Let $f=(f_1,f_2,f_3):\mathbb{R}^2\to\mathbb{R}^3$ continuously differentiable, $\det\begin{pmatrix} D_1f_1 & D_2f_1 \\ D_1f_2 & D_2f_2 \end{pmatrix}\not=0$.
How to prove: In every point $(a_1,a_2)$ exists a neigbourhood $W$ of $(a_1,a_2,0)$ and $V$ of $(f_1(a_1,a_2),f_2(a_1,a_2),f_3(a_1,a_2))$, such that there is a diffeomorphism $g:V\to W$ such that $(g\circ f)(x_1,x_2)=(x_1,x_2,0)$.
It looks like inverse function theorem, but I don't know how to apply it in detail. Could you help me? Regards
Edit With mathcounterexample help, (I first thought that implicit function theorem is necessary, but it's the inverse function theorem as he said), for avery $(x_1,x_2)\in\mathbb{R}^2$ exists an open neigbourhood $U_{(x_1,x_2)}\subseteq\mathbb{R}^2$ and an an open neigbourhood $W_{f_1(x_1,x_2),f_2(x_1,x_2)}$ such that $F:U_{(x_1,x_2)}\to W_{f_1(x_1,x_2),f_2(x_1,x_2)}$ is bijective and it's inverse function is a diffeomorphism. But how I don't know, how to continue.
