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2.8.2 Example 2.2
Equation (2.75) is verified here as an example.
Consider three vectors represented by the column matrices
Let the same vectors be rotated by the rotation operator represented by
(2.99)
(2.100)
Since a rotation operator retains the cross product relationship, the new vectors also satisfy an equation similar to Eq. (2.98). That is,
Using Eqs. (2.99)–(2.101), Eq. (2.102) can be manipulated as follows:
(2.103)
Hence, it is seen that
Equations (2.99) and (2.104) imply the following mutual correspondence, which verifies Eq. (2.75) when
(2.105)
2.8.3 Example 2.3
Equation (2.76) is verified here as a follow‐up to Example 2.2.
Starting with Eq. (2.75), the powers of
Thus, for all k ≥ 0, it happens that
On the other hand, the following Taylor series expansion can be written for
Using Eq. (2.106) and noting that
Hence, as the verification of Eq. (2.76), Eq. (2.108) implies that
(2.109)
2.8.4 Example 2.4
In this example, Eq. (2.87) will be verified for ijk = 123 as explained below. However, it can be verified similarly for any other distinct (unrepeated) triplet of indices as well.
For ijk = 123, Eq. (2.87) becomes
Let
