So in my Calc 3 class we use Shifrin's "Multivariable Mathematics", and his discussion on Differential Forms and Integration on Manifolds is impossible for me to follow.
Can someone recommend resources so I can understand this chapter?
Edit: I'm lost in the notation and a lot of the linear algebra, the section "The Multilinear Setup" was very confusing. How can scalars form a basis for Rn? ({dx1,...dxn})
Edit2: (I'm trying here) My question can be condensed to, "Throw resources at me that you found helpful in understanding differential forms that is meant for an undergraduate to be able to follow"
Omar,
This is tough for most students the first time through. Here are two suggestions:
(1) You might look at Steve Weintraub's book, which is quite concrete.
(2) You might find my lectures from the UGA course based on that text helpful, with examples and a bit more interaction with students. (Note that the last ten minutes or so of the original lecture on differential forms got deleted, and so there is a redo of that lecture one year later labeled Day 24 (complete).)
But I do need to respond to the one question you asked. $dx_1,\dots,dx_n$ are not scalars. They are linear maps from $\Bbb R^n$ to $\Bbb R$. Namely, $dx_i(\mathbf v) = \mathbf v\cdot\mathbf e_i$, i.e., the $i$th component of $\mathbf v$ when you write it as we did starting page 1 of the course.
By the time you get into computational stuff with line integrals and surface integrals, it will just start to make more sense. Keep on going!