Аналитическая геометрия. Часть II. Аналитическая геометрия пространства. Шурыгин В.В. - 24 стр.

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detA detB = detA detB
>
= det(AB
>
)
2
(a, b, c)(a, b, c) =
(a, a) (a, b) (a, c)
(b, a) (b, b) (b, c)
(c, a) (c, b) (c, c)
.
V
P (a,b,c)
P (a, b, c) a b c
G(a, b, c) =
(a, a) (a, b) (a, c)
(b, a) (b, b) (b, c)
(c, a) (c, b) (c, c)
,
V
P (a,b,c)
=
p
det G(a, b, c).
e
1
e
2
e
3
E
3
ε
123
=
q
det (g
ij
).
< a, b >< c, d >=
(a, c) (a, d)
(b, c) (b, d)
, ε
12
=
q
det (g
ij
).
E
3
{e
1
, e
2
, e
3
} E
3
a
a
i
= (a, e
i
) i = 1, 2, 3
x = x
k
e
k
(x, e
i
) = a
i
i = 1, 2, 3 (g
ij
) {e
k
}
(x, e
i
) = (x
k
e
k
, e
i
) = x
k
g
ki
x
i
x
g
ki
x
k
= a
i
                                     (2+ -+9,0C0)2 -+ :0())4 +,)C)B
detA detB = detA detB > = det(AB > ) O
12)13O -2+3 )+ 9 ,09+D (0-2 09)-290 <Š 2
  NJƒLpŸfMLf hmëflhk k rIhLokhŸ¾Mh¡ pLpqflf jhhIiLMnq'
  1)CA 03 F+410 3913)2-3 (0-24 -1(0)4 F+41 <Š~
                                      (a, a) (a, b) (a, c)
                                                                                        
                 (a, b, c)(a, b, c) = (b, a) (b, b) (b, c) .                        <—
                                      (c, a) (c, b) (c, c)
0 4+s)2 -,+17:+9027-3 , 9(-1) +*å)40 V                   ,0011)1),B
,)C0 P (a, b, c) O ,+-2+)++ 0 9).2+0E a O b  c  9+C3 +*+:0())
                                                         P (a,b,c)


                                                                 
                                  (a, a) (a, b) (a, c)
                                                      
                   G(a, b, c) =  (b, a) (b, b) (b, c)  ,
                                  (c, a) (c, b) (c, c)
,+1(0)4~
                                                                        
                                                                    <›
                                        p
                         VP (a,b,c) =       det G(a, b, c).
    (0-2+-2O ,4)33 F+41 Šˆ . 2+D.) 9).2++9 e O e O e ,09++
+2++4+90++ *0:-0 9 E3 O ,+1(0)4
                                                        1 2 3


                                        q
                              ε123 =        det (gij ).
  Î' eIhLokfifMLf ikHg jhpJg rIhLokfifML¡'
  4)A2-3 +()9C) 001+ F+41 ,)CC )+ ,.20 C13 .+-++
,+:9)C)3 0 +)2+90+D ,1+-.+-2~
                                                                  q
                            (a, c) (a, d)
          < a, b >< c, d >=               ,               ε12 =       det (gij ).
                            (b, c) (b, d)
™'° NonLlMJf mnoLpJ k E '
                                    3
?-27 {e , e , e } ; ).+2+D *0:- 9 E  -3.D 9).2+ a +C+:0(+
+,)C)13)2-3 -.0134 ,+:9)C)34 ai = (a, ei) O i = 1, 2, 3  )D-29B
            1 2 3                                 3


2)17+O ,-27 x = xk ek ; ).+2+D 9).2+O C13 .+2+++ (x, ei) = ai O
                                                                           
i = 1, 2, 3 O 0 (gij ) ; 402‡0 -.013++ ,+:9)C)3 9 *0:-) {ek } +C0
                                                                     (
(x, ei ) = (xk ek , ei ) = xk gki O  .++C02 xi 9).2+0 x +C+:0 + 0E+C32B
-3 : --2)4 09)D gkixk = ai O 402‡0 .+2++D )9+sC)0
                                             ژ

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