I am stuck at the following exercise:
Let $f:D \subset \mathbb{R}^k \rightarrow \mathbb{R}^n$ with $k \le n$ be a regular parametrisation and let $\sigma: \mathbb{R}^k \rightarrow D$ be a diffeomorphism. Prove that $\hat{f} = f \circ \sigma$ is a regular parametrisation.
I realise that I only have to show that $rk(D\hat{f}) = k$, but I do not know how to do that. Things would be easy if a diffeomorphism was linear, but we can not assume that here. Could you give me a hint?