Invisibility and Maxwell's Equations

We investigate the origin of the form invariance of Maxwell’s Equations and see how this relates to the theory of transformation optics, in which a particular geometry is related to the equivalent electromagnetic properties required in Euclidian space to create the geometry. This idea is used to reformulate transformation optics in the more natural language of differential geometry and to derive the electromagnetic properties associated with arbitrary transformations using these techniques. The theory of transformation optics is then generalised to non-linear electromagnetic media and to a spacetime formulation.