Electrodynamics in differential forms

[toqp]

(Ok, post edited. It should be ready for reading.)

I'm attending an electrodynamics course and the notation is in differential forms. The course material, however, is not yet finished so it's very coarse. We're supposed to have an introduction to differential forms as the course proceeds, but I'm having trouble understanding the concepts.

For example, there's a following excercise:

Homework Statement

(the first G in *G looks different, it's like a handwritten G, but I don't know how to display it)
$$*G=\frac{1}{2}G_{\alpha \beta}\ dx ^\alpha \wedge dx^\beta$$

when

$$G_{\alpha \beta}=\left[ \begin{array}{cccc} 0 & D_1 & D_2 & D_3 \\ -D_1 & 0 & -\frac{1}{c}H_3 & \frac{1}{c}H_2 \\ -D_2 & \frac{1}{c}H_3 & 0 & -\frac{1}{c}H_1 \\ -D_3 & -\frac{1}{c}H_2 & \frac{1}{c}H_1 & 0 \end{array} \right]$$

Find *G? Do you know what it looks like as a matrix?

The Attempt at a Solution

I'm having trouble understanding the wedge product here, and especially how it translates to matrix.

$$*G=G_{00}\ dx^0 \wedge dx^0 + G_{10}\ dx^1 \wedge dx^0 + G_{20}\ dx^2 \wedge dx^0 + ... + G_{33}\ dx^3 \wedge dx^3$$

Now dx ^ dy = - dy ^ dx and G_ii=0 so then for example:

$$G_{01}\ dx^0 \wedge dx^1 + G_{10}\ dx^1 \wedge dx^0= G_{01}\ dx^0 \wedge dx^1 - G_{10}\ dx^0 \wedge dx^1= D_ 1\ dx^0 \wedge dx^1 - (-D_1 \ dx^0 \wedge dx^1)= 2D_1 \ dx^0 \wedge dx^1$$

=>

$$*G=2D_1 \ dx^0 \wedge dx^1 + 2D_2 \ dx^0 \wedge dx^2 + 2D_3 \ dx^0 \wedge dx^3 + \frac{2}{c}H_3 dx^2 \wedge dx^1 - \frac{2}{c}H_2 dx^3 \wedge dx^1 + \frac{2}{c}H_1 dx^3 \wedge dx^2$$

But does it make any sense at all, and how does it translate to a matrix?

Do I just use the indices of dxⁱ as the indices of the matrix, so that for example dx¹ ^ dx² represents the element of the matrix at row 1 and at column 2?

$$*G_{\alpha \beta}=\left[ \begin{array}{cccc} 0 & 2D_1 & 2D_2 & 2D_3 \\ 0 & 0 & 0 & 0 \\ 0 & \frac{2}{c}H_3 & 0 & 0 \\ 0 & -\frac{2}{c}H_2 & \frac{2}{c}H_1 & 0 \end{array} \right]$$

How completely lost am I?

 

[Jhero]

Thank you for sharing your question and attempt at solving the exercise. It seems like you are on the right track in understanding the concept of differential forms and their relation to matrices. Let me try to provide some clarification and additional guidance.

First, the notation used in differential forms can seem confusing at first, but it is a very powerful tool for describing vector fields and their transformations. The wedge product, denoted by the symbol "^", is used to express the exterior product of two differential forms. This product is anti-symmetric, meaning that changing the order of the terms results in a change of sign. This is why you have the negative sign in your calculations. So, it is important to keep this in mind when working with differential forms.

Second, the matrix notation used in your exercise is a compact way of writing the components of the differential form *G. Each element of the matrix corresponds to a specific term in the differential form, as you correctly identified. The indices of the matrix correspond to the indices of the differential form. So, for example, in the matrix *G_{\alpha \beta}, the element at row 1 and column 2 corresponds to the term 2D_1 dx^0 \wedge dx^1 in the differential form *G.

Finally, it is important to remember that the matrix notation is just a compact way of writing the differential form. It does not change the fundamental properties of the form, such as anti-symmetry. So, when writing the matrix, you still need to keep the anti-symmetry in mind and make sure that the elements are arranged in the correct order.

In summary, it seems like you have a good understanding of the concept of differential forms and their relation to matrices. Keep practicing and working through exercises, and you will become more comfortable with the notation and concepts. If you have any further questions, please do not hesitate to ask. Good luck with your studies!