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What I've been looking for is a way to derive this transformation
without using tensors or four vectors. Today, I found a way on pp.
231-232 of Eyges' text _The Classical Electromagnetic Field_ (Dover,
1980). The method involves Lorentz transforming the components of \/ x
E = -(1/c)dB/dt and imposing form invariance. I'm in the process of
carrying out the whole derivation to make sure I understand it.
I agree what geometric methods are probably the most elegant, but I
don't understand them. I'm just now beginning to think I understand
tensors.
Matter & Interactions, presents a very nice discussion of
reference frames and fields. The authors specifically demonstrate how a
static E field in one inertial frame is seen as a combined E and B
field in another inertial frame. They do so using the Lorentz
transformation of E and B, but they do not present an actual derivation
of the transformation. They don't present a derivation of the standard
Lorentz transformation either.