310 Chapter 10Functions in 3 dimensions
EXAMPLE 10.13A circular helix lies on the curved surface of a (right) circular
cylinder, as shown in Figure 10.9.
The cartesian coordinates, in parametric form, are
x 1 = 1 a 1 cos 1 t, y 1 = 1 a 1 sin 1 t, z 1 = 1 bt, where aand bare
constants, and tis a parameter. In cylindrical polar
coordinates, ρ 1 = 1 ais the constant radius of the
cylinder, φ 1 = 1 tdescribes the rotation around the axis
of the cylinder, andz 1 = 1 btgives the displacement
parallel to the axis. The quantity 2 πbis the pitch of
the helix, the displacement parallel to the axis made
in one revolution about the axis.
Figure 10.9 shows a right-handed helix, with
clockwise circulation around the axis.
0 Exercises 38, 39
Confocal elliptic coordinates
These coordinates (also called prolate spheroidal coordinates) are useful for two-
centre potential problems.
The coordinates ξand ηare best visualized, as in Figure 10.10, in terms of the
distances r
Aand r
Bfrom the foci A and B of an ellipse (the ellipse is defined by
r
A1 + 1 r
B1 = 1 constant):
The figure might be a representation of a diatomic molecule with nuclei at A and B,
and an electron at P.
ξη=
,=
rr −
a
rr
a
AB AB22
∇=
−
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
222 2221
11
a()
() ()
ξη
ξ
ξ
ξη
η
ηη
ξη
ξηφ
+
−
−−
∂
∂
()
()()
22222211
dav=−ddd
32 2()ξη ξηφ
ha ha ha
ξηφξη
ξ
ξη
η
= ξη
−
−
,=
−
−
,= −−
2222222211
()( 11 ))
ξη=→ ,11102∞=−→+, φ= →π
xa=−− ,=−− ,=ξηφ ξηφξηya za
22 2211 cos 11 sin
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Figure 10.9