Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 42 стр.

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(42)
ϕ(u) = a
ij
u
i
u
j
(
)
V
n
A
n
a
i
0
j
0
u
i
0
u
j
0
= (a
ij
p
i
i
0
p
j
j
0
)u
i
0
u
j
0
= a
ij
(p
i
i
0
u
i
0
)(p
j
j
0
u
j
0
) = a
ij
u
i
u
j
.
e
A = (a
αβ
)
(n+1)
A = (a
ij
)
(n + 1)
e
A
e
Φ
A
n+1
Φ a
αβ
x
α
x
β
= 0
A
n
π
n
A
n+1
ϕ : A
n
→ π
n
⊂ A
n+1
.
ϕ
C
: A
C
n
→ π
C
n
⊂ A
C
n+1
.
A
n+1
V
n+1
= V
n
⊕ R
Q ∈ A
n+1
π
n
V
n+1
= V
n
⊕R Q = 0
π
n
= V
n
⊕ 1 {O; e
i
} A
n
≡ π
n
{Q; e
α
} = {Q; e
i
, e
n+1
=
−→
QO} A
n+1
   ¬[YZ˜š›]Y –^ ¸imemb‚ }fjkbmbat (42) abkjfajbibj gibghaimo‚bg efmp
g|fjlgkjbat vggf nabji€
   ¬[YZ˜š›]Y µ^ õgfd}ogu ϕ(u) = a uiuj abkjfajbibg ( bmljkahadg gi
kc|gfj hahimdc ggf abji) gefm motmiht vkjnfjiasbjt `gfdj bj kmvp
                  v  n          n  ij


igfbgd efghifjbhikm Vn w jhhg†aafgkjbbgd h j``abbcd efghifjbp
hikgd An €
   £—™œ¤œšY[¥˜š›—^ E .*3B-” „-3:26A •²²‹ .-62)5*:
                  0   0                    0   0               0         0
           ai0 j 0 ui uj = (aij pii0 pjj 0 )ui uj = aij (pii0 ui )(pjj 0 uj ) = aij ui uj .


   @5730½5 Ae = (a ) 8B68*7/8 :5730½*” 1B5D3570),-” „-3:A  -.3*D*
6*,,-” ,5 ,*1-7-3-: (n+1) :*3,-: B*17-3,-: .3-/735,/7B* < ,5 8B68*7/8
                    αβ


:5730½*” 5,56-90),-” :5730½* A = (aij ) D68 ,*1-7-3-” 90.*3.-B*3C,-/70
B (n + 1) :*3,-: 5„„0,,-: .3-/735,/7B*  5 0:*,,-  :5730½5 Ae 5//-½00
3-B5,5 / 1-,2/-: Φe B An+1  ,5.35B68‡ˆ*” 1-7-3-9- 8B68*7/8 .-B*3C,-/7>
Φ < 35B,*,0* ¨7-9- 1-,2/5 0:**7 B0D aαβ xα xβ = 0 < 5//:-730: ¨72 1-,
/7321½0‡ .-D3-+,** <
   ’-:*/70: 5„„0,,-* .3-/735,/7B- An 151 90.*3.6-/1-/7> πn B ,*1-7-
3-* 5„„0,,-* .3-/735,/7B- An+1  7< * < -/2ˆ*/7B0: ,*1-7-3A” 0E-:-3„0E:
                                   ϕ : An → πn ⊂ An+1 .

’30 ¨7-:
                                                   C
                                  ϕC : A C    C
                                         n → πn ⊂ An+1 .
   15)*/7B* .3-/735,/7B5 An+1 :-“,- BE87>  ,5.30:*3  Vn+1 = Vn ⊕ R <
  56**  BA+*3*: 0 E5„01/032*: 7-)12 Q ∈ An+1  ,* .30,5D6*“5ˆ2‡ 90
.*3.6-/1-/70 πn < /62)5* .3-/735,/7B5 Vn+1 = Vn ⊕R :-“,- BE87> Q = 0
0 πn = Vn ⊕ 1 < *.*32 {O; ei} B .3-/735,/7B* An ≡ πn /-.-/75B0: 3*.*3
{Q; eα } = {Q; ei , en+1 = QO} B .3-/735,/7B* An+1 <
                           −→




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