10.5 The Laplacian operator 305
of electromagnetic theory) and quantum mechanics (as in Schrödinger’s wave
mechanics, see Chapter 14).
EXAMPLE 10.10Evaluate ∇
2fforf(r) 1 = 1 e
−rin (i) spherical polar coordinates and
(ii) cartesian coordinates.
(i) Becausef(r)is a function of the radial coordinate only it follows that
Then
(ii) In cartesian coordinates,
Then, by the chain rule,
The derivatives with respect to yand zare obtained in the same way. Then
so that, becausex
21 + 1 y
21 + 1 z
21 = 1 r
2,
We note that the use of cartesian coordinates in this case naturally involves, via
the chain rule, a transformation to spherical polar coordinates.
0 Exercises 25–27
∇= + =−+
2 −2222
f 1
r
df
dr
df
dr
r
e
r∇=
∂
∂
∂
∂
∂
∂
=−
++
222222222233
f
f
x
f
y
f
z
r
xyz
r
()
++
df
dr
xyz
r
df
dr
222222∂
∂
=
∂
∂
=,
∂
∂
=
∂
∂
f
x
df
dr
r
x
x
r
df
dr
f
x
x
x
r
df
dr
22==− +
1
232222r
df
dr
x
r
df
dr
x
r
df
dr
rxyz
r
x
x
r
r
y
y
r
r
z
z
r
=++ ,
∂
∂
=,
∂
∂
=,
∂
∂
() =
222122∇=
∂
∂
∂
∂
∂
∂
2222222f
f
x
f
y
f
z
∇= + = − =−
2 −− −2222
1
2
f
df
dr
r
df
dr
e
r
e
r
e
rr r∂
∂
=,
∂
∂
=,
∂
∂
==−,
∂
∂
==
−fff
r
df
dr
e
f
r
df
dr
rθ φ
00
2222ee
−r