Maxwell Equations in Tensor notation

534
1
[tex]
0
= \epsilon^{\alpha \beta \gamma 0} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}
= {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta} - {\partial F_{\beta\alpha}\over\partial x^\gamma} - {\partial F_{\gamma\beta}\over\partial x^\alpha} - {\partial F_{\alpha\gamma}\over\partial x^\beta}
= 2 \left( {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta} \right)
[/tex]
because [tex]F_{\alpha\beta}[/itex] is antisymmetric.
 
Why was delta set to 0?
 

HallsofIvy

Science Advisor
41,626
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What "delta" are you talking about? There was no "delta" in your original question nor in Adriank's response.
 
What "delta" are you talking about? There was no "delta" in your original question nor in Adriank's response.
Actually, delta is the fourth index on the anti-symmetric epsilon tensor in both Karliski's post and in the wikipedia equation he linked to. It is basically an error, we should have delta = 0 as in adriank's post, but if delta is not zero then we just get four copies of the same equation.
 

Fredrik

Staff Emeritus
Science Advisor
Gold Member
10,730
403
The "equation" in #1 is actually four equations, one for each value of [itex]\delta[/itex]. The one with [itex]\delta=0[/itex] is the scalar equation

[tex]\nabla\cdot\vec B=0[/tex]

The one with [itex]\delta=i\neq 0[/itex] is the ith component of the vector equation

[tex]\nabla\times\vec E+\frac{\partial\vec B}{\partial t}=0[/tex]

(Edit: ...except for a factor of two).
 
Last edited:
534
1
The equation
[tex]0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}[/tex]
is written with a [tex]\delta[/tex] only on one side, which means we can "plug in" any specific value for it. So I put [tex]\delta = 0[/tex] and explicitly wrote out the sum [tex]\epsilon_{\alpha\beta\gamma0}F_{\alpha\beta,\gamma}[/tex].
 

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