Page 21 - Introduction to Continuum Mechanics

6 indicia! Notation

representing

'
We note that x j = a jmx m , j= 1,2,3, is the same as Eq. (2A2.3) and e j = Q mje m, j=1,2,3 is the
same as Eq. (2A2.4). However,

a i = b j
is a meaningless equation. The free index appearing in every term of an equation must be the
same. Thus the following equations are meaningful
a i + k i = c i
a i + b ic jd j = 0
If there are two free indices appearing in an equation such as

then the equation is a short-hand writing of 9 equations; each has a sum of 3 terms on the
right-hand side. In fact,
T 11 = A 1mA 1m = A 11A 11 + A 12 A 12 +A 13 A l3

T 12 =A 1mA 2m =A 11A 21 +A 12A 22 +A 13A 23
T 13 = A 1mA 3m = A 11A 31 + A 12A 32 + A 13A 33

+
T 33 = A 3mA 3m = A 31A 31 A 32A 32 + A 33A 33
Again, equations such as
T ij = T ik
have no meaning,

2A3 Kronecker Delta
The Kronecker delta, denoted by d ij, is defined as

That is,