I'm taking an introductory E&M course, and we're currently covering what our text calls "Galilean Electromagnetism" (i.e. the transformation of electric and magnetic fields between non-relativistic inertial frames).
Our text "derives" the transformation(s) in the following way (given mostly verbatim):
Consider two inertial frames, call them $A$ and $B$.
Suppose that frame $B$ moves with velocity $\vec{V_{BA}}$ w.r.t. frame $A$. Suppose further that there exists a charge $+q$, call it $C$, moving with velocity velocity $\vec{V_{CA}}$ w.r.t frame $A$/$\vec{V_{CB}} = 0$ w.r.t frame $B$. Lastly, suppose that there exists a uniform magnetic field $\vec{B_A}$ in the region of the charged particle, as viewed from frame $A$.
The Lorentz Force law gives the force on $C$ due to $B$ as viewed by frame $A$ as $q\vec{v_{CA}} \times \vec{B_A}$. By the invariance of Newton's laws under the Galilean transformation, we may equate this to the force on $C$ as viewed in frame $B$.
"But there can be no magnetic force on the charged particle as viewed in frame $B$, as the particle has no velocity w.r.t. frame $B$. So, the force observed must be that of an electric field!"
We then have, by the Lorentz Force law, $q\vec{v_{CA}} \times \vec{B_A} = q\vec{E_B}$. Clearly, then, in general $\vec{E_B} = \vec{E_A} + \vec{v_{CA}} \times \vec{B_A}$.
Particularly coming from a mathematical background, this all seems very "hand-wavy" and, at times, arbitrary to me.
Why must it be an electric field that is measured in frame $B$?
Clearly this can be determined empirically, however I feel as though I'm missing some deeper fundamental understanding.
