  In this paper the proofs are given that the electric and magnetic fields are
properly defined vectors on the four-dimensional (4D) spacetime (the 4-vectors
in the usual notation) and not the usual 3D fields. Furthermore, the proofs are
presented that under the mathematically correct Lorentz transformations (LT),
e.g., the electric field vector transforms as any other vector transforms,
i.e., again to the electric field vector; there is no mixing with the magnetic
field vector B, as in the usual transformations (UT) of the 3D fields. The
derivations of the UT from some well-known textbooks are discussed and
objected.
