# Electrodynamics in differential forms

1. Oct 8, 2008

### toqp

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:

1. The problem statement, all variables and given/known data
(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?

3. 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 Gii=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?

Last edited: Oct 8, 2008