Mathematical language of field theory

Abstract

A general theory of the electromagnetic field, or of any other kind of field, requires a development of the field concept from the intuitive and rather pictorial forms used in Chapter 1 into an instrument of great precision. Such a theory will consist of equations of some kind that are sufficient to define all the spatial and temporal variations of a field when the sources of the field are known, or when some other equivalent information is given. The appropriate language in which to express these field equations is the mathematical language of vector analysis. It is difficult to state Maxwell’s equations of the electromagnetic field or to discuss their physical meaning without making use of this language. A familiarity with vector analysis is a fundamental requirement for the approach to electromagnetism adopted in this book. To distribute an account of this language throughout the text might be considered a gentle way of introducing this necessary mathematics, but for a direct approach to Maxwell’s equations such a procedure is not possible; to relegate it all to an appendix would hardly do justice to its vital importance. For these reasons, therefore, an account of vector analysis will be given here and discussion of Maxwell’s equations will be deferred until this necessary mathematical support has been established.