"Introduction to Smooth Manifolds" - differential of a function query

Question

I'm trying to understand the differential $df$ as an approximation to $\Delta f$, in Lee's Introduction to Smooth Manifolds (p282-283 - see image below). He says “let $p$ be a point on $M$” but then, in the diagram, labels $p$ in $U$ but not on $M$. Why does he do that?

Answer 1

Lee uses a chart $\phi : V \to U \subset \mathbb R^n$ around $p$ and considers $f \circ \phi^{-1} : U \to \mathbb R$. In Fig. 11.2 he should have written $\phi(p)$ instead of $p$, but working with a chart means that we may consider without loss of generality the special case $M = U$ and $p \in U$. Lee explicitly says "we can think of $f$ as a function on an open $U$". This is what Fig. 11.2 shows.

Answer 2

Paul Frost's answer is completely correct. But I also want to add that what's really going on here is an example of using a coordinate chart to "identify" an open subset of $M$ with the corresponding open subset of $\mathbb R^n$. I discuss this at some length starting at the bottom of page 15.