Page 24 - Introduction to Continuum Mechanics

Part A Manipulations with the Indicia! Notation 9

(b) Multiplication

If

and

then,

It is important to note that pq # a mb mc md m. In fact, the right hand side of this expression
is not even defined in the summation convention and further it is obvious that

Since the dot product of vectors is distributive, therefore, if a = a ie i and b = b ie i, then

In particular, if e 1e 2e 3 are unit vectors perpendicular to one another, then e i . e j = so that

(c) Factoring
If

then, using the Kronecker delta, we can write

so that (i) becomes

Thus,

(d) Contraction
The operation of identifying two indices and so summing on them is known as contraction.
For example, T ii is the contraction of T ij,