Maxwell Equations in Tensor notation

[Karliski]

http://en.wikipedia.org/wiki/Formul...s_in_special_relativity#Maxwell.27s_equations

Why does
0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}<br />
reduce to
<br /> 0 = {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta}<br />
?

 

[adriank]

<br /> 0<br /> = \epsilon^{\alpha \beta \gamma 0} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}<br /> = {\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}<br /> = 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)<br />
because F_{\alpha\beta}[/itex] is antisymmetric.

 

[Karliski]

Why was delta set to 0?

 

[HallsofIvy]

What "delta" are you talking about? There was no "delta" in your original question nor in Adriank's response.

 

[confinement]

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]

The "equation" in #1 is actually four equations, one for each value of \delta. The one with \delta=0 is the scalar equation

\nabla\cdot\vec B=0

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

\nabla\times\vec E+\frac{\partial\vec B}{\partial t}=0

(Edit: ...except for a factor of two).

 

[adriank]

The equation
0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}
is written with a \delta only on one side, which means we can "plug in" any specific value for it. So I put \delta = 0 and explicitly wrote out the sum \epsilon_{\alpha\beta\gamma0}F_{\alpha\beta,\gamma}.