Geography Reference
In-Depth Information
K 2
K=
{
KL }
=
=
,
(1 E 2 sin 2 U )(1 E 2 sin 2 U cos 2 V ) 1 / 2
A 1 (1
cos U cos V sin U sin VE 2 (1 E 2 sin 2 U cos 2 V ) 1 / 2
A 1 (1
E 2 )
E 2 )
(20.24)
cos U sin U sin VE 2 (1 E 2 sin 2 U cos 2 V ) 1 / 2
A 1 (1
(1 E 2 (1 sin 2 U sin 2 V ))(1 E 2 sin 2 U cos 2 V ) 1 / 2
A 1 (1
E 2 )cos V
E 2 )
1
E 2 sin 2 U cos 2 V (2
E 2 (1 + sin 2 U cos 2 V ))
2 A 1 (1 − E 2 )
tr[ K ]
2
h =
=
,
k =det[ K ]= (1 E 2 sin 2 U cos 2 V ) 2
,
(20.25)
A 1 (1
E 2 )
E 2 cos 2 V )
(1 −E 2 sin 2 U cos 2 V ) 3 / 2
A 1 cos V sin U (1
−A 1 cos U sin V
(1 −E 2 sin 2 U cos 2 V ) 3 / 2
A 1 (1 −E 2 )cos V cos U
(1 −E 2 sin 2 U cos 2 V ) 3 / 2
A 1 (1 E 2 )sin U sin V
(1 −E 2 sin 2 U cos 2 V ) 3 / 2
J= {J 2
KL } =
.
(20.26)
A 1 cos V (1 E 2 sin 2 U )
(1
A 1 E 2 sin U cos U sin V cos 2 V
(1
−E 2 sin 2 U cos 2 V ) 3 / 2
−E 2 sin 2 U cos 2 V ) 3 / 2
Eigenvalues :
1st eigenvalue of K: κ 1 = 1
E 2 sin 2 U cos 2 V/A 1 ;
E 2 sin 2 U cos 2 V ) 3 / 2 /A 1 (1
E 2 ) .
2nd eigenvalue of K: κ 2 =(1
Christoffel symbols M
:
KL
1
11
( U, V )= E 2 sin U cos U cos 2 V (3
E 2 (3
sin 2 U sin 2 V ))
,
E 2 sin 2 U cos 2 V )(1
(1
E 2 )
1
12
( U, V )=
− E 4 sin 2 U cos 2 U cos 2 V )
sin V (1 − E 2
,
E 2 sin 2 U cos 2 V )(1
(1
E 2 )cos V
1
22
( U, V )= E 2 sin U cos U (1
E 2 sin 2 U )
E 2 ) ,
(20.27)
E 2 sin 2 U cos 2 V )(1
(1
2
11
( U, V )
= sin V cos V (1 E 2 (1 + 2 sin 2 U cos 2 V )+ E 4 sin 2 U cos 2 V (2 sin 2 U sin 2 V ))
(1
E 2 sin 2 U cos 2 V )(1
E 2 )
2
12
( U, V )= E 2 sin U cos U cos 2 V (1
E 2 (1 + sin 2 U sin 2 V ))
,
E 2 sin 2 U cos 2 V )(1
(1
E 2 )
2
22
( U, V )= E 2 sin 2 U sin V cos V (3 E 2 (2 + sin 2 U ))
(1
.
E 2 sin 2 U cos 2 V )(1
E 2 )
 
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