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I have a problem. I want to prove a necessary condition in a theorem. I know that a smooth transformation is diffeomorphism around the origin. Can I show that its jacobian is nonsingular at the origin?

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- Thread starter mby110
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I have a problem. I want to prove a necessary condition in a theorem. I know that a smooth transformation is diffeomorphism around the origin. Can I show that its jacobian is nonsingular at the origin?

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quasar987

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Yes: say F is a smooth map that is locally invertible at some point x_0 and it's inverse is differentiable there, then apply the chain rule to [itex]F\circ F^{-1}=id[/itex] and [itex]F^{-1}\circ F=id[/itex] to conclude that [itex]dF_{x_0}[/itex] is invertible and its inverse is [itex]d(F^{-1})_{F(x_0)}[/itex].

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