Use Theorem 3-14 to prove Theorem 3-13 without the assumption that g1 (x) ≠ 0.
Answer to relevant QuestionsIf g: Rn → Rn and detg1 (x) ≠ 0, prove that in some open set containing we can write g = to gn 0 ∙ ∙ ∙ o g1, 0.., where is of the form gi(x) = (x1, ∙ ∙ ∙ Fi (x) , ∙ ...Prove that a k-dimensional (vector) subspace of Rn is a k-dimensional manifold.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.If w is a (k- 1) -form on a compact k-dimensional manifold M, prove that ∫Mdw =0. Give a counter-example if M is not compact.If there is a nowhere-zero k-form on a k -dimensional manifold M, show that M is orientable
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