Integration on chains in Spivak's calculus on manifolds

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Discussion Overview

The discussion revolves around the integration of k-forms over k-chains as presented in Spivak's "Calculus on Manifolds." Participants explore the implications of Spivak's approach, particularly regarding the structure of chains, the independence of integrals over chains with identical images, and the geometric interpretation of reparametrizations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant notes that Spivak's treatment of chains lacks certain structures, such as orientation and injectivity, leading to confusion about the independence of integrals over chains with identical images.
  • Another participant questions whether two k-chains with the same image necessarily yield the same integral, providing a counterexample where one chain is a multiple of another.
  • A participant corrects their earlier terminology, clarifying that they meant k-cube instead of k-chain, defining a k-cube as a smooth map from [0,1]^k to a subset of R^n.
  • Counterexamples are discussed, such as one curve tracing a circle and another tracing the same circle twice, challenging the intuition about integrals over chains with identical images.
  • Participants express curiosity about the geometric implications of reparametrizations that satisfy the condition det p'(x) >= 0, seeking clarity on what such a reparametrization looks like.
  • A new participant introduces a homework problem involving the computation of area over a chain representing a square with a semicircle, indicating a lack of clarity on how to define the appropriate chain for this scenario.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Spivak's definitions and the independence of integrals over chains. There is no consensus on the geometric interpretation of reparametrizations or the resolution of the homework problem presented.

Contextual Notes

Participants highlight limitations in understanding the definitions and implications of chains and reparametrizations, particularly regarding injectivity and the geometric meaning of the determinant condition.

Who May Find This Useful

Students and educators interested in advanced calculus, particularly those studying integration on manifolds and the theoretical foundations of differential forms and chains.

quasar987
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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 1-1 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 k-form over a k-chain is whether or not the result is independent of the chain. More precisely, if c and d are two k-chains with identical images, does

\int_c\omega=\int_d\omega

as intuition demands??

I found a little guidance in answering this in the person of problem 4-25 (Independence of parametrization), but that's not entirely satisfying, because after all, coudn't it be that there is no 1-1 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?!)
 
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quasar987 said:
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 1-1 requirement and so on), and as a result, I find that things get a little weird.
Isn't a chain a formal linear combination of parametrized regions? The parametrization on a region gives it an orientation.

More precisely, if c and d are two k-chains with identical images, does

\int_c\omega=\int_d\omega

as intuition demands??
If we set d = 2c, then wouldn't they have the same image and different integrals?
 
Oops, I wrote k-chain everywhere where I should have written k-cube.

His k-cube on A (subset of R^n) is a smooth map c:[0,1]^k-->A.
 
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...)
 
Last edited:
Mmmh, true.

And about problem 4-25? It reads,

"Let c be a k-cube and p:[0,1]^k-->[0,1]^k a bijection with det p'(x) >= 0 everywhere. If w is a k-form, then

\int_c\omega=\int_{c\circ p}\omega"

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?
 
Hello everybody!
I am an exchange student in Canada and one of the course I choose is Calculus on Manifolds. This is a interesting 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 semi-circle 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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