  In this paper it is exactly proved that the standard transformations of the
three-dimensional (3D) vectors of the electric and magnetic fields E and B are
not relativistically correct transformations. Thence the 3D vectors E and B are
not well-defined quantities in the 4D spacetime and, contrary to the general
belief, the usual Maxwell equations with the 3D E and B are not in agreement
with the special relativity. The 4-vectors E^a and B^a, as well-defined 4D
quantities, are introduced instead of ill-defined 3D E and B. The proof is
given in the tensor and the Clifford algebra formalisms.
