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

Chapter 2 - Transformations and Second-Rank Tensors: Further Developments

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