Suppose $x_1,..,x_k$ form a local coordinate on a neighborhood $V$ of $x$ in $X\subset \bf{R^n}.$ Show that, there exists smooth function $g$ defined on an open set $U$ in $\bf{R^k}$ such that $V$ may be taken to be $\{(x,g(x))\in \bf{R^n}:x\in U\}.$
This is basically implicit function theorem. But, since I am an beginner , I want someone to check my solution.
Consider, the function, $h:V\to\mathbb{R}^k$ defined by, $h=(x_1,...,x_k)$ which is smooth. Since, $x_1,..,x_k$ form a local coordinate on $V$ so $h$ is a submersion therefore by submersion theorem for each $x\in V$ there exists open sets $U(\subset\mathbb{R}^k)$ and smooth function $g_1:U\to X$ ( where $g_1(u)=(g_2(u),g_3(u))\in\mathbb{R^k}\times\mathbb{R}^{n-k}$) such that $h(g_1(u))=u $,so $g_2(u)=u$ hence if we take $g(u)=\pi_2\circ g_1 (u)=g_3(u)$ then the problem is solved.