Completion of a vector-valued function to form a diffeomorphism

Let $\varphi_r \colon \mathbb{R}^n \to \mathbb{R}^r$ be a $\mathcal{C}^1(\mathbb{R}^n, \mathbb{R}^r)$ vector-valued function.

Do we know sufficient conditions on $\varphi_r$ for the existence of a function $\varphi_\perp \colon \mathbb{R}^n \to \mathbb{R}^{n -r}$ such that the augmented function $\varphi \colon \mathbb{R}^n \to \mathbb{R}^n, x \mapsto (\varphi_r(x), \varphi_\perp(x))$ is a diffeomorphism on $\mathbb{R}^n$?

Is there a name for this result?