I am reading "Analysis on Manifolds" by James R. Munkres.
Definition. Let $S$ be a subset of $\mathbb{R}^k$; let $f:S\to\mathbb{R}^n$. We say that $f$ is of class $C^r$ on $S$ if $f$ may be extended to a function $g:U\to\mathbb{R}^n$ that is of class $C^r$ on an open set $U$ of $\mathbb{R}^k$ containing $S$.
If $g$ exists, we can define partial derivatives of $f:S\to\mathbb{R}^n$ at $a\in S$.
But the author didn't define partial derivatives of $f:S\to\mathbb{R}^n$ at $a\in S$.
I guess the partial derivative of $f:S\to\mathbb{R}^n$ at $a\in S$ with respect to an aribitrary vector $e_j$ is not useful.
So, the author didn't define it.
Am I right?