ABSTRACT
This chapter is on a generalization of the change of variables formula found in Chapter 14. In this section, Ω will be a Lebesgue measurable set in ℝ n and h : Ω → ℝ m will be Lipschitz. In the case of the earlier change of variables formula, the mapping was from a subset of ℝ n , U, to a subset of ℝ n and Lebesgue measure was involved in both U and h (U). Here m can be larger than n and instead of Lebesgue measure on h (U), we use Hausdorff measure. Also, h is only assumed to be Lipschitz, not C 1. It turns out all the earlier results have a pleasing generalization in this setting. This generalization also makes possible the consideration of surface measure and the divergence theorem. First, we give a simple extension property of Lipschitz maps.
