40
2 Vector and Tensor Analysis
Fig. 2.6 Cylindrical polar
components of the vector
u = u x e x depend on location
e r
e T
e r
e T
u
u
e x
e y
Fig. 2.7 Spherical polar
coordinate system
I
T
r
x
y
z
e r
e I
e T
Example 2.7 In terms of Cartesian base vectors, the unit base vectors for spherical
polar coordinates (r, θ, φ) are (Fig. 2.7)
¯
e r = e x sin θ cos φ + e y sin θ sin φ + e z cos θ
¯
e θ = e x cos θ cos φ + e y cos θ sin φ − e z sin θ
¯
e φ = −e x sin φ + e y cos φ.
(2.54)
For clarity, bars are added to the spherical base vectors to designate the “rotated”
system. Determine the matrix [Q ij ] for the transformation from Cartesian to
spherical coordinates.
Solution
With the Cartesian and spherical coordinates representing the unrotated (unbarred)
and rotated (barred) systems, respectively, Eq. (2.52) yields
Q ij
=
⎡
⎢
⎣
e x · ¯
e r e x · ¯
e θ e x · ¯
e φ
e y · ¯
e r e y · ¯
e θ e y · ¯
e φ
e z · ¯
e r e z · ¯
e θ e z · ¯
e φ
⎤
⎥
⎦
(e i e j )
=
⎡
⎢
⎣
sin θ cos φ cos θ cos φ − sin θ
sin θ sin φ cos θ sin φ cos φ
cos θ
− sin θ
0
⎤
⎥
⎦
(e i e j )
.