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. . – .: ISBN 5-7416-0147-7
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2011 2011
3
..................................................................................
8
.................................................................................
10
1. § 1.1. § 1.2. § 1.3. § 1.4. § 1.5. § 1.6. § 1.7. § 1.8. 2. § 2.1. § 2.2. § 2.3. § 2.4. § 2.5. § 2.6. § 2.7. § 2.8. § 2.9.
.
.......... ............................................ ........................................ ...................... ......................... .................. ............................................................ ............................. ....................................................
.................................... .......................... ......................................................................... .................................................................... ......................................................................... .................................................................... ................................... ................................................................ ....................... ...............................................................
12 12 21 24 28 34 38 44 47 54 54 57 59 61 65 68 70 72 75
4
. 3. § 3.1. § 3.2. § 3.3. § 3.4. § 3.5. 4. § 4.1. § 4.2. § 4.3. § 4.4. § 4.5. § 4.6. 5. § 5.1. § 5.2. § 5.3. § 5.4. § 5.5. § 5.6. 6. § 6.1 § 6.2 § 6.3. § 6.4. § 6.5. § 6.6. § 6.7. § 6.8.
.......................................... ........................................... .............. ................................... ........... .................................................... ............................................................ ................ ............................... ....... ................. .... .................. .............................. .......................................... . .................................. ..................... .......... ........ ................................................... ....................... ...................................................... ..................................... ................................. ................................................ ...................................................... m n ............................................................. ................... . ...
79 79 84 93 103 107 119 119 124 127 130 138 141 147 147 158 161 169 184 189 191 191 192 199 205 208 213 216 227
5 7. § 7.1. § 7.2. § 7.3. § 7.4. § 7.5. 8 § 8.1. § 8.2. § 8.3. § 8.4. § 8.5.
.................................... ................
235 235
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251 254 267 267 269 275 283
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§ 8.6. § 8.7. 9. § 9.1. § 9.2. § 9.3. § 9.4. § 9.5. § 9.6. 10. § 10.1. § 10.2.
............................................................ .................................. ............................... ......................................................................... ........................................................................ .............................................................. .............................. .................................. ................... . ................................................................
325 325 329 339 348 353 354 356 356 360
6
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362 368
§ 10.5. ……...... ........................................................... .......................... ...............................
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.................................. ............ .................................................................... ........................................ . ......................... .....................
410 413
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{O, g1 , g 2 , g 3 }.
)
{O, e1 , e2 , e3 } , 1.7.2.
, (
)
-
.
{O, g1 , g 2 , g 3 } , M
r, –
O,
M.
r 1.7.3.
OM {O, g1 , g 2 , g 3 }.
M
M M
1.7.4.
{O, g1 , g 2 , g 3 }. .
45
1.
1.7.1.
{O , g 1 , g 2 , g 3 } M
N,
MN .
MN .
.
. 1.7.1 . 1
OM
1
ON
2
2
3
3
OM MN
MN
.
ON
ON OM . MN
1
1
2
2
3
3
.
. 1.7.1
1.7.2.
{O, g1 , g 2 , g 3 }
M1
-
M2, 1
OM 1
2
1
OM 2
3
M,
,
2 3
M1M
MM 2 .
.
46 ,
.
,
1,
-
M
. 1.7.2).
M.
OM 1 M
OM1 M 1 M
-
OMM 2
OM ; OM MM 2
MM 2 ,
M1M
,
OM 2 ,
(OM 2 OM )
OM OM 1 ,
OM 1 1
OM 1
OM 2 .
1
M . 1.7.2 1
1
1 OM
2
3
1
:
1.7.2 .
2
1 3
.
47
1.
§ 1.8.
, .
-
, ,
,
. :
{O, g1 , g 2 , g 3 } “ ”
” {O , g1 , g 2 , g 3 } (
“ ”
“
. 1.8.1).
OO ,
,
.
”
“
-
1.5.1 :
g1
11
g1
21
g2
31
g3 ,
g2
12
g1
22
g2
32
g3,
g3
13
g1
23
g2
33
g3 ,
2
g2
3
OO
1
g1
(1.8.1)
g3.
“
” “
1.8.1.
”
: 1
,
1
11 1
12
2
13
3
2
21 1
22
2
23
3
2
3
31 1
32
2
33
3
3
,
.
(1.8.2)
-
48 .
M
“
{O, g1 , g 2 , g 3 }
”
1 2
,
“
”
3 1
{O , g 1 , g 2 , g 3 } –
2
.
3
“
”
“
M.
OM
OM
1
g1
1
g1
1
(
2
g2
2
2
11
g2
g1
g3
3
3
21
g3
g2
31
g3 )
(
12
g1
22
g2
32
g3 )
(
13
g1
23
g2
33
g3 ) .
3
OM , O M
OM . 1.8.1
”
:
OO O M OO ,
49
1. 1
g1
2
1
g2
1
g3
3
11 1
o,
12
2
13
3
1
,
2
2
21 1
22
2
23
3
2
3
3
31 1
32
2
33
3
3
,
.
{g1 , g 2 , g 3 }
o,
, 1
2
,
,
-
0
3
12
2
13
3
1
,
1
11 1
2
21 1
22
2
23
3
2
3
31 1
32
2
33
3
3
,
.
.
(1.8.2) 1.8.1.
{O, g1 , g 2 , g 3 } {O , g 1 , g 2 , g 3 } .
,
«
»
(1.8.1)
(1.8.2)
. , “
S ,
(1.8.2), ”
“ “
”, ”
50 1
“
”
,
“
2
”
3
“
”
. 11
12
13
21
22
23
31
32
33
S 1.8.2.
-
{g1 , g 2 , g 3 } {g1 , g 2 , g 3 } .
1.8.2.
det
11
12
13
21
22
23
31
32
33
0.
. 11
12
13
21
22
23
31
32
33
-
-
{g1 , g 2 , g 3 } 1.6..3 .
{g1 , g 2 , g 3 } . .
51
1.
: 1.8.1.
” {O, g1 , g 2 }
” {O , g 1 , g 2 } (
“
1.8.2).
.
.
-
,
“
g1
OO
”
“
”,
1 g1 . 2
g2
.
“ -
, “
”
”:
S
. 1.8.2 ,
“
1 2
{O, e1 , e2 }
2
2
1
”
g1
2 g2 ,
g2
g1 2 g 2 .
0 2
1 , 2 “
1
1
2
0
.
”
1,
2 2.
{O , e1 , e2 } . . 1.8.3.
,
52
e1
e1 cos
e2 sin
e2
e1 sin
e2 cos
:
S
cos sin
sin cos
1
OO
, ,
,
2
“
” “
”
1
1
cos
2
sin
1
,
2
1
sin
2
cos
2
.
. 1.8.3 -
“
OO
”
O. ,
,
, .
. 1.8.4.
. 1.8.4
,
e1
e1 ,
-
e2
53
1.
, . 1
1
cos
2
sin
1
2
1
sin
2
cos
2
,
, .
. 1.8.3
. 1.8.4,
,
a 1.8.3.
b
,
a
b
-
. .
S ,
, , det S
1,
-
det S (
), .
det S
1
1,
54
2
§ 2.1. l 2.1.1.
b
.
b
l. M, l,
2.1.2.
,
l–
M
. 2.1.1
M*, M
-
. –
,
-
, .
a
l
-
2.1.3.
Prl a ,
l,
-
55
2.
a
l,
2.1.3.
–
a. b,
-
b
e
{e }
|b| l(
. 2.1.1). -
2.1.4.
a
Prl a
l
{ e } 4. a
2.1.5.
b
-
, .
a l
l
a
a.
. 2.1.2
l
,
a
a cos ,
e.
. 2.1.2
4
,
: .
,
,
,
-
56
1.1 . :
Prl (a1 a 2 )
Prl a1 Prl a 2 .
. 2.1.3.
. 2.1.3 1.2 .
,
-
:
Prl ( a ) ,
1.1
Prl a .
1.2
: :
Prl ( .
1
a1
2
a2 )
1
Prl a1
1.1
1.2
2
Prl a 2 . -
57
2.
,
2.1 .
l
2.2 .
l
2.1
(a1 a 2 )
l
a
l
a.
2
a2 )
a1
l
a2 ;
2.2 , l
(
1
a1
1
l
a1
2
l
a2 .
, .
§ 2.2.
a 2.2.1.
b
, . , , .
a
(a , b ) . (a, b )
a
,
| a | | b | cos ,
b
,
a. b
2.1.5, 0
, ,
b:
–
.
(a , b )
b
b
o,
.
58
1 .
(a , b )
a 2 .
a
0
o
b
o
b
,
;
(a , b )
(b, a)
(
3 .
(a1 a 2 , b )
).
(a1 , b ) (a 2 , b ) (
).
.
b
o,
3
(a1 a 2 , b ) b b
b
.
a1
o,
(a1 a 2 )
b b
b b
(a1 , b ) (a 2 , b ).
a2
.
4 .
( a, b )
( a, b ) ;
5 .
(a , a ) | a |2
0
a ; | a|
(a, a) ;
(a , a)
,
0
a
o
); 6 .
a
o
b
o cos
(a, b) | a || b |
a
b.
,
–
-
59
2.
§ 2.3.
{ g1 , g 2 , g 3 } a
g1
1
2
3
( a, b )
(
g2
g3
3
b,
a
b
1
g1
2
-
g2
3
g3 .
4
g1
1
2
g2
( g1 , g1 )
3
1
g3 , 2
1
g1
( g1 , g 2 )
2
1
1
1
3
2
1
( g 2 , g1 )
2
2
(g2 , g2 )
2
3
1
( g 3 , g1 )
3
2
(g3 , g2 )
3
g2
3
g3 )
( g1 , g 3 ) 3
3
(g2 , g3 )
(g3 , g3 )
3
(
1
j
( g j , g1 )
j
2
(g j , g2 )
j
3
(g j , g3 ) )
j 1 3
3 j
i
( g j , g i ).
j 1 i 1
{e1 , e2 , e3 }
-
,
(ei , e j ) ij
1, i 0, i
ij
–
j, j, . -
( a, b )
1
1
2
2
3
3,
60 : 2 1
a
a
o
b
2 2
2 3
o 1
cos
2 1
1
2
2 2
2 3
2
3 2 1
3 2 2
2 3
.
,
cos 1
–
1 1
2
2
3
2 1
3
2 2
:
2 3
2 1
2 2
2 3
i
;
i
-
,
2.3.1.
. . 1
OM 2
{O, e1 , e2 , e3}
2 3
1
OM 1
.
2
,
3
1.7.1,
M1M 2 | M 1M 2 |
(
1
(
1
1
) e1 (
1
)2
2
(
2
2
) e2 (
2
)2
(
3
3
3
) e3
3
)2 .
61
2.
§ 2.4.
2.4.1.
{a , b , c }
,
(
-
a
)
b
c
-
.
{a , b , c }
. -
2.4.2.
a
b
c, | c | | a | | b | sin
1 .
,
–
,
a
b.
c
2 .
a
b. {a , b , c }
3 .
.
,
( , ), .
a [a, b].
2.4.2
,
b
-
62 1 .
[ a, b ]
,
a
b.
a
2 .
b
, .
1 .
[a, b ]
[b, a] (
,
sin ).
2.4.2 2 .
[ a, b ]
-
[a, b ] (
-
[ a, b ]
,
[a, b ]
-
a b 3 .
0 ).
[a b , c ] [a , c ] [b , c ] (
). .
a
b,
-
2.4.1.
l.
a
-
b
l
-
l
.
63
2.
2.4.1 ,l
(a b)
,l
a
,l
b.
. 2.4.1.
e
p
1
. 2.4.1
[ p, e ]
o,
-
2.4.2.
p
,
e,
e
. 2
.
,
p
O –
e –
e.
p p e, , (
. 2.4.2)
e,
,
-
64
[ p, e ] e
p e sin
p cos(
1.
[ p, e ]
Pr 2
,e
),
2
,
e
Pr
p,
e
p
-
p
,
e.
. 2.4.2 .
. 3.
c
o,
3 2.4.1, 2.4.2
.
c 1.1
o, § 2.1
-
65
2.
c
[a b, c ] | c | [ a b,
] |c |
|c| | c |(
,c
2
|c |( 2
a Pr
(Pr
,c
c
| c | ( [ a,
2
Pr ( a b ) c
b ))
c
c
a
Pr
,c
c
2
c
] [b ,
|c|
,c
b)
Pr c
] ) [ a , c ] [ b , c ].
|c |
.
§ 2.5.
{g1 , g 2 , g 3 } , g1 , g 2 , g 3 a
1
1
,
g1
g1
2
g2
3
g3
b
g1
1
2
g2
3
2
g2
1 1 [ g1 , g1 ] 2 1 [ g 2 , g1 ]
3
g3 ,
1
g1
1 2 [ g1 , g 2 ] 2
2[ g2 , g2 ]
-
a
,
2
[a, b ] [
,
2
g2
3
g3 ]
1 3 [ g1 , g 3 ] 2
3[g 2 , g3 ]
b,
3
g3 .
-
66 3 1 [ g 3 , g1 ] 3
2 [g3 , g 2 ]
3
3 3[g3 , g 3 ]
3 i [ g j , g i ].
j j 1i 1
f1 , f 2
f1
[ g 2 , g 3 ]; f 2
f3
-
[ gi , g j ]
:
[ g 3 , g 1 ]; f 3
[ g 1 , g 2 ].
[a , b ]
-
, 3-
2(
[ a, b ] (
2
det
det
.
3
1.1.1),
3
2
) f1 (
2
3
2
3
f1
f2
f3
1
2
3
1
2
3
1
f 1 det
3
1
1
3
f 2 det
1
3
,
e1 ,
f2
e2 ,
f3
1
2
2
1
2
1
2
1
) f3
f3
.
{e1 , e2 , e3 }
f1
) f2 (
3
e3 .
,
2.4.2,
67
2.
:
[a, b ] det
e1
e2
e3
1
2
3
1
2
3
.
.
a 2.5.1.
b
,
-
,
det
2
3
2
3
det
1
2
3
1
2
3
1
3
1
3
det
1
2
1
2
0,
.
2.5.2.
a
,
S
det 2
2
3
2
3
S
det 2
det
1
3
1
3
1
2
1
2
det 2
.
b,
-
1
2
1
2
,
68
§ 2.6. (
)
2.6.1.
a, b
c,
(a, b, c ) , 2.6.1.
-
( [ a , b ], c ) .
(a, b, c )
,
a, b
c.
a, b, c
-
,
, ,
–
.
.
a
b,
.
a
b,
-
(a , b , c ) |[ a, b ]|
c, [a ,b]
S = | [ a, b ] |
,
a
b,
|
c | | c | | cos | [ a,b ]
-
69
2.
–
S, (
.
V
. 2.6.1)
(a , b , c ) . ,
(a, b, c ) | [ a , b ] | | c | cos ,
. 2.6.1 .
.
: 1 . (a , b , c )
(c, a, b)
(b , c , a )
(b , a , c ) 2 . ( a, b, c) 3 . ( a1
(c, b, a)
(a, c , b) ;
(a , b , c ) ;
a2 , b , c )
( a1 , b , c )
( a2 , b , c ) , -
2.6.1.
70 ,
,
, -
.
§ 2.7.
a, b
{ g1 , g 2 , g 3 } a
1
g1
,c
[ a, b ]
2
1
g2
3
g1
det
2
g3 , b
g2
2
3
2
3
3
1
g1
2
1
3
1
3
g2
3
g3
c,
,
-
g3 .
f 1 det
f 2 det
f1 , f 2 , f 3
1
2
1
2
f3 ,
§2.5.
(g k , f j )
,
( g1 , g 2 , g 3 ), k
j,
0,
k
j,
1
3
1
3
(a , b , c ) (a, b, c ) ([a, b ], c )
3
det
1
2
1
2
(
1
2
det
2 2
)(g , g , g ) 1
3
2
3
det
3
det
1
2
3
1
2
3
1
2
3
( g1 , g 2 , g 3 ) ,
71
2.
,
, 3-
. (
1.1.1.) .1 .
2.6.1
. -
1.6.3. 2 .
( e1 , e2 , e3 )
1,
( a , b, c )
det
3 .
1
2
3
1
2
3
1
2
3
.
f1 , f 2 , f 3
§ 2.5
-
{ f1, f2 , f3 } 2.7.1.
{g1 , g 2 , g 3 } ). .
,
f1, f 2 , f3
. 1
1
,
2
f1
2
f2
2
( f2, g j )
,
3,
3
,
f3
o.
g j , j [1,3] , 1
( f1 , g j )
3
( f3 , g j )
0, j
[1,3] . (2.7.1)
72
( fi , g j ) , i
,i 0, i
( fi , g j )
j , j
[1,3] , j
0.
[1,3] ,
-
( f i , g i ) , i [1,3]
-
g1 , g 2 , g 3
.
( fi , g j ) , i
j ,
. (2.7.1), ,
i
0, i
[1,3] ,
f1 , f 2 , f 3 . .
§ 2.8.
a, b 2.8.1.
c
[ a , [ b , c ]] .
2.8.1.
[ a , [ b , c ]]
b( a, c )
c ( a, b) ,
a, b, c .
73
2. .
,
a, b , c
,
,
,
(a, b )
,
x
-
(a, c )
.
[ a , [ b , c ]].
-
x
[b, c ] ,
a.
1º.
( x , [ b , c ])
( x, b, c )
0
,
{ x, b , c} x
b
,
c,
.
( x , a)
2º.
1.4.1,
( b
0
c , a)
,
0
(b , a)
( c , a)
r,
3º.
0. -
:
x)
) r (
b ) ( r , b)
(a , r , [ b , c ])
0
(r , c)
0.(
,
c; .
. 2.8.1.)
(a ,[ r , [ b , c ]]) .
,
( r , b)
0,
74
2.8.1.
( a , r , [ b , c ]) (r, b
0
,
( r , a , [ b , c ]) c)
( r , b)
,
[ r , [ b , c ]] [ r ,[b, c ] ] (a , r , [ b , c ] )
b
( r , [ a , [ b , c ]])
[ r ,[ b , c ] ]
r b c cos(
2
(r , c)
( r , x)
( r , c ).
b,
b. ( a, b ) .
r b c sin sin( { r ;[ b , c ]}) )
(r, c) b
(r, c) ,
75
2.
r
[b, c ]
(a , r , [ b , c ] )
.
(a, b ) ( r , c ) .
(a , r , [ b , c ] ) , (r, c)
(a, b ) ( r , c )
,
(a, b ) .
,
.2º,
,
(a, c ) . .
4(
.§
. 4.5).
§ 2.9.
, .
,
,
)
(
. ,
a
b, 1
{g1 , g 2 , g 3 }
2 3
76 1 2
,
-
3
k
i
;
k , i 1, 2,3 ,
-
1
1
1
2
1
3
2
1
2
2
2
3
3
1
3
2
3
3
(2.9.1)
,
, . ,
k
i
; k , i 1, 2,3 ,
)
-
(
, . ,
, ,
2.9.1, .
{e1 , e2 , e3 }
S
{e1 , e2 , e3 }
11
12
13
21
22
23
31
32
33
.
§ 1.8, 3
et
pt p 1
ep; t
1,2,3 ,
-
77
2. 3
3
s
s 1,2,3; i;
si
s
st
i 1
–
it
(
; s 1,2,3.
. § 2.3),
{e1 , e2 , e3 } 3
(ei , et )
t
t 1
it
3
si e s , s 1
3
{e1 , e2 , e3 }
3
(
-
3
pt e p ) p 1
si
pt
(e s , e p )
s 1 p 1
3
3 si
pt
sp
si
s 1 p 1
st
; t
1,2,3 .
s 1
3
,
si
st
it
; i, t
1,2,3
s 1
S .
