  We use the method of field decomposition, a technique widely used in
relativistic magnetohydrodynamics, to study the small velocity approximation
(SVA) of the Lorentz transformation in Maxwell equations for slowly moving
media. The "deformed" Maxwell equations derived under the SVA in the lab frame
can be put into the conventional form of Maxwell equations in the medium's
comoving frame. Our results show that the Lorentz transformation in the SVA up
to $O(v/c)$ ($v$ is the speed of the medium and $c$ is the speed of light in
vacuum) is essential to derive these equations: the time and charge density
must also change when transforming to a different frame even in the SVA, not
just the position and current density as in the Galilean transformation. This
marks the essential difference of the Lorentz transformation from the Galilean
one. We show that the integral forms of Faraday and Ampere equations for slowly
moving surfaces are consistent with Maxwell equations. We also present Faraday
equation the covariant integral form in which the electromotive force can be
defined as a Lorentz scalar independent of the observer's frame. No evidences
exist to support an extension or modification of Maxwell equations.
