Modern Geometry — Methods and Applications pp 145-233 | Cite as

# 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*.

## Keywords

Vector Field Transformation Rule Algebraic Theory Orthogonal Transformation Finite Subgroup## Preview

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