Maxwell's equations are derived starting with a mathematical structure which is then given physical meaning with the minimum of assumptions. From these, all the laws of electromagnetism, such as Ampère’s law, Faraday’s law of electromagnetic induction, Coulomb’s law and the Biot-Savart law are derived. It is shown how the equations can be applied to electromagnetic phenomena in material media. It is demonstrated that, in the presence of electromagnetic fields, there is an electromagnetic energy density in the vacuum.