1
1
2
2
3
{e1 , e2 , e3 } ,
3
: 3
3 i
3
3
(
i
i 1
i 1
si s 1
3
1
2
2
3
st
3
t
it
t t 1
t
.
3 i
i 1 t 1 3
i
3
t)
t 1
3
i 1 t 1
1
3
i )(
t
si s 1
st
78 , . ,
,
-
2
3
3
3
1
1
1
2
2
, 3 , 1 . 2
.
79
3.
3
,
. ,
-
,
-
,
. 3
– .
3, 4
– 5
x,
-
y z,
.
§ 3.1. {O, g1 , g 2 } r0 ,
L,
a.
-
80
a 3.1.1.
L. L
3.1.1.
r
r0
a,
-
–
. .
r –
a
,
r r0 (
. 3.1.1) ,
1.4.2
r r0
a. :
r
r0
a
(
,
).
.
. 3.1.1 -
L.
x
r g
ax
a g
ay
,
y
x0
r0
,
g
.
y0
81
3.
3.1.2.
Ax
By
C
0,
A
B
0.
.
r r0 det
ay (x Ax
By C
C
a y x0
x x0
y y0
ax
ay
x0 ) a x ( y
0, A
B
y0 )
0, A
ay ; B
ax ,
, .
A a
-
0.
0,
a x y0 ,
a
B
,
0
o. .
Ax By C 3.1.3.
A
B
0, .
.
Ax
By x0
C y0
0, ,
A B 0. Ax0 By0 C
0.
0,
82 ,
A( x x0 ) B( y
y0 ) x0
r0 3.1.2
0. a
y0
,
B .
A
r0
,
a, y0 ) 0 .
A( x x0 ) B( y
,
-
-
. .
:
3.1.1–3.1.3
, ,
,
-
, : "
?"
3.1.4.
A1 x B1 y C1 A2 x
0,
A1
B1
0
B2 y C 2 0,
A2
B2
0 ,
, ,
A1
A2 ; B1
B2 ; C1
C2 .
0,
83
3. .
A2 x B2 y C 2 0 . 1 1 A2 x B2 y C 2 A1 x B1 y 1
B2 y C 2
( A1 x B1 y C1 )
C1 0,
A1 x B1 y C1 0 . A1 x B1 y C1 0
0,
A2 x
1
,
0.
.
A1 x B1 y C1
0
A2 x B2 y C 2
0 -
. (
A1
A2 ; B1
0,
3.1.2)
,
B2 . ,
A2 x ,
B2 y C1 0 C1 C2 .
A2 x B2 y C2
0
,
. :
,
(
P sec(
0) .
. . § 4.6)
-
84
§ 3.2. {O, g1 , g 2 }
-
. . 1 . , -
a
r2 r1
-
x2
x1
y2
y1
,
-
r r1
r2
x1
(1
r
) r1
( r2 r1 )
r1
r2 .
y1
, ,
x2
:
x x1 x 2 x1
y2
y y1 ; ( x2 y 2 y1
x1 )( y 2
y
y1
x,
y2
y1
x
x1
y,
x2
x1.
y1 )
0;
,
x det x1 x2
y y1 y2
1 1 1 r1
3.2.1.
r3
x3 y3
0.
x1 y1 ,
, r2
x2 y2
85
3.
,
x1 det x 2 x3
y1 1 y2 1 y3 1
0.
2 .
,
r0
x0 y0
-
, -
n
nx ny
. 3.2.1
.
a
r r0
x x0 y
y0
x
r
,
y (
3.2.1).
n
r r0
(n , r r0 )
( n, r )
d,
d
0, ( n , r0 ) .
.
86 -
( n, r ) r0
d
(n , r r0 ) , r0
,
0, d
-
n.
( n, n )
{O, e1 , e2 }
-
nx ( x x0 ) n y ( y
nx x
ny y
d,
Ax
By
y0 )
d
0,
n x x0
C
n y y0 . -
0,
,
n,
A
n g
B
,
-
.
n 3.2.1.
3 .
-
L. -
-
{O, e1 , e2 } Ax ,
By C
0, A
B
0
A2
B2 .
87
3.
A
cos A
x cos
2
B
y sin
2
B
; sin A
2
B
2
C
; A
2
B2
,
0. .
3.2.2.
. 3.2.2
, ,
Ax By C
0, A
B
(
,
y
),
Ax
By
0
C
0,
x -
A
B
0. ,
L : (n, r ) d P ( . . 3.2.3).
P
,
P
88
. 3.2.3 \
M
, 3.2.2.
R
P ( 0 ( M M
M L.
n, M
M
P ), 0 ),
P ,
3.2.1.
( n, R ) d .
, -
89
3. .
M M M
P ,
0
,
M
( n, R ) .
n.
L,
( n, OM ) d , ( n , R) d
( n , OM ( n, n)
M M)
( n , OM ) ( n , M M )
d .
.
( n, R ) d
M M
( n, OM ) d ,
( n , R)
( n , OM ) ( n , M M )
n
o
,
n,
,
( n , OM
d
( n, n) , M
0 ,
M M) d.
P.
.
{O, g1 , g 2 }
3.2.1.
L,
( n, r r0 ) ,
r1
x1 y1
.
0.
-
90
MK
1 .
n,
r
2 .
K ,
r1
n. ( .
.
3.2.4).
( n, r1
n r0 ) ( n,r1 r0 ) 2
0. .
| n| 3 .
MK , . 3.2.4
| MK | | (r1 r0 ,
n
) |.
| n| 4 .
.
Ax
n
By C
0, A
B
0,
,
A .
B
| MK |
A( x1
x0 ) B( y1 A2
By 0
B2 r0
, , Ax0
y0 )
C
| MK |
L ,
0, Ax1
By1 A2
.
B2
C
.
-
91
3.
,
3.2.3.
,
.
,
,
3.2.2.
A1 x B1 y C1
0
A2 x B2 y C 2
0.
1)
,
( A1 x B1 y C1 )
,
( A2 x B2 y C 2 )
0
, 2)
,
,
( A1 x B1 y C1 )
( A2 x B2 y C 2 )
0 .
.
x
r
1 .
,
y
,
A2 x
B2 y
C2 ,
( A1 x
B1 y
C1 ) . r
0,
,
. ,
( A2 x ( A1 x
B2 y B1 y
C 2 )( A1 x C1 )( A2 x
B1 y C1 ) B2 y C 2 )
r , ,
,
.
0
92
A1 x
2 .
B1 y C1
0
A2 x
B2 y
C2
0 – ,
,
( A1 x B1 y C1 ) ( A1
( A2 x B2 y C 2 )
A2 ) x ( B1
B2 ) y ( C1
0.
C2 )
0
,
A1
B1
0,
A2
A1
,
B2
A2
0
B1
B2
,
A1x
0 0.
:
A1
A2
0,
B1
B2
0.
0
B1 y C1
A2 x
,
(3.2.1)
B2 y C 2
, ,
0 .
.
3.1.4 0,
,
A1
A2
B1
-
B2 . det
1.6.2
A1
A2
B1
B2
0. ,
det
A1
A2
B1
B2
0
1.1.2
3.2.1 ,
. ,
0, 0.
-
93
3.
,
A1
A2
B1
B2
0.
.
3.2.4.
( A1 x B1 y C1 )
( A2 x B2 y C 2 )
0 -
.
§ 3.3. {O, g1 , g 2 , g 3 } r0
S,
p
S
-
q. p
q
-
3.3.1.
S. S
3.3.1.
r
r0
p
q,
– .
.
r – r r0
, (
. 3.3.1).
p, q
94 1.4.3
1.4.1
r r0
q,
p
,
,
r
r0
(
p , ).
( ,
-
q, )
. 3.3.1
.
-
x r
S.
px
y , p g
py g
z
pz
qx q
qy , g
.
qz -
3.3.2.
Ax
By
Cz
D
0,
A
B
C
0.
.
r r0 , p 1.6.3
q
-
95
3.
x
x0
y
px qx
det
y0
z
py qy
z0 pz qz
A( x x0 ) B( y y 0 ) C ( z Ax By Cz D 0,
0.
z0 )
0,
A, B
C
1.1.1
D
A
det
py qy
pz ;B qz
C
det
px qx
py , qy
Ax0
By0
px qx
det
pz ; qz
Cz 0 , ,
,
, .
p
C
q
A, B -
2.5.1. .
Ax 3.3.3.
A
B
C
By Cz
D
0
.
.
,
Ax
By
Cz
D
0,
0,
A
B
C
0
96
C x
0 DA B2 C 2
A2
det
y
A2
DB B2 C 2
0
C
C
0
C
z
DC B2 C 2
A2
0,
B A
0 x det
DA A B2 2
y
B 0
DB A B2 2
z
A
0
0
1
p 0 p
0,
q
C
C g
0
0
q g
B
C
0, ,
C
0,
A B
p
A g
0
,
0 q
0 . g
1 ,
. .
-
97
3.
{O, g1 , g 2 , g 3 } 3.3.1.
,
,
-
:
r1
x1
x2
x3
y1 ; r2
y 2 ; r3
y3 .
z1
z2
z3
,
.
r2 r1
r3 r1
.
,
r r r1
-
.
r r1 , r2 r1
r3 r1
,
( r r1 , r2 r1 , r3 r1 ) (
§ 2.7)
x
x1
y
y1
z
z1
det x 2
x1
y2
y1
z2
z1
x3
x1
y3
y1
z3
z1
{O, g1 , g 2 , g 3 } 3.3.2.
0,
0.
-
,
r0
x0 n
y0
z0
nx
ny
T
-
nz
T
.
98
r
.
r r0
, ,
. (r
,
r0 , n )
n
-
0. {O, e1 , e2 , e3 }
nx ( x
x0 ) n y ( y A
, , D
nx x0
nx ; B n y y0
Ax
y 0 ) n z ( z z0 ) ny; C
nz
0 ,
-
nz z0 ,
By Cz
D
0. -
3.3.1.
Ax By Cz
D
{O, e1 , e2 , e3 } A B 0,
C
A B C
n
0,
.
n 3.3.2.
-
( r r0 , n )
0.
A B C
3.3.3.
Ax
By Cz
-
D
0, A
B
C
0.
99
3.
.
3.3.3.
{O, e1 , e2 , e3 }
M
x y z
r
.
1 .
( r r0 , n )
K
M
MK
,
n
r
r
. 3.3.2.) K
2 .
r0 )
,
| n|
| MK | |(r
r0 ,
n
) |.
. 3.3.2
| n| .
.
-
n.(
n r0 )
,
2
0.
.
,
( n, r
( n ,r
-
0, ,
100
A n B C Ax By Cz D 0 . | A( x x 0 ) B( y y0 ) C ( z | MK | A2 B 2 C 2 ,
Ax0
,
A
r0 Cz 0
,
B
C
| Ax
By A
A1
D
,
0,
0,
| MK |
3.3.4.
By0
z0 ) |
B1
C1
2
Cz B
2
C A2
0
D| 2
.
B2
C2
0,
A1 x B1 y C1 z D1 0, A2 x B2 y C 2 z D 2 0 , . .
.
,
0,
A1
A2 ; B1
B2 ; C1 C2 A1 x B1 y C1 z D1 0, A2 x B2 y C 2 z D2 0
101
3.
D1
A1 x B1 y C1 z
D1
A1 x
D2
B1 y C1 z
D1 D2 D2 –
0, 0. ,
,
. .
A1 x B1 y C1 z D1 0 A2 x B2 y C2 z D2 0 .
. -
x
,
0
z
0.
,
-
,
x
x
0,
B1 y C1 z
D1
0
B2 y C2 z
0, D2
0. 0
B1
,
B2 ; C1 z
z A1 x
B1 y
,
B1 A1 .
C2 . 0,
z
0, D1
B2 A2 .
0
A2 x
B2 y
0, D2
0,
102 3.3.2.
A1 x
B1 y C1 z
A2 x
B2 y C 2 z
D1
0, A1
D2
B1
0, A2
C1
B2
0
C2
0 ,
-
,
0, A1
,
A2 ; B1
B2 ; C1
C 2 ; D1
D2 . -
,
3.3.4.
. , ,
3.3.5.
-
A1 x
B1 y C1 z
A1
B1
A2 x
B2 y C 2 z
D2
A2
B2
0,
( A1 x
D1
C1
0,
0
C2
B1 y C1 z
( A2 x B 2 y C 2 z
0,
D1 ) D2 )
0,
0. ,
3.3.6.
.
103
3.
P, 3.3.7.
A1 x
B1 y C1 z
A1
B1
A2 x
B2 y C 2 z
D2
B2
0
A2
C1
D1 0,
C2
A3 x B3 y C 3 z
D3
A3
0,
B3
0,
C3
0,
0,
,
( A1 x
B1 y C1 z
D1 )
( A2 x B2 y C 2 z
D2 )
( A3 x B3 y C 3 z
D3 )
0,
0 ,
-
P. ,
3.2.1
.
§ 3.4.
{O, g1 , g 2 , g 3 } .
r
1. -
,
x
y
z
T
104
ax a
ay az
x0 r0
y0 , z0
a
,
r r0
-
r
a.
r0
2.
r
-
r0
a x x0
ax ,
y
y0
ay ,
z
z0
az , -
x x0 ax
y
y0 ay
z
z0 az
, .
axa y az ,
3.
§ 3.2 (1 ).
a
,
0
105
3. -
r2 r1
x2 y2 z2
(r2 r1 )
r
x1 y1 , z1
x1 r1
y1
r
r1
(1
) r1
r2 .
z1
-
x2 r2
x x1 y y1 z z1 , x 2 x1 y 2 y1 z 2 z1 ( x2 x1 )( y 2 y1 )( z2 z1 ) 0 .
y2 z2
4.
1-
(n1 , r )
-
d1
(n 2 , r )
n2 –
n1
d2 ,
, ,
d1
d2 –
.
r0 ,
,
-
, :
(n1 , r r0 )
0,
(n2 , r r0 )
0.
A1 x
B1 y C1 z
D1
0,
A2 x
B2 y C 2 z
D2
0.
106 5.
-
-
a 2-
-
r r0 ,
[ a , r r0 ] [a, r ]
b,
b
o
[ a , r0 ] .
{O, e1 , e2 , e3 }
-
e1
e2
e3
det a x
ay
az
x
y
z
a y z az y b
bx ,
a z x a x z by , a x y a y x bz .
, , .
ax ,
,
az
ay
0 0,
, , ,
-
ax , ay
az
bx
a y z0
a z y0 ,
by bz
a z x0 a x y0
ax z0 , a y x0 .
,
-
107
3.
,
d
R
r
r0
-
a
-
, ,
S– , ,
.
. 3.4.1
. 3.4.1
S
d
a
[ R r0 , a ] . a
§ 3.5.
. , ,
, ,
-
, .
( r r01 , n1 )
3.5.1.
( r r02 , n2 )
0
0 -
n1
n2 .
108
( r r0 , n )
3.5.2.
r
r0
a
0
,
–
2
n
a.
3.5.1–3.5.3 . 3.5.1
1 .
r
r01
a1
r
r02
a2 .
a1 2 .
r01
a1
r
r02
a2 .
r0
a
(n1 , r )
d1 ,
( n2 , r )
d2.
a 2 .
0,
,
[a1 , a2 ] o . 0.
1 .
r
,
a2 .
(a1 , a 2 ) r
0,
[n1 , n2 ] .
[ a , [n1 , n2 ]]
o.
109
3.
( a , n1 , n 2 ) r
r0
0.
a
(n1 , r )
d1 ,
( n2 , r )
d2.
r
r01
a1
,
r
r02
a2 .
r01 r02 2 .
[a 1 , a 2 ]
a1
[a1 , a 2 ] r01
a1
r
r02
a2 .
r r
r02
r01 a2 .
a1
[a1 , a 2 ]
a1 .
o.
o
(r01 r02 , a1 , a 2 )
-
a2
o
[r01 r02 , a1 ]
r
0,
0
1 .
0.
o
(r01 r02 , a1 , a 2 )
0.
110 3.5.2
-
r r
r02
r01 p2
p1
0,
1 .
-
q1
[ p1 , q1 ]
[ p2 , q2 ]
(r01 r02 , p1 , q1 )
q2 .
,
0.
2 . [[ p1 , q1 ],[ p2 , q 2 ]]
(r01 r02 , p1 , q1 )
r
r01
p1
q1
[ p1 , q1 ]
r
r02
p2
q2 .
(r01 r02 , p1 , q1 )
r
r01
p1
q1
r
r02
p2
q2 .
r
r02
r01 p2
p1 q2 .
,
[ p2 , q2 ] 0.
[[ p1 , q1 ],[ p2 , q 2 ]] (r01 r02 , p1 , q1 )
-
r
0.
0,
1 .
2 .
q1
o
( [ p1 , q1 ],[ p 2 , q 2 ])
0.
o 0.
111
3.
-
r
r0
( n, r )
d.
r
p
r0
p
( n, r )
d.
q
q
-
r
r0
p
( n, r )
d.
( p, n )
0
( q, n)
0
( p, n )
0
( q, n)
0
[ p, q ] 2 .
d.
( n , r0 )
d.
0,
1 .
q
( n , r0 )
,
n.
[[ p, q ], n ]
o.
3.5.3
-
;
1 .
r
r01
a
r
r02
p
,
q.
a
(r01 r02 , p, q )
;
p 0.
0,
q
112
( a , p, q)
2 .
0,
(r01 r02 , p, q ) 0,
1 .
r
r01
a
r
r02
p
r
r01
( n, r )
r
r01
( n, r )
2 .
a
[ a ,[ p , q ]] ( a , n)
0
( n, r0 )
d.
r01
( n, r )
o.
d.
a
( a , n)
0,
(r0 , n)
d.
d.
0,
1 .
r
,
[ p, q] .
a q.
0.
a
a
d.
2 .
,
n.
[a, n ]
o.
1 .
(n1 , r )
d1 ,
(n 2 , r )
d2 ( n, r )
;
;
n d.
2 .
0,
n1
[ n ,[ n1 , n2 ]]
n2 . o.
,
113
3.
,
3.5.1–3.5.3 . 5
.
,
,
(n1 , r )
d1 ,
(n 2 , r )
d2, r
r0
,
a.
,
-
a
[ n1 , n 2 ] ,
r0
n1 r0
,
det
n1
(n1 , r0 )
d1 ,
(n 2 , r0 )
d2
(n1 , n1 )
(n1 , n2 )
(n2 , n1 ) (n2 , n2 )
5
n2 ,
,
,
det
d1
(n1 , n 2 )
d2
( n2 , n2 )
, , ,
,
, 3.5.2.)
n2 .
.(
.,
-
114
(n1 , n1 )
det
d1
(
.
1.1.2).
(n 2 , n1 ) d 2 ,
n1
-
n2
0. .
r
a
r0
,
r0
a,
.
-
,
n1
,
-
n2
[ a , r0 ]
[ a , n1 ] .
-
, 2.8),
(
.§
2
n2
[ a , n1 ] [ a , [ a , r0 ]]
d1 d2
( a , r0 ) a ( a , a ) r0
d2 ,
( a , r0 ) a
(n1 , r0 )
d1
,
(n2 , r0 ) .
r0
a r0 .
a
. .
(n, r ) 3.5.1.
r
r0
d0
-
a.
.
115
3. .1 .
,
(n, a )
,
0,
, .
,
(n, a )
0. r
2 .
r0
n,
r – -
, (
3.5.1
– . 3.5.1). -
,
( n , r0
d ( n , r0 )
, r
,
r0
a)
d
d0 .
( n , r0 )
a.
( n, a )
( n, a )
R 3.5.2.
r
r0
a. ,
-
.
R
.1 .
,
(
rx
-
.
| R rx | .
. 3.5.2).
116
rx
2 .
