Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.3 CARTESIAN TENSORS

O x 1

x 2

x′ 1

x′ 2

θ

θ

θ

Figure 26.1 Rotation of Cartesian axes by an angleθabout thex 3 -axis. The

three angles markedθand the parallels (broken lines) to the primed axes

show how the first two equations of (26.7) are constructed.

coordinate systems, the transformation matrix is given by

Lij=e′i·ej.

We note that the product of two rotations is also a rotation. For example,

suppose thatx′i=Lijxjandx′′i=Mijx′j; then the composite rotation is described

by

x′′i=Mijx′j=MijLjkxk=(ML)ikxk,

corresponding to the matrixML.

Find the transformation matrixLcorresponding to a rotation of the coordinate axes

through an angleθabout thee 3 -axis (orx 3 -axis), as shown in figure 26.1.

Takingxas a position vector – the most obvious choice – we see from the figure that
the components ofxwith respect to the new (primed) basis are given in terms of the
components in the old (unprimed) basis by

x′ 1 =x 1 cosθ+x 2 sinθ,

x′ 2 =−x 1 sinθ+x 2 cosθ, (26.7)

x′ 3 =x 3.

The (orthogonal) transformation matrix is thus

L=





cosθ sinθ 0

−sinθ cosθ 0

001



.

The inverse equations are

x 1 =x′ 1 cosθ−x′ 2 sinθ,

x 2 =x′ 1 sinθ+x′ 2 cosθ, (26.8)

x 3 =x′ 3 ,

in line with (26.5).