Cylindrical Coordinates (r,θ,z )
The transformation from rectangular coordinates (x, y, z )to cylindrical coordi-
nates (r, θ, z )is given by
x=x(r, θ, z ) = rcos θ, y =y(r, θ, z) = rsin θ, z =z(r, θ, z ) = z
so that a general position vector is given by r =r (r, θ, z ) = rcos θˆe 1 +rsin θˆe 2 +zˆe 3 In
cylindrical coordinates the coordinate surfaces are
r (r 0 , θ, z ) = r 0 cos θˆe 1 +r 0 sin θˆe 2 +zˆe 3 a cylinder
r (r, θ 0 , z ) = rcos θ 0 ˆe 1 +rsin θ 0 ˆe 2 +zˆe 3 a plane perpendicular to z−axis
r (r, θ, z 0 ) = rcos θˆe 1 +rsin θˆe 2 +z 0 ˆe 3 a plane through the z−axis
These surfaces are illustrated in the figure 7-26.
Figure 7-26.
Coordinate surfaces and coordinate curves for cylindrical coordinates (r, θ, z).
The coordinate curves are
r (r 0 , θ 0 , z ), lines perpendicular to plane z= 0
r (θ 0 , z 0 ), lines emanating from the origin
r (r 0 , θ, z 0 ), circles of radius r 0 in the plane z=z 0
The vectors
∂r
∂r = cos θ
ˆe 1 + sin θˆe 2 , ∂r
∂θ = −rsin θ
ˆe 1 +rcos θˆe 2 , ∂r
∂z =
ˆe 3 are
tangent vectors to the coordinate curves and the vectors
ˆer=∂r
∂r = cos θ
ˆe 1 + sin θeˆ 2 , ˆeθ=^1
r
∂r
∂θ =−sin θ
eˆ 1 + cos θˆe 2 , ˆez=∂r
∂z =
ˆe 3 (7 .107)
