TITLE: "Introduction to Smooth Manifolds" - differential of a function query
QUESTION [6 upvotes]: 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?

REPLY [10 votes]: 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.