# Question: Let g1 g2 R2 R be continuously

Let g1, g2: R2 → R be continuously differentiable and suppose D1 g2= D2 R1.. As in Problem 2-21, let

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a. Let g: Rn → Rn be a linear transformation of one of the following types:If M is a k-dimensional manifold with boundary, prove that ∂M is a (k - 1) -dimensional manifold and M - ∂M is a k=dimensional manifold.a. If M is a k-dimensional manifold in Rn and k < n, show that M has measure 0. b. If M is a closed -dimensional manifold with boundary in Rn, show that the boundary of M is ∂M. Give a counter-example if M is not ...Let g: A →Rp be as in Theorem 5-1. If f: Rn → R is differentiable and the maximum (or minimum) of f on g-1 (0) occurs at , show that there are , such thatGeneralize Theorem 5-6 to the case of an oriented (n - 1) -dimensional manifold in Rn. The generalization is w Є A n-1(Mx) defined byPost your question