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Let $F:\mathbb{R}^{2}\to \mathbb{R}^{2}$ be a smooth map where its Jacobian $$ Jf(x, y) = \det\begin{pmatrix}\frac{\partial F_{1}}{\partial x} & \frac{\partial{F_{1}}}{\partial y} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial F_{2}}{\partial y}\end{pmatrix} $$ is nonzero everywhere. Is it true that $F$ is injective? Also, if it is true, if $F$ is homeomorphism, then is it diffeomorphism?

This is true for smooth maps $F:\mathbb{R}\to \mathbb{R}$, but I believe there exists counterexamples for higher dimensional case. But I can't find it.

Seewoo Lee
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  • I found that second assertion is true - if $F$ is a smooth homeomorphism s.t. $JF$ is nowhere vanishing, then $F$ is a diffeomorphism since it's inverse is locally smooth everywhere by inverse function theorem. – Seewoo Lee Apr 23 '18 at 11:54
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    Look at $z\mapsto e^z$. For this function $J=e^{Re(z)}\neq0$ but it is not injective since $e^{2\pi i}=1=e^0$. –  Apr 23 '18 at 12:06
  • @deyore Thank you very much! – Seewoo Lee Apr 23 '18 at 12:16

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