# Tensors: The Algebraic Theory

• B. A. Dubrovin
• A. T. Fomenko
• S. P. Novikov
Part of the Graduate Texts in Mathematics book series (GTM, volume 93)

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• B. A. Dubrovin
• 1
• A. T. Fomenko
• 2
• S. P. Novikov
• 3
1. 1.c/o VAAP-Copyright Agency of the U.S.S.R.MoscowUSSR
2. 2.3 Ya KaracharavskayaMoscowUSSR
3. 3.L. D. Landau Institute for Theoretical PhysicsAcademy of Sciences of the U.S.S.R.MoscowUSSR