where c 1 , c 2 , c 3 are constants. These curves are, respectively, circles of radius c 1 sin c 2 ,
meridian lines on the surface of the sphere, and a line normal to the sphere. These
curves are illustrated in figure 8-18. The surfaces r =c 1 , θ =c 2 , and φ=c 3 are,
respectively, spheres, circular cones, and planes passing through the z-axis.
The unit tangent vectors to the coordinate curves and scale factors are given by
ˆeρ= sin θcos φˆe 1 + sin θsin φˆe 2 + cos θˆe 3 ,
ˆeθ= cos θcos φˆe 1 + cos θsin φeˆ 2 −sin θˆe 3 ,
ˆeφ=−sin φˆe 1 + cos φˆe 2 ,
h 1 =hρ= 1
h 2 =hθ=ρ
h 3 =hφ=ρsin θ.
The element of arc length squared is
ds^2 =dρ^2 +ρ^2 dθ^2 +ρ^2 sin^2 θ dφ^2 ,
and the metric components of this space are given by
gij =
1 0 0
0 ρ^20
0 0 ρ^2 sin^2 θ
.
Note that the spherical coordinate system is an orthogonal system.
Example 8-17.
An example of a curvilinear coordinate system which is not orthogonal is the
oblique cylindrical coordinate system (r, θ, η )illustrated in figure 8-19
Figure 8-19. Oblique cylindrical coordinate system.
