324 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
Finally, we here have to constrain the general solution x(l, b) of the Laplace–Beltrami equation
(compare with (15.29)) to the ellipsoidal Korn–Lichtenstein equation y l = −
G 11 /G 22 x b (compare
with (15.27)), in particular
y l = −r(b)x b = −(r 0 + r 1 b + r 2 b
2 + r 3 b
3 + O(4))x b ,
(15.65)
y l = y 10 + 2y 20 l + y 11 b + 3y 30 l
2 + 2y 21 lb + y 12 b
2 + 4y 40 l
3 + 3y 31 l
2 b + 2y 22 lb
2 + y 13 b
3 + O(4) =
= −r 0 x 01 − r 0 x 11 l − 2r 0 x 02 b − r 1 x 01 b − r 0 x 21 l
2
− 2r 0 x 12 lb − r 1 x 11 lb−
−3r 0 x 03 b
2
− 2r 1 x 02 b
2
− r 2 x 01 b
2
− r 0 x 31 l
3
− 2r 0 x 22 l
2 b − r 1 x 21 l
2 b − 3r 0 x 13 lb
2
−
−2r 1 x 12 lb
2
− r 2 x 11 lb
2
− 4r 0 x 04 b
3
− 3r 1 x 03 b
3
− 2r 2 x 02 b
3
− r 3 x 01 b
3 + O(4) ,
(15.66)
Alternatively, we here have to constrain the general solution to the ellipsoidal Korn–Lichtenstein
equation y b = −
G 22 /G 11 x l (compare with (15.27)), in particular
y b = s(b)x l = (s 0 + s 1 b + s 2 b
2 + s 3 b
3 + O(4))x l ,
(15.67)
y b = y 01 + y 11 l + 2y 02 b + y 21 l
2 + 2y 12 lb + 3y 03 b
2 + y 31 l
3 + 2y 22 l
2 b + 3y 13 lb
2 + 4y 04 b
3 + O(4) =
= s 0 x 10 + 2s 0 x 20 l + s 0 x 11 b + s 1 x 10 b + 3s 0 x 30 l
2 + 2s 0 x 21 lb + 2s 1 x 20 lb+
+s 0 x 12 b
2 + s 1 x 11 b
2 + s 2 x 10 b
2 + 4s 0 x 40 l
3 + 3s 0 x 31 l
2 b + 3s 1 x 30 l
2 b + 2s 0 x 22 lb
2 +
+2s 1 x 21 lb
2 + 2s 2 x 20 lb
2 + s 0 x 13 b
3 + s 1 x 12 b
3 + s 2 x 11 b
3 + s 3 x 10 b
3 + O(4) .
(15.68)
Corollary 15.2 (Korn–Lichtenstein equations solved in the function space of bivariate polynomials).
If a polynomial (15.29)–(15.32) of degree n fulfills the Korn–Lichtenstein equations (15.27) with respect
to an ellipsoid-of-revolution and subject to the n − 1 constraints given by (15.52)–(15.64), then the
following mixed coefficient relations hold.
n = 1 :
y 10 = −r 0 x 01 , y 01 = s 0 x 10 .
(15.69)
n = 2 :
2y 20 = −r 0 x 11 , y 11 = −2r 0 x 02 − r 1 x 01 , y 11 = 2s 0 x 20 , 2y 02 = s 0 x 11 + s 1 x 10 .
(15.70)
n = 3 :
3y 30 = −r 0 x 21 , 2y 21 = −2r 0 x 12 − r 1 x 11 , y 12 = −3r 0 x 03 − 2r 1 x 02 − r 2 x 01 ,
y 21 = 3s 0 x 30 , 2y 12 = 2s 0 x 23 + 2s 1 x 20 , , 3y 03 = s 0 x 12 + s 1 x 11 + s 2 x 10 .
(15.71)
n = 4 :
4y 40 = −r 0 x 31 , 3y 31 = −2r 0 x 22 − r 1 x 21 , 2y 22 = −3r 0 x 13 − 2r 1 x 12 − r 2 x 11 ,
y 13 = −4r 0 x 04 − 3r 1 x 03 − 2r 2 x 02 − r 3 x 01 , y 31 = 4s 0 x 40 , 2y 22 = 3s 0 x 31 + 3s 1 x 30 ,
3y 13 = 2s 0 x 22 + 2s 1 x 21 + 2s 2 x 20 , 4y 04 = s 0 x 13 + s 1 x 12 + s 2 x 11 + s 3 x 10 .
(15.72)
In general:
y l =
∞
n=1
n−1
i=0
(n − i)y n−i,i l
n−i−1 b
i = −
∞
n=1
n−1
i=0
i
j=0
(i − j + 1)r j x n−1−i,i−j+1 l
n−i−1 b
i =
= −r(b)x b ,
y b =
∞
n=1
n−1
i=0
(n − i)y n−i−1,i+1 l
n−i−1 b
i =
∞
n=1
n−1
i=0
i
j=0
(n − i)s j x n−i,i−j l
n−i−1 b
i =
= s(b)x l .
(15.73)
End of Corollary.