( a , R rx ) rx
r0
0
a,
, ,
-
, . 3.5.2
rx
( R r0 , a )
r0
|a | ( R r0
( R r0 , a ) |a|
| R r0 | 2
a , R r0
2
( R r0 , a ) |a|
( R r0 , a ) 2 |a|
a
2
a)
2
.
2
, 2
p
2
2
( p, q ) 2
q
[ p, q ]
-
[ R r0 , a ] .
§ 3.4
|a| r 3.5.3.
r
r02
a2 .
r01
a1
117
3. .
a1
1 .
a2
, . 3.4.1
,
S
| [r02 r01 , a1 ] |
| a1 |
| a1 |
-
.
a1
2 .
a2
-
, , ,
-
r01 , r02 (
-
. 3.5.3). ,
-
a1 , a 2 . 3.5.3
r02 r01 ,
,
-
, , ,
-
(r02 r01 , a1 , a 2 ) – .
,
| (r02 r01 , a1 , a 2 ) | |[a1 , a 2 ]|
.
118
[ p, r ]
( n, r )
q.
-
3.5.4.
R –
.
.
n,
[ n ,[ p, r ]] [ n , q ] .
-
R
§ 2.8,
p ( n , R) R ( n , p ) [ n , q ] . R
( n , R)
,
( n , p)
0,
, ,
R
p [ n, q ] ( n , p)
.
119
4.
4
§ 4.1. {O , g1, g 2 }
,
(
,
). ,
L
4.1.1.
r r g
Fx ( ), Fy ( ) –
F( ) ( Fx ( ) , Fy ( )
,
,
),
r
1.
F( )
L.
r0 ,
2.
,
0
r0
L,
,
F( 0) .
-
120
G ( x , y)
0,
-
x
Fx ( )
y
Fy ( )
,
.
1 . 4.1.1.
r
r0
a,
a –
, .
x
x0
ax
y
y0
ay
-
,
(
Fx ( )
x0
ax
Fy ( )
y0
ay
,
Ax By C 0 , G ( x , y ) Ax By C .
r0 –
,
),
(
A
,
B
),
0,
2 .
x0
R
x
x0
R cos
y
y0
R sin
,
y0
[0,2 ) ,
Fx ( )
x0
R cos
Fy ( )
y0
R sin
,
[0,2 ) ,
( x x0 ) 2 ( y y0 ) 2 R 2 , G( x, y) ( x x0 ) 2 ( y y0 ) 2
R2 .
-
121
4.
, 4.1.2. m k
x
pk
y
qk
pk
0,
qk –
-
k 0
,
-
k
.
N
max { pk
qk }
k [ 0,m ]
4.1.3.
,
-
4.1.2,
k,
0.
k
,
-
, .
x
3y
2
y
x2
0
(N
1)
0
(N
2)
xy 1 0
(N
2)
(N
3)
4.1.2 ).
-
x3
”
4.1.1.
y3
xy
0
. .
L
G ( x , y)
{O, g1 , g 2 }
0
{O, g1 , g 2 } . (1.8.2),
:
N. ,
122
x
11
x
12
y
1
y
21
x
22
y
2
L “
G(
11 x
1,
12 y
, ,
” 21 x
22
y
2
N
4.1.2, “
)
0. N ,
”
-
.
{O, g1 , g 2 } {O, g1 , g 2 } ,
N
N
N
N .
.
,
:
-
. 1 .
-
4.1.4.
x
x
0,
y
0,
y 3
-
0 , 3.
x2
2 .
y2
4
0
2
. .
{O, g1 , g 2 , g 3 } .
-
123
4.
,
L
4.1.4.
r x y z Fx ( ), F y ( ), Fz ( ) –
F( ) ( Fx ( ) Fy ( ) , Fz ( ) ,
,
),
r
1)
F( )
-
L,
-
L,
r0 ,
2) ,
0
r0
,
F( 0 ) .
G ( x, y, z ) 0, H ( x, y , z ) 0, x
Fx ( )
y
Fy ( ) ,
z
Fz ( )
,
,
,
G 2 ( x, y, z )
H 2 ( x, y, z )
-
0.
1 .
-
4.1.3.
x .
2
y
2
0, z
124 2 .
2 a
R
:
x
R cos ,
y
R sin ,
z
a
(
,
),
z R cos , a z y R sin . a
x
§ 4.2.
{O, g1 , g 2 , g 3 }
, ,
–
,
:
.
,
S
4.2.1.
r
F( , ) ( Fx ( , ) Fy ( , ) ,
r g
Fz ( , )
Fx ( , ) , Fy ( , ) , Fz ( , ) –
-
, ,
,
,
),
-
125
4.
1)
,
r
F( , )
S,
r0 ,
2) 0
,
S,
,
0
r0
F(
0
,
0
).
G( x, y, z) 0 , x
Fx ( , ),
y
Fy ( , ), ,
z
Fz ( , ).
4.2.1.
.
x0 y0
R
z0
(x
x
x0
R cos sin ,
y
y0
R sin sin ,
z
z0
R cos ,
x0 ) 2
(y
y0 ) 2
( z z0 ) 2
0
2 ,
0
,
R2 .
, -
126 , 4.2.2. m k
x
pk
q
y kz
rk
0,
pk , q k
,
k
rk –
k 0
.
N 4.2.3.
max{ pk
qk
k [ 0,m]
rk }
-
, (
k, -
4.2.2),
0.
k
,
-
, .
4.2.2 ).
x2
y2 1 0
x2
y2
4.2.1.
z2
R2
0
(N
2)
(N
2)
. .
4.1.1. ,
:
.
-
127
4.
§ 4.3. {O, g1 , g 2 , g 3 }
r
F( ),
,
-
. ,
4.3.1.
,
-
a. , ,
-
. .
r
F( )
NM ( .
. 4.3.1),
,
NM ,
a,
,
r( , )
-
F( )
a,
,
(
,
).
Fx ( ) F( )
Fy ( ) g
ax a
ay , g
az
x Fx ( ) ax
y Fy ( ) ay
z Fz ( ) . az
Fz ( )
128 ,
-
4.3.1.
-
3, , ,
,
-
,
-
,
x
3 cos
y
3 sin
3 cos ,
F( )
z
0
3 sin
;a
0
x2
z,
N
y2
9
,
2.
. 4.3.1
0 . 1
,
,
-
. 4.3.2
129
4. 4.3.2.
(
), , ,
A (
)
r0 . , , . .
r
F( ) (
NM
(r0 F ( )) , , r( , )
(1
.
NM , .
4.3.2)
,
) F( )
r0 ,
-
,
(
,
).
x0 r0
y0 , g
x Fx ( ) x0 Fx ( )
z0
y Fy ( ) y0
-
Fy ( )
z Fz ( ) . z 0 Fz ( ) ,
4.3.2.
-
3, , ,
,
-
130 -
,
0 r0
0 , 1 (
x 3 cos 3 cos
.
4.3.1):
y 3 sin 3 sin
z , 1
[0,2 ) .
,
x2 9
, N
y2 9
( z 1) 2
0,
2.
§ 4.4.
{O, e1 , e2 }
L. 4.1.2
4.4.1.
,
4.1.3
L ,
Ax 2
2 Bxy Cy 2 A, B
2 Dx 2 Ey
F
0, (4.4.1)
C
,
y
,
x -
L. 4.4.1 ,
-
131
4.
{O , e1 , e2 } , . 4.4.1.
,
a
b
0, p
( (
0) )
-
: 4 . 4. 1
-
-
0 x2 a2
y2 b2
1
x2 a2
y2 b2
0
-
0
0 2
y
a2
x
2
y
0
x
x2 a2
y2 b2
1
x2 a2
y2 b2
0
x2 a2
y2 b2
1
2
y
a2
x
y
2
2 px
132
det
A B
B2
AC
B C .
1 .
,
: B
B
,
0
A C.
0,
-
A C,
4.4.1.
,
,
e1
e2 ; e2
e1 ; OO
o,
-
,
x
y;y
x
§ 1.8.
,
,
det
B 0,
2 .
B
C
B
B
A
det
A B .
B C
0,
4
. 140. -
{O , e1 , e2 } , 0
O
4
,
,
xy . (
.
-
. § 1.8):
e1 e2
e1 cos e1 sin
e 2 sin
OO
o,
e 2 cos {O, e1 , e2 }
{O , e1 , e2 }
133
4.
x y
x cos x sin “
y sin , y cos . ”
“
”,
-
4.4.1
y sin ) 2
A( x cos 2 B( x cos
y sin )( x sin
y cos ) 2 2 D( x cos y cos ) F 0,
C ( x sin 2 E ( x sin
Ax
2
y cos )
2B x y
2
Cy
2D x
y sin )
2E y
F
0 .
,
A cos 2
A 2B
2 A sin cos 2 B sin A sin 2
C
2
2 B cos
,
2
2C sin cos , C cos 2
2 B sin cos
B 0 2 B cos 2 ,
C sin 2
2 B cos sin
.
,
( A C ) sin 2
0,
,
2B ; A C
tg 2
1 2B , arctg 2 A C
A C,
A C,
4
.
,
tg 2 ,
B
0,
A C tg B
1 0, B .
0,
-
134 3 .
,
A C
.
,
1 ctg 2 2
1
0
sin 2
A
( A2BC )
2
1 sin 2 2
4
2B 4B 2
(A C)2
; cos 2
A C 4B 2
( A C) 2
A C 1 cos 2 1 cos 2 A B sin 2 C 2 2 A C A C A C B sin 2 cos 2 2 2 2 A C A C 2B B 2 4B 2 ( A C ) 2 4B 2 ( A C ) 2
A C 2
1 4B 2 2
( A C) 2 .
,
A C 2
C
det
A 0
1 4B 2 2
( A C)2 .
0 C
AC
(
AC
B2
A C 2
1 (4 B 2 ( A C ) 2 ) 4 A B det , B C
)
2
.
135
4.
-
A
. A C.
C
,
4 .
B
,
0
Ax 2
Cy 2
2 Dx 2 Ey F 0.
D A
(
A x
)
2
x
x
(
E2 C
F,
e1
e1 ,
e2
e2 ,
2
P | | A
(
C y
E C
)
0
2
| C |2 ,
) ( y
P | | C
E2 C
F.
,
-
x
x
y
y
D , A E , C ,
1;
)
2
2
0;
| A |2 ,
-
0
2
y 2
C
D2 A
2
D E e1 e2 , A C Cy 2 P ,
OO
:
2
D2 A
P
Ax
0. A
,
0,
0
P
0,
P
0,
136
x2 a2 x2 a2
y2 b2 y2 b2
x2 a2 x2 0; 2 a
y2 b2 y2 b2
0;
x2 a2 x2 1; 2 a
y2 b2 y2 b2
1;
1
0,
1
0. -
(
. 4.4.1),
1 x
0, A 0( x y ), 2 Dx 2 Ey F 0
0. 0 (
5 .
C
Cy
AC
, !).
2
E C
(
C y D
-
y .
)
E2 C
2
F
2 Dx, C
A
-
0.
0
(
C y 2
y 0, E y C
E C
a2; y
)
2
2
E2 C 0; y
F, 2
a2 .
D
( ,
p 0.
) ,
2
2D 1 E2 F x C 2D C y 2 2 px , y 2
(
(
0, ,
)), 2 px ,
137
4.
( 4.4.1),
.
:
e1
e1 ,
e2
x y
e2 ,
OO
x, y.
o,
. .
1 .
4.1.1
,
, 4.4.1
-
, .
2 .
,
-
, 4.4.1,
. . § 5.4),
(
, , 4.4.1,
-
. 3 . , –
-
4 .
4.4.1 , , , .
138 1.
§ 4.5.
{O, e1 , e2 , e3 }
. 4.2.2 ,
4.5.1.
4.2.3
S ,
A11 x 2
A22 y 2
A33 z 2
2 A12 xy 2 A13 xz
2 A23 yz
2 A14 x 2 A24 y 2 A34 z
(4.5.1)
A44
0,
A11 ; A22 ; A33 ; A12 ; A13 ; A23 , , ,
x, y
z
S. (4.5.1)
, ,
-
. 4.5.1.
{O , e1 , e2 , e3 } , :
-
139
4.
,
y 2 b2 z 2 b2
x 2 a2
x2 a2 z
x
1
y2 b2
2
y,z
1
x2 a2
x2 a2 z
y2 b2 z2 b2
y2 b2
0
0
x2 a2 z
y2 b2
1
x2 a2 z
y2 b2
1
-
a2 x2 a2 z
y2 b2
x 2 a2 x2 0
y 0
y,z y,z
2
2 px
z
x2 y2 a2 b2 z2 0 c2
140
-
-
x2 a2
y2 b2
z2 c2
x2 a2
1
y2 b2
x2 a2
2z
y2 b2
z2 c2
-
x2 a2
a
0, b
y2 b2
0, c 0,
-
x2 a2
2z
p
1
y2 b2
z2 c2
1
0.
.
,
,
4.4.1, , 4.5.1
,
-
§ 12.2.
2.
141
4.
§ 4.6. ,
.
.
{ , }, OM ,
( OM , OP ) ,
0, 0
2 .
O OP –
, .
. 4.6.1 (
. 4.6.1).
-
. :
x y
cos , sin ,
,
x2 x
cos x
2
y
2
y2 ; y
; sin x
.
2
y2
.
142 ,
-
,
x2
y2
1,
1. ,
1
, –
,
-
–
(1
cos )
p
0,
0
p
0 –
, -
, p
0
1,
:
1
1.
-
4.6.2.
. 4.6.2.
-
143
4.
,
(1
cos )
-
0
p
.
x2
y2
,
x
cos x
,
p
x
y 2 (1
x2
x
2
y
2
2
)
y2
,
y2
p
0, 0
,
0
2
(1
)x
2
2
2 px
p .
1,
1,
. :
p
(x
1
1
)2 2
,
2
1
p2
y2
2 2
(1
)
4
. 4.4.1,
0
,
1
-
1–
,
-
.
(1
cos )
p
0,
(
)
-
,
, ,
4.4.1. ,
p
0
0 –
p
0 ,
.
0
,
p
0
cos
1–
144
. 4.6.3.
,
-
4.6.1.
(1
cos )
p
0
p
0,
0,
. ,
, ,
,
4.6.3.
,
,
.
145
4.
{ , , }, ( OM ,
(Ox, OP) ,
0; 0
(OM , Oz ) ,
2 ; 0
.
-
, .
-
, -
1. -
. 4.6.4 :
x y
cos sin , sin sin ,
z
cos ,
. 4.6.4),
146
x2
y2
x
z 2 ; cos x
2
y
x
z
cos x
,
2
y2
y
; sin
2
z2
2
y2
;
.
-
, . -
{ , , h} . 4.6.5),
OM ,
(Ox, OP) ,
. 4.6.5
0; 0
2 ;h
(
,
). :
x
cos ,
y z
sin , h,
x2 x
cos x
2
y
2
y2 ; y
; sin x
2
y
2
;h
z.
147
5.
5
§ 5.1.
5.1.1. ji
C i [1, n] , j
m n ( [1, m] )
A jk
i
j
m l (
[1, m] , k
l n [1, n] ),
-
B
[1, l ] )
(
k
ki
-
[1, l ] ,
l ji
jk
ki
i
[1, n] , j
[1, m] .
k 1
C
–
m n
C
A
–
l,
B .
. 5.1.1. , .
5.1.1.
A
1 .
2 1,
B –
2 2,
C
2 1.
148
C
A
B
11
12
11
11 11
12 21
21
22
21
21 11
22 21
11
12
...
1i
...
1n
11
12
...
...
...
...
...
1l
21
22
...
2i
...
2n
21
22
...
...
...
...
...
2l
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
j1
j2
... jl
...
...
...
...
...
...
...
...
...
...
...
...
...
...
m1
m2
...
...
...
...
...
ml
...
...
...
...
...
...
l1
l2
...
li
...
12
...
21 ...
11
j1
1i
...
22
...
2i
...
2n
...
...
...
...
...
...
ji
...
j2
.
ln
1n
l
jn
...
...
...
...
...
...
m1
m2
...
mi
...
mn
ji
jk ki k1
. 5.1.1
A
2 .
C C
B A
B – 1 2,
2 2, 1 2.
11
11
12
21
22
12
11 11
21 12
12
11
22
12
.
149
5.
3 .
A
,
B
C
2 2, 2 2.
, : 1)
,
A
B
B
A ,
C )
( A
2)
A ( B
B ) C ,
3)
-
A ( B
C )
A
B
A
C .
,
-
, . ,
(
,
)
A
E A .
. § 1.1) 1
A 5.1.2.
A , A
1
A
A
A
1
E .
150 .
A ,
,
0 6.
det A
,
A ,
det A
0,
5.1.3.
,
det A
,
–
. ,
5.1.1.
0,
-
. .
A
,
A
1
A
1
A
:
E
1
A
A
1
A
1
A
1
A
1
A A
2 1
.
E
2 1
,
E
2
E
O . A
A ,
1
A ( A
1
A
,
1 1
1 1
A A
A
1 2
)
A
1 1
O
1 1
,
-
O
E , 1 1
A
1 2
O .
.
6
n 6.
-
151
5.
A
,
det A
1 det A
1
A
12
21
22
1
A
0,
11
22
12
21
11
.
(5.1.1) 7
n
-
:
det ( A
B )
1
1
det A
det ( A ) det ( B ) ;
det A
,
det A
0.
,
-
5.1.2. 11 1
12
2
21 1
22
2
A 1
x
b
;
2
2
b , ; A
A
1
n 1.1.9,
11
12
21
22
,
1
) –
b . -
(1.8.2)
5.1.3.
7
A
1
,
2
(
x
x
1
2
n
6.
152
g1
g1
g2
S
T
g3
1
;
g2 g3
S
S
2 3
–
1
1
2
2
3
3
,
.
5.1.1.
B )T
( A
B
T
T
A
.
.
A
, ,
B
-
, ,
-
. ik
C
A
,
kj
,
A , B
ij
B
.
,
-
5.1.1, l ij
ik
kj
.
k 1
,
,
-
1.1.8, l T ij
ji
jk k 1
,
l
l T kj
ki k 1
5.1.1, . .
T ik
T ik k 1
T kj
,
153
5.
,
g1
g1
g2
S
T
g3
g1
g2
g3
g1
g2 g3
g2
g3 S . -
.
Q 5.1.2.
x
nn-
-
Q
,
. .
Q
11
12
...
1n
21
22
...
2n
... n1
... n2
... ...
...
x
.
nn
0 ... 1 , ... 0
i.
154
0
1i
0
...
...
...
Q 1
ii
0
...
...
...
0
ni
0
i
-
. .
5.1.2.
A
B
( A
B )
1
1
B
1
A
.
.
( A
1 .
1
-
x
n
( A x
B )
B )
A
B
2 .
1
x
c
(
c .
c
,
.
, 5.1.1
5.1.2).
,
, 1
A B
1
x
B 1
A
c
x
c .
3 .
( A
B )
1
x
c
B
1
-
A
1
x
c ,
155
5.
-
(( A
1
B )
1
B
1
A
) x
o ,
5.1.2
x
,
( A
B )
1
B
1
A
1
.
. . 1 T
( A
)
( A T) 1 .
5.1.1.
Q , 5.1.4.
Q
1
-
T
Q ,
.
,
-
,
.
Q
-
5.1.2
-
5.1.3.
det Q
1.
.
Q
1
Q
Q , Q
det 2 Q
T
Q
1,
Q Q
T
T
E .
,
156 ; -
;
det E
-
1.
.
Q , 5.1.4.
det Q cos sin
sin cos
1 –
,
,
det Q cos sin
sin cos
-
1 –
.
.
Q
Q Q ,
T
11
12
21
22
,
11
12
11
21
21
22
12
22
-
E
,
11
2 11
2 12
21
12
11 22
21 2 21
12 2 22
22
1 0 0 1
.
