Some notation

TENSORS
26.1 Some notation
Before proceeding further, we introduce the summation convention for subscripts,
since its use looms large in the work of this chapter. The convention is that
any lower-case alphabetic subscript that appears exactly twice 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) ai xi stands for a1 x1 + a2 x2 + a3 x3 ;
(ii) aij bjk stands for ai1 b1k + ai2 b2k + ai3 b3k ;
(iii) aij bjk ck stands for 3j=1 3k=1 aij bjk ck ;
(iv)
∂v1
∂v2
∂v3
∂vi
stands for
+
+
;
∂xi
∂x1
∂x2
∂x3
(v)
∂2 φ ∂2 φ ∂2 φ
∂2 φ
stands for
+ 2 + 2.
∂xi ∂xi
∂x21
∂x2
∂x3
Subscripts that are summed over are called dummy subscripts and 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, aij bjk ckl cannot, and must
not, be replaced by aij bjj cjl or by ail blk ckl , but could be replaced by aim bmk ckl
or by aim bmn cnl . 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 delta δij , which is defined by
#
1 if i = j,
δij =
0 otherwise.
When the summation convention has been adopted, the main use of δij is to
replace one subscript by another in certain expressions. Examples might include
bj δij = bi ,
and
aij δjk = aij δkj = aik .
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(26.1)