322 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
First, let us here consider the ellipsoidal vectorial Laplace–Beltrami equations which are defined
by (15.28), namely
∆ lb x =
G 22 /G 11 x l
l
+
G 11 /G 22 x b
b
= 0 ,
∆ lb y =
G 22 /G 11 y l
l
+
G 11 /G 22 y b
b
= 0 ,
(15.41)
sx ll + (rx b ) b = sx ll + r b x b + rx bb = 0 ,
(15.42)
sy ll + (ry b ) b = sy ll + r b y b + ry bb = 0 ,
(15.43)
x(l, b) = x 0 + x 10 l + x 01 b + x 20 l
2 + x 11 lb + x 02 b
2 + x 30 l
3 + x 21 l
2 b + x 12 lb
2 + x 03 b
3 +
+x 40 l
4 + x 31 l
3 b + x 22 l
2 b
2 + x 13 lb
3 + x 04 b
4 + O(5) ,
(15.44)
x l (l, b) = x 10 + 2x 20 l + x 11 b + 3x 30 l
2 + 2x 21 lb + x 12 b
2 +
+4x 40 l
3 + 3x 31 l
2 b + 2x 22 lb
2 + x 13 b
3 + O(4) ,
(15.45)
x ll (l, b) = 2x 20 + 6x 30 l + 2x 21 b + 12x 40 l
2 +
+6x 31 lb + 2x 22 b
2 + O(3) ,
(15.46)
sx ll (l, b) = (s 0 + s 1 b + s 2 b
2 + O(3))x ll =
= 2s 0 x 20 + 6s 0 x 30 l + 2s 0 x 21 b + 2s 1 x 20 b + 12s 0 x 40 l
2 + 6s 0 x 31 lb+
+6s 1 x 30 lb + 2s 0 x 22 b
2 + 2s 1 x 21 b
2 + 2s 2 x 20 b
2 + O(3) ,
(15.47)
x b (l, b) = x 01 + x 11 l + 2x 02 b + x 21 l
2 + 2x 12 lb + 3x 03 b
2 +
+x 31 l
3 + 2x 22 l
2 b + 3x 13 lb
2 + 4x 04 b
3 + O(4) ,
(15.48)
x bb (l, b) = 2x 02 + 2x 12 l + 6x 03 b+
+2x 22 l
2 + 6x 13 lb + 12x 04 b
2 + O(3) ,
(15.49)
r b x b (l, b) = (r 1 + 2r 2 b + 3r 3 b
2 + O(3))x b =
r 1 x 01 + r 1 x 11 l + 2r 1 x 02 b + 2r 2 x 01 b + r 1 x 21 l
2 + 2r 1 x 12 lb+
+2r 2 x 11 lb + 3r 1 x 03 b
2 + 4r 2 x 02 b
2 + 3r 3 x 01 b
2 + O(3) ,
(15.50)
rx bb (l, b) = (r 0 + r 1 b + r 2 b
2 + O(3))x bb =
2r 0 x 02 + 2r 0 x 12 l + 6r 0 x 03 b + 2r 1 x 02 b + 2r 0 x 22 l
2 + 6r 0 x 13 lb+
2r 1 x 12 lb + 12r 0 x 04 b
2 + 6r 1 x 03 b
2 + 2r 2 x 02 b
2 + O(3) .
(15.51)