the covariant form of Maxwell's equations
Maxwell's four equations, in the vector-calculus form of Vol I, look like four different laws about divergences and curls. Written covariantly, they collapse into just two tensor equations that are manifestly the same in every inertial frame — the deepest expression of the fact that electromagnetism and relativity are one theory.
The two INHOMOGENEOUS equations (Gauss's law and the Ampere-Maxwell law), the ones with sources, become the single line d_mu F^(mu nu) = mu_0 J^nu: the four-divergence of the field tensor equals the four-current. The two HOMOGENEOUS equations (Faraday's law and the no-magnetic-monopoles law) become d_mu (dual F)^(mu nu) = 0, equivalently the Bianchi identity d^alpha F^(beta gamma) + d^beta F^(gamma alpha) + d^gamma F^(alpha beta) = 0. That second pair is automatic once you write F = dA from a four-potential. Both equations are built from tensors, so if they hold in one inertial frame they hold in all — which is what 'manifestly covariant' means.
This form is not just elegant compression. It makes Lorentz invariance obvious rather than a miracle to be checked; it shows electromagnetism as the prototype gauge field theory; and taking the four-divergence of the source equation gives d_nu d_mu F^(mu nu) = 0, which — since F is antisymmetric — forces d_nu J^nu = 0. Charge conservation is thus not an extra assumption but a consequence of the equations' very structure.
Setting the free index nu = 0 in d_mu F^(mu nu) = mu_0 J^nu reproduces Gauss's law, div E = rho/epsilon_0; setting nu = 1, 2, 3 reproduces the Ampere-Maxwell law with its displacement-current term. Two four-dimensional lines unpack into all four of the familiar three-dimensional equations.
All four Maxwell equations = d_mu F^(mu nu) = mu_0 J^nu (sources) plus d_mu (dual F)^(mu nu) = 0 (structure).
The homogeneous pair is an identity once F = d^mu A^nu - d^nu A^mu, so only the source equation carries dynamical content; this is also why magnetic monopoles are absent in standard electromagnetism. Covariance was already hidden in Maxwell's 1860s equations — special relativity made it manifest, it did not add it.