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Integration on chains in Spivak's calculus on manifolds 
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#1
Aug1708, 11:58 PM

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I would like to discuss this chapter with someone who has read the book.
From looking at other books, I realize that Spivak does things a little differently. He seems to be putting less structure on his chains (for instance, no mention of orientation, no 11 requirement and so on), and as a result, I find that things get a little weird. For instance, the first thing I asked myself after reading the definition of the integral of a kform over a kchain is whether or not the result is independent of the chain. More precisely, if c and d are two kchains with identical images, does [tex]\int_c\omega=\int_d\omega[/tex] as intuition demands?? I found a little guidance in answering this in the person of problem 425 (Independence of parametrization), but that's not entirely satisfying, because after all, coudn't it be that there is no 11 p such that c o p = d? If c and d are not injective for instance, the obvious p(t) := c^1(d(t)) fails. And that det p'(x) >= 0 condition... what does it say about p? What characterize reparametrizations p with det p'(x) >= 0? (Does injectivity implies that the determinant does not chance sign? locally okay, but globally?!) 


#2
Aug1808, 12:14 AM

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#3
Aug1808, 01:37 AM

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Oops, I wrote kchain everywhere where I should have written kcube.
His kcube on A (subset of R^n) is a smooth map c:[0,1]^k>A. 


#4
Aug1808, 02:37 AM

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Integration on chains in Spivak's calculus on manifolds
Well, it's easy enough to construct counterexamples in the same spirit. For example, c could be a curve tracing out a circle on the Euclidean plane, and d could be another curve that traces out the same circle twice.
(Of course, I doubt your intuition ever really demanded that these be the same....) 


#5
Aug1808, 11:09 AM

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Mmmh, true.
And about problem 425? It reads, "Let c be a kcube and p:[0,1]^k>[0,1]^k a bijection with det p'(x) >= 0 everywhere. If w is a kform, then [tex]\int_c\omega=\int_{c\circ p}\omega[/tex]" The proof is direct... what I'm wondering is say I want to reparametrize c with a p as in the exercise. What does a p with det p'(x) >= 0 looks like? What does det p'(x) >= 0 says about p geometrically or otherwise? 


#6
Nov2309, 03:34 PM

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Hello everybody!
I am an exchange student in Canada and one of the course I choose is Calculus on Manifolds. This is a intersting as it is difficult but I deal with it! Anyway, I try to compute an aera over the chain but I can't find the right chain! I have to compute the area on R2 of a square with a semicircle on its top (I hope it is easy to understand). If you could give me any to start with or hint, I'd be glad because for now I don't even have an idea. Hope to hear from you, Alex 


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