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B3NR4Y
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Homework Statement
We are using Gaussian units. To begin, the vector Hk is the magnetic field, which relates to a second rank antisymmetric tensor, Fij.
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]
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]
The Attempt at a Solution
The last problem says that the tensor Fij 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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