Because the inclusion $\iota:N\hookrightarrow M$ has constant rank $n$, the <constant rank theorem> gives coordinates $y^1,\ldots,y^n$ on $N$ and $x^1,\ldots,x^m$ on $M$ in which
$$
\iota(y^1,\ldots,y^n)=(y^1,\ldots,y^n,0,\ldots,0).
$$
Since $\iota$ is an embedding, the ambient chart can be shrunk so that it meets no other local sheet of $N$. It then satisfies
$$
\psi(V\cap N)=\{x\in\psi(V):x^{n+1}=\cdots=x^m=0\},
$$
and $(x^1,\ldots,x^n)$ restricts to the required chart on $N$. This is a <slice chart for an embedded submanifold>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 2 / b / Solution

{wiki}

If a smooth map has constant rank $r$ near a point, there are local coordinates in which it is
$$
(x^1,\ldots,x^m)\longmapsto(x^1,\ldots,x^r,0,\ldots,0).
$$


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 Incoming links: Constant rank theorem