I'm getting very muddled by the notion(s) of integration used in Lee's book. At first I thought the 4th chapter on the integral of 1-forms was separated from the main treatment of integration in chapter 10 simply because the author wished to present an application early on in the text to stop things getting too terse, but after closely looking at the two definitions I'm starting to get paranoid that these integrals are subtly different.
In chapter 4 our integral requires a curve $\gamma : [a,b] \rightarrow M$ and a $1$-form to be integrated $\omega$ that eats vectors from $T_{\gamma(t)}[a,b]$. Our "abstract manifold" is really just the interval $[a,b]$. What confuses me is we have this freedom to choose the manifold $M$ which $\gamma$ maps into.
We define $\displaystyle \int_\gamma \omega:=\int_a^b\gamma^*\omega$
Compare this to chapter 10 where we are integrating in an $n$ manifold $N$, and we integrate the $n$-form $\omega \in T^* N$ by taking the pullback of an inverse chart map $\varphi^{-1}$.
Here we have $\displaystyle \int_U \omega:=\int_{\varphi(U)}{(\varphi^{-1})}^*\omega$
At first glance it looks obvious, the curve is just like the inverse chart map for a 1-D manifold, job is done. However, if I want to do the "chapter 10 integrals" in 1-D then $N = [a,b]$ my chart maps $\varphi : U\subset N \rightarrow \varphi(U) \subset \mathbb{R}$, in particular, note how since $N$ is a 1-d manifold, my chart can only map to subsets of the real line by definition of being a smooth manifold of dimension 1. There is no freedom like in chapter 4 to get exotic curves in $\mathbb{R}^2, \mathbb{R}^3$ or any $M$.
What's going on here? Is there some freedoms that you can get in 1-D that don't exist in higher dimensions? I certainly remember my vector calculus courses letting me do line integrals in any number of dimensions I wanted, Lee's example 4.18 certainly shows a nice curve in $\mathbb{R}^3$, while the chapter 10.22 example has a 2-form integrated on sphere which is calculated by pulling back with a map $F:D \subset \mathbb{R}^2 \rightarrow \mathbb{S}^2$, like we are trapped in the $\mathbb{R}^n$ of the same dimension as the manifold!
Even the name line integral begins to confuse me, are integrals of differential forms meant to generalise integrals from calculus such as
$\hspace{15mm}\displaystyle \int {\bf{v}} \cdot d{\bf{r}} \hspace{10mm}\text{or}\hspace{10mm} \int f({\bf{r}})ds \hspace{10mm}\text{or}\hspace{10mm} \int f(x)dV$?
Chapter 4 makes it sound like differential forms give us $n$-dimensional analogues on curved spaces to integrals like the left, while chapter 10 sounds more like an n-dimensional curved analogue of the right most integral, but I thought integrating functions was done by the Riemannian volume form? And that still leaves me wondering how to generalise the middle to get a "curved scalar hypersurface integral", of the kind you might see on the RHS of an $n$-dimensional divergence theorem, say. I think this needs a Riemannian metric, but am unsure how it could be used, since problem 4.3 rules out the existence of any differential form $\omega$ such that $\omega "=" ds$ generalises the concept of a line element.
I would appreciate any assistance in clearing up this huge error in my understanding.