Both of those forms incorporate the displacement current that their namesake added to correct Ampere's law.
A derivation of these statements using only Newton's second law and the canonical position-momentum commutation relation was published by Freeman Dyson in 1989 and credited to Richard Feynman. In the theory of differential forms, they assert that the field strength tensor is closed and exact, and they give the exterior derivative of its Hodge star in terms of a pseudo three-form. In empty space, applying the curl-of-the-curl identity to these statements yields two vectorial (*) wave equations. Their relativistic formulation sets the d'Alembertian of the four-potential equal to a constant times the four-current. One can convert between their integral and differential forms using the divergence theorem and Stokes' theorem. Both of those forms incorporate the displacement current that their namesake added to correct Ampere's law. For 10 points, name this set of four differential equations that govern classical electromagnetism.
A derivation of these statements using only Newton's second law and the canonical position-momentum commutation relation was published by Freeman Dyson in 1989 and credited to Richard Feynman.
Maxwell's equations