Let $M$ be a smooth manifold of dimension $n$. A subset $S$ of $M$ is said to be a regular submanifold of $M$ of dimension $k$ if, for every point $p \in S$, there exists a coordinate chart $(U,\phi)$ of $M$ around $p$ such that $S \cap U$ is given by the vanishing of (last) $n-k$ coordinates.
In this case, we call $(U,\phi)$ an adapted coordinate chart around $p$ relative to $S$. I am curious about the terminology "adapted coordinate chart". Is it because $S$ has "adapted" the coordinate chart of $M$ in some sense?