Page 33 - Introduction to Continuum Mechanics

18 Tensors

It is easily seen that by this definition T + S is indeed a tensor.
To find the components of T + S, let

Using Eqs. (2B2.2) and (2B4.1), the components of W are obtained to be

i.e.,

In matrix notation, we have

2B5 Product of Two Tensors

Let T and S be two tensors and a be an arbitrary vector, then TS and ST are defined to be
the transformations (easily seen to be tensors)

and

Thus the components of TS are

i.e.,

Similarly,

In fact, Eq. (2B5.3) is equivalent to the matrix equation:

whereas, Eq. (2B5.4) is equivalent to the matrix equation:

The two matrix products are in general different. Thus, it is clear that in general, the tensor
product is not commutative (i.e., TS * ST).
If T,S, and V are three tensors, then
(T(SV))a = T((SV)a) = T(S(Va))