10.6 Other coordinate systems 309
whose coordinate lines (and surfaces) are mutually perpendicular (orthogonal) at
their point of intersection. The cartesian and spherical polar systems are the best-
known examples. Such coordinate systems have the following properties.
(i) The (infinitesimal) distance between two points on a coordinate line is
ds
i1 = 1 h
i1 dq
i(10.23)
where
(10.24)
(ii) The volume element is
dv 1 = 1 ds
1ds
2ds
31 = 1 h
1h
2h
3dq
1dq
2dq
3(10.25)
(iii) The Laplacian operator is
(10.26)
For the cartesian and spherical polar coordinates,
cartesian: h
x1 = 1 h
y1 = 1 h
z1 = 11
spherical polar: h
r1 = 1 1, h
θ1 = 1 r, h
φ1 = 1 r 1 sin 1 θ
The following are two of the more widely-used alternative coordinate systems.
Cylindrical polar coordinates
These coordinates are the plane polar coordi-
nates in the xy-plane plus the z-coordinate, and
are useful for the description of systems with
cylindrical symmetry.
x 1 = 1 ρ 1 cos 1 φ, y 1 = 1 ρ 1 sin 1 φ, z 1 = 1 z
ρ 1 = 101 → 1 ∞, φ 1 = 101 → 12 π, z 1 = 1 −∞ 1 → 1 +∞
h
ρ1 = 1 1, h
φ1 = 1 ρ, h
z1 = 11
dv 1 = 1 ρ 1 dρ 1 dφ 1 dz
∇=
∂
∂
∂
∂
+
∂
∂
∂
∂
22222211
ρρ
ρ
ρ
ρφ z
∇=
∂
∂
∂
∂
∂
∂
2123 12311 23121
hhh q
hh
hq q
hh
h
∂∂
∂
∂
∂
∂
∂
hh
hq
231233
h
x
q
y
q
z
q
iiii2222=
∂
∂
∂
∂
∂
∂
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.P(ρ,φ,z)
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Figure 10.8