Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 31

If we take their scalar product, multiply by sinθ, and integrate over a sphere, we
find that
2π

π

S n · S k sinθ d θ d φ

l

0 0

2π π

u n u l P n P l sinθ e i ( m + k )φ d θ d φ =

0 0

⎣

P n P l ⎦

2π π

0 v

dP l

dP n
d θ

mk
sin 2

m

n v k

sinθ e i ( m + k )φ d θ d φ. +

d θ −

(1.199)

l

θ 0

From the orthogonality of Legendre functions (1.180), the first integral on the
right is

1) m 4 π
2 n

− k u n u l .

n m

( −

1 δ

l δ

(1.200)

+

After integrating over φ, the second integral on the right gives

⎣

P n P l ⎦

π

dP l

dP n
d θ

m 2
sin 2

m n v k

2πδ

− k v

d θ +

sinθ d θ. (1.201)

l

θ

0

Further integration by parts of the first term yields

P l π sinθ P l sinθ d θ

dP n
d θ

π
0 −

dP n
d θ

1
sinθ

d
d θ

m n v k

2πδ

− k v

sinθ

l

0

π (1.202)

m 2
sin 2

P n P l sinθ d θ + .

θ

0

Since sinθ vanishes at both limits, and on replacing

sinθ n ( n P n ,

dP n
d θ

m 2
sin 2

1
sinθ

d
d θ

by − + 1) − (1.203)

θ

in accordance with Legendre's equation (B.1), we are left with

1) π
0

n v k

m

P n P l sinθ d θ 2πδ

− k v

n ( n +

l

(1.204)

1) m 4 π
2 n

n m n v l .

= ( −

1 δ

l δ − k n ( n

+ 1)v

+

The spheroidal vector fields S n and S k

then obey the orthogonality relation

l

2π π

S n · S k sinθ d θ d φ

l

0 0

1 u n ( r ) u l ( r )

n v l ( r ) δ

1) m 4 π
2 n

m n m

= ( − + n ( n + 1)v

l δ − k .

(1.205)

+