Understanding Stokes Theorem and is the variation of the metric a tensor?

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Discussion Overview

The discussion revolves around two mathematical questions related to the variation of the metric in the context of differential geometry and the application of Stokes' theorem. The scope includes theoretical aspects of tensor analysis and mathematical reasoning related to diffeomorphisms and action variations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the variation of the metric is a tensor, suggesting it may not transform like one and seeks a sensible expression for the covariant derivative of this variation.
  • The same participant expresses confusion about Stokes' theorem, asking if it applies only to covariant divergences or also to partial and exterior derivatives, indicating a need for clarification in rewriting the variation of an action.
  • Another participant inquires if the original poster has found answers to their questions, indicating ongoing interest in the topic.
  • A later reply humorously notes the age of the original post and suggests providing insights for readers, indicating a desire to enhance the discussion for others.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants are exploring questions and expressing confusion without definitive answers or resolutions.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the transformation properties of the variation of the metric and the specific applications of Stokes' theorem, which remain unresolved.

haushofer
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Hi, I have 2 little questions and hope to find some clarity here. It concerns some mathematics.

1) Is the variation of the metric again a tensor? I have the suspicion that it's not, because i would say that it doesn't transform like one. How can i get a sensible expression then for the covariant derivative of the variation of the metric? I need this to calculate the variation of the Riemanntensor, induced by a diffeomorphism multiplied by some test function.

2) I have the idea that I don't quite understand Stoke's theorem. Does it only apply for covariant divergences, or also for partial derivatives and/or exterior derivatives? I'm a little confused here :( I need this to rewrite the variation of an action.

Thanks in forward,

Haus.
 
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Hi Greg,

Yes, I did, why do you ask? I didn't recognize the podt until I saw that it's 12 years old :D
 
haushofer said:
Hi Greg,

Yes, I did, why do you ask? I didn't recognize the podt until I saw that it's 12 years old :D
Would you be interested in writing a couple sentences on each to give insight for those guests reading this thread? :smile:
 

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