Every submanifold of $\mathbb {R}^n $ is locally a graph I'm trying to do this Pollack exercise. I managed to do item (a), but I'm stuck on item (b) I don't know what function to set to $g_i$. I thought about using that as $X$ is submanifold for all $p \in X$ exists chart $(V, \psi)=(V, r_1,..,r_N)$ such that $X \cap V = \{q \in V | r^{k+1}(q)=...=r^N(q)=0 \}$, but i don't know how to proceed

 A: It is not true.
Take for example $X = \{x_1,\ldots,x_N) \in \mathbb R^N \mid x_1 = \ldots = x_{N-k} = 0 \}$. This is a $k$-dimensional submanifold of $\mathbb R^N$. But there cannot be any $V \subset X$ such that $V$ is the graph of some function $g : U  \to \mathbb R^{N-k}$ defined on an open $U \subset \mathbb R^k$. In fact, the graph of such $g$ is the set
$$\operatorname{graph}(g) = \{(a,g(a)) \mid a \in U\}.$$
Since $U$ is open, there must be some $a \in U \setminus \{0\}$. Then certainly $(a,g(a)) \notin X$, thus $(a,g(a)) \notin V$.
So what can be done? If you reflect upon this example, you see that the problem is that one wants that the first $k$ coordinates of $x \in V$ to form a point of $U$. But that is not essential. You can prove that there exists a permutation $\sigma$ of $\{1,\ldots,N\}$ such that the linear automorphism
$$\phi_\sigma : \mathbb R^N \to \mathbb R^N, \phi_\sigma(x_1,\ldots,x_N) = (x_{\sigma(1)},\ldots,x_{\sigma(N)}) $$
has the property
$$\phi_\sigma(\operatorname{graph}(g)) = V.$$
Only in this more general sense one can say that that a submanifold $X$ is locally a graph.
Update:
I did not properly read (a) and (b) and missed that $x_1,\ldots,x_N$ denote the standard coordinate functions. (a) says that there exist an open neigborhood $V$ of $x$ in $X$ and indices $i_1,\ldots, i_k$ such that the restrictions $x_{i_r} \mid_V$ form a local coordinate system. Thus, working with a suitable permutation of $\{1,\ldots, N\}$, we may w.l.o.g. assume that $i_r = r$. And that was the assumption in (b).
So let us prove (b). Let $p :  \mathbb R^N \to  \mathbb R^k$ denote the projection onto the first $k$ coordinates and $q: \mathbb R^N \to  \mathbb R^{N-k}$ the projection onto the last $N-k$ coordinates. The assumption means that $p_V : V \stackrel{p}{\to} p(V) =  U$ is a diffeomorphims onto an open $U \subset \mathbb R^k$. We claim that $V$ is the graph of the smooth map $g = q \circ p_V^{-1} : U \to  \mathbb R^{N-k}$.
For $\xi \in V$ we have $\xi = (p(\xi),q(\xi))$ with $p(\xi) \in U$. Since $p_V$ is a bijection, we have $\xi = p_V^{-1}(p(\xi))$. Therefore
$$\xi = (p(\xi),q(\xi)) = (p(\xi),q(p_W^{-1}(p(\xi)))) = (p(\xi),g(p(\xi))).\tag{1}$$
This shows
$$V = \{(p(\xi),g(p(\xi))) \mid \xi \in V \} = \{(a,g(a)) \mid a \in p(W) = U \} = \operatorname{graph}(g) .$$
