The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism. The field tensor was first used after the 4dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.
Contents 
The electromagnetic tensor F^{μν} is commonly written as a matrix:
or
From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:
If one forms an inner product of the field strength tensor a Lorentz invariant is formed:
The product of the tensor with its dual tensor gives the pseudoscalar invariant:
where is the completely antisymmetric unit pseudotensor of the fourth rank or LeviCivita symbol. Caution: the sign for the above invariant depends on the convention used for the LeviCivita symbol. The convention used here is = +1.
Notice that:
More formally, the electromagnetic tensor may be written in terms of the 4vector potential :
Where the 4vector potential is:
To derive all the elements in the electromagnetic tensor we need to define the derivative operator:
and the 4vector potential:
where
Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:
As an example, the x components are just
Using the definitions we began with, we can rewrite these two equations to look like:
Evaluating all the components results in a secondrank, antisymmetric and covariant tensor:
Thus, for example,
and
Compare with the matrix above.
Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism. Consider the electrostatic equation
which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation
that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4π times the current density.
These two equations for electricity reduce to
where
The same holds for magnetism. If we take the magnetostatic equation
which tells us that there are no "true" magnetic charges, and the magnetodynamics equation
which tells us the change of the magnetic field with respect to time plus the curl of the electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to
where the comma indicates a partial derivative
Classical electromagnetism and Maxwell's equations can be derived from the action defined:
where
This means the Lagrangian is
The far left and far right term are the same, because μ and ν are just variables of integration after all. The two middle terms are also the same, so the Lagrangian is
We can then plug this into the EulerLagrange equation of motion for a field:
The second term is zero, because the Lagrangian in this case only contains derivatives. So the EulerLagrange equation becomes:
That term in the parenthesis is just the field tensor, so this finally simplifies to
That equation is just another way of writing the two homogeneous Maxwell's equations as long as you make the substitutions:
where and take on the values of 1, 2, and 3.
The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of (nongravitational) physical laws being recognised after the advent of special relativity. This theory stipulated that all the (nongravitational) laws of physics should take the same form in all coordinate systems  this led to the introduction of tensors. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell's equations of electromagnetism may be written using the field tensor as:
The second equation implies conservation of charge:
In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way:
where the semicolon represents a covariant derivative, as opposed to a partial derivative. The elegance of these equations stems from the simple replacing of partial with covariant derivatives. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):
The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity, from to incorporate the creation and annihilation of photons (and electrons).
In quantum field theory, it is used for the template of the gauge field strength tensor. That is used in addition to the local interaction Lagrangian, nearly identical to its role in QED.
