2.5 Coordinate Transformation
37
Solution
Let Q ij and ¯
Q ij represent the respective components of Q relative to an unrotated
Cartesian basis {e i } and a rotated basis {¯ e i }. We can write
Q = Q ij e i e j = ¯
Q ij ¯
e i ¯
e j .
(2.44)
Equations (2.37) and (2.43) 2 give
Q ij = e i · Q · e j = e i · (¯ e k e k ) · e j = e i · ¯
e k δ kj
= e i · ¯
e j
¯
Q ij = ¯
e i · Q · ¯
e j = ¯
e i · (¯ e k e k ) · ¯
e j = δ ik e k · ¯
e j
= e i · ¯
e j ,
(2.45)
which show that Q ij = ¯
Q ij .
With this background, transforming the components of vectors and tensors
between different coordinate systems is relatively straightforward. It really is just a
matter of writing the vector or tensor in one system and using Eqs. (2.43) to extract
the components in the other system.
To derive explicit transformation relations, we again consider the case of two
Cartesian coordinate systems (with base vectors e i and ¯
e i ) that differ by a rigidbody rotation and write
a = a i e i = ¯
a i ¯
e i
T = T ij e i e j = ¯
T ij ¯
e i ¯
e j .
(2.46)
The components relative to the rotated system are given by
¯
a i = a · ¯
e i = (a j e j ) · ¯
e i = a j (e j · ¯
e i )
¯
T ij = ¯
e i · T · ¯
e j = ¯
e i · (T kl e k e l ) · ¯
e j = T kl (¯ e i · e k )(e l · ¯
e j ),
(2.47)
and using Eq. (2.45) 1 gives
¯
a i = Q ji a j
¯
T ij = Q ki Q lj T kl .
(2.48)
These equations show that a Q ji is needed to transform each subscript to the new
basis. In this context, the Q ji are called coordinate transformation coefficients.
Although the above relations were derived for transformation between two
Cartesian coordinate systems, they can be applied more generally to any two
orthogonal coordinate systems, the only type considered in this book. Because
the coordinate curves for each system are mutually orthogonal at every point in