Two, Three and Four-Dimensional Electromagnetics Using Differential Forms

The exterior calculus of differential forms provides a mathematical framework for electromagnetic field theory that combines much of the generality of tensor analysis with the computational simplicity and concreteness of the vector calculus. We review the pedagogical aspects of the calculus of differential forms in providing distinct representations of field intensity and flux density, physically meaningful graphical representations for sources, fields, and fluxes, and a picture of the curl operation that is as simple and intuitive as that of the gradient and divergence. To further highlight the benefits of differential forms notation, we demonstrate the flexibility of the calculus of differential forms in responding to changes in the dimensionality of the underlying manifold on which the calculus is defined. We develop Maxwell’s equations in the case of two space dimensions with time (2+1), and in a four-dimensional (4D or spacetime) representation with time included as a differential basis element on an equal footing with the spatial dimensions. The 2+1 case is commonly treated in textbooks using component notation, but we show that Maxwell’s equations and the theorems and principles of electromagnetics can be expressed in a fundamentally two-dimensional formulation. In the 4D representation, graphical representations can be given to illustrate four-dimensional fields in a way that provides intuition into the interplay between the electric and magnetic fields in wave propagation. These results illustrate the usefulness of differential forms in providing the physical insight required for engineering applications of electromagnetics.