Let
be a set of
orthonormal basis vectors (i.e.
are mutually orthogonal
unit vectors). For any vector
,
 |
(2.8.3.1) |
Let
. Then
 |
(2.8.3.2) |
or
 |
(2.8.3.3) |
Taking the inner product of both sides of this eqn. with
,
we obtain
 |
(2.8.3.4) |
where
 |
(2.8.3.5) |
is called the component of
with respect to the
basis
.
For computation purposes, eqn. (2.8.3.4) is written as
![$\displaystyle \left\{\begin{array}{c} b_1\\ b_2\\ b_3\end{array}\right\} = \l...
...\end{array}\right]\left\{\begin{array}{c} a_1\\ a_2\\ a_3\end{array}\right\}.$](img316.gif) |
(2.8.3.6) |
Analogous to the representation (2.8.3.1) for vector
, we have
the following representation for second-order tensor
.
 |
(2.8.3.7) |
Because of (2.8.2.2), 
need not equal 
. In
order to see that (2.8.3.7) is equivalent to (2.8.3.5), we
evaluate
.
which is equivalent to
or (2.8.3.4).
It is clear from (2.8.3.7) that the components of
depend
upon the choice of basis
. Let
 |
(2.8.3.9) |
where
is an orthogonal matrix (i.e.
).
Then
Hence
 |
(2.8.3.11) |
and in matrix notation,
![$\displaystyle [T] = [Q]^T[T^\prime][Q],$](img333.gif) |
(2.8.3.12) |
and since
is orthogonal,
![$\displaystyle [T^\prime] = [Q][T][Q]^T.$](img334.gif) |
(2.8.3.13) |
The transpose
of a second-order tenseor
is defined
by
 for every vector and |
(2.8.3.14) |
The components of
and
are related by
 |
(2.8.3.15) |
