Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k
Complete a set of functions to obtain a system.
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differential-geometry
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0Your manifold $M$ is given somehow. What are the data describing $M$ in the neighborhood of $p$? – 2012-05-20
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1If $F'(p)$ 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. – 2012-05-20
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0Concerning 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