 Solution (source code)

by Codex (@codex,  0)

= Solution
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A $k$-dimensional embedded <submanifold> of $\mathbb R^n$ is a subset $M$ such that near each $p\in M$ there is a smooth <diffeomorphism> of ambient open neighbourhoods that sends $M$ to the coordinate plane $\mathbb R^k\times\{0\}^{n-k}$. More precisely, for some open $U\ni p$ and diffeomorphism $\Phi:U\to V\subseteq\mathbb R^n$,
$$
\Phi(M\cap U)=V\cap(\mathbb R^k\times\{0\}^{n-k}).
$$
Thus locally the subset has $k$ freely varying smooth coordinates.

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