157
5.
2 11
2 12
1,
11 21 12 22 2 2 1, 21 22
0,
, 5.1.3,
det Q
,
det Q
1. 11
(
2 11
(
11
2 12
2 21
) (
22
)2
22
(
12
2 22 12
21
2 11 2 11
2 21
1; 0 cos 11 , sin 21
11
)2
0, 22 , 21 .
2 12
1;
12
,
21
) 2(
11
0
1.
1, 22
12
21
,
2 21
2 22
1
1,
Q , ,
2 11
det Q .
2 21
1
)
1. .
0
158 5.1.1.
. .
§ 1.8
S –
,
:
cos sin
sin cos
cos sin
sin cos
,
–
.
S
5.1.4.
.
§ 5.2.
. ( ,
, ,
,
) , .
Aˆ ,
, 5.2.1.
,
, .
-
159
5.
Aˆ y
Aˆ x,
x
,
y
.
x,
:
y x–
y.
–
–
5.2.2.
, .
,
, , y
( x ), x
:
-
.
x
1 . 5.2.1.
y,
-
x
l,
,
y
Pˆrl l x
l.
Aˆ Pˆrl .
,
[ , ]
2 .
f( )
-
f ( ) – f ( ) Aˆ
d f ( ), d d . d
,
160
x
3 .
-
x – .
, ,
.
[ , ]
4 .
f( )
f ( )d ,
(f)
f ( )d
,
-
[ , ]. Aˆ , 5.2.3.
)P
Q, -
, P Q.
:
Aˆ : P
Q.
M
M
Q,
-
M
-
M .
M–
Aˆ : P 5.2.4.
Aˆ M (
Aˆ (M ) ),
M M,
M
P
Q
,
Q .
-
161
5.
Aˆ
P P.
5.2.5.
5.2.6.
Aˆ M
M
Bˆ M
M
(
-
)
.
M
Bˆ Aˆ M .
,
, . ,
Aˆ : P
5.2.7.
Aˆ : P 1
P,
P,
,
M
-
P
Aˆ 1 ( Aˆ M )
M.
Aˆ Aˆ .
P,
5.2.8.
, P,
Aˆ ,
Aˆ . Aˆ
P,
, -
Aˆ . § 5.3.
{O, g1 , g 2 }
M ,
M ,
5.2.6,
162
Aˆ M .
M
x
rM
y
g
x
rM
x
,
y
g
x
-
y
x
Fx ( x , y )
y
Fy ( x , y )
y
,
-
Fx ( x , y ) Fy ( x , y )
x y rM
Aˆ rM
{O, g1 , g 2 } . ,
Fx ( x , y )
5.3.1.
-
Fy ( x, y ) .
rM
ArM
-
,
{O, g1 , g 2 }
x y
Aˆ Aˆ
g
x
11
x
12
y
1
y
21
x
22
y
2
g
x y
1
,
2 11
12
21
22
, .
163
5.
A( A)
-
{O, g1 , g 2 } .
rM
5.3.2.
ArM
-
,
-
5.3.1 ,
,
1
2
1
2
0,
0. Aˆ
.
: 5.3.1.
A,
-
, ”,
“ “
Aˆ 1
2
g
”, 1
0
0
2
–
-
, ;
, ; .
A 5.3.1.
:
-
164
ˆ (r 1 . A 1
Aˆ r1 Aˆ r2
r2 )
ˆ( r ) 2 . A
Aˆ r
r1 , r2 .
r, .
.
, ,
-
.
1º
Aˆ (r1 r2 )
11
12
21
22
11
12
21
22
x1 y1
(
x2 ) y2
x1 y1
11
12
21
22
x2 y2
Aˆ r1 Aˆ r2 .
.
A
-
5.3.2.
ˆ (r 1 . A 1 2 . A(
r2 )
r)
Aˆ r1 Aˆ r2
Ar
r1 , r2 ,
r, , .
.
r
x g1 y g 2
Ar
x g1
y g2 –
,
x g1 Aˆ g1
y g2
11
g1
A ( x g1
21
g2
y g2 ) Aˆ g 2
x A g1
12
g1
y A g2 .
22
g2 ,
165
5.
x g1 y g 2
x Aˆ g1 y Aˆ g 2 (
11
x
11
x
12
y
y
21
x
22
y
x
12
,
y ) g1 (
x
22
g1
g2
21
y) g 2 .
x
11
12
x
y
21
22
y
.
.
a,
,
ax
a g
ay
-
Aˆ a
{g1 , g 2 } ,
Aˆ a g
5.3.1
ax ay
11
12
21
22
ax ay
5.3.2
.
. -
5.3.1.
A
{g1 , g 2 } A g1
A g2 . -
5.3.2.
, .
166 ,
,
5.3.1.
: 1 .
-
ˆ Bˆ : A
A
2 .
Aˆ
g
Bˆ g .
g
1
,
A, A
1
1
A
g
g
.
, .
{O, g1 , g 2 }
5.3.3.
A
A
g
{O, g1 , g 2 }
.
S
g
1
A
S
S ,
g
{O, g1 , g 2 }
–
-
{O, g1 , g 2 } .
.
-
A
Aˆ
r –
Aˆ
r g
r g
{O, g1 , g 2 }
r g
g
,
,
-
g
S –
-
g
{O, g1 , g 2 }
,
167
5.
r
S
r
r
g
S
g
r
.
g
g
1.8.2
S
1
S
(
),
-
S
r
,
S
Aˆ
r
S
r
r
g
g
S
g
1
Aˆ g
g
,
Aˆ
r
r g
g
r
,
( g
g
5.1.2)
Aˆ
S
1
Aˆ
g
S . g
.
det Aˆ
. g
5.3.3. .
,
5.3.3
S det Aˆ
det ( S
1
Aˆ
g
S )
det S
1
det Aˆ det S
g
1 det S .
det Aˆ
g
det S g
det Aˆ . g
-
168 -
,
5.3.2.
x
3y
2
0.
.
M
-
x0
r0
,
y0
M
–
M –
-
x0
r0
y0
. . 5.3.1
. 5.3.1
x
M
,
3y
2
0
,
-
M.
x
3y
2
0 ,
x
x0
1
y
y0
3
.
,
M x0
x0
y0 x0
y0 3 , 3y0 2 0
,
x0 y0
9 3 1 x0 y0 , 10 10 5 3 1 3 x0 y0 . 10 10 5
-
169
5.
, ,
9 10 3 10
x0 y0
Aˆ
3 10 1 10 9 10 3 10
e
1 5 , 3 5
x0 y0
3 10 1 10
.
§ 5.4. ,
(
A: P
P)
,
.
x y
5.4.1.
P
Aˆ det
g
11
12
21
22
11
12
21
22
A
g
x
1
y
2
,
,
,
0,
.
170 5.4.1
det A
0
g
).
, . .
5.3.3
-
,
det A
,
g
0
-
. .
5.4.2.
,
.
.
det Aˆ
g
(
. § 5.1),
g
x
x
1
y
y
2
x y
x y
.
,
, g
1.1.2
Aˆ
1
Aˆ
0,
171
5.
Aˆ
,
,
x
Aˆ
y
1 g
x
1
y
2
Aˆ
1
,
1
1 g
.
2
2
.
,
-
5.3.2,
5.4.1.
. , ,
? -
5.4.3.
, ,
-
. .
g1
11
g1
det Aˆ
21 T
x
11
x
12
y
1
y
21
x
22
y
2
g2; g2 det Aˆ
12
g1
det
22
, ,
g2 .
11
12
21
22
0,
172
g2
g1
(
1.6.2)
. 1.8.2
5.3.1,
,
{ g1 , g 2 }
,
-
Aˆ ,
{g1 , g 2 } {g1 , g 2 }
-
{ g1 , g 2 } Aˆ .
S
g
,
, ,
,
-
,
. .
, . 5.4.4.
. .
x
x0
p
y
y0
q
,
p
q - (
)
,
x y
Aˆ
x
1
y
2
.
-
173
5.
-
x
(
11 0
x
12
y0
1
) (
y
(
21 0
x
22
y0
2
) (
,
11
p
12
q
11
21
21
.
p
12
p
p
q) ,
22
22
q
q) . 0,
, 11
p
12
q
0
21
p
22
q
0
,
det ) p ,
1.1.2 (
11
12
21
22
0
q 0 .
.
5.4.5.
, . .
,
-
. ,
. ,
-
. ,
.
, .
-
174 ,
. .
5.4.6.
. .
xi
M i ; i 1,2,3 (
yi
M i ; i 1, 2,3
. 5.4.1)
. 5.4.1
xi yi
y2 y3
y1 y2
,
.
x2 x3
,
1,
,
x1 x2
175
5.
x2 x3
x2 x3
x1 x2
y2 y3
x1 x2
y1 y2
.
x
11
x
12
y
1
y
21
x
22
y
2
( x2 11 ( x 3 11
( x3 11 ( x3
11
x1 ) x2 ) x2 ) x2 ) y2 y3
,
( y2 12 ( y 3 12
, ,
y1 ) y2 )
( y3 y2 ) y2 ) 12 ( y 3 12
y1 y2
.
.
,
, . :
M1 M 2 M3M2 ( x3 ( x3
( x2
x1 ) 2
( y2
y1 ) 2
( x3
x2 ) 2
( y3
y2 ) 2
x2 ) 2
( y3
y2 ) 2
x2 ) 2
( y3
y2 ) 2
( x3
(x2
x1 ) 2
( y2
y1 ) 2
( x3
x2 ) 2
( y3
y2 ) 2
.
x2 ) 2
( x3
x2 ) 2
M 1M 2 . M 2M3
( y3 ( y3
y2 ) 2 y2 ) 2
176 ,
5.4.6
, -
.
5.4.7.
,
, . .
M 1M 2 M 3M 3 ,
.
,
M 3M 4 M4 M2 . , 5.4.6 M 4 M 2 M 3 M 3 . 5.4.2).
M4M2M3 M3 – ,
M2 M4
. 5.4.2
M3 M3
.
-
177
5.
5.4.7
M1 M 2
M1 M 2
M 1M 2
M 1M 2 .
M3M4
M3 M2
M 3M 2
M 3M 4
.
-
5.4.8.
, . .
. 5.4.3
{g1 , g 2 }
O.
5.4.3 .
O , {O , g1 , g 2 } .
-
178
M
x
y,
-
M
x
y (
. 5.4.3),
5.4.6
O M1
OM 1
x ;
x g1
g1
OM 2
O M2 y .
y g2
g2 .
.
1.8.3
,
-
. 5.4.2.
n P,
.
a
b
[a, b ] [a, b ]
n
-
0,
,
n.
,
0,
,
,
179
5.
S –
1 . 5.4.9.
S –
det
S
11
12
21
22
-
S.
2 . ,
det
11
12
21
22
0,
det
,
-
11
12
21
22
0.
.
.
g1
-
,
g2 ,
Aˆ g1 ,
Aˆ g1
11
g1
21
g2
g2
Aˆ g 2
5.3.2,
11
12
g1
,
12
,
A, Aˆ
g
11
12
21
22
.
22
21
g2 , 22
180 (
. §2.4)
-
g1
,
S
[ g1 , g 2 ] ,
, ,– S
[g1 , g 2 ] [
11
g1
(
11
22
det A
S
g2 ,
21
g2 ,
12
12
21
[ g1 , g 2 ] . 22
)[ g1 , g 2 ]
,
S
g
g1
-
g2] det Aˆ
g
[ g1 , g 2 ] ,
(
5.4.2)
{g1 , g 2 } , det A det A
g
g
0 0.
,
, ,
-
,
. .
, 5.4.10.
-
4.4.1: -
;
-
, .
-
181
5. .
. 1 .
5.4.6
5.4.8 , ,
, -
. , .
, , .
2 .
, ,
,
, .
. 5.4.6. 3 .
.
,
. 4 .
, , 5.4.5
5.4.6
-
,
. . 4.4.1
5.4.1
, ,
:
x
2
y
2
x
2
y
2
1; x 0; y
2
2
y
2
1;
1 0; y
2
2x
0; y
2
(5.4.1)
0.
182 , ,
-
(5.4.1), . . :
"
".
§ 9.4 (
9.4.1),
.
5.4.11.
, . .
.
p p
q
q e
-
.
,
e
(
)
p e
11
12
11
12
21
22
21
22
,
183
5.
q e
11
12
11
12
21
22
21
22
.
q
p
{e1 , e2 } (
11
( ( (
11 11
21
12
22
21
12
)(
12
22
21
22
U
11
)(
)
2
(
)
2
0,
2
)
12 21
22
2 11
2V
)
2 12
U
0 2 21
2
2 22
)
0.
:
U
1)
V
0, .
0
U
2)
V
– 3)
p
q
( )
2
.
U
,
2V U
( )
( )
1, 2
U. .
0,
0, 0,
1 0, V U
V2 U2
1
-
184
§ 5.5. P 5.5.1.
-
Q x
*
y
*
Qˆ
e
x
1
y
2
Qˆ
,
e
11
12
21
22
. , ,
det Q
5.1.3
1,
e
§ 5.4
det Q
1.
e
-
, .
-
. ,
-
, 5.5.1.
, . .
P
{e1 , e2 }
{e1 , e2 }
S .
-
5.1.1,
-
S Q
1
T
S , {e1 , e2 } ,
Qˆ e
1
T
Qe.
185
5.
{e1 , e2 } , Q,
Qˆ
5.3.3,
1
S
Qˆ
S .
e 1
e
Qˆ e .
5.1.1
S 1
Qˆ
1
( S
5.1.2,
Q e,
Qˆ
S )
e
1
S
1
1
Qˆ
( S
1
T
Qˆ
S
Qˆ
S
S )T
T
T
Qˆ ( S e
( S
1
Qˆ
e
Qˆ e Qˆ
1
)
1
e
1 e
1
( S
e
S
-
T
)T T
Qˆ .
S )T e
T
Qe
e
,
{e1 , e2 } . .
5.5.2.
: 1) 2)
; ;
3)
.
.
Qˆ
1 .
Qˆ e
186
{O, e1 , e2 } .
,
§ 2.3,
a 1
a e
-
b 1
b e
2
-
2 T 1
(a, b)
1
1
2
2
1
a
2
b . e
2
e
a
b,
Qˆ e ,
-
T
(Qˆ a , Qˆ b )
Qˆ a
Qˆ b e
( Qˆ
e
a ) T Qˆ e
T
Qˆ
a
T
e
Qˆ
b
e
e T
e 1
Qˆ
a
b
e
e
Qˆ
e
b
e
e
e
T
a
T
E
b
e
(Qˆ a , Qˆ b )
a e
(a, b)
a, b
( a , b ).
b e
e
, .
187
5.
2 . ,
Qˆ a
(Qˆ a , Qˆ a )
3 .
( a, a )
a
a .
2
, .
.
,
-
, . -
5.5.3.
. .
1 .
5.3.1 5.3.1
5.3.2,
-
5.3.1,
,
{e1 , e2 } . 2 .
5.4.11 ,
( , )
{ 1, 2},
-
188
Aˆ
-
{e1 , e2 } .
1
1
e1
2
; e2
1
;
2
1
,
,
;
1
2
.
2
2
, {e1 , e2 } – 3 .
.
Qˆ ,
,
{e1 , e2 } {e1 , e2 } ,
-
,
-
Qˆ e .
0
1
1
0
(
Aˆ
e1
;
e2
2
2
{e1 , e2 }
e1
e
e2
T
1
e
0
Aˆ
0
Qˆ
2
T e
e1
T
Qˆ
;
e2
T e
Aˆ
1
T e
2
)e
o . o
1
e2
1
0
0
2
Qˆ
T
, e
,
e1 e2
,
189
5.
Aˆ
Qˆ
e
e
1
0
0
2
.
,
-
"(
.
5.3.1).
.
§ 5.6. G 5.6.1.
,
x
y
-
,
xy , : 1) 2)
x ( yz ) ( xy ) z ; x G
xe
3) 1
x x
, xy
yx
,
x 1,
x ,
,
e, x;
ex
-
e.
x, y
G,
. , 5.6.1.
:
1)
, e–
0.
e–
1.
2) ,
190 3) . 4) .
191
6.
6
§ 6.1. ,
1, 2, 3, ..., n .
(
)
{k 1 , k 2 , k 3 ,..., k n } .
,
n!.
ki
, 6.1.1.
kj
-
(
i
),
ki
(k1 , k 2 ,..., k n ) .
A
12
13
...
1n
21
22
23
...
2n
33
...
3n
... n1
32
... n2
... n3
... ...
... nn
-
kj.
{k 1 , k 2 , k 3 ,..., k n } (3,1,4,2) 3 .
,
11
31
,
j
ij
; i, j
[1, n].
192 ( 6.1.2.
)
A
n n
det A , det A
( 1)
( k1 , k 2 , k 3 ,..., k n ) 1k1
2 k2
...
nk n
,
{ k1 , k 2 , k 3 ,..., k n }
{k 1 , k 2 , k 3 ,..., k n } – ,
A . , ,
-
n!. 6.1.2
,
-
. 6.1.2 6.1.1.
1.1.9
1.1.10.
§ 6.2. 6.2.1.
. .
-
B ( 1)
A
T
( m1 , m2 ,..., mn )
1m1 2 m2 ... nmn ,
193
6.
,
,
det A
k mk
T
mk k
( 1)
,
( m1 , m2 ,..., mn ) m11
{ m1 ,m 2 ,..., mn }
m2 2
...
mn n
.
, ( m1 , m2 ,..., mn )
( 1)
1k1
,
2k2
...
nk n
1, 2, 3, ..., n –
,
k1 , k 2 , k 3 ,..., k n –
.
,
: km i
i, i -
(m1 , m 2 ,..., m n ) (k1 , k 2 ,..., k n ) . , mi mj mi
i
mj
km j ,
j,
k mi
i : k mi
k mi
i
j
i, ,
km j
,
mi
mj .
,
det A
,
.
T
( 1)
( k1 , k 2 ,..., k n ) 1k1
2k2
...
nk n
det A .
{ k1 ,k 2 ,..., k n }
.
6.2.1.
6.2.1 .
-
A
, 6.1.2,
-
194 , . 6.2.1. , ik i
11
12
13
1n
21
22
23
2n
31
32
33
3n
n1
n2
n3
nn
-
jk j
,
” -
“ ,
. 6.2.1
. ,
“
”
, 6.1.2, , .
-
, ,
6.2.1.
-
, ,
.
6.2.2.
. .
, .
( 1)
( k1 , k 2 ,..., k n ) 1k1
2k2
...
{ k1 ,k 2 ,..., k n }
, .
nk n
,
195
6.
{k1 , k 2 , ... k i , k i 1 , k i 2 ... , k n } . ki
ki+1,
-
{k1 , k 2 ,...ki 1 , k i 2 ,..., k n } ,
, ,
ki
ki
.
1
, ,
,
-
. ,
, l
,
l+l+1
,
( 1)
2l 1
1,
-
. .
, 6.2.2.
,
.
.
,
,
,
,
,
.
.
.
k 6.2.3
"
, -
" ,k
).
" .
-
196 .
A ik
ik
ik
k
,
i=1,2,...,n.
( k1 ,k 2 ,..., k n )
( 1)
...
ik
1k1
2k 2
...(
1k1
2k 2
...
1k1
( 1)
( k1 , k 2 ,..., k n )
( 1)
( k1 , k 2 ,..., k n )
2 k2
( k1 ,k 2 ,..., k n )
( 1)
-
1k1
2 k2
...
nk n ik
ik
...
ik
... ik
)...
nk n
nk n
...
nk n
.
n!
det A ,
k
det A
det A A
ik
det A
,
-
k
-
A
ik
-
, i=1,2,...,n.
.
6.2.3.
