Page 45 - Intro to Tensor Calculus

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Transformations Form a Group

A group G is a nonempty set of elements together with a law, for combining the elements. The combined
elements are denoted by a product. Thus, if a and b are elements in G then no matter how you define the
law for combining elements, the product combination is denoted ab.The set G and combining law forms a
group if the following properties are satisfied:
(i) For all a, b ∈ G,then ab ∈ G. This is called the closure property.
(ii) There exists an identity element I such that for all a ∈ G we have Ia = aI = a.
(iii) There exists an inverse element. That is, for all a ∈ G there exists an inverse element a −1 such that
aa −1 = a −1 a = I.
(iv) The associative law holds under the combining law and a(bc)= (ab)c for all a, b, c ∈ G.
2
For example, the set of elements G = {1, −1,i, −i},where i = −1 together with the combining law of
ordinary multiplication, forms a group. This can be seen from the multiplication table.

× 1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
-i -i i 1 -1
i i -i -1 1

The set of all coordinate transformations of the form found in equation (1.2.30), with Jacobian different
from zero, forms a group because:
(i) The product transformation, which consists of two successive transformations, belongs to the set of
transformations. (closure)
(ii) The identity transformation exists in the special case that x and x are the same coordinates.
(iii) The inverse transformation exists because the Jacobian of each individual transformation is different
from zero.
(iv) The associative law is satisfied in that the transformations satisfy the property T 3 (T 2 T 1 )= (T 3 T 2)T 1 .
When the given transformation equations contain a parameter the combining law is often times repre-
sented as a product of symbolic operators. For example, we denote by T α a transformation of coordinates
having a parameter α. The inverse transformation can be denoted by T α −1 and one can write T α x = x or
x = T α −1 x. We let T β denote the same transformation, but with a parameter β, then the transitive property
is expressed symbolically by T α T β = T γ where the product T α T β represents the result of performing two
successive transformations. The first coordinate transformation uses the given transformation equations and
uses the parameter α in these equations. This transformation is then followed by another coordinate trans-
formation using the same set of transformation equations, but this time the parameter value is β. The above
symbolic product is used to demonstrate that the result of applying two successive transformations produces
a result which is equivalent to performing a single transformation of coordinates having the parameter value
γ. Usually some relationship can then be established between the parameter values α, β and γ.