Vector and tensor fields

Abstract

In this first chapter we concentrate on the algebra of vector and tensor fields, while postponing ideas that are based on the calculus of fields to Chapter 2. Our starting point is a consideration of vector fields in the familiar setting of three-dimensional Euclidean space and how they can be handled using arbitrary curvilinear coordinate systems. We then go on to extend and generalize these ideas in two different ways, first by admitting tensor fields, and second by allowing the dimension of the space to be arbitrary and its geometry to be non-Euclidean.¹ The eventual goal is to present a model for the spacetime of general relativity as a four-dimensional space that is curved, rather than flat. While some aspects of this model emerge in this chapter, it is more fully developed in Chapters 2 and 3, where we introduce some more mathematical apparatus and relate it to the physics of gravitation.

We use the term non-Euclidean simply to mean not Euclidean. Mathematicians sometimes restrict the term to describe the geometries that arise as a result of modifying Euclid’s parallel postulate.