How to calculate the matrix of a form?

[Abhishek11235]

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.

 

[Orodruin]

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.

 

[Abhishek11235]

So,how to find matrix given in screenshot for 2 forms?

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.

 

[Orodruin]

Its just a matrix containing the components $\omega_{ij}$ in the appropriate positions.

 

[Abhishek11235]

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.

 

[Orodruin]

Are you familiar on how to write down the components of a rank 2 tensor in a matrix?

 

[Abhishek11235]

Are you familiar on how to write down the components of a rank 2 tensor in a matrix?

Yes

 

[Orodruin]

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.