Can I show that its jacobian is nonsingular at the origin?

[mby110]

Hi
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?

 

[quasar987]

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 $F\circ F^{-1}=id$ and $F^{-1}\circ F=id$ to conclude that $dF_{x_0}$ is invertible and its inverse is $d(F^{-1})_{F(x_0)}$.