Introduction to vector and tensor analysis

third invariance above, ie. that the i’th component of the image u = T · c of T
operating on c, is always obtained as
ui = Tijcj, (3.27)
irrespective of the chosen coordinate system, we can tranform the components
of u from one (unprimed) to another (primed) coordinate system and see if these
equal the transformed component matrix of T times the transformed component
triplet of c:
[T]E ′ ,ij[c]E ′ ,j
= T ′
ij c′ j
= (aikajlTkl)(ajmcm) Transf. rules for components
= aikTkl(ajlajm)cm
= aikTklδmlcm
= aik(Tklcl)
= aik[u]E,k
= [u]E ′ ,i
Orthogonality (summing over k)
We shall return to other important invariances in chapter 5
4.2.4 Active and passive transformation
Tranf. rules for components
(4.15)
Comparing the transformation of the components of a vector upon a basis transformation,
Eq. (4.4) with the matrix representation of a tensor operating on
a vector, Eq. (3.27), it is clear that these two operations are defined in the
same manner algebraically. Since the first equation only expresses the change
of components due to a change of basis (the vector itself remains unchanged) it
is referred to as a passive transformation as opposed to the second case which
reflects an active transformation of the vector. Here, we shall show a simple
algebraic relation between the two type of transformations. Let O represent an
orthogonal tensor, ie. a tensor for which
O t · O = 1,
where 1 as usual denotes the identity tensor. In any orthonormal basis the
matrix representation of this identity will be that of Eq. (4.12). Further, let a
new basis system E ′ = {e ′ 1 , e′ 2 , e′ 3 } be defined by
e ′ i = O · ei
where E = {e1, e2, e3} is the old basis. We shall use the short-hand notation
E ′ = O(E)
Due to the orthogonality property of O, E ′ will be ortonormal iff E is.
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