I am really lost in the notation of my book and am looking for some conceptual help.
So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, which maps from $\mathbb{R}^n \to \mathbb{R}$? That's what I get from definition 11 and remark 12.
I don't get remark 13 since I never really understood the application of bases when we first covered it.
Moving on, definition 14 states that the differential of f is the differential 1-form $df$ given by $df_x = Df(x)$. This means that for every value of $x\in U$ maps to a linear function $df_x$, which maps from $\mathbb{R}^n \to \mathbb{R}$?. Since $df_x=Df(x)$, $df_x$ is a linear map from $\mathbb{R}^n \to \mathbb{R}$ such that $\lim_{h \to 0} \|f(x+h)-f(x)-df_x(h)\|/\|=0$.
Remark 15 (not included) just notes that the definition from point 14 is, in a sense, a mere change of notation.
I'm having trouble with definition 16. I'm a bit confused with why we suddenly switch from working with $df_x$ to $dx_i$. It made sense when we were taking the differential 1-form of a function, but you can do it for a point too? I'm having trouble wrapping my mind around this part. From definition 11, since the differential 1-form is $dx_i$, we have that $dx_i_x$ is the linear map associated with the point $x\in U$. I just don't even know what that means anymore.
I am not required to complete exercise 17, so I am just treating it as a remark. I don't know what to make of $dx_i=d\pi _i$ though since the special differential 1-form $dx$ is a mystery to me.
Remark 18 seems to be consistent with intuition.
I am not required to complete exercise 19, although I think it may be required in order to complete exercise 20, which I am required to complete. However, since I don't understand definition 16, I'm afraid I don't have any meaningful way to get started on either exercise 19 or 20.
Ultimately, the purpose of me posting this on Stack Exchange is to develop a solution for exercise 20. I included everything else though since I clearly need some conceptual help before I can proceed with the exercise. Help, please?
That $x_i$ notation is really confusing, indeed. Basically, as Exerc.17. states, we have $$dx_i\ =\ d\pi_i$$ where $\pi_i:\Bbb R^n\to\Bbb R$ is the $i$th projection.
This is a linear function, and, as such, it maximally approximates itself in every point, hence for each point $P$, we have $$(d\pi_i)_P\ =\ \pi_i\,,$$ which justifies both Def.16 and Exerc.17. It has not much to do with the actual coordinates of $P$, that's why I avoided to use $x$ for the general point.
The notation $dx_i$, however, is traditional.
Note that in matrix form, $dx_1=(1,0,0,\dots),\ dx_2=(0,1,0,0,\dots)$, etc. at each point $P$.