Reduction of the classical electromagnetism to a two-dimensional curved surface

The reduction of the three-dimensional classical electromagnetism is performed in a twofold way. In the first case the ordinary two-dimensional electromagnetism is obtained with sources in the form of conserved electric currents flowing along the surface. The electric field is a two-vector tangent to the surface and magnetic field is a scalar quantity. In the second approach the reduced theory is that of the two-vector magnetic field and a scalar electric one. The only source coupled to the fields is now a scalar subject to no conservation law. In the redefined theory this scalar source is may be converted into an eddy magnetic current flowing in the surface. No magnetic monopoles appear. Our results can find some applications in the electrodynamics of thin layers and of metal-dielectric interfaces.