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Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k<m,$ 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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Your manifold $M$ is given somehow. What are the data describing $M$ in the neighborhood of $p$? –  Christian Blatter May 20 '12 at 14:40
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If $F'(p)$ has rank $k$ at some point, then the derivative matrix has a nonzero $k\times k$ minor, which will remain nonzero in some neighborhood. This shows that the rank of $F'$ will be at least $k$ in some neighborhood (i.e. rank is lower semicontinuous). But since $k$ is also the maximal possible rank for a map into $\mathbb R^k$, you get $\mathrm{rank}\, F'=k$ in a neighborhood. –  user31373 May 20 '12 at 17:04
    
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 $M$. –  user31373 May 20 '12 at 17:07

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