For manifolds $X, Y \subset \mathbb{R}^{n}$, define the terms tangent space to $X$ at a point $x \in X$ and derivative $d f_{x}$ of a smooth map $f: X \rightarrow Y$. State the Inverse Function Theorem for smooth maps between manifolds without boundary.

Now let $X$ be a submanifold of $Y$ and $f: X \rightarrow Y$ the inclusion map. By considering the map $f^{-1}: f(X) \rightarrow X$, or otherwise, show that $d f_{x}$ is injective for each $x \in X$.

Show further that there exist local coordinates around $x$ and around $y=f(x)$ such that $f$ is given in these coordinates by

$$\left(x_{1}, \ldots, x_{l}\right) \in \mathbb{R}^{l} \mapsto\left(x_{1}, \ldots, x_{l}, 0, \ldots, 0\right) \in \mathbb{R}^{k},$$

where $l=\operatorname{dim} X$ and $k=\operatorname{dim} Y$. [You may assume that any open ball in $\mathbb{R}^{l}$ is diffeomorphic to $\mathbb{R}^{l}$.]