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The standard classic special relativistic transformation of theelectromagnetic (EM) field under proper Lorentz transformations is revisited.As to the pure Lorentz-boosts, popular treatments on EM transformationcontemplate ideal geometries generating special static charge and steadycurrent distributions and in conjunction, invoke parallel and perpendicular (tothe boost-velocity) components of the fields so engendered; the outcomessubsequently being suitably generalized. We demonstrate ab initio, the EM fieldtransformation under arbitrary oblique uniform relative boost velocities from abroader, rigorous yet remarkably lucid perspective, exploiting the antisymmetryof the EM-field tensor and symmetry of the general Lorentz-boost transformationpreemptively eliminating the labor of performing complicated matrixmultiplications. Gratifyingly, this instructive exercise manifestly encompassestime varying fields and yields the transformation relations in acoordinate-free form. Further, the EM field transformation under proper,instantaneous 3D rotations is demonstrated to transform as a passive rotationof the coordinate axes, where the electric and magnetic fields transformindependently as purely electric and magnetic fields, respectively. The presentdidactical exercise, amenable to be incorporated in graduate courses onrelativistic electrodynamics, should serve as an excellent illustration on howpowerful symmetry arguments simplify the otherwise tedious calculations whileendowing complete generality to the connections obtained.

Novel, first-principles approach to deriving Electromagnetic field transformations under oblique Lorentz-Boosts and arbitrary Spatial Rotations