User:Wikieditoroftoday/Solutions To Physics Textbooks/Physical Properties of Crystals (9780198511656)/Chapter 2

Exercise 2.1
p.35

Problem: Prove that the square length $$p_i p_i$$ is an invariant (does not change by a transformation of axes).

Solution:

$$p_i p_i = a_{ji} p_j' a_{ki} p_k'$$

with the relation (6) on p.34 $$a_{ji}a_{ki}=\delta_{jk}$$ ($$\delta_{jk}$$ is the Kronecker delta) it follows that

$$p_i p_i = \delta_{jk} p_j' p_k' = p_i' p_i'$$

Exercise 2.2
p.46

Exercise 2.3
p.46

Exercise 2.4
p.47

Exercise 2.5
p.47

Exercise 2.6
p.48