中，替换是由“不定积分”进行的，即由形式为t H c + .t +∫_a〜tg =：G（t）的Riemann可积分在[a，b]上的函数进行andcis任何常数。我们证明了这个定理在G是内射的情况下的多维概括，这是基于Riemann原语与Lipschitz函数相同的事实，这些Lipschitz函数几乎在（a，b）处都是可微分的。我们证明了Sard引理的Lipschitz函数的概化，这些函数几乎可以在所有地方强微分，这使我们能够将所有证明都保留在我们力求达到的黎曼理论的框架内。
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