Let An be a closed set contained in (n, n + 1). Suppose that f: R →R satisfies ∫Arf = (−1)n/n and f = 0 outside Un An.. Find two partitions of unity Φ and Ψ such that ∑ǿЄΦ∫Rǿ∙f and ∑ǿЄΦ∫Rψ∙f converge absolutely to different values.
Answer to relevant QuestionsUse Theorem 3-14 to prove Theorem 3-13 without the assumption that g1 (x) ≠ 0.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) ...If M is an -dimensional manifold-with-boundary in Rn, define μ as the usual orientation of M x = Rnx (the orientation μ so defined is the usual orientation of M. If xЄ∂M, show that the two definitions ...a. Show that Theorem 5-5 is false if M is not required to be compact. b. Show that Theorem 5-5 holds for noncom-pact M provided that w vanishes outside of a compact subset of M.a. If c: [0, 2π] x [-1, 1] → c: [0 , 2π] x [-1, 1] → R3 is defined by c (u,v) = (2 eos (u) + vsin (u/2) eos (u), 2sin (u) + vsin (u/2) sin (u), veos (u/2)).
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