Exterior Calculus and Differential Forms?

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Discussion Overview

The discussion revolves around the topic of exterior calculus and differential forms, specifically focusing on sinusoidal solutions to a 1-form field on a pseudo-Riemannian manifold. Participants explore the implications of various differential operators and their relationships to solutions of differential equations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the appropriateness of the forum for discussing exterior calculus and differential forms.
  • Another participant confirms that the forum is suitable for such discussions.
  • A participant expresses interest in finding sinusoidal solutions to a 1-form field, A, on a pseudo-Riemann manifold and questions whether the equation *d*d*A=0 yields the most general case.
  • The same participant introduces the Laplace-Beltrami operator, represented as (d + δ)², and discusses its potential application in the context of the equation (*d*d* + d*d*)A=0.
  • Further, the participant notes that after expanding over time and spatial indices, the equation simplifies to the d'Alembertian, ∇²A = 0, and questions which operator is the most general.
  • The participant also mentions the possibility of considering higher-order equations of the form (*d*d*d*d + d*d*d*d* + etc.)A=0.
  • Another participant seeks a forum where they can engage with individuals knowledgeable in the field of exterior calculus and differential forms.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants express varying levels of inquiry and seek clarification on different aspects of the topic without resolving the questions posed.

Contextual Notes

The discussion includes references to specific operators and equations, but the implications of these operators and their generality remain unresolved. There is also a lack of clarity regarding the definitions and assumptions underlying the equations discussed.

Phrak
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Would this be the right forum to pose questions on this topic?
 
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Phrak said:
Would this be the right forum to pose questions on this topic?

yes, it is.
 
thanks for responding robphy!

I'm looking for sinusoidal solutions to a 1-form field, A on a psuedo-Riemann manifold (-,+,+,+).

*d*d*A=0 yields a set of solutions, but I don't know if it's the most general case.

There's an operator (d + \delta)^2, where \delta = *d* called the Laplace-Beltrami that might apply as (*d*d* + d*d*)A=0.

After some very tedious expansion over time and spatial indices it collapses to the deAlembertian,
\box{A} = 0.

Which operator is most general?

Even 4th and higher order equations are available as (*d*d*d*d + d*d*d*d* + etc.)A=0.
 
Where might I find a physics forum where I could address individuals who are actually capable in this field?
 

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