Another doubt about real functions on manifolds Well, some days ago I've asked here how do we describe functions on manifolds. My idea was that it could be done using the coordinate functions of a chart: if $(x,U)$ is a chart for a manifold $M$ then we can define one function $f : U \to \Bbb R$ as a combination of the $x^i$ functions. Now I have one doubt that seems very silly (the answer is probably obvious, and I'm failing to see it).
Now here comes my doubt: let $C^{\infty}(U\subset M,\Bbb R)$ be the set of all smooth functions defined in the subset $U$ of a manifold $M$ of dimension $n$. I've defined a $k$-combination to be a map
$$c:\prod_{i=1}^k C^{\infty}(U,\Bbb R) \to C^{\infty}(U,\Bbb R)$$
so for instance, for $k=2$ the map $c(f,g)=\lambda f + \sin \circ g$ would be a $2$-combination of $f$ and $g$. Now, let $k = n$, then trivially by the definition we have that:
$$c(x^1,\dots,x^n)\in C^\infty(U,\Bbb R)$$
My question is: do we have that for any $f \in C^\infty(U,\Bbb R)$ there exists a unique $n$-combination of the functions $x^i$ such that $f = c(x^1,\dots, x^n)$? In other words, do we have that any function defined on $U\subset M$ is a suitable combination of the coordinate functions?
Thanks very much in advance!
 A: I believe the answer to your question is yes. Suppose $f: U \subseteq M \rightarrow \mathbb{R}$ where $M$ is a manifold. Let $p \in U$ then, by the definition of a manifold, there exists (at least one) $(V,x)$ a coordinate chart with $V \subseteq U$. If $V$ is too large we can construct a new chart by intersection and shrink the domain as needed. Note
$$ f|_V = f \circ x^{-1} \circ x $$
which means the formula you desire exists locally at $p$. Now, I may not be able to write a formula for $f$ in terms of coordinates on all of $U$ since it is conceivable that $U$ needs to be covered by several charts.
But, perhaps, the real question you are asking, is how can we define a function in terms of something besides a coordinate chart on a manifold. The answer there is usually given in terms of the explicit structure of the set as a point set. For example, $x+y+z=1$ gives a plane hence $f(x,y,z) = 2x+2y+z$ is some function on the plane not (yet) given in coordinate chart on the plane( which I have not stated). However, it is a simple exercise to choose parameters for the plane as in so doing construct charts which then could be used to formulate $f$. I know this is possible by the general argument I give at the outset of this post.
