Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

26.1 Some notation

Before proceeding further, we introduce thesummation conventionfor subscripts,

since its use looms large in the work of this chapter. The convention is that

anylower-casealphabetic subscript that appearsexactlytwice in any term of an

expression is understood to be summed over all the values that a subscript in

that position can take (unless the contrary is specifically stated). The subscripted

quantities may appear in the numerator and/or the denominator of a term in an

expression. This naturally implies that any such pair of repeated subscripts must

occur only in subscript positions that have the same range of values. Sometimes

the ranges of values have to be specified but usually they are apparent from the

context.

The following simple examples illustrate what is meant (in the three-dimensional

case):

(i)aixistands fora 1 x 1 +a 2 x 2 +a 3 x 3 ;

(ii)aijbjkstands forai 1 b 1 k+ai 2 b 2 k+ai 3 b 3 k;

(iii)aijbjkckstands for

∑ 3

j=1

∑ 3

k=1aijbjkck;

(iv)

∂vi

∂xi

stands for

∂v 1

∂x 1

+

∂v 2

∂x 2

+

∂v 3

∂x 3

;

(v)

∂^2 φ

∂xi∂xi

stands for

∂^2 φ

∂x^21

+

∂^2 φ

∂x^22

+

∂^2 φ

∂x^23

.

Subscripts that are summed over are calleddummy subscriptsand the others

free subscripts. It is worth remarking that when introducing a dummy subscript

into an expression, care should be taken not to use one that is already present,

either as a free or as a dummy subscript. For example,aijbjkcklcannot, and must

not, be replaced byaijbjjcjlor byailblkckl, but could be replaced byaimbmkckl

or byaimbmncnl. Naturally, free subscripts must not be changed at all unless the

working calls for it.

Furthermore, as we have done throughout this book, we will make frequent

use of the Kronecker deltaδij, which is defined by

δij=

{

1ifi=j,

0otherwise.

When the summation convention has been adopted, the main use ofδijis to

replace one subscript by another in certain expressions. Examples might include

bjδij=bi,

and

aijδjk=aijδkj=aik. (26.1)