The transformation equations (8.68) are obtained from the geometry in figure 8-19.
These equations are
x=rcos θy=rsin θ+ηcos α
z=ηsin α,which for α= 90◦reduces to the transformation equations for cylindrical coordinates.
The unit tangent vectors are
ˆer= cos θˆe 1 + sin θeˆ 2
ˆeθ=−sin θˆe 1 + cos θˆe 2ˆeη= cos αˆe 2 + sin αˆe 3 ,and the metric components of this space are
gij =
1 0 sin θcos α
0 r^2 rcos θcos α
sin θcos α r cos θcos α 1
.Orthogonal Curvilinear Coordinates
The following is a list of some orthogonal curvilinear coordinates which have
applications in many different scientific investigations.
Cylindrical coordinates (r, θ, z ) :
x=rcos θ 0 ≤θ≤ 2 π
y=rsin θ r ≥ 0
z=z −∞ < z < ∞
ds^2 =h^2 rdr^2 +h^2 θdθ^2 +h^2 zdz^2
hr= 1 , h θ=r, h z= 1gij =
h^2 r 0 0
0 h^2 θ 0
0 0 h^2 z
(8 .77)