Is electromagnetism frame invariant in classical physics? In chapter 1.2 of Modern Physics for Scientists and Engineers 2nd ed. by Taylor, I read that electromagnetism works the same regardless or the point of origin, but then in 1.4 it says that electromagnetism is not frame invariant (referring to unaccelerated frames) in classical physics. What detail am I missing here which makes this appear contradictory?
 A: I have not read the book, so I can't assure you this is exactly what the author meant, but it is the possibility I find more likely.
Maxwell's equations are invariant under Lorentz transformations, which means they do not change under relativistic changes of reference frames. Notice the detail that this change must be relativistic: if we consider Galilean transformations instead, i.e., the changes of reference frame commonly used in classical, non-relativistic mechanics, then the Maxwell equations are not invariant. This is briefly discussed in Wikipedia, for example.
It is also worth pointing out that while Maxwell's equations are invariant under relativistic changes of reference frames, the fields themselves are not. One of the most common examples is the problem of a conducting wire with zero net charge and current flowing in some direction due to the motion of electrons with speed $v$ (the net charge can be zero due to the presence of protons standing still on the wire). If a test particle moving also with velocity $v$ alongside the wire is studied in this reference frame, there is no electric force acting on it, since the wire is electrically neutral and hence the electric field vanishes. However, if we change reference frames to the test particle's reference frame, its velocity is now zero and hence it can't feel a magnetic force. How can these results be consistent? It turns out that Lorentz contraction leads the wire to having a negative net charge, and hence an electric field occurs on this reference frame, even though it vanished in the previous one. This example is discussed in much more detail, e.g., on Griffiths' Introduction to Electrodynamics, Sec. 12.3. I believe Purcell's Electricity and Magnetism discusses it as well, although I haven't checked.
In summary, Maxwell's equations are Lorentz-invariant, but not Galilei-invariant.  Furthermore, the fields themselves are not Lorentz invariant.
Notice also that, in principle, a theory can be independent of origin without being independent of reference frame. Choosing an origin means the theory is invariant under translation, while choosing a reference frame means the theory is invariant under a transformation known as a boost. In principle, it is not necessary for both invariances to be present (though I can't think of a counterexample right now), but in the particular case of Electromagnetism they both are indeed present.
