GLOBAL GEOMETRY OF ELECTROMAGNETIC SYSTEMS

Engineering calculations of electromagnetic systems require information about both local fields and global system parameters. This information depends on the local geometry, the global topology and the relation between the two. Maxwell's differential equations describe the local geometry with its four-dimensional space-time metric. These equations are greatly simplified when they are expressed in terms of differential forms, in which physical and geometrical information is combined. The paper shows the geometrical significance of the conservation of electric charge and of the gauge invariance of the potentials. Global topology is explored in terms of the interaction of differential forms and the structure of the manifold containing the system. It is shown that this depends on integral relationships which cannot be inferred from the differential equations. It is also shown that the gauge invariance of the potentials is related to the phase properties of state functions in quantum mechanics, a relationship which provides a further link between geometry and physical interaction.