The development and derivation of the Maxwell equations are always considered as one of the major triumphs of human beings. It is because it links magnetism to electricity, and to electromagnetic waves. This chapter discusses the mathematical theory employed in solving Maxwell’s equations. The tensor or vector form of Maxwell equations has become the standard and they are much more compact than the original 20 equations by Maxwell. The idea behind Maxwell’s theory came from a Faraday law that predicts a changing magnetic field can induce an electric current in a wire loop. If Maxwell’s theory is correct, electromagnetic waves must be generated by varying electric and magnetic fields coupled through electromagnetic induction predicted by the Faraday law as well as the reverse mechanism Maxwell proposed. Maxwell’s equations can be written at a microscopic level and macroscopic level. The integral form of Maxwell’s equations involves the ideas of flux and circulation.