Tensors: The Algebraic Theory

Abstract

The fact that many physical entities find mathematical expression as numerical functions of points in space, will by now be familiar to the reader; the distance from a fixed point to a variable point is one among many examples. If we have several such entities, then their mathematical counterparts form a collection of functions from the points of space to the numbers (or, in other words, a (single) vector-valued function on the points). Thus for instance to fully determine the position of a point in 3-dimensional space, we need the values at that point of (at least) three functions (and of course we call these values “co-ordinates” of the point): each co-ordinate x ⁱ is a function of the point, and together they form an ordered triple (x ¹, x ² , x ³) which specifies the point completely. In Chapter 1 we encountered various kinds of coordinate systems (i.e. triples of functions); for example in the plane we introduced Cartesian co-ordinates x ¹, x ², and polar co-ordinates r, φ, where x ¹ = r cos φ, x ² = r sin φ; and in space Cartesian co-ordinates, cylindrical co-ordinates r, z, φ, and spherical co-ordinates r, θ, φ.