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

If $X$ is tangent to $N$ and $f|_N=0$, then the restriction of $f$ to every curve in $N$ is zero, so $(Xf)|_N=0$.

Conversely, use a <slice chart for an embedded submanifold>. The functions $x^{n+1},\ldots,x^m$ vanish on $N$. Writing $X=X^i\partial_i$, the hypothesis gives
$$
Xx^\alpha=X^\alpha=0\quad\text{on }N,\qquad \alpha>n.
$$
Thus $X$ has no normal component and is tangent to $N$. Equivalently, $X$ preserves the <vanishing ideal of an embedded submanifold>.

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

If $N^n\subset M^m$ is embedded and $p\in N$, there are smooth coordinates $(x^1,\ldots,x^m)$ around $p$ in which
$$
N=\{x^{n+1}=\cdots=x^m=0\}.
$$
The first $n$ coordinates restrict to a chart on $N$.


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 Incoming links: Slice chart for an embedded submanifold