Use Fubini's Theorem to derive an expression for the volume of a set in R3 obtained by revolving a Jordan measurable set in the yz -plane about the -axis.
Answer to relevant QuestionsIf A = [a1, b1] x . x [an, bn] and f: A → R is continuous, define f: A → R bya. Suppose that f: (0, 1) → R is a non-negative continuous function. Show that ∫ (0, 1) exists if and only if lim Є→ ∫ c 1-c f exists. b. Let An = [1 - 1/2n, 1 - 1/2n +1] Suppose that f: (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) ...a. If f is a differentiable vector field on M C Rn, show that there is an open set AЭM and a differentiable vector field F on A with F(x) = F (x) for xЄM. b. If M is closed, show that we can choose A = Rn.An absolute k-tensor on v is a function Vk →R of the form |w| for w Є Ak (V). An absolute k-form on M is a function such that n (x) is an absolute k-tensor on Mx. Show that ∫Mn can be defined, even if M is ...
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