(Bx, By, Bz). Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed meaning this number does not change from one frame of reference to another. Pseudoscalar invariant: The product of the tensor with its Hodge dual gives a Lorentz invariant: where is the rank-4 Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is . Determinant: which is proportional to the square of the above invariant. Trace: which is equal to zero. Significance This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively: and reduce to the inhomogeneous Maxwell equation: , where is the four-current. In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively: which reduce to Bianchi identity: or using the index notation with square brackets for the antisymmetric part of the tensor: Relativity 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 physical laws being recognised after the advent of special relativity. This theory stipulated that all the 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 simpler presentation of physical laws. The inhomogeneous Maxwell equation leads to the continuity equation: implying conservation of charge. Maxwell's laws above can be generalised to curved spacetime by simply replacing partial derivatives with covariant derivatives: and where the semi-colon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime): Lagrangian formulation of classical electromagnetism Classical electromagnetism and Maxwell's equations can be derived from the action: where is over space and time. This means the Lagrangian density is The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is Substituting this into the Euler–Lagrange equation of motion for a field: So the Euler–Lagrange equation becomes: The quantity in parentheses above is just the field tensor, so this finally simplifies to That equation is another way of writing the two inhomogeneous Maxwell's