#### B3NR4Y

Gold Member

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**1. 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

[tex] F_{ij} = \frac{\partial A_{j}}{\partial x_{ i }} - \frac{\partial A_{ i }}{\partial x_{j}} = \partial_{ i } A_{j} - \partial_{j} A_{ i } [/tex]

where [itex] \nabla \times \vec{A} = \vec{H}[/itex]

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.[tex] F_{ij} = \frac{\partial A_{j}}{\partial x_{ i }} - \frac{\partial A_{ i }}{\partial x_{j}} = \partial_{ i } A_{j} - \partial_{j} A_{ i } [/tex]

where [itex] \nabla \times \vec{A} = \vec{H}[/itex]

**2. Homework Equations**

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

**3. The Attempt at a Solution**

The last problem says that the tensor F

_{ij}can be written as the following matrix:

[tex]

\left(\begin{array}{ccc}

0 & H_{z} & -H_{y} \\

-H_{z} & 0 & H_{x} \\

H_y & -H_x & 0 \end{array}\right) [/tex] and the hint says that

[tex] (curl \, \vec{A})_{i} = \epsilon_{ijk} \partial_{j} A_{k} = H_{i} [/tex] So, [itex] F_{12} = \partial_{1} A_2 - \partial_2 A_1 [/itex]. Do I just et the partials from the definition of the curl of A being H?

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