Maxwell Equations in Tensor Notation

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The discussion centers on the formulation of Maxwell's equations in tensor notation, highlighting a 4-vector equation presented in Griffith's Introduction to Electrodynamics as a simple and elegant representation. The conversation questions whether this formulation adequately captures the homogeneous Maxwell equations, particularly regarding Gauss' Law and the Ampere-Maxwell Law. The role of Faraday's Law and the divergence of the magnetic field is also examined, with an emphasis on defining the magnetic field as the curl of a vector potential to ensure its divergence-less nature. An alternative formulation using the Faraday tensor is proposed, which necessitates an understanding of differential geometry. The discussion underscores the complexity and elegance of expressing electromagnetic laws in tensor notation.
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2A\mu=-\muoJ\mu

Griffith's Introduction to Electrodynamics refers to this 4-vector equation as "the most elegant (and the simplest) formulation of Maxwell's equations." But does this encapsulate the homogeneous Maxwell Equations? I see how the temporal components lead to Gauss' Law, and I'm assuming, though I haven't shown it to myself, that the spatial components lead to the Ampere-Maxwell Law. What about Faraday's Law and the divergence of B?
 
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The other two laws are basically obtained by definition of the E and B fields. For example, by defining B as the curl of a vector potential, it is then divergence-less by definition.

I would say that the "most elegant" way to formulate Maxwell's equations is by using the Faraday tensor (F\equiv dA, where d is the exterior derivative) :

dF=0

d*F=4\pi*J

But this requires a little bit of differential geometry to understand.
 

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