.
6.2.4.
,
-
. .
,
, ,
(
-
6.2.3) ,
. .
6.2.2.
197
6.
n n 6.2.4.
,
det ( A B )
det A det B .
.
C
1 .
A
B .
A ,
C
ij
,
B pq .
kl
-
n
,
5.1.1,
pq
pj
jq
j 1
det C
det
11 11
12
21
21 11
22
21
... ...
1n
n1
2n
n1
nn
n1
... n1 11
n2
21
...
... ... ... ...
... ... ... ...
11 1n 21 1n
n1 1n
1n
nn
2n
nn
nn
nn
-
1, 2, 3, ..., n , .
[i1 , i 2 , i3 ,..., i n ] . (
6.2.3)
det C 1i1
i11 [ i1 , i2 ,...,in ]
i1 1 [ i1 , i2 ,...,in ]
i2 2
i2 2
...
...
.
in n
in n
det
2i1
1i2 2i2
...
...
ni1
ni2
det A
[ i1 ,i2 ,...,in ]
... ... ... ... .
1in 2in
... nin
198
[i1 , i 2 , i3 ,..., i n ] ( {i1 , i 2 ,..., i n } )
, n
n ,
-
n! .
6.2.2 2 .
,
A
,
{i1 ,i2 ,...,in }
A ,
,
-
, 6.2.2
.
A
{i1 ,i2 ,...,in }
,
,
ik ; k
[1, n] . , ,
{i1 , i2 ,..., in } , A
,
,
{i1 ,i2 ,...,in }
det A
{i1 ,i2 ,...,in }
( 1)
( i1 ,i2 ,...,in )
det A
.
det C ,
3 .
det C
det A
( 1) {i1 ,i2 ,...,in } T
det A det B ,
( i1 ,i2 ,...,in ) i11 i2 2
...
in n
-
199
6.
6.2.1
det ( A B )
det A det B .
.
§ 6.3.
A j1 , j 2 ,..., j k ,
n 1 k
i1 , i 2 ,..., i k n.
-
k,
-
,
6.3.1.
i1 , i 2 ,..., i k
j1 , j 2 ,..., j k , j , j2 ,..., jk
M i 1, i
k-
1 2 ,...,ik
.
n k, 6.3.2.
,
i1 , i 2 ,..., i k j1 , j 2 ,..., j k ,
, 2 ,..., j k M i j1, i, j,..., i 1 2
,
k
j , j2 ,..., jk . 1 2 ,...,ik
M i 1,i
A i
j ij
,
(n 1) (n 1) .
.
,
A A
-
200
A 6.3.3.
-
j
Mi
ij
.
A –
6.1.2 –
(n 1)!
,
,
ij
.
det A
ij
Dij
.
Dij
6.3.4.
ij
.
,
6.1.2 n
n
det A
ij
Dij
j 1
kj
Dkj
j
[1, n]
i
[1, n], (6.3.1)
k 1
-
, ,
Dij
6.3.1.
( 1)i
j
j
Mi .
.
1 .
6.1.2
det A
11
( 1)
(1,k 2 , k3 ,..., k n ) 2k2
...
nk n
3 k3
...
nk n
,
{1, k 2 , k 3 ,..., k n }
D11
( 1)
( k 2 ,..., k n ) 2k 2
,
{ k 2 ,..., k n }
,
(1, k 2 , k 3 ,..., k n )
(k 2 , k 3 ,..., k n ) ,
201
6.
D11
-
A
n 1,
-
.
D11
,
1 M 1.
A ,
2 .
A
,
i
i 1
, j
,
A ( 1) i
det A
1 j 1
det A
( 1) i
j
det A . (
,
,
6.2.3)
(6.3.1)
Dij .
( 1) i
Dij ,
j
-
D11 .
,
ij
A ,
ij
j
Mi
-
j 1
,
.
3 .
-
ij
1
M 1.
4 . j
Mi
M Dij
.
1 1
D11
( 1) i j Dij , j
( 1) i j M i .
-
202 i6.3.1.
n
( 1) k
det A
i
i ki
Mk
k 1 n
i
( 1) k i M ki M k .
det A k 1
A 6.3.2. n ij
Dis
,
js
i 1
det A (
js
1, j 0 , j
s – s
-
. § 2.2).
.
6.3.4
det A
1j
D1 j
j
2j
s. 1 j D1s
D2 j j s
2j
D2 s
nj
Dnj ,
.
...
nj
Dns
s ,
.
s
j 6.2.2.
.
203
6.
A
,
6.3.2.
1
A
( 1) i j M
i j
ij
; i, j [1, n] .
.
A ij
ij
A
; i, j
B ,
[1, n] .
B ,
pq
–
,
5.1.1
6.3.2, n pq
j
( 1) j q M q
n pj
n
pj
jq
j 1
pj
pq
Dqj
j 1
j 1
1
B
1
pq
; i, j
A
[1, n] .
1.1.4
A
B
B
,
A
E ,
5.1.2
B
A
5.1.1, 1
.
.
(5.1.1).
I
i1 i2 ... ik ,
J
j1
j2
...
6.3.1,
jk ,
204
j1 , j2 ,..., jk
6.3.3 ).
( 1) I
det A
J
j1 , j 2 ,..., j k
2 ,..., j k M i1j1,i,2j,..., i k M i1 ,i 2 ,..., ik .
{i1 ,i2 ,..., i k }
,
-
i1 , i 2 ,..., i k . n6.3.1.
x a det a
a ... a a ... a x ... a . ... ... ... ... ... a a a ... x
n
.
1 .
:
a x a
,
x
a (n 1) .
-
, ( 6.2.4
n
(x
, .
6.2.3):
1
1
1
...
a
x
a
... a
1
a (n 1)) det a a x ... a . ... ... ... ... ... a
2 .
a
, ,
a
...
x ,
a,
-
205
6.
1 0 (x
n
1 x
a
1
...
1
0
...
0 0
a(n 1)) det 0
0
...
...
a ... ... ...
0
0
0
x
.
...
... x
a
n 1
3 . 6.3.1 ,
(x
n
a) n
a (n 1))( x
1
.
§ 6.4. n
n
-
: 11
1
12
2
...
1n
n
1
,
21
1
22
2
...
2n
n
2
, (6.4.1)
............................................... n1
1
n2
2
...
nn
n
n
,
n ji
i
j
; j
[1, n]
-
i 1
A ji
6.4.1.
x ,
b ,
A x
b –
i
{ 1,
2
,...,
n
j
.
} , -
.
206
6.4.1
(6.4.1) -
, ).
det A
,
i i
i
0
; i 1,2,..., n ,
–
,
A
-
i
-
b :
i
det
11
12
21
22
...
... 1 ... ... 2 ... ... ... ... ... n ...
...
n1
n2
1n 2n
...
.
nn
i.
1 .
, 1
x
(6.4.1)
2
... n
n ji i 1
i
j
; j
[1, n] .
,
-
207
6.
j [1, n] D jk
-
j, n
n
n
D jk ( j 1
ji
i
)
i 1
j
D jk
k
[1, n] .
j 1
(
-
) n
:
n
n
( i 1
ji D jk )
i
j
j 1
D jk .
j 1 ik
6.3.2),
,
(
, n
n j D jk
ik
k
i
k
k
,
i 1
j 1 k
,k
k
k k
,
[1, n] . , k
,
,
[1, n] .
(6.4.2)
, k
k
0
,k
[1, n]
,
,
(6.4.2), 2 .
.
,
{
k i
,i
[1, n] } .
,
i
(6.4.1):
208 n
1
i
n
n ji (
ji i 1
i 1
1
n
1
k Dki ) k 1
n k(
k 1
ji
Dki )
i 1
n kj
k
j
, j
[1, n].
k 1
6.3.2. .
§ 6.5.
A
m n. min{ m, n} .
1 k
A k
, k. k
k
, ,
(k
k,
-
1) k (c .
6.3.1). ,
6.5.1.
A ,
-
rg A . , ,
6.5.2.
.
( 6.5.3.
) ,
,
.
209
6.
nm
:
11
12
21
a1
22
; a2
...
...
m1
1n 2n
; ... ; an
...
m2
mn
0
1 2
b
...
; o
0 ...
.
0
m
(
)
,
,
-
b
,
a1 , a 2 , ... , a n
,
1
,
2
, ... ,
n
,
n
b
,
i
ai .
i 1
(
)
6.5.1
(
)
-
.
). .
r.
1 . ,
A . i j .
-
210
...
1r
1j
...
...
...
r1
...
rr
rj
i1
...
ir
ij
11
...
det
,
r 1
r. 2 .
, i1
M
D1
i2
D2
...
Dr
ir
0 –
M
ij
D1 ,..., Dr –
, ,
,
ij
1
s
i1
2
0, i.
...
i2
Ds , s M
r
ir
-
,
i.
[1, r ]
.
a1 , a 2 ,..., a n
6.5.4.
-
, 1
n
,
2
,...,
n,
,
n i
o , (
ai
i 1
i
0 ).
i 1
( 6.5.1.
)
,
, . 1.4.1.
.
6.5.2.
,
.
211
6. .
k
,
-
, ,
: 11
12
21 1
22 2
... m1
...
...
k
m2
1k
0
2k
0
...
...
.
0
mk
,
11 21 1
1, k 1
1k
...
... m1
2k k
...
1n
2,k 1
0
...
...
0
m ,k 1
mk
2n
... mn
. .
, 6.5.2.
,
( .
.
, n. 6.5.2 .
.
)
-
212 :
.
6.5.1 . n 1
an
,
ai .
i
-
i 1
i 1,2,
(
,n 1) i
.
i
an
, ,
-
. .
6.5.3
.
). .
1 .
, .
r
0.
,
r r
-
. ,
, .
2 .
k
r
, .
A
-
R
.
A .
r,
A
-
213
6.
, R
r
A
k
,
,
,
A
6.5.2.
.
§ 6.6.
m n m
n
-
:
... 1n n 1, ... 2 n n 21 1 22 2 2, .............................................. ... mn n m1 1 m2 2 m, 11 1
12
2
(6.6.1)
n ji
i
,
j
j
[1, m] ,
i 1
A m n
x ji
i
,i
[1, n] ,
b ,
A x
, j
,j
-
[1, m] . { 10 ,
6.6.1.
b
0 2
,...,
(6.6.1), .
-
0 n
}
214 0 1 0 2
x0
.
... 0 n
(6.6.1) (6.6.1). (6.6.1) ,
6.6.2.
,
-
–
.
6.6.3.
A
11
12
...
1n
21
22
...
2n
...
...
m1
m2
... ...
-
... mn
(6.6.1),
A b
11
12
...
1n
1
21
22
...
2n
2
...
...
m1
m2
... ...
–
... mn
m
. (6.6.1)
6.6.4.
,
0 j
j
– .
[1, m] , -
215
6.
(6.6.1) 6.6.1
,
, –
.
). .
(6.6.1) {
1
,
2
,...,
n
}, -
: 1
a1
2
a2
... T
ai
1i
2i
an
n
mi
,i
b ,
[1, n] . -
, , .
,
) rg A
6.5.3 (
rg A b .
.
r.
, ,
-
6.5.1 ( r
b
)
i
ai ,
-
i 1 r
b
n i
i 1
2 ,...,
r ,0,...,0} ,
0 ai . i r 1
,
{ 1,
ai
(6.6.1)
.
216 . 6.6.1.
Ai x
Bi y
Ci
0, i
[1, n] ,
,
rg
A1
B1
A2
B2
...
...
An
Bn
rg
A1
B1
C1
A2
B2
C2
...
...
...
An
Bn
Cn
.
§ 6.7. § 6.6 (6.6.1)
, , .
(6.6.1)
, . (6.6.1) .
6.7.1.
(6.6.1) . .
xi
i 1 i 2
...
,
i
[1, k ] –
i n
,
A xi
o
i
[1, k ] .
217
6. k
y
xi .
i i 1
n
A y
A
i
xi
i 1
n i(
A xi )
o .
i 1
.
6.7.2.
-
(6.6.1) (6.6.1). .
x –
,
y –
,
A
x
o ,
A
y
b . -
A ( x
y )
A x
A y
o
b
b .
.
6.7.3.
(6.6.1) (6.6.1). .
x
y –
A x A( x
b , A y y )
.
,
A x
b. A y
-
b
b
o .
218 1 .
6.7.1–6.7.3.
,
,
-
. 2 .
, ,
,
, ,
,
.
3 .
,
,
, ,
6.7.1,
-
,
-
6.5.4.
n
(6.6.1) 6.7.1.
rg A
. .
1 .
(6.6.1)
... 1n n 1 21 1 22 2 ... 2n n 2 ............................................... 11 1
m1 1
12
2
m2
2
...
mn
,
r
A|b , .
min{ m, n} ,
n
m
219
6.
6.5.1 (
m r
)
-
r , ,
,
, r
. , 1,
r
1
,...,
,
n.
(
r
),
r 1
(
,
11 1 21 1
r
2 ,...,
12 22
2 2
... ...
,
,...,
n
,
-
–
).
1r
r
1
1r 1
r 1
...
2r
r
2
2r 1
r 1
...
1n
n
2n
, n
,
...................................................................................... r1 1
r2
2
...
1 ,...,
r 1
(
rr
r
n
r
r ,r 1
r 1
...
r ,n
n
. -
n r
6.4.1) : j n r 11
1 det M
j
...
1
1, r k
k
...
1r
...
...
...
... ,
n r r1
...
r ,r k
r
k
k 1
j
...
k 1
[1, r ] , M
–
.
...
rr
(6.7.1)
220 2 .
, k
(6.7.1),
0 ; k [1, n r ] ,
-
(6.6.1). . (
-
6.2.3) :
n r j
jk
k
, (6.7.2)
k 1
j
[1, r ] ;
r i
... ... ...
11
1 det ... M
jk
r1
j
[1, r ] , k
i
,i
[1, n r ] ,
1,r k
... r ,r k
... ... ...
1r
... , rr
[1, n r ] . j
,
(6.7.2)
1
11
12
1, n r
r 1
2
21
22
2,n r
r 2
r
r1
r2
r ,n r
n
.
(6.7.3)
221
6. 1
11
12
1, n r
2
21
22
2,n r 1
r1
r
3 .
1
0
0
r 2
0
1
0
n
0
0
1
2
{ 11 ,
1 2 1
{ 12 ,
2 2
...
3
2 r
2
3
,
n r
0,
n r
1 r
, ... , 0,
, ... ,
2
r ,n r
r 1
1,
1
r2
, 1, 0, ... , 0} . 1, 3 ... 2
n r
0
, 0 , 1, ... , 0} . ,
-
, 1
{
n r 1
,
n r 2
...
, ... ,
n r r
n r 1
0,
n r
1
, 0, 0, ... , 1} . .
4 .
n r
,
(6.6.1) .
,
, 1 1 2 1
1 2 2 2
...
1 r 2 r
...
...
...
...
... ... ... ...
n r 1
n r 2
...
n r r
0
...
1
0 ... 0
0
1 ... 0 0 ... 1
.
(6.7.4)
222 ,
,
,
,
n r,
,
,
,
,
n r,
,
n r,
. .
(6.6.1)
6.7.1.
n rg A
,
n –
(6.6.1),
A –
(6.6.1), .
(6.7.4)
-
. (6.6.1) -
6.7.2.
, . .
{ 1,
2
,...,
(6.6.1).
... ...
} (n r 1) n
n
0
... ...
0
0
1
...
0 , (6.7.5)
...
...
...
... ...
n r r
0
0
...
1 1 1 2 1
2 1 2 2 2
...
r 1 r 2 r
...
...
...
n r 1
n r 2
...
r 1
r 2
1
n
1
223
6.
,
,
n r.
,
,
r
,
(
n r
(6.7.3)) , ,
.
, (6.7.5),
r n r. n r, , ,
n r.
,
6.5.1 ( r
(6.7.5) , , (6.6.1)
,
),
-
, -
,
2
1 1 1 2
2 1 2 2
n r 1 n r 2
...
...
...
...
r
1 r
2 r
1
1 r 1 r 2
...
1 0 ... 0
n
i
i
[1, n r ] –
2
0 1
...
n r r n r
... 1
... 0 .
.
(6.6.1) 6.7.1.
0 0
,
224
2
1 1 1 2
2 1 2 2
...
...
...
r
1 r
2 r
1
1 r 1 r 2
...
2
1 0
n
...
... ...
0 1
0 r 0 r 1 0 r 2
n r r n r
0 0
,
...
... 1
... 0
... 0
0 1 0 2
n r 1 n r 2
0 n
0 1 0 2
... 0 r 0 r 1 0 r 2
... 0 n
(6.6.1),
i
i
[1, n r ] –
. .
x0 –
(
,
, (6.6.1),
.
x
x0 .
y x
.
x –
6.6.3 (6.6.1)
y
)
y
x0 .
-
225
6.
6.7.1
6.7.2
n m
(6.6.1) 6.7.2.
,
rg A
n.
, (6.6.1)
det A
, -
,
0.
,
,
(6.6.1) ,
6.7.3 ).
y
1
2
,
T
...
-
m
11
1
21
2
...
m1
m
0,
12
1
22
2
...
m2
m
0,
............................................... 1n
1
2n
...
2
A
mn
T
y
m
0
o )
-
m i
0
i
-
i 1
b
T
y
0 ).
.
(6.6.1)
x
,
-
b
A
x .
226
b A
b
T T
y
T
y
,
o .
( A x )T y
y
x
T
A
T
y
x
T
o
0.
.
b
T
y
0
A
T
y A
o . T
y
-
o T
A b
T
o,
y
0
y
,
.
6.7.1
,
6.7.2,
m rg A
T
A m rg b
T
rg A
T
A rg b
T
,
-
rg A
,
rg A | b ,
6.6.1
A
x
b .
.
“
”(
.
10.6.4
10.6.5).
-
227
6.
§ 6.8.
. 6.7.3 ,
,
6.7.4
, -
,
. ( )
. ,
(6.6.1), .
,
, . “
” (
i
.
,
0
ij
j ),
.
m n (n
m) ,
.
a11
a12
a13
... a1, m
2
a1, m
1
a1, m
a1, m
1
...
a1n
0
a 22
a 23
... a 2 ,m
2
a 2 ,m
1
a 2 ,m
a 2 ,m
1
...
a 2n
0 ...
0 ...
a 33 ...
... a 3, m ... ...
a 3, m ...
a 3, m ...
... ...
a3n ...
0
0
0
...
0
0
0
0
...
0
2
a 3, m ... am
1
1,m 1
0
am
1,m
a mm
am
1
1,m 1
a m, m
1
... a m ...
:
1, n
a mn
228 -
(
); (
-
); -
(
,
-
); -
( );
-
,
-
). ( )
. , . -
i
:
A
j
m n S
n n,
1
,
n
,
E
i
j-
; -
A
i
A
0
S
m m )
i ;
2
,
-
E (
-
229
6.
-
i
S
A
j
m n,
3
-
m
E
-
, ,
i
j
(
A ).
i (
.
S S
8.4.3)
A , rg ( S det S
-
A )
1 , det S
1
,
-
rg A . 0
2
det S
1,
3
A
. , ,
6.8.1.
, .
A 6.8.2.
S A ,
S
A
T
A ,
.
230
.
,
S
,
,
A
.
S 1
A
E
E
,
S
A
-
A
E
A
1
S
A
1
A
1
A
S
E .
,
A
1
-
B
A| B (
,
B A
A ),
.
-
B . . : 6.8.1. 1
3
1
2
2
2 2
5
1
4
2
2 3
3
4
3
4
3 3
2 3
4 4
5
3 6
5 5 5
7, 2, 23, 12.
231
6. .
1 .
1 3 0 5
1 2 1 4
1 1 2 3
1 1 2 3
1 7 3 2 . 6 23 1 12
2 .
