# “Introduction to Smooth Manifolds” - differential of a function query

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? 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.
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.
• I usually think charts as injections to Euclidean space. i.e. I consider $U\subset M$ as a subset of $\Bbb R^n$ simultaneously. Is this correct? – C.F.G Feb 8 at 19:03