Let $f: X \rightarrow Y$ be a smooth map between manifolds without boundary.

(i) Define what is meant by a critical point, critical value and regular value of $f$.

(ii) Show that if $y$ is a regular value of $f$ and $\operatorname{dim} X \geqslant \operatorname{dim} Y$, then the set $f^{-1}(y)$ is a submanifold of $X$ with $\operatorname{dim} f^{-1}(y)=\operatorname{dim} X-\operatorname{dim} Y$.

[You may assume the inverse function theorem.]

(iii) Let $S L(n, \mathbb{R})$ be the group of all $n \times n$ real matrices with determinant 1. Prove that $S L(n, \mathbb{R})$ is a submanifold of the set of all $n \times n$ real matrices. Find the tangent space to $S L(n, \mathbb{R})$ at the identity matrix.