304 Chapter 10Functions in 3 dimensions
and, after integration over the angles (see equation (10.13)),
(ii)
0 Exercises 23, 24
10.5 The Laplacian operator
The Laplacian operator in two dimensions was discussed in Section 9.6. The operator
in three dimensions is:
in cartesian coordinates,
(10.18)
in spherical polar coordinates,
(10.19a)
(10.19b)
The transformation from cartesian to spherical polar coordinates is achieved in
the same way as that described in Example 9.18 for the two-dimensional case. The
operator in spherical polar coordinates is usually quoted in the slightly more compact
form (10.19a), and the expanded form (10.19b) is obtained by use of the product rule
of differentiation. For example,
(10.20)
The Laplacian operator occurs in the equations of motion concerned with wave
motion and potential theory in both classical mechanics (as in Maxwell’s equations
11
2
222222r
r
r
f
r
rr
f
r
r
f
r
∂
∂
∂
∂
=
∂
∂
∂
∂
=
∂
∂
∂
∂
22f 2
r
r
f
r
=
∂
∂
∂
∂
∂
∂
∂
∂
222222 2221 1
r
rr
rr rθ
θ
θ
θ
cos
sin sin θθ φ
∂
∂
22∇=
∂
∂
∂
∂
∂
∂
∂
∂
222211
r
r
r
r
r sinsin
θ
θ
θ
θ
∂
∂
1
2222r sin θ φ
∇=
∂
∂
∂
∂
∂
∂
2222222xyz
=×!××=
1
32
5
2
3
25
05060π
π
a
aa
r
a
re dr d
praz2050502021
32
=
−π
ππZZ Z
∞20cos sinθθθ ddφ
ψ θ
2205221
32
praza
= re
−π
20cos
r
a
erdr
aa
a
sra10302
303040443
2
3
2
==×
!
=
−Z
∞22
0()