Exterior Calculus and Differential Forms?

In summary, the conversation is about finding sinusoidal solutions to a 1-form field on a psuedo-Riemann manifold and determining which operator is the most general. The Laplace-Beltrami operator is mentioned as a possibility, but further discussion is needed to find the best solution. The individual is also seeking a physics forum with knowledgeable individuals in this field.
  • #1
Phrak
4,267
6
Would this be the right forum to pose questions on this topic?
 
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  • #2
Phrak said:
Would this be the right forum to pose questions on this topic?

yes, it is.
 
  • #3
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.
 
  • #4
Where might I find a physics forum where I could address individuals who are actually capable in this field?
 

1. What is Exterior Calculus and Differential Forms?

Exterior calculus is a mathematical framework used to study differential forms, which are mathematical objects that generalize the concept of a vector field. It is a powerful tool for solving problems in geometry, physics, and engineering.

2. How is Exterior Calculus different from traditional calculus?

Traditional calculus deals with functions of one or more variables, while exterior calculus focuses on the properties of differential forms. It allows for a more general and elegant treatment of multivariate calculus problems.

3. What are the applications of Exterior Calculus and Differential Forms?

Exterior calculus has a wide range of applications in mathematical fields such as differential geometry, topology, and algebraic geometry. It is also used in physics to study and solve problems in areas such as electromagnetism, general relativity, and fluid mechanics.

4. What are the key concepts in Exterior Calculus?

The key concepts in exterior calculus include differential forms, exterior derivative, wedge product, Hodge star operator, and the exterior derivative of a differential form. These concepts are used to define and solve problems in the framework of exterior calculus.

5. Is knowledge of traditional calculus necessary to understand Exterior Calculus and Differential Forms?

While traditional calculus forms the basis of exterior calculus, it is not necessary to have a thorough understanding of traditional calculus to learn and understand exterior calculus. However, a basic knowledge of multivariable calculus and linear algebra is helpful in understanding the concepts and applications of exterior calculus.

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