. )
, ,
.
,
-
, ,
, ,
(
3 ),
.
. -
:
5)
( ,
,
1
: .
1 0 0 0
1 1 1 1
1 2 2 2
1 2 2 2
1 6 6 6
7 23 ; 23 23
) , ,
. –
.
-
232
1 0 0 0
1 1 0 0
1 2 0 0
1 2 0 0
1 6 0 0
7 23 ; 0 0
)
, ,
-
, . 3 . ,
. .
)
,
, 2(
-
6.6.1
); ) 6.7.1 n
rg A
5 2
3
-
. 4 . , . , , – 1
,
: 2
7
2
23
3
2
3
4
2
4
5
,
6 5.
(6.8.1)
233
6.
( 1) , . (6.8.1)
-
16 23 0 . 0 0
,
1
1
2
0
2
0
2
7,
2
23.
3
2
4
2
3
4
5
6
, 5
,
6.7.1.
1 2 1 0 0 1
2
1,
2
2.
234
1 2 0 1 0
5 6 0 0 1
–
-
. : 1
1
1
5
16
2
2
2
6
23
3
1
1
2
0
3
0
0 ;
4
0
1
0
0
5
0
0
1
0
1,
2,
3.
,
,
:
, , ,
.
235
7.
7
§ 7.1. x, y, z ,
, 7.1.1.
8
,
, -
,
x, y
1 . ,
x
y,
-
,
) x
y y x; ) x ( y z) ( x
y) z ; o,
)
x
,
x o x
)
-
x;
( x) , x ( x) o .
, 8
(x y) x, y
« « .
x
y » (x
-
y)
x
y»
236
x
2 .
, -
x , ) 1x x ; ) ( )x
:
( x) .
3 . : ) (
)x ) ( x y)
.
1 .
“
x x
x; y x, y , .
;
” .
2 .
-
, ( 9
7.1.1.
1 .
. § 5.6).
: .
2 .
.
3 .
n
.
4 .
, n.
m n.
5 .
9
,
.
237
7.
6 . C[a,b] – [a,b].
,
7 .
m n
.
,
-
7.1.1.
(n, r )
, .
,
,
.
,
,
-
,
7.1.2.
.
7.1.3.
R ? .
. 1 .
“
”
.
, . 2 .
“
”
,
” , , “1”.
“ -
238 7.1.1.
. .
o1 ,
1
)
o2 .
7.1.1
,
o1 o1
o2
o1
o2
o1
o2 .
o2 . .
x
0x
7.1.2.
-
o.
.
x = 1x = (0 + 1) x = 0 x + 1x = 0 x + x. x = 0x + x x, , 0x o .
y,
.
7.1.3.
. .
x
y1
y2 . ,
x
y1
o
x
y2 o . y2 ,
,
1
)
-
239
7.
y2
(x
y1 )
y2 -
.
,
,
y 2 ( x y1 ) y 2 y1 .
( y2
x)
y1
o
y1
y1 .
.
x 7.1.4.
( 1) x . .
7.1.2– 7.1.3
o = 0 x = (1 1) x = 1x + ( 1) x = x + ( 1) x . ,
x
( 1) x . .
§ 7.2
, n
1 . 7.2.1.
i
xi
i 1
x1, x 2 ,..., x n
-
. 2 .
x1 , x 2 ,..., x n
,
1
,
2
,...,
n
,
-
n
,
,
i i 1
xi
o.
240
x1 , x 2 ,..., x n
3 .
,
-
n i
xi
o
,
i 1 1
...
2
0.
n
7.2.1.
,
-
, . .
“
”
x1 , x 2 ,..., x n
7.2.2.
1.4.1, ”.
“
,
-
-
x1 , x 2 ,..., x n . .
,
k
n
x1 , x 2 ,..., x n . k 1
,
2
,...,
k
,
,
i
xi
o.
i 1 k
n i xi
i 1
0 xi
o,
i k 1
x1 , x 2 ,..., x n . .
-
241
7.
7.2.2.
n
,
1 . 2 .
; ,
n 1
,
n
,
-
. n-
7.2.3.
n
,
,
n
.
n
-
n n
dim(
). n
7.2.1.
. . n
{g1 , g 2 ,..., g n }
x.
,
-
{g1 , g 2 ,..., g n , x}
, 7.2.1
x
-
g1 , g 2 ,..., g n . . . n
x
n i gi
i 1
x
i
gi .
,
i 1 n
( i 1
i
i
)gi
o,
,
242 ,
g1 , g 2 ,..., g n
. .
.
,
, .
,
. -
7.2.1 . 7 . 2. 1
-
-
2 -
.
3
, .
n n-
nc
1 0 0 ; ... 0
0 1 0 ; ... 0
0 0 1 ; ... ... 0
0 0 0 . ... 1
243
7.
n+1
n+1
,
P1 ( ) 1 ; P2 ( ) 2
P3 ( )
n
; 3
; P4 ( )
;
... ; n 1
Pn ( ) nm
; Pn 1 ( )
n
.
nm
m n, m n
,
, 10
1.
.
, [a,b]
n r .
m n r
10
n ,
,
{ 1, cos , cos 2 , 7.2.2.
, cos n } ,
n 1
244
§ 7.3.
, 7.3.1.
,
,
x, y
: 1) x
y
,
x
2)
.
7.3.1
:
, ,
, -
, . 1 .
,
7.3.1.
, ,
.
2 .
, -
n,
C[ , ] . 3 .
n n
r
n r. 4 . : ) )
; , .
245
7. 1
7.3.2.
-
2
. 1 .
1
x
x
x
1
2
2
,
.
-
1 1
2
.
2 .
1
1
-
2
x
,
-
.
2 1
-
2
2.
1
3 .
1
2
x1 x1
,
x2
1
x2 2.
1
-
2
2.
1
4 .
1
x1 x2
-
2
,
x2
, 2
1 1
,
{o} .
2 2.
x1
-
2
1
246 ,
1
7.3.1.
.
2 1
7.3.2.
dim(
1
2
2
) dim(
1
) dim(
2
) dim(
1
2
).
.
1 .
1
2
{g1 , g 2 ,..., g k }
k.
-
{g1, g 2 ,..., g l } {g1, g 2 ,..., g m } x
1
2. 1
2
{g1, g 2 ,..., g k , g1, g 2 ,..., g l , g1, g 2 ,..., g m } . 2 .
,
{g1, g 2 ,..., g k , g1, g 2 ,..., g l , g1, g 2 ,..., g m }
-
. ,
,
-
: m
k
l i gi
jgj
p
,
o.
(7.3.1)
p 1
j 1
i 1
gp
~ x
m pgp p 1
2
,
247
7.
,
,
~ x
k
l
m
(
pgp
jg j)
i gi
~ x
,
1.
j 1
i 1
p 1
1
,
2
,
-
(7.3.1)
0 , i [1, l ] ;
i
0 , p [1, m] .
p
{g1, g 2 ,..., g k } j
–
,
0, j [1, k ]
,
(7.3.1),
.
,
{g1, g 2 ,..., g k , g1, g 2 ,..., g l , g1, g 2 ,..., g m } –
.
3 .
2
,
{g1, g 2 ,..., g k , g1, g 2 ,..., g l , g1, g 2 ,..., g m } 2.
1 1
2
dim(
1
2
)
l
k
dim(
m 1
(k
l ) (k
) dim(
2
m) k
) dim(
1
2
.
7.3.1.
dim(
1
2
x x1
x2
,
)
dim( (
x1
1
x2
1
1
) dim(
2
) 2,
,
{g1 , g 2 ,..., g l , g1 , g 2 ,..., g m } 1
2
.
2
)
).
248
7.3.3.
{ x1, x 2 ,..., xk }
-
L {x1 , x 2 ,..., x k } . ,
n,
-
7.3.2.
{1, ,
2
,..., n } C[ , ] .
-
{x1 , x 2 ,..., x k } L{x1, x2 ,..., xk } ,
, -
k
x
i
xi
i 1
, 7.3.3.
L{x1, x2 ,..., xk } , m ,
m –
{x1, x 2 ,..., xk } . .
1 .
,
-
k
x
i i 1
i
)
xi (
,
249
7.
7.1.1,
.
2 .
{x1 , x2 ,..., xk }
m
k.
,
x1, x 2 ,..., x m . m
xj
ji
xi ;
j
[m 1, k ]
i 1
-
x1 , x 2 ,..., x m . 3 .
l (l
,
m) .
y1 , y 2 ,..., y l ,
l ,
-
x1, x 2 ,..., x m , m
yj
ji
xi ;
j
[1, l ].
i 1
y1 , y 2 ,..., y l : l
l
m
j yj j 1
m
j j 1
i 1
l
(
ji x i i 1
ji
x1 , x2 ,..., x m
) xi
o. , -
i l ji j 1
j
j 1
j
0, i
[1, m].
250 (
6.7.1)
l rg
l m
ji
0
,
rg
,
,
m.
ji
m
,
l
,
l rg
l m 1,
ji
y1 , y 2 , ..., y l ,
o.
.
, 7.3.4.
x x0 ,
x0 ,
x – ,
( ) .
-
.
1 . .
dim( )
2 .
k,
k
-
. ,
x
7.3.1.
,
z –
x (1 .
y -
)y ,
251
7.
§ 7.4.
1
7.4.1.
,
2
,...,
n
n
x
i
gi
i 1
(
)
-
n
x {g1 , g 2 ,..., g n } . ,
x
7.2.1 n
-
{g1 , g 2 ,..., g n }
n-
,
x
-
{g1 , g 2 ,..., g n } : 1
x
2 g
...
.
n
n
n
.
n
” {g1 , g 2 ,..., g n }
:“
{g1 , g 2 ,..., g n } n
x: x
n i
i 1
gi
x
i i 1
gi .
“
”
252 ,
,
“
“
”
-
”: n
gj
ij
gi ; j
[1, n].
(7.4.1)
i 1
S , j
( j [1, n])
7.4.2.
-
ij
“
”
“
-
”,
{g1, g 2 ,..., g n }
{g1, g 2 ,..., g n } .
, 1.8.2 1
7.4.1.
,
2
,...,
n
1
,
2
,...,
-
n
n i
j,
ij
i [1, n] ,
-
j 1
,
ij
–
S . .
(7.4.1) n
n i gi
n
x
i 1
j 1 n i
ij j 1
j
)g i
o.
n ij g i
j j 1
n
( i 1
n
jgj
i 1
n
( i 1
ij j 1
j
)gi ,
253
7.
( ,
)
,
.
, n i
ij
i
j
[1, n] .
j 1
.
,
,
“ “
”
” T
S , “
-
”
“
”
.
x
,
x
g
g
-
(7.4.1)
-
, n i1
i [1, n] ,
j1 ,
ij j 1
x
S
g
x
g
(
. §5.1).
,
g1
g1
g2
S
...
T
gn g1
g2
... g n
g2 ... gn
g1
g2
... g n S .
, .
254 n
n
x
i gi
y
i
i 1
gi ,
i 1
: 1 .
n
: n
n i gi
,
x
y
i
i 1
gi ,
i 1
x
y
x
y
g
g
.
n
: x
2 .
y
(
i
i
)gi ,
i 1
x
y
3 .
x
g
y g.
g
: n
n
x
i gi i 1
(
i
)gi ,
i 1
x
g
x
g
.
,
-
( ), .
§ 7.5. :
P2 ( )
,
2, .
255
7.
:
(
2 1
(
1
1 2)
1
( 2
(
2
) (
2
2
2)
1
)
(
2
(
2
2)
1
) (
)
)
(
,
) 2.
: 1
2
1
2
1
2
1
2
1
2
1
2
;
.
,
,
,
,
-
, . . 1
7.5.1.
2
,
Fˆ : x, y
1
2
,
,
1
1 . Fˆ ( x
Fˆx Fˆx.
y)
2 . Fˆ ( x)
F
Fˆy ; -
1
2
.
256
F
, ),
( )
1
(F
2
); )
-
2
(F
1
).
7.5.1 -
1
).
,
2
. .
dim(
1
)
dim(
2
).
,
x
-
1
2,
,
,
1
n
,
dim(
1
)
dim(
.
2
)
2
-
.
m,
1
2
n n
1
2,
.
, -
n
n
1.
m , , .
, n
m.
257
7. 7.1.2.
R (
R
, -
7.1.3) :
x
ln( y )
y
e x ; x R; y R
.
7.5.1
n
n n
,
-
,
n
§§ 6.5–6.7
.
, 7.5.2.
n
,
.
. n
n 6.5.3 (
-
, ).
n
k 7.5.1.
,
,
,
min{n, k } .
,
, 7.5.2.
det S
0.
.
, det S
0,
rg S .
n
258
g1 , g 2 ,..., g n , {g1 , g 2 ,..., g n } – .
, .
,
-
,
S x
g
y
S
g
x
S
g
y
T
-
x
g
S
1
x
g
g
.
,
7.5.3.
S ,
-
.
g1 g2 ...
g1 T
g2
T
gn g1
x
...
g
T
x g,
gn g2
gn
g1
g2
x
gn S ,
S x
g
7.4.1. n
{g1 , g 2 ,..., g n } , n
x
i i 1
gi .
g
259
7.
m 7.5.4.
n n ji
0, j
i
[1, m]
i 1
n
.
.
, 6.7.2
-
n ji
0, j
i
n
[1, m] ,
-
i 1
n 1
2
T
...
n
.
n
,
, –
-
.
7.5.5.
m
n n ji
i
j
, j
[1, m]
i 1
n
.
.
,
7.5.4.
260
g1 , g 2 , g 3
, 7.5.1.
3
x ,
1 3 , g1 1
x
.
:
1
2
1 , g2
1
1
0
3 g3
0 . 1
g1 , g 2 , g 3
1 . 3
-
,
7.2.2), .
-
7.5.1
3
1 2 3 rg 1 1 0
3,
1 0 1 ,
1 2 3 det 1 1 0
4
0.
1 0 1 2 .
x 1,
2,
x
3.
1 3 1
1 g1
1 1
1 1
2 g2
2 2
1 0
3 g3 ,
3 3
0 . 1
261
7.
3 . ,
2
1
2
1
3
1,
3
3,
2
1
1,
3
(
,
–
6.4.1
2, x
1
1,
2
– § 6.8), ,
1. { g1 , g 2 , g 3 },
3
-
2 xg
1 . 1
3
7.5.2.
{g1 , g 2 , g 3 } ,
,
{g1 , g 2 , g 3 } , :
1 g1
1 ,
2 g2
1
0
16 g2
5 9
1 , 22
g3
7 . 8
g3
3
7
0 , g1
3 ,
1
3
262 .
x ,
1 .
x
x
x
, {g1 , g 2 , g 3 }
:
.
{g1 , g 2 , g 3}
(
7.4.2
x
G
7.4.1)
x
x
G
F
x
,
F
-
g1 , g 2 , g 3 g1 , g 2 , g3 , G
1
2
3
1
1
0
1
0
1
F
7
16
22
3
5
7 .
3
9
8
S {g1 , g 2 , g 3 }
x
S x
{g1 , g 2 , g 3} ,
x
F
x
.
x 1
G
F
G,
x
x
x
,
x
S
G
G
x
,
, 1
F
.
x
,
5.1.2
S
G
1
F .
,
263
7.
2 . 1
1 2 3 1 1 0 1 0 1 ,
7
16
3
5
7 ,
3
9
8
,
22
, 1
G
§ 6.8,
F ,
2 3 4 1 2 3 . 1 3 4
S
, -
3,
7.5.3.
:
x1 ( )
1 2
x2 ( )
1 8
x3 ( )
10
2
3 3,
6
2
5 3,
5
2
8 3,
y1 ( ) 1 4
.
2
5 3,
y2 ( )
3 2
6
2
3 3,
y3 ( )
4 2
5
2
8 3.
1 .
7.4.1 .
x
x
1 1
2
x2
1
x ,
3 3
-
264 –
2
y
y
1 1
2
(
y2
3
y3 . , 7.5.4).
.
-
1 2 1
2
3
0.
4 3 4
,
1
1
1
2
2 1
1
,
2
1
3
1
4
3
,
0
8 2
10 3
6 5
,
8
–
3
5
,
-
1 2 1
2
3
4 3 4
1
2
3
4
(
1
1 2 1 3
2
1 8 6 5
3
0 10 ) 5 8
0,
265
7. 1 1
,
2
,
3
,
2
1
8
1
2
2
3
2
,
3
, -
4
2
10
,
3
4
0,
6
3
5
4
0,
5
3
8
4
0.
,
,
,
§ 6.8,
4
1
7
1
2 1 3
2
4
0
4 2
0
;
1,
2
,
,
5
, 1
,
2
,
3
,
4
,
,
1
4 7
1
2
2
4
1
5
2
,
4
2:
22 11 ,
0, 0.
3
1 1
9 6
2
14
0,
3
7
2
1
4
2
0. -
266
4
1
7
2
2
4
1
22 11
1 1
0,
3
5
2
9 6
2
14
7
2
1 3 4
, 1
2
) 1
2 6 . 7 2
4
0, 0,
2 6 , 7 2
2
dim(
0,
3
1
,
4
1
,
2
267
8.
8
§ 8.1.
x 8.1.1.
y
.
,
, ,
y
Aˆ x . y
x,
x –
y.
§ 5.2, ,
, ,
.
,
-
,
,
,
, .
8.1.2.
y Ax x, x1 , x2 1 . A( x1
x 2 ) Ax1 ˆ ( x) 2 . A Aˆ x .
,
Ax 2
-
268 1 .
2-
-
8.1.1. 1
a11
a12
1
2
a21
a 22
2
,
1
x
-
2 1
y
.
2
2 . ,
-
.
Pn ( )
3 .
-
.
,
1 ,2
8.1.1.
3
-
.
A, 8.1.2.
x a
a o,
.
.
?
A–
,
269
8.
§ 8.2.
8.2.1.
A B : Aˆ x Bˆ x . Aˆ Bˆ .
x
,
A C,
A
B B,
-
x
Ax
Bx . -
8.2.1.
. .
x, y Cˆ ( x
;
y)
–
,
C
A
B,
y ) Bˆ ( x y) Aˆ y Bˆ x Bˆ y ( Aˆ x Bˆ x) ( Aˆ y Bˆ y ) Cˆ x ( Aˆ Bˆ ) y ( Aˆ Bˆ ) x
A( x Aˆ x
Cˆ y.
.
O x
8.2.2.
,
-
.
270
A,
, 8.2.3.
-
Aˆ ,
,
-
x ˆ ). ( Ax , . :
Aˆ Bˆ ( Aˆ Bˆ ) Aˆ Oˆ
Aˆ ; Cˆ Aˆ ( Bˆ Cˆ ) ; Aˆ ; Aˆ ( Aˆ ) Oˆ .
Bˆ
A 8.2.4.
Aˆ ,
,
x ( Aˆ x ) . 8.2.2.
,
( Aˆ ) (
(
) Aˆ ( Aˆ Bˆ )
) Aˆ ; 1Aˆ Aˆ Aˆ ; Aˆ
Aˆ ;
Bˆ .
.
.
x
:(
,
) Aˆ x Aˆ ((
) x)
Aˆ ( x
x)
Aˆ x
Aˆ x .
271
8.
, 8.2.1.
,
-
. .
7.1.1, 8.2.1–8.2.4
8.2.1, 8.2.2.
A 8.2.5.
B
Aˆ Bˆ ,
,
-
x
-
Aˆ ( Bˆ x ) . 8.2.2.
-
,
Aˆ ( Bˆ Cˆ ) ( Aˆ Bˆ )Cˆ ; Aˆ ( Bˆ Cˆ ) ( Aˆ Bˆ )Cˆ Aˆ Cˆ Bˆ Cˆ .
Aˆ Bˆ Aˆ Cˆ ;
.
