Maxwell's Equations in 4-D Space

[B3NR4Y]

Homework Statement

We are using Gaussian units. To begin, the vector H_k is the magnetic field, which relates to a second rank antisymmetric tensor, F_ij.

a.) Prove
$$F_{ij} = \frac{\partial A_{j}}{\partial x_{ i }} - \frac{\partial A_{ i }}{\partial x_{j}} = \partial_{ i } A_{j} - \partial_{j} A_{ i }$$

where $\nabla \times \vec{A} = \vec{H}$​

I'm only posting this first part because I feel like once I get it everything will roll out and I'll be able to answer it all.

Homework Equations

In the last problem, we proved that $H_{k} = \frac{1}{2} \epsilon_{ijk} F_{ij} \iff F_{ij} = \epsilon_{ijk} H_{k}$

The Attempt at a Solution

The last problem says that the tensor F_ij can be written as the following matrix:
$$\left(\begin{array}{ccc}<br /> 0 & H_{z} & -H_{y} \\<br /> -H_{z} & 0 & H_{x} \\<br /> H_y & -H_x & 0 \end{array}\right)$$ and the hint says that
$$(curl \, \vec{A})_{i} = \epsilon_{ijk} \partial_{j} A_{k} = H_{i}$$ So, $F_{12} = \partial_{1} A_2 - \partial_2 A_1$. Do I just et the partials from the definition of the curl of A being H?

 

[phyzguy]

There are several problems with what you've written. First of all, in 4D space-time, the matrix representing F_ij is a 4x4 antisymmetric matrix, not 3x3 as you've written it. It contains both the electric field and magnetic field. Secondly, even what you have written for the space part of F_ij is wrong, since it should contain H_x, H_y, and H_z. Normally in 4D space-time, time is x₀, and the space components are x_1,2,3.
So F₁₂ = ∂₁ A₂ - ∂₂ A₁ = ∂_x A_y - ∂_y A_x = H_z. Try working through it again.

 

[B3NR4Y]

There are several problems with what you've written. First of all, in 4D space-time, the matrix representing F_ij is a 4x4 antisymmetric matrix, not 3x3 as you've written it. It contains both the electric field and magnetic field. Secondly, even what you have written for the space part of F_ij is wrong, since it should contain H_x, H_y, and H_z. Normally in 4D space-time, time is x₀, and the space components are x_1,2,3.
So F₁₂ = ∂₁ A₂ - ∂₂ A₁ = ∂_x A_y - ∂_y A_x = H_z. Try working through it again.

This is the first part of my problem, I should have been more clear, sorry. Later on in the problem it extends the tensor to a 4x4, antisymmetric tensor, but I have trouble proving the first part. I only posted the first part because I felt like I could get the rest if I could get the first part. I mistyped as it was early in the morning, I'll fix my first post, the matrix is:
$$\left(\begin{array}{ccc}<br /> 0 & H_{z} & -H_{y} \\<br /> -H_{z} & 0 & H_{x} \\<br /> H_y & -H_x & 0 \end{array}\right)$$

What I noticed, is that $(\nabla \times \vec{A})_{ i } = \epsilon_{ijk} \partial_{j} A_{k} = H_{i} \rightarrow (\nabla \times \vec{A})_{ 1 } = \partial_{2} A_{3} - \partial_{3} A_{2} = H_{1}$, can I use this in my tensor?

 

[phyzguy]

Yes, this looks correct now.