Question about Stokes' Thm and Boundaries of Surfaces

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Discussion Overview

The discussion revolves around the application of Stokes' Theorem in the context of differential forms, specifically examining the 1-form \(\eta\) on \(\mathbb{R}^2 - \{0\}\) and its implications when considering extensions to \(\mathbb{R}^3 - \{0\}\). Participants explore the significance of the C'' condition for surfaces and the challenges of defining forms in higher dimensions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the integral \(\int_{\gamma}\eta=2\pi\) suggests that \(\gamma\) is not the boundary of any 2-chain of class C'' in the punctured plane, raising questions about extending the discussion to \(\mathbb{R}^3 - \{0\}\).
  • Another participant points out that the 1-form \(\eta\) can be interpreted as a form on \(\mathbb{R}^2 - \{0\}\) or \(\mathbb{R}^3\) minus the z-axis, emphasizing the need for a continuous definition on the z-axis.
  • A different viewpoint suggests that one could consider \(\eta\) as remaining in the xy-plane without involving the z-axis, prompting a challenge regarding the definition of a 1-form in \(\mathbb{R}^3 - \{0\}\).
  • Concerns are raised about the necessity of specifying values for \(\eta\) on all points of \(\mathbb{R}^3 - \{0\}\) to maintain its status as a 1-form.
  • One participant expresses confusion about the importance of the C'' condition for the surface, contrasting it with the C' condition which relates to the Jacobian in variable changes.
  • Another participant suggests that it is necessary to demonstrate that \(\eta\) cannot be extended to a closed 1-form in \(\mathbb{R}^3 - \{0\}\), proposing a method involving the unit sphere and reducing the problem to the planar case.

Areas of Agreement / Disagreement

Participants express differing views on the implications of extending the 1-form to higher dimensions and the significance of the C'' condition. There is no consensus on the necessity of the C'' condition or how to approach the extension of \(\eta\) in \(\mathbb{R}^3 - \{0\}\).

Contextual Notes

Participants note limitations in defining the 1-form continuously across different dimensions and the implications of the punctured plane on the application of Stokes' Theorem. The discussion also highlights unresolved questions regarding the nature of the surfaces involved and the conditions required for their classification.

Poopsilon
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So I'm going over Rudin's chapter on differential forms in his Principles of Mathematical Analysis and I'm looking at Example 10.36 which gives the 1 form \eta = \frac{xdy-ydx}{x^2+y^2} on the set \mathbb{R}^2-{0} and then parametrizes the circle \gamma(t)=(rcos(t),rsin(t)) for fixed r>0 and 0≤t≤2pi.

Nowd\eta=0yet direct computation shows that \int_{\gamma}\eta=2\pi.

Thus by Stokes' Theorem we can then conclude that gamma is not the boundary of any 2-chain of class C'' in the punctured plane.

Now I understand that because the origin is not included we can't just use the disk of radius r as our 2-surface with boundary equal to gamma (I think this is related to the Cauchy Residue Theorem) but what if we extended ourselves to \mathbb{R}^3-{0} than we could parametrize some sort of cone-like 2-surface with boundary equal to gamma which we could probably make C'' which would then by Stokes' Theorem force the integral to be \int_{\gamma}\eta=0 thus contradicting the integral given above.

Also what is up with this C'' requirement? I can't figure out why its important.
 
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The formula for eta as it stands can be interpreted as a 1-form on R² - 0 or R³ - "z axis". If you want a 1-form on R³ - 0, you need to specify in addition what eta is to be on "z-axis" - 0. But you're going to have trouble doing that in a continuous fashion!
 
Can't I just leave it is as so it doesn't even involve the z axis just sits inside the xy-plane?
 
We if you're going to talk about a 1-form on R³ - 0, you better tell what its value is on each point of R³ - 0. Otherwise, that's not a 1-form on R³ - 0.

It's like if you say "consider the function f: R-->R defined by f(0)=0". That's nonsense: the formula f(0)=0 only defines a map f:{0}-->R.
 
Ah Sorry you are completely right the interpretation I will have to have is R³ - 'z-axis' which would disallow any type of cone structure with boundary gamma. Could you tell me why the C'' condition is important for my surface, I mean C' I understand because you need to be able to take the Jacobian upon changing variables, but why C''?
 
I don't know about the C² thing.
 
I think that technically you would need to show that eta has no extension to a closed 1 form in R^3 - 0.

Try pulling such a form back to the unit sphere via the inclusion map and then reducing the problem to the planar case.
 

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