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Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k a collection of functions such that on a point $p$, we have that $dy^1|_p,\dots,dy^k|_p$ is linearly independent on $T^*_pM$. Show that there exist smooth functions $y^{k+1},\dots,y^m$ such that $y^1,\dots,y^m$ is a coordinate system for $M$ on a neighborhood of $p$?

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    Concerning 2), I think that after "straightening out" the given collection of maps (so that they become $x^1,\dots,x^k$ on $\mathbb R^m$), you should take $x^{k+1},\dots,x^m$ and reverse the process to plant them back onto $M$.2012-05-20

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