Finally, we here have to constrain the general solution x(l, b) of the Laplace–Beltrami equation
(compare with (15.29)) to the ellipsoidal Korn–Lichtenstein equation y l = −
G 11 /G 22 x b (compare
with (15.27)), in particular
y l = −r(b)x b = −(r 0 + r 1 b + r 2 b
2 + r 3 b
3 + O(4))x b ,
(15.65)
y l = y 10 + 2y 20 l + y 11 b + 3y 30 l
2 + 2y 21 lb + y 12 b
2 + 4y 40 l
3 + 3y 31 l
2 b + 2y 22 lb
2 + y 13 b
3 + O(4) =
= −r 0 x 01 − r 0 x 11 l − 2r 0 x 02 b − r 1 x 01 b − r 0 x 21 l
2
− 2r 0 x 12 lb − r 1 x 11 lb−
−3r 0 x 03 b
2
− 2r 1 x 02 b
2
− r 2 x 01 b
2
− r 0 x 31 l
3
− 2r 0 x 22 l
2 b − r 1 x 21 l
2 b − 3r 0 x 13 lb
2
−
−2r 1 x 12 lb
2
− r 2 x 11 lb
2
− 4r 0 x 04 b
3
− 3r 1 x 03 b
3
− 2r 2 x 02 b
3
− r 3 x 01 b
3 + O(4) ,
(15.66)
Alternatively, we here have to constrain the general solution to the ellipsoidal Korn–Lichtenstein
equation y b = −
G 22 /G 11 x l (compare with (15.27)), in particular
y b = s(b)x l = (s 0 + s 1 b + s 2 b
2 + s 3 b
3 + O(4))x l ,
(15.67)
y b = y 01 + y 11 l + 2y 02 b + y 21 l
2 + 2y 12 lb + 3y 03 b
2 + y 31 l
3 + 2y 22 l
2 b + 3y 13 lb
2 + 4y 04 b
3 + O(4) =
= s 0 x 10 + 2s 0 x 20 l + s 0 x 11 b + s 1 x 10 b + 3s 0 x 30 l
2 + 2s 0 x 21 lb + 2s 1 x 20 lb+
+s 0 x 12 b
2 + s 1 x 11 b
2 + s 2 x 10 b
2 + 4s 0 x 40 l
3 + 3s 0 x 31 l
2 b + 3s 1 x 30 l
2 b + 2s 0 x 22 lb
2 +
+2s 1 x 21 lb
2 + 2s 2 x 20 lb
2 + s 0 x 13 b
3 + s 1 x 12 b
3 + s 2 x 11 b
3 + s 3 x 10 b
3 + O(4) .
(15.68)
Corollary 15.2 (Korn–Lichtenstein equations solved in the function space of bivariate polynomials).
If a polynomial (15.29)–(15.32) of degree n fulfills the Korn–Lichtenstein equations (15.27) with respect
to an ellipsoid-of-revolution and subject to the n − 1 constraints given by (15.52)–(15.64), then the
following mixed coefficient relations hold.
n = 1 :
y 10 = −r 0 x 01 , y 01 = s 0 x 10 .
(15.69)
n = 2 :
2y 20 = −r 0 x 11 , y 11 = −2r 0 x 02 − r 1 x 01 , y 11 = 2s 0 x 20 , 2y 02 = s 0 x 11 + s 1 x 10 .
(15.70)
n = 3 :
3y 30 = −r 0 x 21 , 2y 21 = −2r 0 x 12 − r 1 x 11 , y 12 = −3r 0 x 03 − 2r 1 x 02 − r 2 x 01 ,
y 21 = 3s 0 x 30 , 2y 12 = 2s 0 x 23 + 2s 1 x 20 , , 3y 03 = s 0 x 12 + s 1 x 11 + s 2 x 10 .
(15.71)
n = 4 :
4y 40 = −r 0 x 31 , 3y 31 = −2r 0 x 22 − r 1 x 21 , 2y 22 = −3r 0 x 13 − 2r 1 x 12 − r 2 x 11 ,
y 13 = −4r 0 x 04 − 3r 1 x 03 − 2r 2 x 02 − r 3 x 01 , y 31 = 4s 0 x 40 , 2y 22 = 3s 0 x 31 + 3s 1 x 30 ,
3y 13 = 2s 0 x 22 + 2s 1 x 21 + 2s 2 x 20 , 4y 04 = s 0 x 13 + s 1 x 12 + s 2 x 11 + s 3 x 10 .
(15.72)
In general:
y l =
∞
n=1
n−1
i=0
(n − i)y n−i,i l
n−i−1 b
i = −
∞
n=1
n−1
i=0
i
j=0
(i − j + 1)r j x n−1−i,i−j+1 l
n−i−1 b
i =
= −r(b)x b ,
y b =
∞
n=1
n−1
i=0
(n − i)y n−i−1,i+1 l
n−i−1 b
i =
∞
n=1
n−1
i=0
i
j=0
(n − i)s j x n−i,i−j l
n−i−1 b
i =
= s(b)x l .
(15.73)
End of Corollary.