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Let Id be the identity map in R" and let f : R → R be the function f(x) = ||||². Prove that the function G := fId : R" → R" is a differentiable bijection that satisfies: • DG(0) = 0, • DG(p) has non-zero determinant for all other p. The inverse of G is not differentiable at 0.

Let Id be the identity map in R" and let f : R → R be the function f(x) = ||||². Prove that the function G := fId : R" → R" is a differentiable bijection that satisfies: • DG(0) = 0, • DG(p) has non-zero determinant for all other p. The inverse of G is not differentiable at 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I have the following question from multivariable analysis, please if able explain in steps, thank you in advance:

 

Let Id be the identity map in R" and let f : R → R be the function f(x) = ||||2. Prove that the function G := fId: R → R" is a differentiable bijection that satisfies: • DG(0) = 0, • DG(p) has non-zero determinant for all other p. . The inverse of G is not differentiable at 0.
Transcribed Image Text:Let Id be the identity map in R" and let f : R → R be the function f(x) = ||||2. Prove that the function G := fId: R → R" is a differentiable bijection that satisfies: • DG(0) = 0, • DG(p) has non-zero determinant for all other p. . The inverse of G is not differentiable at 0.
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