.
x, y
,
,
Aˆ Bˆ ( x y ) Aˆ ( Bˆ x Bˆ y ) Aˆ ( Bˆ x) Aˆ ( Bˆ y ) ( Aˆ Bˆ ) x
( Aˆ Bˆ ) y. -
.
( Aˆ ( Bˆ Cˆ )) x ,
Aˆ ( Bˆ Cˆ x )
Aˆ ( Bˆ (Cˆ x)) ,
Aˆ Bˆ (Cˆ x)
Aˆ ( Bˆ (Cˆ x)) ,
,
(( Aˆ Bˆ )Cˆ ) x
. . .
272 :
( ,
,
-
),
AB
B A. AB
8.2.6.
BA
A
B. .
8.2.1.
n k
Pn ( )
:
k k 0
A,
-
B –
,
.
AB
.
Aˆ Pn ( )
k
(
k
)
k 1
k
k
,
k 1
k
.
k 0
n
n
(
k
k 1 k
)
k 1
A( Bˆ Pn ( ))
k
k 1
)
k 0
Bˆ ( Aˆ Pn ( ))
k
n
n
Bˆ Pn ( )
n
d n ( d k 0
d Pn ( ) d
Pn ( )
B A.
d ( d k
n
k
k
k
k
k 1
k 0
n
n k 1 k 0
)
(k 1) k 0
k k
,
k k
273
8.
( Aˆ Bˆ
n
n
Bˆ Aˆ ) Pn ( )
(
k
(k 1)
k
) (
k 0
k
k k
)
k 0
n k
Pn ( ).
k k 0
,
-
.
8.2.1
,
-
Aˆ Bˆ Bˆ Aˆ
-
.
E 8.2.7.
(
)
-
x
, ,
Eˆ x
x
x
. :
Aˆ Eˆ
Eˆ Aˆ
Aˆ Aˆ ,
E. B
8.2.8.
A 1
A ,
AB
BA
E .
f ( ), 8.2.1.
[ , ] f
(k )
( )
Aˆ f
0;k df d
0,1, 2, ... , Bˆ f
f ( )d
-
274 –
. ,
d d
Aˆ Bˆ f
df d d
Bˆ Aˆ f .
f ( )d
Eˆ f
f( )
f( )
f ( a)
Eˆ f .
f( )
1 . .
O Oˆ x
, .
o
x
( Bˆ Oˆ ) x ,
,
B
,
Bˆ (Oˆ x )
o
Bˆ Oˆ
,
x
,
Eˆ
-
B. 2 .
,
,
.(
,
-
5.1.1.) 3 .
-
AB
E BA
E ,
,
-
, n k
Pn ( )
k k 0
B
A
B,
275
8.
, n k
A
-
k k 0 n k 1
.
k k 1
§ 8.3.
n
{g1 , g 2 ,..., g n } m
A § 7.2
n
x
,
{ f1 , f 2 ,..., f m } .
1
n
x
i
gi ,
x
2 g
i 1
m
.
... n
Aˆ x ,
y
-
Aˆ n
y
Aˆ x
n
Aˆ (
i
gi )
i
i 1
Aˆ g i .
i 1
m
-
m
Aˆ g i
ki
fk
i
[1, n] ,
,
k 1 m
y
n
( k 1
ki i 1
i
) fk .
,
276 1
y
,
2 f
–
...
-
m
m
y
,
k
fk .
,
-
k 1
n k
ki
i
;k
[1, m] .
i 1
.
Aˆ :
n
, m
m n
ki
m n, 8.3.1.
.
-
Aˆ g i :
Aˆ
fg
11
12
...
1n
21
22
...
2n
...
...
m1
m2
... ...
...
,
mn
A
{g1, g 2 ,..., g n }
n
{ f1 , f 2 ,..., f m }
m
.
277
8. n k
ki
i
;k
[1, m]
i 1
y
Aˆ
f
fg
x
(8.3.1)
g
: n k1
ki
i1
;k
[1, m] .
i 1
8.3.1.
Aˆ : n m n
m
.
.
,
Aˆ :
n
m
m n.
8.3.1 , 1
11
12
...
1n
1
2
21
22
...
2n
2
... m
Aˆ :
n
m
,
m1
... m2
... ...
... mn
... n
.
.
...
278 1 .
c
,
8.3.1.
Oxy . Aˆ g1
1 g1 0 g 2 0 g 3
Aˆ g 2 Aˆ g
0 g1 1 g 2 0 g 3 ,
A
1 0 0 0 1 0 . 0 0 0
g
0 g1 0 g 2 0 g 3
3
Aˆ :
n
,
n
,
{g1 , g 2 ,..., g n } , § 1.1
n
n.
,
-
§ 5.1
-
. 1 .
Aˆ
:
Aˆ
Bˆ
Bˆ
. g
g
Aˆ Bˆ
8.2.1 n
x Aˆ
x g
Aˆ x
: g
Bˆ x , Bˆ x g
Aˆ
5.1.2,
Bˆ g
Aˆ Bˆ
n
x
g
.
, ,
,
g
Aˆ
Bˆ g
,
. g
279
8.
2 .
A
:
B
A
g
B
g
g
. n
n
Aˆ g i
,
Bˆ g i
ki g k
ki
gk
k 1
k 1
,
( Aˆ Bˆ ) g i
Aˆ g i
n
Bˆ g i n
n ki
gk
ki
k 1
gk
(
k 1
3 .
ki
ki
)g k .
k 1
Aˆ
:
Aˆ
g
g
.
n
Aˆ g i
ki
gk
,
k 1 n
( Aˆ ) g i
Aˆ ( g i )
(
ki
)g k .
k 1
4 .
AB
:
A
g
g
B
g
.
n
( Aˆ Bˆ ) g i
Aˆ ( Bˆ g i )
Aˆ (
ki
gk )
k 1
ˆ ki Ag k k 1
jk g j
ki k 1
n
n
n
n
j 1
ji
gj,
j 1
n ji
jk
ki
k 1
5.1.1.
,
-
280 5 .
1
A
:
1
A
g
g
.
,
. 8.2.8
A 1A
AA
1
3 ,
A
E, ,
1
A
g
A
,
1 g
A
1
-
A
g
g
A
1
E
g
A
g
g
,
-
. -
8.3.1.
n
m
m n.
,
.
8.3.1 n
m
m n. (
,
),
.
.
Aˆ
,
Aˆ :
n
m
–
fg
.
{g1 , g 2 ,..., g n }
n
{g1 , g 2 ,..., g n } ,
7.5.1
281
8. m
G ,
{ f1 , f 2 ,..., f m }
–
{ f1 , f 2 ,..., f m } , Aˆ
F .
Aˆ
fg
.
fg
Aˆ 8.3.2.
{g1, g 2 ,..., g n }
-
fg
{ f1 , f 2 ,..., f m } Aˆ
{g1, g 2 ,..., g n }
fg
{ f1 , f 2 ,..., f m } Aˆ
fg
1
F
Aˆ
fg
G .
.
{g1, g 2 ,..., g n } x – ,
7.3.1
{g1, g 2 ,..., g n }
x
G
g
x
g
Aˆ , y f
1
x
2 g
... n
F
y
2 g
y – -
1
; x
-
... n
,
f
,
282 1
1
2
y
f
;
...
2
y
f
.
...
m
m
-
y
Aˆ
f
fg
x
y
g
Aˆ
f
x fg
g
,
,
y
F
f
F
1
1
y Aˆ
1
F
f
Aˆ
-
x fg
G
x
fg
g
.
g
,
(
Aˆ
fg
F
1
Aˆ
fg
G ) x
o ,
g
x 5.1.2
g
. .
8.3.2.
{g1 , g 2 ,..., g n } Aˆ
g
{g1, g 2 ,..., g n } S
1
Aˆ
g
S .
n
283
8. 8.3.3.
n
.
.
8.3.2
det Aˆ det( S
1
det ( S
g 1
Aˆ
g
S )
Aˆ
(det S
det S det S
S ),
g
1
)(det Aˆ )(det S ) g
1
1
,
det S
0,
,
det Aˆ
det Aˆ . g
g
.
,
,
8.3.2 ,
–
.
§ 8.4. , ,
, .
A x
Ax .
, .
,
-
284 “ ?”
-
A –
,
8.4.1.
-
.
Aˆ x
1 .
x
-
. 2 .
,
n
,
{g1 , g 2 ,..., g n } ,
-
rg Aˆ . g
.
Ax y1, y 2
Ax1 A
x1
x2
,
Aˆ x 2 Aˆ ( x1 x 2 ) Aˆ ( x )
.
.
y1
Ax 2
,
y2 .
:
y1
Aˆ x1 Aˆ x
y2
y .
n
{g1 , g 2 ,..., g n } .
x ,
-
A Aˆ g1 , Aˆ g 2 ,..., Aˆ g n , { Aˆ g1 , Aˆ g 2 ,..., Aˆ g n } .
–
-
285
8.
{ Aˆ g1 , Aˆ g 2 ,..., Aˆ g n } , k. ,
7.4.1, *
rg A
,
k,
7.5.2
,
k.
g
.
A 8.4.1.
n
.
rg Aˆ .
A rg Aˆ
rg Aˆ
n
-
g
8.4.1.
. 8.4.2.
A,
,
dim(
-
).
.
,
8.4.1. .
A 8.4.2.
B .
286 .
AB . 8.4.2
B.
,
AB
,
A, ,
,
-
AB
-
A. .
A
,
8.4.3.
B rg ( A
B )
rg ( B
A )
rg B .
A
B
.
A
B
-
.
det A 8.4.2
0,
1
A , rg ( A
, – rg B
rg ( A
rg( A B ) .
1
A
B )
B )
rg( B A )
rg B ,
rg ( A
rg B .
B ).
287
8.
B
. 1 .
,
A det A rg ( A
B )
B
B
0 rg B
rg ( B
A )
rg B . B
,
B
-
,
B
-
A
,
B
A ,
A ,
B
,
rg B
rg B .
2 .
.
:
1 0
0 0
0 0
0 0
0 1
0 0
.
,
ker Aˆ .
A 8.4.2.
x
,
,
Ax
o.
288 n
rg Aˆ r , ker A dim( ker Aˆ ) n r .
8.4.4.
-
.
,
ker A
r
-
7.4.1.
Aˆ
{g1 , g 2 ,..., g n } Aˆ g
8.4.1 rg A
.
ij
g
.
1
n
x
x
2 g
A
... n
n ij
0; i
j
[1, n].
j 1
,
n ij
0; i
j
[1, n]
j 1
A, ,
n
rg A
n r.
g
.
,
6.7.1,
289
8.
§ 8.1,
, ,
. § 7.5 ,
-
. .
.
Aˆ x ,
y
x
,
y
8.4.3.
(
Aˆ x1
),
x1 y
Aˆ x ,
x
,
x2 ,
x1 , x 2
-
Aˆ x 2
.
y
.
Aˆ x ,
y
x
,
y
8.4.4.
( ),
-
. ,
,
,
,
.
, n
n n
, 8.3.1.
8.4.1 .
A {g1, g 2 ,..., g n } , -
290 8.4.1
Aˆ 8.4.5.
g
n,
-
Aˆ g1 , Aˆ g 2 ,..., Aˆ g n
{g1 , g 2 ,..., g n } ,
A {g1 , g 2 ,..., g n }. , n
Aˆ : –
m
m
ker Aˆ
o .
,
, -
291
8.
, 8.4.5.
,
,
. .
{g1 , g 2 ,..., g n }
1º. n
Aˆ : rg Aˆ Aˆ
fg
m
Aˆ
y
g
rg Aˆ y
6.6.1 (
f
m
y
f
,
m.
rg Aˆ
A
,
.
n.
fg
)
,
A 2º.
fg
-
m.
fg
x
{ f1 , f 2 ,..., f m }
,
6.4.1 (
Aˆ
fg
x
g
o
),
-
f
,
A
.
. .
,
, ,
,
(
.
-
5.2.4). ,
, , . ,
-
292 7.5.1 (
),
-
, .
Pr ,
1 . 8.4.1.
,
,
, 3
1
,
,
, .
,
,
,
-
. ,
-
{O, e1 , e2 , e3 }
,
-
, T
e1
1 1 1 . e
r
,
x r
y , e
r
(r,e ) 2
e
z
e
-
293
8.
x x y z
y 3 y 3 y 3
x x
z z ,
-
z
Pr
Pr e
,
1 1 1 1 1 1 1 . 3 1 1 1 e1
, 1
,
,
8.4.5,
Pr Pr ee
1 1 1 1. 3 Aˆ
2 .
11
12
21
22
11
12
21
22
-
.
,
11
12
0
21
22
0
.
-
294
Aˆ :
3
3
8.4.1.
1 2 3 Aˆ
2 3 4 . 3 5 7 .
, -
. 1
x
.1 .
1
y
2
2
3
-
3
Aˆ x . Aˆ x o ,
y
–
x,
–
Aˆ x
0
1
1
2
2 .
3
1
1
2
2
3
3
0,
2
1
3
2
4
3
0,
3
1
5
2
7
3
0,
,
Aˆ 1 2 , 1 ,
.
,
295
8.
,
-
,
1 2 3
1 2 3
rg 2 3 4
rg 0 1 2
3 5 7
0 0 0
–
2
3
.
Aˆ
2.
,
y
y
,
Aˆ x x
.
-
y
1 2 3
1
1
2 3 4
2
2
3 5 7
3
3
,
, 1,
, 2,
3
. 6.6.1 (
,
, ), .
1 2 3 rg 2 3 4 3 5 7 1 2 3 rg 0 1 2 0 0 0
1 2 3 1
2
1 1
2 2
3
1 2 3 rg 2 3 4 3 5 7
2
296 , 1
2
0,
3
,
, ,
,
A 1
1 2
1
1
1
0
2
0
3
1
y
,
2
,
1 1
,
-
2
3
0.
, 3
3
,
-
.
§ 8.5.
8.5.1.
A, x 1 . 8.5.1.
Oxy , ,
: Aˆ x
-
. . 8.5.1
297
8.
Oz ( .
. 8.5.1).
2 .
-
f( ), ( , )
, n
1
{e 1
,
2
,...,
n
,e
2
, ... , e
–
n
-
},
,
.
A, 8.5.1.
n
11
...
1r
...
...
...
...
rr
r1
-
{g1 , g 2 ,..., g n } ,
1, r 1
...
1n
...
...
...
r ,r 1
...
rn
0
...
0
r 1,r 1
...
r 1,n
...
...
...
...
...
...
0
...
0
n,r 1
...
{g1 , g 2 ,..., g r } A.
,
nn
-
298 .
A
. .
{g1, g 2 ,..., g r }
-
, 8.3.1 . r
,
k
gk
,
k 1 r
Aˆ (
k
gk )
k 1 r
r
ˆ k ( Ag k )
r k
k 1
k 1
r i 1
7.4.1
gi
r
r
(
ik i 1
k ) gi
ik k 1
i
.
gi
i 1
–
,
.
-
. .
-
A,
-
{g1, g 2 ,..., g r } . ,
,
, ,
, .
r
Aˆ g k
-
n ik
i 1
0g i ; k
gi
[1, r ]
i r 1
8.4.5 .
-
.
299
8.
, 8.5.1.
A A 1.
x
.
,
–
A,
,
, y
A A
Aˆ x
,
. -
1
x, y
x
1
A y,
-
A 1. " .
",
f 8.5.2.
A, ,
,
Aˆ f
f . A,
-
f. ,
,
, f
Aˆ f
ker( Aˆ
Eˆ ) .
-
Eˆ ,
300
A,
, n
,
n
{g1 , g 2 ,..., g n } , Aˆ g1 1 g1 ;
Aˆ g 2
2 g2
Aˆ g n
; ... ;
n
gn .
,
Aˆ g k
0 g1
0 g2
k
,
-
0 gn ; k
gk
7.2.1,
[1, n] . ,
8.3.1,
,
, -
A 1
Aˆ
0 ... 0
f
0 2
... 0
... ... ... ...
0 0 , ... n
. ,
A
,
8.5.2.
f ,
f
A2
AA 2
.
-
301
8.
Aˆ f
.
f ,
-
Aˆ
Aˆ 2 f
Aˆ ( Aˆ f )
Aˆ ( f )
2
f.
n n
{g1, g 2 ,..., g n } , n
n
f
-
f
i
gi ,
i 1
Aˆ
A
Aˆ f Aˆ
g
f
f
g
k j
g
. n
f g
,
11 1
12
2
...
1n
n
1
,
21 1
22
2
...
2n
n
2
,
................................................... n1 1
(
n2
11
)
1
21 1
(
22
2
...
nn
n
n
,
12
2
...
1n
n
0,
)
2
...
2n
n
0,
........................................................ n1 1
n2
2
... (
nn
)
n
0.
(8.5.1)
302 , (8.5.1),
, 6.7.2, (8.5.1).
,
: det 11 21
det
...
kj
12
...
1n
...
2n
...
n1
0
kj
22
...
det Aˆ
0.
...
...
n2
,
(8.5.2)
nn
Eˆ
8.5.3.
0
g
-
,
det Aˆ
Eˆ
g
– n
A, 8.5.2.
n
.
.
.
Aˆ Aˆ
,
Eˆ , Eˆ .
, ,
8.3.3,
.
{g1 , g 2 ,..., g n } {g1 , g 2 ,..., g n } det Aˆ .
:
Eˆ
g
det Aˆ
Eˆ
g
.
303
8.
n
-
, 6.1.2
(8.5.2). n
,
: (8.5.2),
-
,
-
(8.5.1) , ,
. n
8.6.1 ,
8.6.2.
, .
,
, ,
f( )
-
e
(
-
–
) ,
df d
-
f.
§ 8.6. n
8.6.1.
.
-
304 .
n
-
, 11
,
, .
.
8.6.1 .
Oxy
,
k .
cos sin
sin cos
cos
-
, . § 5.5).
(
det
-
2
0
i sin .
2 cos
1 0,
k
,
. n
8.6.2.
-
, . .
, (8.5.1)
i, f
u
wi ,
. ,
u
(8.5.1), n
w–
n
11
,
.
,
.
,
-
305
8.
, :
u
Aˆ ((
u
w. i ) w)
w
(
.
Aˆ f w,
Aˆ w
i )w ,
-
f
, –
-
, .
-
ˆf : A Aˆ (u
wi )
(
f . i)(u
w i) ,
A
( Aˆ u ) ( Aˆ w) i
( u
w) ( u
w) i, ,
Aˆ u Aˆ w
u
w,
u
w. A
,
-
, u
Aˆ ( u
Aˆ u
w) (
w,
( u Aˆ w )u (
w) ) w.
( u
w)
.
8.6.1.
A,
-
1
2 2
2
1 2 .
3
2 3
306 :
1 .
A
,
-
. (8.5.1) – (8.5.2).
(
.
1.1.1),
1 det
2 2 3
2 2
1 2
(1
)(
3
2
3
1) 2
2( 2
1
6 6) 2( 4 3 3 ) 2
(
1)(
1).
, –
2 12
1
1
i
i –
2
-
.
2 .
.
1,
1
,
(8.5.2)
2
2 2
1
0
2
2 2
2
0 .
3
2 2
3
0 -
,
12
.
1
1
1
1
1
0
1
0
0
2
0 .
3
0
3.
,
2
3
307
8.
f1 ,
,
-
1,
1
3 .
2
1
0
2
1
3
1
0.
i,
(8.5.1)
1 i
2
2
1
0
2
1 i
2
2
0
3
2
3 i
3
0
13
,
1 i.
,
1
1 i 1 i
1
0
2
1 i
2
2
0
3
2
3 i
3
0
. ,
1 0
13
1 i
1 i
1
0
1 3i
2i
2
0 .
3
0
3.
