A Geometric Approach to Differential Forms by David Bachman

[sponsoredwalk]

I wish I read this book properly earlier, I read the first three chapters but I was just so
stubborn that I wouldn't allow myself to learn a thing further about differential forms until
I found out how to express forms & the wedge product in terms of mappings.
Well I got help on PF on how to express forms via the dual spaces & found out how to express
the wedge product as a mapping from the first few pages of Cartan's book on differential
forms, so now I've spent the past week back reading Bachman's book & can almost smell
the generalized Stokes' theorem!

First off, I love the pedagogical approach of trying to define things in different ways, for
example the comments about defining the product of forms in such a way so as to take us
out of the world of forms, the comment about trying to define φ⋀ψ⋀ω as φ⋀(ψ⋀ω), the
analysis, & re-analysis, of the 5 steps in defining the integral of a form & the problems in
the definition, the comment about needing to define dω as an alternating sum over all
variations (Halmos makes a similar comment in his FDVS book about making k-linear forms
symmetric in general only by summing over all permutations, small comments like this have
big consequences!).

As for me posting here, well I really can't find help online & don't want to start a thread
asking questions that possibly depend on the previous material in this book & get no
responses. I just want to post:
1) a possible small mistake in the book,
2) a small question,
3) my derivation of Stokes theorem for rectangles as it's done in the example in
__ the book, with an (n - 1) form.
__ I'm really hoping for is some patient person to follow all the subscripts & try to find
__ some ridiculous error on my part & explain the subscript jump at the very end.

1) First off, I think there's a small mistake the example of a 2-cell.
Bachman defines an n-cell as the image of a map σ : [0,1]ⁿ → ℝ^m, but then
in the example of a 2-cell he refers to as given earlier in the book where the example is
φ(x,t) = ... where a ≤ x ≤b, 0 ≤ t ≤1, it seems to me that unless I'm understanding this
wrong I don't think he can call this map φ a 2-cell, as he has defined a 2-cell, unless a = 0,
b = 1. Is that correct?
That might have confused me to no end had I not read some of Lang's explanation of
forms & remembered he defines a simplex as the image of a rectangle, not a cube, a
rectangle in ℝⁿ & does not restrict anything to [0,1]. It's a minor point but more than
likely it's me that's wrong & I'd like to find out why!

2) I don't think anything needs to be modified to define everything in terms of simplexes
as it's done in the book, right?

3) I ask this only for notational purposes really.
If ω is an (n - 1) form on ℝⁿ, where ƒ_i : ℝⁿ → ℝ, then
ω = ƒ₁dx₂⋀dx₃⋀...⋀dx_n + ... + ƒ_ndx₁⋀dx₂⋀...⋀dx_{n - 1}
and
dω = ∂₁ƒ₁dx₁⋀dx₂⋀dx₃⋀...⋀dx_n + ... + (-1)^n-1∂_nƒ_ndx₁⋀dx₂⋀...⋀dx_{n - 1}⋀dx_n

Now, supposing S is a simplex in ℝⁿ with a rectangular lattice of points {p_{i_1,...,i_n}} in S,
define the vectors $$V^{k}_{i_1,i_2,...,i_n} \ = \ p_{i_1,i_2,...,i_k + 1,...,i_n} \ - \ p_{i_1,i_2,...,i_k,...,i_n}$$ for (k ∈ ℕ_n).

Then

$$d \omega_{p_{i_1,i_2,...,i_n}} (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) \ = \ \frac{\partial f_1}{\partial x_1}dx_1 \wedge dx_2 \wedge ... \wedge dx_n (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) + ... + (-1)^{n-1}\frac{\partial f_n}{\partial x_n}dx_1 \wedge dx_2 \wedge ... \wedge dx_n (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n})$$

Using the idea in the book of the volume of the cube of height t having a base dx₁⋀dx₂(V₁,V₂),

$$d \omega_{p_{i_1,i_2,...,i_n}} (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) \ \approxeq \ \frac{f_1(p_{i_1+1,i_2,...,i_n}) \ - \ f_1(p_{i_1,i_2,...,i_n}) }{t_1}t_1 dx_2 \wedge ... \wedge dx_n (V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) + ... + (-1)^{n-1}\frac{f_n(p_{i_1,i_2,...,i_n+1}) \ - \ f_n(p_{i_1,i_2,...,i_n}) }{t_n}t_n dx_1 \wedge ... \wedge dx_{n - 1} (V^{1}_{i_1,i_2,...,i_n},...,V^{n-1}_{i_1,i_2,...,i_n})$$

So

$$d \omega_{p_{i_1,i_2,...,i_n}} (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) \ \approxeq \ [f_1 (p_{i_1+1,i_2,...,i_n})dx_2 \wedge ... \wedge dx_n (V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) \ - \ f_1(p_{i_1,i_2,...,i_n})dx_2 \wedge ... \wedge dx_n (V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) ] + ... + (-1)^{n-1} [f_n(p_{i_1,i_2,...,i_n+1})dx_1 \wedge ... \wedge dx_{n - 1} (V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_{n-1}}) \ - \ f_n(p_{i_1,i_2,...,i_n})dx_1 \wedge ... \wedge dx_{n - 1} (V^{1}_{i_1,i_2,...,i_n},...,V^{n-1}_{i_1,i_2,...,i_n})]$$

or

$$d \omega_{p_{i_1,i_2,...,i_n}} (V^{1}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n}) \ \approxeq \ [\omega(V^{2}_{i_1+1,i_2,...,i_n},...,V^{n}_{i_1+1,i_2,...,i_n}) \ - \ \omega(V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n})] \ + \ ... \ + \ [(-1)^{n-1} [ \omega(V^{1}_{i_1,i_2,...,i_n + 1},...,V^{n-1}_{i_1,i_2,...,i_n + 1}) \ - \ \omega(V^{1}_{i_1,i_2,...,i_n},...,V^{n-1}_{i_1,i_2,...,i_n})]]$$

Now we want to integrate dω over S, so we sum over all the i_i's by distributing
the summation to each component of the last monster latex thing I posted, summing first
with respect to the appropriate component so that each term is over the "top" & "bottom"
of S (i.e. the boundary points) as is done in the book.

Taking the limit as the lattice points in the simplex S go to zero, we find that

$$\int_S d \omega \ = \ \int_{\partial S} \omega$$

Hopefully that's right, not too sure how the
$$f_1 (p_{i_1+1,i_2,...,i_n})dx_2 \wedge ... \wedge dx_n (V^{2}_{i_1,i_2,...,i_n},...,V^{n}_{i_1,i_2,...,i_n})$$
terms become
$$\omega(V^{2}_{i_1+1,i_2,...,i_n},...,V^{n}_{i_1+1,i_2,...,i_n})$$
the subscripts go from i_i to i_i+1 on the V's in the book,
but other than that I think it's okay...

Thanks so much anybody who read through that