Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 346 стр.

UptoLike

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒË°}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ

äÓºmkp
|¹¯ËËãËÓÒË
¯
°m
E
n
ÈÓ ËÓϺ¯ Ò¹È
),( pq
q
p
jjj
iii
...
...
2
21
1
α
 Ë
1
p
 ËÓϺ¯ Ò¹È
)1,1(
+ pq
q
p
jjjj
ii
...
...
210
2
β
ÓÈÏ©mÈË°«énoyszjzvuwvlt¹zq¹rvkjéqjtztv
mv qtlnrxj
i
1
y zntovéj
q
p
jjj
iii
...
...
2
21
1
α
 Ë°ãÒ m }Èκä ÈÏÒ°Ë ÒäËË äË°º
¯ÈmËÓ°mº
q
p
q
p
jjj
iii
ji
jjjj
ii
...
...
...
...
2
21
01
210
2
1
αβ
γ
=

ÈÈ
¯
Æ
2
E
x {ytljuntzjstuunzéq·nxrqu zntovévu
γ
ij
=
23
35
zntové
i
jk
α
ojljtujzéq|np
34
57
25
13
Ëjpzqujzéq|zntovévk
ijk
α
q
ij
k
α
ËÓÒË
°
 iã« º¹°}ÈÓÒ« ¹Ë¯mºº ÒÓË}°È mº°¹ºãÏËä°« Áº¯ä㺮
m
jkimijk
=
α
γ
α
ºãÈËä
α
γ
α
γ
α
α
γ
α
γ
α
α
γ
α
γ
α
α
γ
α
γ
α
α
α
γ
α
α
γ
α
γ
α
α
α
111 11 11
1
12 11
2
112 11 12
1
12 12
2
121 11 21
1
12 21
2
122 11 22
1
12 22
2
211 21 11
1
22 11
2
212 21 12
1
22 12
2
221 21
23 35 21
22 31 7
24 37 29
25 33 19
33 55 34
32 51 11
=+=+=
=+=+=
=+=+=
=+=+=
=+=+=
=+=+=
=
21
1
22 21
2
222 21 22
1
22 22
2
34 57 47
35 53 30
+=+=
=+=+=
γ
α
α
γ
α
γ
α
.
vã˺mÈËãÓºäÈ¯ÒÈËÓϺ¯È
ijk
α
ÒäËËmÒ
21 29
34 47
719
11 30

°
 iã« ¹ºÓ«Ò« mº¯ºº ÒÓË}°È °ãËË ¹¯ÒäËÓÒ Áº¯äã
mji
mk
ij
k
γ
αα
=
Ë
ij
γ
}ºÓ¯ÈmȯÒÈÓÓ©®äË¯ÒË°}Ò®ËÓϺ¯
äÈ¯ÒÈ }ºº¯ºº º¯ÈÓÈ äÈ¯ÒË ËÓϺ¯È
ij
γ
ÒÒäËËmÒ
23
35
53
32
1
=
ºªºä
 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈ÒË°}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



 
                                                                                                 α i1i22...i p q  Ë p ≥ 1  ‘ËÓϺ¯ ˆÒ¹È
                                                                                                    j1 j ... j
 |¹¯ËËãËÓÒË            ‚°ˆ  m E n  ÈÓ ˆËÓϺ¯ ˆÒ¹È ( q, p )
 ¯
                         (q + 1, p − 1) β •0i
                                                j j1 j 2 ... j q
                                                                   ÓÈÏ©mÈˈ°«énoyszjzvuwvlt¹zq¹rvkjéqjtztv
                                                  2 ...i p

                         mv qtlnrxj i1  y zntovéj α i 1i 2...i
                                                                           j j ... jq
                                                                                         Ë°ãÒ m }ÈÎºä ­ÈÏÒ°Ë ÒäËˈ äË°ˆº
                                                                           12     p

                         ¯ÈmËÓ°ˆmº β •0i 1...i2              = γ i1 j 0 α i 1i
                                              j j j ... j q                 j j 2 ... j q
                                                                                            
                                         2      p                           1 2 ...i p
            
            
                                                                                                                          2 3
 ~ÈÈÈ                 Æ E 2  x {ytljuntzjst€u unzéq·nxrqu zntovévu γ ij =                                             zntové
 ¯                                                                                                                3 5
                                                                    3 4
                                                                    5 7
                         α •i jk ojljtujzéq|np                            Ëjpzqujzéq|€zntovévk α ijk q α ijk • 
                                                                    2 5
                                                                    1 3
            
 cËËÓÒË                ° iã« º¹‚°}ÈÓÒ« ¹Ë¯mºº ÒÓË}°È mº°¹ºã ςËä°« Áº¯ä‚㺮
                                  α ijk = γ imα •mjk ºã‚ÈËä
                                                      α111 = γ 11α11
                                                                  1
                                                                     + γ 12 α11
                                                                             2
                                                                                = 2 ⋅ 3 + 3 ⋅ 5 = 21
                                                      α112 = γ 11α12
                                                                  1
                                                                     + γ 12 α122 = 2 ⋅ 2 + 3 ⋅ 1 = 7
                                                      α121 = γ 11α 21
                                                                   1
                                                                      + γ 12 α 21
                                                                               2
                                                                                  = 2 ⋅ 4 + 3 ⋅ 7 = 29
                                                      α122 = γ 11α 22
                                                                   1
                                                                      + γ 12 α 22
                                                                               2
                                                                                  = 2 ⋅ 5 + 3 ⋅ 3 = 19
                                                                                                                       
                                                      α 211 = γ 21α11
                                                                   1
                                                                      + γ 22 α112 = 3 ⋅ 3 + 5 ⋅ 5 = 34
                                                      α 212 = γ 21α12
                                                                   1
                                                                      + γ 22 α12
                                                                              2
                                                                                 = 3 ⋅ 2 + 5 ⋅ 1 = 11
                                                      α 221 = γ 21α 21
                                                                    1
                                                                       + γ 22 α 21
                                                                                2
                                                                                   = 3 ⋅ 4 + 5 ⋅ 7 = 47
                                            α 222 = γ 21α 22
                                                          1
                                                             + γ 22 α 22
                                                                      2
                                                                         = 3 ⋅ 5 + 5 ⋅ 3 = 30 .
                                                                                               21                            29
                                                                                               34                           47
                                  vã˺mȈËã ÓºäȈ¯ÒȈËÓϺ¯È α ijk ÒäËˈmÒ                                            
                                                                                                7                           19
                                                                                               11                           30
                         
                         ° iã« ¹ºÓ«ˆÒ« mˆº¯ºº ÒÓË}°È °ãË‚ˈ ¹¯ÒäËÓ҈  Áº¯ä‚ã‚
                                  α •ij•k = α •i mk γ mj  Ë γ ij   }ºÓˆ¯ÈmȯÒÈӈө® äˈ¯ÒË°}Ò® ˆËÓϺ¯
                                  äȈ¯ÒÈ }ºˆº¯ºº º­¯ÈˆÓÈ äȈ¯ÒË ˆËÓϺ¯È γ ij  Ò ÒäËˈ mÒ
                                             −1
                                    2 3                5 −3
                                                  =         ºªˆºä‚
                                    3 5               −3  2