Introduction to vector and tensor analysis

4.2.1 Cartesian coordinate transformation
A particular case of an orthonormal basis transformation is the transformation
between two cartesian coordinate systems, C = (O, R) and C ′ = (O ′ , R ′ ). Here,
the form of the vector transformation, Eq. (4.4), can be directly translated to
the coordinates themselves. To show this we employ the simple relation between
coordinates and the position vector, Eq. (2.9), valid in any cartesian system.
The displacement vector between any two points is given by ∆r = ∆xiei. This
vector –or its differential analogue dr = dxiei – is a proper vector independent
of the coordinate system. Following relation between the coordinate differentials
must then be satisfied
dxi and dx ′ j
Multiplying with e ′ j
or equivalently
we obtain
dr = dxiei = dx ′ j e′ j
∂x ′ j
dx ′ j = dxiei · e ′ j
= e ′ j · ei = aji, (4.9)
∂xi
where aji is defined as in Eq. (4.3). Since the rhs. is constant (constant basis
vectors) the integration of Eq. (4.9) gives
x ′ j = ajixi + dj. (4.10)
Here, dj is the j’th coordinate of the origin, O, of the unprimed coordinate
system as seen from the primed coordinate system, d = (x ′ )C ′(O). In matrix
notation Eq. (4.10) takes the form
x ′ = A · x + d, A = (aij), (4.11)
which is identical to Eq. (4.6) except from the optional displacement.
4.2.2 The orthogonal group
Considering Eq. (4.4) and Eq. (4.5) as a set of two matrix equations we note
that the transpose matrix A t operates as the inverse of A
A · A t = 1, (4.12)
where 1 is the identity matrix, (1)ij = δij. In index notation
(A · A t )ij = aikakj = δij
(4.13)
A matrix with the above property is called orthogonal, and the set of of all
orthogonal 3×3 matrices constitutes a continuous group 3 called O(3). According
to Eq. (4.12)
det(A · A t ) = det(A) 2 = 1,
3 Recall, that a mathematical group (G, ∗) is a set G with a binary operator, ∗, that satisfies
following axioms:
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