14. If is a closed rectangle, show that is Jordan measurable if and only if for every there is a partition of such that , where consists of all subrectangles intersecting and consists of allsubrectangles contained in .
Answer to relevant QuestionsIf A is a Jordan measurable set and ε > 0, show that there is a compact Jordan measurable set C C A such that ∫ A − C1 < ε.If is continuous, show thata. Let g: Rn → Rn be a linear transformation of one of the following types:Prove a partial converse to Theorem 5-1: If MCRn is a k-dimensional manifold and xЄM, then there is an open set A C Rn containing and a differentiable function g: A →Rn-k such that A∩M = g-1 (0) and g1 (y) ...Let f: A → Rp be as in Theorem 5-1. a. If x Є M = g-1(0), let h: U → Rn be the essentially unique diffeomorphism such that goh (y) = (y n – p + 1 . yn) and h (0) = x. Define f: Rn- p → f: Rn-p ...
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