How to Express the Maxwell Field Using Wedge Products?

[waddles]

Homework Statement

The problem is to write it in terms on coordinate basis using the wedge product,
$$F=F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$$
from the basis with the tensor prdouct.

Homework Equations

The EM field strength tensor can be written, with a coordinate basis,
$$F=F_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$$

The Attempt at a Solution

My initial thought was write the field strength tensor in terms of is antisymmetric parts,
$$F=\frac{1}{2}\left(F_{\mu\nu}dx^{\mu}\otimes dx^{\nu}-F_{\nu\mu}dx^{\nu}\otimes dx^{\mu}\right)$$
I can't get any further though. Any insights would be appreciated.

 

[mark726]

Hello,

Thank you for posting your question on the forum. As a fellow scientist, I would be happy to offer some insights on this problem.

Firstly, let's review some key concepts related to the wedge product and tensor product. The wedge product is a mathematical operation used to construct multilinear forms from vectors. It is denoted by the symbol ∧ and is defined as follows: If v1, v2, ..., vn are vectors in a vector space V, then the wedge product of these vectors is given by v1 ∧ v2 ∧ ... ∧ vn. This operation results in a multilinear form, which can be thought of as a function that takes in n vectors and outputs a scalar value.

On the other hand, the tensor product is an operation used to construct new tensors from existing ones. It is denoted by the symbol ⊗ and is defined as follows: If T and S are tensors, then the tensor product of T and S is given by T ⊗ S. This operation results in a new tensor that has a higher rank than the original tensors.

Now, let's consider the EM field strength tensor. As you have correctly mentioned, it can be written in terms of its antisymmetric parts as follows:

F=F_{\mu\nu}dx^{\mu}\otimes dx^{\nu}-F_{\nu\mu}dx^{\nu}\otimes dx^{\mu}

However, we can also write it in terms of the wedge product as follows:

F=\frac{1}{2}\left(F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}+F_{\nu\mu}dx^{\nu}\wedge dx^{\mu}\right)

Note that the only difference between this expression and the previous one is the use of the wedge product instead of the tensor product. This is because the wedge product captures the antisymmetric nature of the field strength tensor.

Now, if we expand this expression, we get:

F=\frac{1}{2}\left(F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}+F_{\nu\mu}dx^{\nu}\wedge dx^{\mu}\right)
=\frac{1}{2}\left(F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}-F_{\mu\nu}dx^{\nu}\wedge dx