Editing Classical field theory (section)

==== The equations ====
To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential ''A'', and it's this potential which enters the Euler-Lagrange equations. The EM field ''F'' is not varied in the EL equations. Therefore,
<math display="block">\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a} \,.</math>

Evaluating the derivative of the Lagrangian density with respect to the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial A_a} = \mu_0 j^a \,, </math>
and the derivatives of the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial(\partial_b A_a)} = F^{ab} \,, </math>
obtains [[Maxwell's equations]] in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are
<math display="block">\partial_b F^{ab}=\mu_0 j^a \, . </math>
while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that ''F'' is the 4-curl of ''A'', or, in other words, from the fact that the [[Bianchi identity]] holds for the electromagnetic field tensor.<ref>{{Cite web| url=http://mathworld.wolfram.com/BianchiIdentities.html|title=Bianchi Identities}}</ref>
<math display="block">6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0. </math>

where the comma indicates a [[partial derivative]].