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with
T
given by
T
=∇
2
T
.
(1.192)
The scalars
L
,
P
,
T
may, in turn, be expanded in spherical harmonics as
⎩
⎭
⎩
⎭
L
P
T
l
n
(
r
,
t
)
p
m
∞
n
P
n
(cosθ)
e
im
φ
,
=
(
r
,
t
)
(1.193)
n
t
m
m
=−
n
(
r
,
t
)
n
=
0
n
with
⎩
⎭
⎩
⎭
⎩
⎭
⎝
⎠
−
m
m
m
l
p
t
l
p
t
l
p
t
2
1
r
∂
n
(
n
+
1)
n
=
r
.
(1.194)
∂
r
2
r
2
n
n
In seismology, it is usual to combine the lamellar and poloidal fields into the
single spheroidal field,
S
n
, with components
u
n
P
n
,v
n
P
n
e
im
φ
,
dP
n
im
sinθ
v
S
n
=
L
n
+
P
n
=
m
n
m
d
θ
,
(1.195)
where
u
n
(
r
,
t
) is the radial spheroidal coe
n
(
r
,
t
) is the transverse spher-
cient andv
oidal coe
cient. Comparison with expressions (1.185) and (1.186) show these to
be related to the lamellar and poloidal coe
cients by
∂
l
n
n
(
n
+
1)
1
1
r
∂
∂
r
rp
n
.
u
n
=
p
n
, v
m
n
r
l
n
+
∂
r
+
=
(1.196)
r
In turn, inversion of these relations allows expression of the lamellar and pol-
oidal coe
cients, in terms of the radial spheroidal and transverse spheroidal coef-
ficients, as
2
u
n
−
n
2
∂
u
n
∂
r
2
rl
n
−
n
(
n
1
)
n
(
n
+
1)v
1
r
∂
+
l
m
l
n
=
=
∂
r
+
,
(1.197)
n
r
2
r
and
v
u
n
n
n
2
1
r
∂
∂
r
2
rp
n
−
n
(
n
+
1)
∂v
−
p
m
p
n
=
=
∂
r
+
r
.
(1.198)
n
r
2
As is the case in the application of Legendre functions to the expansion of scalar
fields in the form (1.181), the usefulness of vector spherical harmonics derives
from their orthogonality under integration over a sphere. Suppose we have two
spheroidal vector fields with radial coe
cients
u
n
, v
n
and
u
k
, v
k
, respectively.
l
l
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