Аналитическая геометрия. Шурыгин В.В. - 38 стр.

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R
n
R
n
n
a b E
n
(a, b) = 0
a ⊥ b a ∈ E
n
(a, a) = 1 {e
i
} i = 1, . . . , n E
n
e
1
, . . . , e
n
(g
ij
)
g
ij
= (e
i
, e
j
) g
ij
= δ
ij
δ
ij
δ
ij
= 0 i 6= j δ
ij
= 1 i = j
(E
n
, g)
E
n
a
1
p
(a
1
, a
1
)
e
1
=
a
1
√
(a
1
,a
1
)
n = 1
e
1
E
1
n > 1 a
2
e
1
a
2
b
2
= a
2
+
λe
1
λ b
2
e
1
(b
2
, e
1
) = (a
2
, e
1
) + λ(e
1
, e
1
) = (a
2
, e
1
) + λ
(b
2
, e
1
) = 0 λ = −(a
2
, e
1
) b
2
e
2
=
b
2
√
(b
2
,b
2
)
e
1
e
2
n = 2
E
2
n > 2
e
1
, e
2
, . . . , e
k
u = λ
1
e
1
+ λ
2
e
2
+ . . . + λ
k
e
k
= 0 (u, e
1
) = λ
1
= 0 (u, e
2
) = λ
2
= 0 . . .
(u, e
k
) = λ
k
= 0
   5OO0,,-* .3-/836,/8b- R \ 6//- 003-b6,,-* / *b170L-b; b*18-3,;     n
.3-/836,/8b-; R \ 9b79*8/9 .30;*3-; nK;*3,-:- *b170L-b6 6OO0,,-:-
                                          n
.3-/836,/8b6=
   {*18-3 a 0 b *b170L-b6 b*18-3,-:- .3-/836,/8b6 E ,6[b6J8/9 ‚b[6K
0;,-ƒ ŠžŠ©Š‹Ÿ‘’Œ\ */70 0N /16793,-* .3-0[b*L*,0* 36b,- ,27J†                                                       n

                                                  )
(a, b) = 0 = ˆ38-:-,67?,-/8? b*18-3-b -+-[,6 6*8/9 /7*L2J}0; -+36[-;†
        {
a ⊥ b = *18-3 a ∈ E \ /16793,M 1b6L368 1-8-3-:- 36b*, *L0,0 *\ 8- */8?
861-M\ )8- (a, a) = 1 \ ,6[b6*8/9 ŽªŒŒ¤‘’= Â6[0/ {e } \ i = 1, . . . , n \ b E \
                                                   n


,6[b6*8/9 ŠžŠŠ’ŒŠ”‘’\ */70 bN-L9}0* b ,*:- b*18-3 e , . . . , e                                       i                               n


9b79J8/9 *L0,0),;0 0 b[60;,- -38-:-,67?,;0= “8- ~1b0b67*,8,- 8-K                                                                     1       n


;2\ )8- ;6830 6 (g ) \ /-/86b7*,,69 0[ /16793,N .3-0[b*L*,0M b*18-3-b
+6[0/6 g = (e , e ) 9b79*8/9 *L0,0),-M† g = δ \ :L* δ · 861 ,6[b6*;*
                                              ij


/0;b-7 P3-,*1*36† δ = 0 \ */70 i 6= j \ 0 δ = 1 \ */70 i = j =
                     ij               i   j                                          ij        ij              ij


   ˜Txw™ušxXWxZ èŠ ”£‰Š’ Ž”‰‹ŒªŠ”Š’ ”Ž‰žŠŠ’ ¡Šžž”Ž (E , g)
                                                       ij                                 ij


Œ’ŽŽž£ ŠžŠŠ’ŒŠ”‘¦ ¬¢Œ§                                                                                                           n


   éuyÍêÍzx™ë¶ztuZ H-/83-08? -38-,-3;03-b6,,M +6[0/ b E ;-|,- /7*K
L2J}0; -+36[-;= H2/8? a · .3-0[b-7?,M ,*,27*b-M b*18-3= H-L*70b                                                               n


*:- ,6 p(a , a ) \ .-72)0; *L0,0),M b*18-3 e = √ = ¼/70 n = 1 \ 8- 1
                                                                                                            a1

b*18-3 e -+36[2*8 -38-,-3;03-b6,,M +6[0/ b E =
                              1       1                                                            1
                                                                                                           (a1 ,a1 )

   ¼/70 n > 1 \ b-[?;*; .3-0[b-7?,M b*18-3 a 861-M\ )8- b*18-3 e
                     1                                                                                 1


0 a 70,*M,- ,*[6b0/0;= Ç6//;-830; 70,*M,2J 1-;+0,6 0J b = a +                                         2                                       1

         2
                             )        )
λe = °-|,- ,6M80 861-* 0/7- λ \ 8- b*18-3 b +2L*8 -38-:-,67*, b*18-32
                                                                                                                                   2       2

    ‡                                                                  H
e = *M/8b08*7?,-\ (b , e ) = (a , e ) + λ(e , e ) = (a , e ) + λ = -~8-;2\
     1                                                                                         2

 1
                                       )
(b , e ) = 0 .30 λ = −(a , e ) = ˆ *b0L,-\ b*18-3 b ,*,27*b-M\ .-~8-;2\
                                                       2    1               2   1          1       1               2       1


.-76:69 e = √ \ .-72)0; Lb6 *L0,0),N b[60;,- -38-:-,67?,N
     2           1
                                           b2
                                                                2       1                                  2


b*18-36 e 0 e = ¼/70 n = 2 \ 8- -,0 -+36[2J8 -38-,-3;03-b6,,M +6[0/
                          2
                                          (b2 ,b2 )

bE =                      1           2

   ¼/70 n > 2 \ .30;*,0; ;*8-L ;68*;680)*/1-M 0,L21 00=
         2


   ¯6;*80; /,6)676\ )8- b/910M ,6+-3 *L0,0),N b[60;,-        -38-:-,67?,N
b*18-3-b e , e , . . . , e 9b79*8/9 70,*M,- ,*[6b0/0;;= ‡*M/8b08*7?,-\*/70
u = λ e + λ e + . . . + λ e = 0 \ 8- (u, e ) = λ = 0 \ (u, e ) = λ = 0 \ . . . \
                          1       2                k

                 1 1              2 2                       k k                      1             1                       2       2
(u, e ) = λ = 0 =
             k                k


                                                                                –Ô

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