represents a straight line parallel to the z-axis which is normal to the xy plane at
the point r=c 1 , θ =c 2 .The curve
r =r (r, c 2 , c 3 ) = rcos c 2 ˆe 1 +rsin c 2 ˆe 2 +c 3 ˆe 3
represents a straight line in the plane z=c 3 ,which extends in the direction θ=c 2.
Figure 8-17. Cylindrical coordinates.
The tangent vectors to the coordinate curves are given by
∂r
∂r = cos θˆe^1 + sin θˆe^2
∂r
∂θ =−rsin θˆe^1 +rcos θˆe^2
∂r
∂z =ˆe^3
and are illustrated in figure 8-17. The element of arc length squared is
ds^2 =dr^2 +r^2 dθ^2 +dz^2
and the metric components of the space are
gij =
1 0 0
0 r^20
0 0 1
.
