How to calculate the matrix of a form?

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Discussion Overview

The discussion revolves around the calculation of the matrix representation of n-forms, particularly focusing on 2-forms and their components. Participants explore the relationship between forms and matrices, and how to express these mathematical objects in terms of their components.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to find the matrix of any n-form, seeking a detailed explanation.
  • Another participant asserts that there is no "matrix of an n-form," clarifying that while a 2-form can be represented as a matrix with components ##\omega_{ij}##, this does not extend to general n-forms which have components with n indices.
  • A participant reiterates the need for a matrix representation for 2-forms, emphasizing the specific component structure of ##\omega_{ij}##.
  • One participant states that the matrix simply contains the components ##\omega_{ij}## in the appropriate positions.
  • A request for a brief explanation is made by a participant who identifies as new to forms and expresses uncertainty about the relevant material in the book.
  • Another participant inquires if others are familiar with writing down the components of a rank 2 tensor in a matrix format.
  • A later reply provides a definition of the wedge product for the dual basis, suggesting a method to work from that point.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the concept of a "matrix of an n-form," with some asserting that only 2-forms can be represented in this way. The discussion remains unresolved as different perspectives on the representation of forms are presented.

Contextual Notes

There are limitations in the discussion regarding the definitions of forms and matrices, as well as the assumptions underlying the representation of n-forms. The mathematical steps for generalizing the matrix representation beyond 2-forms are not fully explored.

Abhishek11235
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This is screenshot from V.I Arnold's book on Classical mechanics. My question is how do we find matrix of any n-form. Detailed answer please.
Screenshot_2019-02-11-16-28-22.jpeg
 

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There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 
So,how to find matrix given in screenshot for 2 forms?
Orodruin said:
There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 
Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
 
Orodruin said:
Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
Can you explain it in brief? I am new to forms. I forgot where it was written in book.
 
Are you familiar on how to write down the components of a rank 2 tensor in a matrix?
 
Orodruin said:
Are you familiar on how to write down the components of a rank 2 tensor in a matrix?
Yes
 
So the wedge product for the dual basis is defined as
$$
dx^i \wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i.
$$
You can work from there.
 

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