
29.3. Other Special Orthogonal Coordinates 1505
Components of the metric tensor:
g
σσ
= a
2
σ
2
− τ
2
σ
2
− 1
, g
ττ
= a
2
σ
2
− τ
2
1 −τ
2
, g
zz
= 1.
2
◦
. Special coordinate system u, v, z:
σ = cosh u, τ = cos v, z = z;
x = a cosh u cos v, y = a sinh u sin v, z = z
(0 ≤ u < ∞, 0 ≤ v ≤ π, −∞ < z < ∞).
Components of the metric tensor:
g
uu
= g
vv
= a
2
(sinh
2
u + sin
2
v), g
zz
= 1.
3
◦
. Laplacian:
∆f =
1
a
2
(sinh
2
u+sin
2
v)
∂
2
f
∂u
2
+
∂
2
f
∂v
2
+
∂
2
f
∂z
2
=
√
σ
2
−1
a
2
(σ
2
−τ
2
)
∂
∂σ
p
σ
2
−1
∂f
∂σ
+
√
1−τ
2
a
2
(σ
2
−τ
2
)
∂
∂τ
p
1−τ
2
∂f
∂τ
+
∂
2
f
∂z
2
.
Remark 29.4. The elliptic cylinder coordinates σ, τ are also used a s elliptic coordinates on the
plane xy.
29.3.4 Conical Coordinates
Transformations of coordinates:
x = ±
uvw
ab
, y
2
=
u
2
(v