Electromagnetics and Differential Forms

Objective

Students will acquire a detailed understanding of how to describe electromagnetics (EM) in terms of differential forms (DF). • How “space” (and “time”) can be modelled by differentiable manifolds; • How a class of physical fields can be represented by DF; • How Maxwell’s equations and constitutive relations translate into the language of DF; • How this continuous representation can be discretized.

Content

In the recent years, the amount of literature that deals with physical models in terms of differential forms (DF) has increased strongly. For instance, DF allow a clear and elegant representation of electromagnetics (EM). The operators grad, curl, and div of vector analysis are replaced by a single operator of the exterior derivative. Similarly, the integral theorems of Gauss and Stokes are replaced by a single integral theorem. Vector analysis is limited to three dimensions, while DF can be applied to any dimensions. This is useful for the relativistic formulations in four dimensions. Since DF can be canonically integrated over appropriate domains they lend themselves naturally to discretizations of the finite integration type. This lecture series provides an introduction into DF calculus, and its relation to vector analysis. Maxwell‘s equations and the constitutive relations are expressed in terms of DF, and the main steps into discretization are outlined briefly.

Resources