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Let $D\subset \mathbb R^2$, open set. We have a smooth map $f: D\times (0,1)\to \mathbb R^3$. There is a smooth map $u$ from some open subset of $\mathbb R^3$ to $ D$ such that $$u(f(x,y,t)) = (x,y)$$ Show that $u_*$ derivative of $u$ is of rank $2$. $t$ is parameter for $(0,1)$ direction and $(x,y)\in D$.

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up vote 2 down vote accepted

chain rule shows $u_* f_* = \operatorname {id}_* = \operatorname {id}$ which means that $u_*$ is surjective, i.e. has full rank

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