  In this paper it will be exactly proved both in the geometric algebra and
tensor formalisms that the usual Maxwell equations with the three-dimensional
(3D) vectors of the electric and magnetic fields, E{bold} and B{bold}
respectively, are not, contrary to the general opinion, Lorentz covariant
equations. Consequently they are not equivalent to the field equations with the
observer independent quantities, the electromagnetic field tensor Fsup{ab}
(tensor formalism) or with the bivector field F (the geometric algebra
formalism). Different 4D algebric objects are used to represent the standard
observer dependent and the new observer independent electric and magnetic
fields. The proof of a fundamental disagreement between the standard
electromagnetism and the special relativity does not depend on the character of
the 4D algebric object used to represent the electric and magnetic fields. The
Lorentz invariant field equations are presented with 1-vectors E and B,
bivectors Esub{HL} and Bsub{HL} and the abstract tensors, the 4-vectors Esup{a}
and Bsup{a}. All these quantities are defined without reference frames. Such
field equations are in a complete agreement with experiments.
