Geology Reference
In-Depth Information
From relations (1.196), the orthogonality relations for two purely lamellar vectors,
L n and L k
, or two purely poloidal vectors, P n and P k
, follow directly,
l
l
π
L n ·
L l sinθ d θ d φ
0
0
δ
r l k
l n
1) m 4 π
2 n
n ( n
+
1)
l
l n l k
n
m
=
(
r +
l δ
k ,
(1.206)
l
+
1
r
π
P n ·
P k
sinθ d θ d φ
l
0
0
n ( n
r rp l
2 n
+
1)
r 2
1
r rp n
1) m
p n p k
n
m
=
(
+
δ
l δ
k .
(1.207)
l
+
1
r 2
The integral over a unit sphere, of the scalar product of two toroidal or torsional
vector spherical harmonics, multiplied by sinθ,gives
π
T n ·
T k
sinθ d θ d φ
l
0
0
P n P l
π
dP l
dP n
d θ
m 2
sin 2
m
k t n t k
=
2πδ
d θ +
sinθ d θ.
(1.208)
l
θ
0
The same integral was encountered before in expression (1.201). By the previous
result, the orthogonality relation for two toroidal or torsional vectors T n and T k
l
becomes
π
1) m 4 π
2 n
T n · T k
1 ) t n t l δ
n
m
sinθ d θ d φ =
1 n ( n +
(
l δ
k .
(1.209)
l
+
0
0
Toroidal or torsional vector spherical harmonics do not interact with lamellar,
poloidal or spheroidal vector spherical harmonics. If we take their scalar product
with any of these, multiply by sinθ and integrate over a unit sphere, the result is
found, in all cases, to depend on the integral
π
P n (cosθ) P m (cosθ)
P n P n d θ =
d
d θ
π 0
= P n ( 1) P n ( 1) P n (1) P n (1).
0
(1.210)
For m 0, we have P ± n (
1) n and P n (1)
±
1)
=
0. For m
=
0, we have P n (
1)
=
(
=
1.
Thus, in all cases, the integral (1.210) vanishes.
In the four orthogonality relations (1.205), (1.206), (1.207) and (1.209), the
radial coe
cients u k
, v k
, l k
, p l and t l are at our disposal. By choosing v k
=
0
l
l
l
l
and u k
=
1 in (1.205), we find the radial spheroidal coe
cient. By choosing
l
u k
0andv k
=
=
1, we find the transverse spheroidal coe
cient. By choosing
l
l
 
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