Geography Reference
In-Depth Information
M
x ( q,p )=
[A m exp( mq )cos mp + B m exp( mq )sin mp +
(D.35)
m =1
+ C m exp(
mq )cos mp + D m exp(
mq )sin mp ]+ x 0 ,
M
[ A m exp( m q )cos m p + B m exp( m q )sin m p +
y ( q,p )=
(D.36)
m =1
m q )sin m p + y 0 ,
+ C m exp(
m q )cos m p + D m exp(
M
x q =
[ mA m exp( mq )cos mp + mB m exp( mq )sin mp−
(D.37)
m =1
mC m exp(
mq )cos mp
mD m exp(
mq )sin mp ] ,
M
m A m exp( m q )sin m p + m B m exp( m q )cos m p
y p =
[
(D.38)
m =1
m C m exp(
m q )sin m p + m D m exp(
m q )cos m p ]
A m = B m ,B m =
A m ,C m =
D m ,D m = C m .
End of Proof.
Lemma D.4 (Fundamental solution of the d'Alembert-Euler equations subject to integrability
conditions of harmonicity, separation of variables).
A fundamental solution of the d'Alembert-Euler equations (Cauchy-Riemann equations) subject
to the integrability conditions of harmonicity type in the class of separation of variables is
x ( q,p ) =
(D.39)
M
M
= x 0 +
[ I m cosh mq + K m sinh mq ]cos mp +
[ J m cosh mq + L m sinh mq ]sin mp ,
m =1
m =1
y ( q,p ) =
(D.40)
M
M
= y 0 +
[ L m cosh mq + J m sinh mq ]cos mp +
[
K m cosh mq
I m sinh mq ]sin mp .
m =1
m =1
End of Lemma.
Proof.
By separation-of-variables ,namelybyusing x ( q,p )= f ( q ) g ( p )and y ( q,p )= F ( q ) G ( p ), the vec-
torial Laplace-Beltrami equation leads to ( D.34 ) and to similar equations for F ( q )and G ( p ).
f
f
+ g
g
f
f
g
g
=: m 2
f = m 2 f
=0
=
(D.41)
g =
m 2 g
f = k m cosh mq + l m sinh mq
g = i m cos mp + j m sin mp .
 
Search WWH ::




Custom Search