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det A det
e
A
` A
n
v
M
0
x
i
= x
i
0
+ tv
i
Φ
a
ij
(x
i
0
+ tv
i
)(x
j
0
+ tv
j
) + 2a
i n+1
(x
i
0
+ tv
i
) + a
n+1 n+1
= 0 ⇐⇒
(a
ij
v
i
v
j
)t
2
+ 2t(a
ij
x
j
0
v
i
+ a
i n+1
v
i
) + F (x
k
0
) = 0
t
F
i
(x
k
0
)v
i
F
i
(x
k
) = 2(a
ij
x
j
+ a
i n+1
) = ∂F/∂x
i
ϕ(v)t
2
+ F
i
(x
k
0
)v
i
t + F (x
k
0
) = 0.
v
ϕ ϕ(v) = 0 `
Φ v
ϕ
a
ij
v
i
v
j
= 0
`
t
1
t
2
` Φ M
1
M
2
A
C
n
( ) M
1
M
2
Φ
M
1
M
2
t
t
1
+ t
2
2
M
0
(x
i
0
)
`
xaw haxbji}fcw egogyaimobcm a gifajimobcm abnmvhc abmfaaw tkp
ot{iht j``abbcda abkjfajbijda zigxg }fjkbmbat äfgdm igxgw abkjp
fajbijda tkot{iht lbjva gefmnmoaimomu det A a det Ae
YXYY«Y\]Y Ú]WYXWYX×\] XÚ WX®Z WX®° ^
5//:-730: .38:2 ` B An / ,5.35B680: B*17-3-: v .3-C-D82
)*3*E 7-)12 M < ³75 .38:58 0:**7 /6*D20* 235B,*,08~ xi = xi + tvi <
68 BA8/,*,08 ** 35/.-6-*,08 .- -7,-*,0 1 90.*3.-B*3C,-/70 Φ .-D
0 0
/75B0: ** 235B,*,08 B 235B,*,0* 90.*3.-B*3C,-/70 < :**:~
aij (xi0 + tv i )(xj0 + tv j ) + 2ai n+1 (xi0 + tv i ) + an+1 n+1 = 0 ⇐⇒
¶H
(aij v i v j )t2 + 2t(aij xj0 v i + ai n+1 v i ) + F (xk0 ) = 0
-¨0½0*,7 .30 t B .*3B- /7*.*,0 B 235B,*,00 ¶H :-,- .3*D/75B07> B
B0D* Fi(xk0 )vi 9D* Fi(xk ) = 2(aij xj + ai n+1) = ∂F/∂xi .-/6* )*9- 235B,*,0*
¶H .30,0:5*7 B0D <
ϕ(v)t2 + Fi (xk0 )v i t + F (xk0 ) = 0. ¶
VWXYZY[Y\]Y^ ¹mvigf v bjlckjmiht jhadeigiasmhvad kmvigfgd vkjnp
fjiasbgu `gfdc ϕ w mhoa ϕ(v) = 0 ftdjt ` bjlckjmiht eftdgu jhadep
igiasmhvgxg bjefjkombat gibghaimobg xaemfegkmfrbghia kigfgxg egp
ftnvj Φ w mhoa mm bjefjkot{qau kmvigf v tkotmiht jhadeigiasmhvad
kmvigfgd vkjnfjiasbgu `gfdc ϕ w jhhgaafgkjbbgu h }fjkbmbamd zigu
xaemfegkmfrbghiaw ig mhiw mhoa aij vivj = 0
YX° [æ«° WX®® ` ]YY \Y]W]«YY \WX[Y\]Y^
¨7-: /62)5* 235B,*,0* ¶ 0:**7 DB5 3**,08 B--+* 9-B-38 1-:
.6*1/,A* t1 0 t2 0 /6*D-B57*6>,- ` .*3*/*15*7 Φ B DB2C 7-)15C M1 0 M2
1-7-3A* B--+* 9-B-38 .30,5D6*57 ACn <
VWXYZY[Y\]Y^ Ôifmlgv ( j abgxnj a kht eftdjt) M M bjlckjmiht rgf p
ngu xaemfegkmfrbghia Φ 1 2
*3*D0,* C-3DA M M /--7B*7/7B2*7 E,5)*,0* .535:*735 t 35B,-* t1 + t2 <
2/7> /*3*D0,- C-3DA 8B68*7/8 7-)15 M0(xi0) ¨7- .3-0EB-6>,58 7-)15
1 2
2
.38:- ` ,5 1-7-32 ,* +A6- ,56-*,- ,01510C -935,0)*,0 < -9D5
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