Observe that this is an orthogonal system where gij = 0 for i=j. The surface
r=c 1 is a cylinder, whereas the surface θ =c 2 is a plane perpendicular to the xy
plane and passing through the z-axis. The surface z=c 3 is a plane parallel to the
xy plane. The cylindrical coordinate system is an orthogonal system.
Example 8-16. The spherical coordinates (ρ, θ, φ )are related to the rectangular
coordinates through the transformation equations
x=x(ρ, θ, φ ) = ρsin θcos φ
y=y(ρ, θ, φ ) = ρsin θsin φ
z=z(ρ, θ, φ ) = ρcos θ
which can be obtained from the geometry of figure 8-18.
Figure 8-18. Spherical coordinate system.
The position vector (8.70) becomes
r =r (ρ, θ, φ ) = ρsin θcos φeˆ 1 +ρsin θsin φˆe 2 +ρcos θˆe 3 ,
and from this position vector one can generate the curves
r =r (c 1 , c 2 , φ ), r =r (c 1 , θ, c 3 ), r =r (ρ, c 2 , c 3 ),
