kind of criss-crossed your question, and interlacing concepts
of physics with hydrodynamics,oscillations and measurement techniques,photons,gravity,thermodynamics ,to help in your
question,since mathemathics can solve everything probably you can rely in this formula for a start,however it can be the other way around it will only contribute to make you more confused;
....q satisfies p Delta q = cos . The resulting functions of the form Phi(cos( p; q) are called zonal. It is useful to expand Phi(p Delta q) in terms of the spherical harmonics Y j on S n (cf. 11, 12] This we can do by employing the famous Addition Theorem for spherical harmonics [11]: P (n 1; p Delta q) n d n ( dn ( X j=1 Y j (p) Y j (q) 3.2) Here, d n ( is the dimension of the space of n 1 dimensional harmonic polynomials homogeneous of degree and n is the volume of S n . This results in the expansion Phi(p Delta q) 1 X =0 dn ....
where = 1 ; Gamma1 ) by x 1 = cos 1 x 2 = sin 1 cos 2 x 3 = sin 1 sin 2 cos 3 : x Gamma1 = sin 1 sin 2 : sin Gamma2 cos Gamma1 x = sin 1 sin 2 : sin Gamma2 sin Gamma1 : The Laplace operator can be written in the form [14] (8:1) Deltap = p rr Gamma 1 r p r 1 r 2 Delta p where Delta is a second order elliptic operator in ; for = 3, and Delta p = 1 sin 2 (sin p ) 1 sin 2 2 p 2 : Consider a surface S ffl : r = 1 fflf ( with jf j C 2 1, and ....
....(and elliptic estimates) kr Delta k Fk L 2 (B) kFk H 2k 1 (B) K 1 kr Delta k Fk L 2 (B) Thus, it suffices to show that (8.14) holds with kFk H s (B) replaced by k Delta k Fk L 2 (B) and kr Delta k Fk L 2 (B) when s = 2k and s = 2k 1, respectively. On the other hand, since [14] Delta X m;n Fnm (r)Y nm ( X n;m ( 1 r Gamma1 r (r Gamma1 r Fnm ) Gamma n(n Gamma 2)Fnm )Y nm ; we have, using the othogonality properties of Ymn , 8:16) k Delta k Fk L 2 (B) X m;n Z 1 ffi r Gamma1 j[ 1 r Gamma1 r (r Gamma1 r ) Gamma n(n ....