Spherical coordinates (ρ, θ, φ ) :
x=ρsin θcos φ, ρ ≥ 0
y=ρsin θsin φ, 0 ≤φ≤ 2 π
z=ρcos θ, 0 ≤θ≤π
ds^2 =h^2 ρdρ^2 +h^2 θdθ^2 +h^2 φdφ^2
hρ= 1, h θ=ρ, h φ=ρsin θgij =
h^2 ρ 0 0
0 h^2 θ 0
0 0 h^2 φ
(8 .78)Parabolic cylindrical coordinates (ξ, η, z ) :
x=ξη, −∞ < ξ < ∞
y=^1
2(
ξ^2 −η^2)
, −∞ < z < ∞
z=z, η ≥ 0
ds^2 =h^2 ξdξ^2 +h^2 ηdη^2 +h^2 zdz^2
hη=hξ=√
η^2 +ξ^2 , h z= 1gij =
h^2 ξ 0 0
0 h^2 η 0
0 0 h^2 z
(8 .79)Parabolic coordinates (ξ, η, φ ) :
x=ξη cos φ, ξ ≥ 0 , η ≥ 0
y=ξη sin φ, 0 < φ < 2 π
z=^1
2(ξ^2 −η^2 )
ds^2 =h^2 ξdξ^2 +h^2 ηdη^2 +h^2 φdφ^2
hξ=hη=√
η^2 +ξ^2 , h φ=ξηgij =
h^2 ξ 0 0
0 h^2 η 0
0 0 h^2 φ
(8 .80)