Page 59 - Intro to Tensor Calculus

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EXERCISE 1.2

I 1. Consider the transformation equations representing a rotation of axes through an angle α.

1 1 2
x = x cos α − x sin α
T α :
2
1
x 2 = x sin α + x cos α
Treat α as a parameter and show this set of transformations constitutes a group by finding the value of α
which:
(i) gives the identity transformation.
(ii) gives the inverse transformation.
(iii) show the transformation is transitive in that a transformation with α = θ 1 followed by a transformation
with α = θ 2 is equivalent to the transformation using α = θ 1 + θ 2 .
I 2. Show the transformation
1 1
x = αx
T α : 2 1 2
x = x
α
forms a group with α as a parameter. Find the value of α such that:
(i) the identity transformation exists.
(ii) the inverse transformation exists.
(iii) the transitive property is satisfied.
I 3. Show the given transformation forms a group with parameter α.
( 1 x 1
x = 1−αx 1
T α : 2
x 2 = x 1
1−αx
I 4. Consider the Lorentz transformation from relativity theory having the velocity parameter V, c is the
speed of light and x 4 = t is time.
1
 1 x −Vx 4
 x = p
 V 2
 1− c 2


 2 2
x = x

T V : 3 3
 x = x

 1
 4 Vx
 4 x − c 2
 x = p V 2

1− c 2
Show this set of transformations constitutes a group, by establishing:
(i) V = 0 gives the identity transformation T 0 .
= T 0 requires that V 2 = −V 1 .
(ii) T V 2 · T V 1
requires that
(iii) T V 2 · T V 1 = T V 3
V 1 + V 2
V 3 = .
1+ V 1 V 2
2
c
~
~
~
I 5. For (E 1 , E 2 , E 3 ) an arbitrary independent basis, (a) Verify that
1 1 1
~
~
~
~
~
~
~ 1
~ 3
~ 2
E = E 2 × E 3 , E = E 3 × E 1 , E = E 1 × E 2
V V V
~
~
~
ij ~
~ j
is a reciprocal basis, where V = E 1 · (E 2 × E 3 ) (b) Show that E = g E i .