Contact Geometry of second order I - Dept. Math, Hokkaido Univ ...

∂ 2 z
∂x1∂x16
∂ 2 z
∂x2∂x16
∂ 2 z
∂x3∂x16
∂ 2 z
∂x5∂x16
∂ 2 z
∂x7∂x16
∂ 2 z
∂x13∂x17
∂ 2 z
∂x10∂x17
∂ 2 z
∂x8∂x17
∂ 2 z
∂x7∂x17
∂ 2 z
∂x8∂x18
− ∂2 z
∂x7∂x18
− ∂2 z
∂x4∂x17
− ∂2 z
∂x3∂x17
= − ∂2 z
∂x4∂x12
= − ∂2 z
∂x4∂x14
= − ∂2 z
∂x4∂x15
= − ∂2 z
∂x6∂x15
= − ∂2 z
∂x8∂x15
=
∂2 z
∂x14∂x18
=
∂z
= ∂2 z
∂x6∂x10
= ∂2 z
∂x6∂x13
= ∂2 z
∂x9∂x13
= ∂2 z
∂x9∂x14
=
∂2 z
∂x10∂x14
∂2z ∂x15∂x19
∂z
= λ −1 (x25 − x24
∂x11 ∂x13
∂2z =
∂x12∂x18
= λ −1 (x25
∂x9
= − ∂2z ∂x12∂x19
∂z
= λ −1 (x25
=
∂2 z
∂x12∂x21
= λ −1 (x25
=
∂2 z
∂x10∂x19
= − ∂2z = x21
∂x8∂x9
= − ∂2z = x22
∂x8∂x11
= − ∂2z = x23
∂x10∂x11
= − ∂2z = x24
∂x11∂x12
= − ∂2z = x25
∂x12∂x13
=
= − ∂2z ∂x15∂x20
∂z ∂z
∂x6
∂z
=
∂x5
=
− x24
∂x10
∂2 z
∂x16∂x21
+ x23
∂z
(= p1)
− x22
∂x14 ∂x15
= − ∂2 z
∂x16∂x22
+ x23
= − ∂2z =
∂x14∂x20
∂z
− x24 + x22
∂x8
∂2 z
∂x14∂x22
− x24
∂2 z
∂x13∂x20
=
∂z
+ x20
∂x7
= λ −1 ∂z ∂z
(−x25 + x23
∂x4 ∂x8
=
∂2 z
∂x10∂x21
= λ −1 (x25
∂z
=
∂x3
= − ∂2 z
∂x6∂x18
= λ −1 (x24
= − ∂2 z
∂x5∂x18
= λ −1 (x24
∂2 z
∂x13∂x22
− x23
∂z
∂x7
=
= − ∂2z ∂x9∂x19
∂z
∂z
∂ 2 z
∂x1∂x11
∂ 2 z
∂x1∂x11
∂z
− x21
∂x12 ∂x15
∂2z ∂x16∂x23
∂z
∂ 2 z
∂x1∂x11
∂ 2 z
∂x1∂x11
∂ 2 z
∂x1∂x11
+ x20
(= −p2)
∂z
(= p3)
∂z
− x21
∂x12 ∂x14
∂2z ∂x15∂x23
∂z
(= −p4)
∂z
− x19
∂x12 ∂x14
∂2z ∂x16∂x24
∂z
− x22
∂x10
= − ∂2z ∂x15∂x24
∂z
+ x20
(= p5)
+ x21
∂z
+ x19
+ x18
+ x18
∂z
∂x16
∂z
∂x16
∂z
∂x16
∂z
∂x15
∂z
),
),
),
− x17
∂x13 ∂x16
(= p6)
∂z
− x19
∂x10 ∂x13
= − ∂2 z
∂x11∂x20
∂z
∂x4
− x23
∂x6
∂z
+ x22
∂x9
= ∂2z =
∂x9∂x21
∂z
∂x3
∂z
− x23
∂x5
∂z
+ x20
∂x9
37
=
∂z
− x21
∂x11
∂2z ∂2z ∂x11∂x22
=
+ x17
∂2 z
∂x16∂x25
∂x15∂x25
∂z
− x19
∂x11
∂z
∂x15
+ ∂z
),
∂x16
+ ∂z
),
∂x15
),
),
),
(= p7)
(= −p8)