1 The exact proof that Maxwell equations with the 3D E

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 and B 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 F 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 objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors EHL and BHL and the abstract tensors, the 4-vectors E a and B. All these quantities are defined without reference frames. Such field equations are in a complete agreement with experiments.