306 Chapter 10Functions in 3 dimensions
EXAMPLE 10.11Evaluate∇
2fforf(r, 1 θ, 1 φ) 1 = 1 re
−r 22sin 1 θ 1 cos 1 φ.
Using the form (10.19a),
The functionfhas the factorized form
f(r, 1 θ, 1 φ) 1 = 1 (re
−r 22)(sin 1 θ)(cos 1 φ) 1 = 1 R(r) 1 × 1 Θ(θ) 1 × 1 Φ(φ)
so that
and, therefore,
Now
Therefore,
∇= + −+
−
−
222222 221
4
22 1
f
r
r
rrcos sin
sin sin
θθ
θθ
=−
R
r
ΘΦ f
1
4
2
Φ
Φ
=cosφ; =−cos =−Φ( )
φ
φφ
d
d
22Θ
Θ
=;
=
−
sin
sin
sin
cos sin
si
θ
θθ
θ
θ
1 θθ
22d
d
d
d nn
cos sin
sin
()
θ
θθ
θ
= θ
−
222Θ
Rre
r
d
dr
r
dR
dr
r
r
e
r=;
=+−
−− 22221
4
2
2
rrr
r
Rr
2221
4
22
=+−
()
∇=
22211
f
r
d
dr
r
dR
dr
d
d
ΘΦ
sin
si
θθ
nnθ
θ
φ
d
d
R
r
d
d
R
r
ΘΦ Φ Θ
222 222sin θ
∂
∂
=
f
Rr
d
φ d
φ
φ
() ()θ
()
Θ
Φ
∂
∂
=
f
Rr
d
θ d
θ
θ
() φ
()
()
Θ
Φ
∂
∂
=
f
r
dR r
dr
()
ΘΦ() ()θ φ
∇=
∂
∂
∂
∂
∂
∂
∂
∂
222211
f
r
r
r
f
r
rf
sin
sin
θ
θ
θ
θ
∂
∂
1
2222r
f
sin θ φ