What does "new coordinates" mean in this context?

Question

The book I'm reading states that we can use the theorem below to say if an immersion is sufficiently smooth, then it's possible to set new coordinates in the neighborhood of every point of the image so that $f$ is locally an inclusion. So what does new coordinates mean in this context?

Immersion theorem: Let $U\subset \mathbb R^m$ be an open set and $f:U\to \mathbb R^{m+n}$ be a strongly differentiable function at $a\in U$.

There are open sets $Z\subset \mathbb R^{m+n}$ and $V\times W\subset \mathbb R^m\times\mathbb R^n$ with $f(a)\in Z$ and $(a,0)\in V\times W$ and a homeomorphism $h:Z\to V\times W$ such that $hf(x)=(x,0)$ for every $x\in V$ and $h$ is strongly differentiable at the point $f(a)$.

If $f\in C^k (k\ge 1)$, it's possible to restrict if necessary $V$, $W$ and $Z$ such that $h$ is a $C^k$ diffeomorphism.

Answer 1

The "old" coordinates are the canonical Cartesian coordinates on $\Bbb R^{m+n}$. The new coordinate functions on $\Bbb R^{m+n}$ are $m+n$ independent, quite possibly non-linear, functions such that the first $m$ are coordinates on $f(U)$ (and reproduce the canonical coordinates on $U$) and the remaining define $f(U)$ as their root set.

In all, this is an almost trivial application of the inverse function theorem, the only non-trivial part is to construct a complement $B$ of $A=Df(a)$ so that the combined map $f(x)+By$ has an invertible derivative at $(x,y)=(a,0)$.