Tensor Basics

As we will see later, practically all mathematical descriptions of physical quantities are tensors of some kind. Thus, in order to understand how to use mathematics to describe physics, it is important to understand what tensors are and how to work with them. In turn, this requires an understanding of some beginning concepts.

Dimensionality

Consider a space with N degrees of freedom. Then every point can be uniquely described by the set {x1, x2, ..., xN}. Each xi represents a specific coordinate. Another way of looking at this is that each xi represents the projection of the point onto the ith axis. Either way, the totality of points corresponding to all values of coordinates within certain ranges constitutes a space of N dimensions. The range of coordinates can be - to , or they may be restricted to a finite range.

In describing physics mathematically, we are usually interested in more than just a single point, or even an unconnected set of points. Instead, we are interested in a locus of points that are associated with a curve. We define a curve as the total collection of points given by the equation


.

(2.1)

Here u is a parameter and fr are N functions. A subspace is defined by the totality of points given by


,

(2.2)

where the u's are parameters and M < N.

Rank

A tensor can be defined as the set of quantities , where each ellipsis denotes a collection of indices. The total number of indices, both superscript and subscript, indicates the rank of the tensor. Thus, T represents a tensor of rank 0, Ta and Ta both represent tensors of rank 1, and Tab, Tab, Tab and Tab are all tensors of rank 2. Each index runs over the dimensionality of the space. The product of these two numbers, rank and dimensionality, yields the total number of components in the tensor. For example, if there are two superscripted indices and three subscripted indices, and the dimensionality of the space is four, then there are a total of 20 components in the tensor.

Range and Summation Conventions

Range convention: When a small Latin index, either superscript or subscript, occurs unrepeated in a term, it is understood to take all the values 1, 2, ..., N, where N is the number of dimensions of the space.

Einstein Summation convention: When a small Latin index is repeated in a term, summation with respect to that index is understood, the range of summation being 1, 2, ..., N.

Repeated indices are often referred to as “dummies” since the summation convention implies that any pair of repeated indices may be replaced by any other pair of repeated indices without changing the formula. In order to avoid confusion, we impose an additional rule

Repeated index rule: The same index must never be repeated more than twice in any single term or product. If this cannot be avoided, the Einstein summation convention must be suspended and all sums must be indicated explicitly.