Asymmetry in Stokes' theorem & Gauss' theorem

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Discussion Overview

The discussion revolves around the asymmetry observed between Stokes' theorem and Gauss' theorem, focusing on the mathematical and philosophical implications of this asymmetry in the context of calculus. Participants explore the definitions and relationships between closed curves, surfaces, and the volumes they enclose, considering both theoretical and applied perspectives.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that a closed surface defines a unique volume, while a closed curve does not uniquely define a 3D volume, leading to questions about the asymmetry between the two theorems.
  • Others argue that there is no asymmetry, suggesting that a closed 2D surface can enclose multiple 3D volumes if certain conditions, such as allowing "bridges," are met.
  • A participant proposes that the asymmetry might relate to the topology of different dimensional spaces, specifically mentioning ##\mathbb{R^3}## and ##\mathbb{R^n}##.
  • Some participants express doubt about the definitions of closed and connected surfaces, questioning whether a non-planar 1D curve can enclose a 2D surface.
  • A later reply introduces the idea that a simple closed curve bounds at most one area, while a non-simple closed curve may bound multiple areas, drawing a parallel to surfaces.
  • Participants discuss examples, such as connecting two spheres, to illustrate their points about the volumes enclosed by surfaces.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether there is an asymmetry between Stokes' theorem and Gauss' theorem. Multiple competing views remain, with some asserting the existence of asymmetry and others denying it, leading to ongoing debate and clarification.

Contextual Notes

Participants highlight the importance of definitions and the potential for confusion regarding closed versus connected surfaces. The discussion also touches on the implications of dimensionality and topology, which remain unresolved.

  • #31
PeroK said:
Also, this is the third post on the subject. It was already answered in the threads mentioned in post #12 above. Here, for example, is a good answer to the question:

https://www.physicsforums.com/threa...d-by-the-same-curve-in-stokes-theorem.989709/
@feynman1 You apparently got your answer. If you still have problems, then we may have communication problems, which is not unlikely since we are restricted to verbal only communication.

My suggestion is: Quote the original theorem you have a question to. Famous theorems are sometimes differently phrased. It looks as if you have trouble to understand the conditions of Gauß law. So the more precise you ask, the better the answers will be.
 
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