308 ,
( i) 1 0
1 i 1 i 3 i
2
1
0
2
0 .
3
0
3 i,
3
:
f2
1
2
2
2
3
3 i
4 .
0.
,
,
,
3
f3
1
2
2
2
3
3 i
i,
0.
f2
,
f3
,
2
,
3
f2 f 3 .) A
5 . ,
0 1 , 1
8.6.2 A
-
1
1,
309
8.
,
u
2 2 3
-
0 0 , 1
w
2
1 2
0
2
1
2
3
3
-
0 ;
1,
2.
1
,
1 1
.,
2
,
2
3
3
0
8.4.1).
,
-
8.6.3.
A, ,
-
A. .
Aˆ f1
Aˆ ( f1
f1
Aˆ f 2
f2 .
,
-
f2 ) Aˆ f1
Aˆ f 2
f1
f2
( f1 .
.
f 2 ),
310 ,
, )
,
8.6.1.
, (
,
,
A. Aˆ
8.6.4.
-
Bˆ , Aˆ
Bˆ
.
.
–
-
Aˆ ,
Aˆ f
f
f
.
Bˆ Aˆ f
Bˆ ( f ) , A
Aˆ ( Bˆ f )
( Bˆ f )
f
-
B
. ,
Bˆ f
f
,
B.
– .
, 8.6.5.
-
, . .
. m
f 1 , f 2 , ..., f m
A, .
-
311
8.
,
f 1 , f 2 ,... , f m , f m 1 ,
m 1
. : -
f 1 , f 2 ,... , f m , f m 1 ... 1 f1 2 f2
fm
m
m 1
fm
o , (8.6.1)
1
, m 1
Aˆ (
Aˆ ... 2 f2
f
1 1
f
1
-
0.
1 1
2
2
(8.6.1): m
fm
f 2 ...
f
m 1 m 1 m
m
fm
) m 1
m 1
fm
1
o. (8.6.2)
,
(8.6.1) -
m 1
(8.6.2), 1
(
1
m 1
) f1
2
(
2
m 1
) f2 m
... (
m
m 1
) fm
o.
,
f1 , f 2 ,..., f m
, 1
(8.6.1)
2
... m 1
,
f1, f 2 , ... , f m f 1 , f 2 ,... , f m , f m 1 . .
m
0,
0. -
312 n
A 8.6.1.
)
n
-
, . n
A, 8.6.6.
n
, ,
A,
, -
,
A. .
8.6.5
-
§ 8.5. –
-
8.6.7.
A, k.
0
1 dim(
)
k.
. n
{g1, g 2 ,..., g m , g m 1,..., g n }
m dim(
)
Aˆ g i
Aˆ
.
0
Eˆ
,
g
g i ; i [1, m] , ,
(
. §8.5),
313
8.
Aˆ
Eˆ
g
0 ...
... ... ...
0 ... 0
... ... ...
0
0 ...
0
0 ... 0
0 0 ...
1,m 1 2,m 1
... m ,m 1
0
... 0
... n,m 1
... ... ...
1n 2n
...
... ... ...
.
mn
... nn
,
det Aˆ
Eˆ
(
g
(
Pn m ( ) ,
) m Pn m ( ).
0
0
)
m
k,
k – det Aˆ
0
1 m
Eˆ g .
,
(
).
.
,
-
,
0
k, 8.6.2.
k, ,
1 1 0 1
.
314 :
.
det
1
1 0
)2
(1
1
0,
1
1, 2
2.
k
0 1
1
0
0 0
2
0
1
x
,
0
0.
, -
(m
dim(
) 1 ),
-
1
2.
8.6.2, 8.6.5
8.6.6
, ,
-
, . n
A
-
8.6.8.
A
, . . n
A ,
,
det A
, n
.
,
-
0. -
315
8. 1
11
12
...
1n
1
2
21
22
...
2n
2
...
...
n
...
n1
6.4.1 (
...
...
...
n2
)
...
nn
.
n
,
-
, ,
(
-
), 6.7.1
( ). .
Q 8.6.2.
k
2
k
Q . Q
1
Q
Q
0
-
,
E . n
A
-
8.6.9
.
– ). .
,
Aˆ
n
{ f1 , f 2 ,..., f n } .
Aˆ
n
Aˆ
k f
k k 0
0.
316
Aˆ
f, ,
(
-
.
8.5.2) n
n
k
(
Aˆ f ) f
k k 0
k
k
( Aˆ
f )
f
k 0
n k k 0
( Aˆ f ( Aˆ f ...( Aˆ
f )...))
f
n
n k(
k
f )
k
(
k 0
k
) f
0 f
o.
k 0
n
x
, 5.1.2 n k
Aˆ
k 0
k
,
Oˆ f .
f
,
{g1 , g 2 ,..., g n } , n k
Aˆ
k 0
n
k g
1
( S
Aˆ
k 0 n
1
k( S k 0
S
1
Oˆ
1
Aˆ
k 0
n k
( S
k
S
f
.
Aˆ
f
k f
S S S )
Oˆ g .
1
f
Aˆ S
S )k S ... S
f
1
1
n
(
k k 0
Aˆ
Aˆ k f
f
S )
) S
.
317
8. :
, .
§ 8.7. ,
,
, -
. § 5.2,
, .
x 8.7.1.
-
f (x ) . f (x ) .
,
1 .
,
n
8.7.1. 1 2
,
...
-
n
n i
i
,
i
,i
[1, n] –
-
i 1
. 2 .
,
3 .
[ 1,1]
x( ) , f ( x)
f ( x) | x |.
x(0) –
318 ”
(x) , x( )
”, .
x( ) ,
4 .
[ , ],
-
f ( x)
,
p( ) –
p ( ) x ( )d ,
[ , ]
.
5 . 11
12
21
22
det
11
12
21
22
11
22
12
21
.
f (x ) 8.7.2.
-
(
),
x, y
: 1 . f (x
y)
2 . f ( x) , 8.7.1.
f ( x) f ( x).
f ( y ). 1 ,3
,
4
5 –
2
. n
n
{g1 , g 2 ,..., g n }
n
x
i i 1
gi .
319
8. n
n
f ( x)
f(
f (gi ) , i
f (gi )
i i 1
i 1 i
n
i gi )
[1, n] –
i
i
,
i 1
,
.
n
f (x ) 8.7.1.
-
{g1, g 2 ,..., g n } f
g
-
,
1
2
n
1
2
n
-
n n
f ( x)
i
i
i 1
f ( x)
f
n
x
g
g
.
(
)
8.4.5,
n n
1
-
. -
n
.
{g1 , g 2 ,..., g n }
{g1 , g 2 ,..., g n } ,
-
n
S
ij
,
gj
ij i 1
gi
j
n
[1, n] .
320
x n
n
x
i
gi
i
i 1
gi ,
-
i 1
f (x ) – n
-
n
f ( x)
i
i
i
i 1
.
i
i 1
i
i
.
-
, n i
f ( gi )
n
f(
n
ki g k )
ki f ( g k )
k 1
k 1
k
ki
,
k 1
. n
{g1 , g 2 ,..., g n }
8.7.2.
f f
1
g
,
2
1
g
, ... ,
n
;k
[1, n] ,
{g1 , g 2 ,..., g n }
,
2
, ...,
-
n
n k
i
ik
ik
–
i 1
S – .
f
g
f
g
S ,
,
n
, (
. §7.3).
-
321
8.
(
)
.
) -
, ,
. n
,
{g1 , g 2 ,..., g n }
-
8.7.2.
p( x ) p q
q ( x)
1
g
p p q
1
2
g
1
g
1
g
2
1
...
2
n
...
n
2
...
n
2
...
n
n
,
.
,
§ 8.2,
, 8.7.3.
,
-
. , 8.7.3.
, )
(
-
.
8.7.1 n,
322 n
.
, n
n
-
,
n n
. n
n
,
,
n.
8.7.4.
n
,
n
. n
{r1 , r2 ,..., rn }; ri
i
f
[1, n] .
n
n
f
,
n
r,
i i i 1
f, 1
f
2 r
.
n
-
f
{g1 , g 2 ,..., g n }
n,
, ij
ri
n
gj.
ri ( g j ) ; i, j [1, n] –
{r1 , r2 ,..., rn } rg
,
n
-
323
8. 8.7.3.
n
{g1 , g 2 ,..., g n } – f
T
(
r
n
{r1 , r2 ,..., rn } –
,
rg
)
1
f
rg
8.7.4.
,
T
f
g
T
f
g
r
rg
ij
(
n
{g1 , g 2 ,..., g n }
n
,
n
f (x ) f
r
f
T g
n
, n
)
n n
,
, n
,
n
. n n
, ,
, n
n
-
.
(
n
-
).
{r1 , r2 ,..., rn }
,
.
E ,
1, i j ; i, j [1, n] , 0, i j {g1 , g 2 ,..., g n } {r1 , r2 ,..., rn }
ij
,
. n
,
.
n
n-
, -
.
324 n
x – n
, n
X(
.
f
1 1 1
n
f
,
f2 )
X ( f1 )
2
f ( x)
1 1
2
X ( f2 )
1
X(f)
,
f 2 ( x) ;
2 n
f
, n
X ( x) : n X ( f ( x)) x ;y
,
(
.
-
,
.
-
,
n
) dim (
,
X( f )
n
,
.
)
f (x ) , , , , y X ( f ( x )) –
.
, n
y
n
f1 , f 2
–
7.5.1 dim(
.
, 2
,
f ( x) n
X( f )
,
8.4.1,
X(f)
,
X( f ) –
,
,
) n.
, n
dim (
) n
n
dim( )
n
,
y
.
, n
X ( f ( x)) x
;y
n
n
n
,
-
n n
,
x( f )
f ( x) ; x
n
n
,
; f
n
.
325
9.
9
§ 9.1. 9.1.1.
x
-
y
B( x , y ) , 1) B( x1 x2 , y )
B ( x1 , y )
B( x2 , y )
x1 , x 2 , y 2) B( x, y1
; y2 )
, , B( x, y1 )
B ( x, y 2 )
x, y1 , y 2
;
, ,
, ).
F (x )
1 . 9.1.1.
G( y) ,
,
B( x , y )
F ( x )G ( y ) .
2 .
B( x, y )
K ( , ) x( ) y ( ) d d
(
x( )
K ( , ) y ( )d
K( , )
)d ,
326
:
[ , ]
,
.
3 . .
n n
.
{g1 , g 2 ,..., g n }
-
B( x , y ) . . n
x
,
i
gi
i 1 n
y
j
gj ,
,
9.1.1,
-
j 1 n
B ( x, y )
B(
n
i 1 n
n
i gi , j 1
n j
i 1
j
gj)
j 1 n
B( g i , g j )
i 1 j 1
9.1.2.
i B( g i , n
i
n
jgj)
ij
i
j
.
i 1 j 1
ij
B( g i , g j )
B( x, y ) {g1 , g 2 ,..., g n } ,
B
g
ij
– .
-
327
9. n
{g1 , g 2 ,..., g n } n
n
( x, y )
n ki
k
k1
x
g
x x
B
g
...
y
g
y
g
y
ki
n
n T 1k
i1
k 1 i 1
2
T
n
i
k 1 i 1
1
n
k 1
ki
i1
i 1
11
12
...
1n
1
21
22
...
2n
2
...
...
...
...
...
n1
n2
...
nn
n
, g
–
. .
S – {g1, g 2 ,..., g n }
9.1.1.
{g1, g 2 ,..., g n } ,
B
S
g
T
B
g
S .
.
n
(
. § 7.3) n
gk
ik i 1
gi , k
[1, n] ,
328 n kl
B( g k , g l )
B(
n ik g i ,
jl
i 1 n
gj)
j 1
n ik
jl
B( g i , g j )
i 1 j 1 n
n
n ik
jl
ij
i 1 j 1
n T ki
i 1
ij
jl
j 1
k , l [1, n] . .
det B
9.1.1.
g
det B
g
det 2 S .
.
9.1.1, 6.2.1
(
-
6.2.4).
,
-
. 9.1.2.
. .
8.4.3
-
S . B( x , y ) 9.1.3.
, x
y
B ( x, y )
B ( y, x ) .
-
329
9. 9.1.2.
n
,
-
. .
B( g i , g j )
ij
i, j
B( g j , g i )
[1, n]. .
ij
ji
i, j
ji
,
[1, n] , n
n
n
B ( y, x )
ji
j
i
j 1 i 1 n
n i
ji
j
j 1 i 1
n ij
i
j
B( x, y ).
i 1 j 1
.
§ 9.2. 9.2.1.
x
( x)
B ( x, x) ,
B(x,y) – ,
,
(
-
). ,
.
330 ,
(x)
B( x, y ), x
-
y
(x
y)
B ( x y, x y ) B( x, x ) B( x, y ) B( y, x) B ( y , y ) ( x) 2 B ( x, y ) ( y) , (x
B( x, y )
y)
( x)
( y)
2
.
n
-
9.2.2.
1 2
(x
y)
( x)
( y )) (x).
n
{g1, g 2 ,..., g n } , n
-
n
( x)
ki
k
i
k 1 i 1
1
...
2
n
11
12
...
1n
1
21
22
...
2n
2
...
...
...
...
...
...
nn
n
n1
x
T g
g
x
g
n2
, n
x .
g
x
–
i
gi
i 1
,
, g
9.1.1.
S
T g
S ,
-
331
9.
,
(x )
-
, .
B( x , y ) 1 ( B( x , y) 2
,
B ( y , x )) ,
-
(x ) ,
B( x , y ) ..
, ij ij
ji
ji
2
ij ji
2
–
, i, j
[1, n] ,
. 3
9.2.1.
B1 ( x, y )
1
3
3
1 1
2
2
2
1
2
2 3
2 1
1 2
3
3
3
2
1
3
0
1
1
3
1
2
2
1
0
3
1 3
2
1
3
0
1
3
1
2
1
0
9.1.2
.
3
1
( x)
2 1
3
2 2
4
1
2
2
2
3
.
,
332
B2 ( x , y )
1 1
3
1 3
3
1
2
2
2
1
1
2
2
2
2
3
3
3
2
1
2
1
2
1
1
2
3
1
2
1
1
0
3
1
2
1
2
3
1 ,
1
1
0
,
3
2
2 1
( x)
2 2
3
1
4
1
2
2
1 3
2
2
3,
( x)
1
2
1
2
3
1 .
1
1
0 -
, .
(x ) 9.2.3.
n
{g1 , g 2 , ..., g n } n
( x)
i
2 i
,
i 1 i
i
[1, n] –
.
,
333
9.
,
,
i
0
,i
[1, n]
1,
,
-
.
9.2.1
n
, .
). .
.
(x ) =
n 1
1 .
2 1
11
.
11
,
1
1,
11
2 .
0,
11
0, ,
-
.
,
-
n 1
, n
,
.
,
11
0.
,
ii ii
,i
,i
[2, n]
.
[1, n]
, ,
1
1
2
,
2
1
12
2
,
3
-
0. 3
, ...,
,
n
n,
334
(x ) = 2 12 12 F ( 1 , 2 , 3 ,..., n )
2
2 2
21
F( 1,
,
2
3
,...,
n
),
1
2
.
3 .
, 1:
n
n
( x)
ik
i
k
i 1 k 1 n
n 2 1
11 (
1i
2
1 i 2
n
i)
ik
i
k
,
i 2 k 2
11
, n
k
i
(
k 1 i 1
n
n
n
n
k
)(
k 1
i
)
(
i 1
k k 1
n
n
(
2 k)
1
2 1 n
2
1
k
n
.
n 1i
1
i i 2 k 2
11 1i
1k
ik
i 2 k 2
1k 2 11
i
k
)
)
i
k
11
n 11 (
i
n
(
( x)
)2
k 2
1i
2 i 2
n
k
k 2 i 2
n 2 1
k
n
k
k 2
11 (
1
(
k 2
n
( x)
n
2
k 2 2 1
)2
n 1i
1 i 2
11
n
2 i)
( i 2 k 2
1i
1k
ik 11
)
i
k
.
335
9.
, –
,
1
,
,
-
n k
ki
i
,k
[2, n] .
i 2
4 .
(x ) n 1
11
1i
(
i
1 i 2
),
11
(9.2.1)
n k
ki
i
;
[2, n] ,
k
i 2
,
11
. -
0,
12 11
11
1n
...
11
11
T
22
...
2n
...
...
...
...
0
n2
...
nn
,
,
T
1
(9.2.1) –
T
.
S
11
0
(
.
:
7.5.3),
.
.
-
336 ,
:
,
,
n
. ( )
.
-
( )
, .
{g1, g 2 ,..., g n }
,
{g1, g 2 ,..., g n } S
g
T g
S
-
S .
S
g
(
. § 6.8).
S
,
g
, 6.8.2 ,
.
g
-
S
, .
,
g
-
, , ,
,
-
337
9. g
x
,
g
S
x
–
g
{g1 , g 2 ,..., g n } {g1 , g 2 ,..., g n } ,
. . 3
9.2.1.
( x)
2
2 1
2 2
4
2 3
8
.
2
4
4
1
1
4
1
2
2
1 3
8
2
3
.
(x ) 1 4 . 4
1º. : ; -
,
,
1
0
0
0
1
0
0
0
1
2
5
1
5
3
0 .
1
0
4
338
,
1
0
0
0
1
0 .
0
1
1
2º. ,
-
,
5. . :
93
0
3
3
0
0
0
3
0
5
1
0 .
3
0
4
5
1
1
3º.
, 31. , -
,
93
0
0
3
0
3
0
3
0
5
1
5 .
0
0
3751
5
1
,
3
0
3
5 ;
1 ;
5
5
1
26
26
339
9. 1
3
1
3 3,
2
5
1
2
5 3,
3
5
1
2
26 3 ,
:
( x)
93
2 1
3
2 2
3751
2 3
.
§ 9.3.
, ,
( n
)
.
-
. 9.3.1.
rg .
(x ) n
9.3.1.
-
. .
9.1.2 . . ,
8.4.3
9.1.1 -
. .
340 . 1 . 9.3.2.
(
-
) n
(x )
-
rg
.
2 . (
-
) n
(x )
-
rg
.
3 .
n
(x ) sgn
rg
9.3.2
rg
.
,
(x ) -
,
).
( . .
(x )
1 .
{g1 , g 2 ,..., g n } n
n
(x )
ij i 1 j 1
i
j
-
n
)
341
9.
{g1 , g 2 ,..., g n }
(x )
{g1 , g 2 ,..., g n } ,
:
m
k
( x)
2 i
i
2 i
i
i 1
;
i k 1
m
n,
0, i
i
[1, m]
q
p
( x)
2 i
i
2 i
i
;
i p 1
i 1
q
n,
0, i
i
[1, q ] .
,
ij
ij
{g1, g 2 ,..., g n } {g1, g 2 ,..., g n } {g1, g 2 ,..., g n } , , n
n s
j;
sj
s
[1, n]
s
sj
j
;
s
[1, n] .
j 1
j 1
(9.3.1)
(x )
2 .
{g1 , g 2 ,..., g n }
{g1 , g 2 ,..., g n } k gk
i gi
k 1 k
2 i
i i k 1
i
2 i
i
2 i
i i 1
i
2 i
i p 1
p i
i p 1
2 i
i 1
q
k
:
q
p
m
i 1
jgj j 1
i 1
2 i
i
i 1
n
n
n
x
2 i
m i i k 1
2 i.
(9.3.2)
342 3 .
.
0 i
i
k
,
p,
x
,
-
[1, k ] ;
0 i
i
[ p 1, n] . k p.
n,
,
(9.3.1),
{ 1,
2
,...,
n
}.
,
,
-
6.7.1 ,
x
,
. , i; i
(9.3.2),
[1, m]
0 i
i
,
i
i; i
-
[1, q ] ,
[1, k ] ; 0 i i 0 ; i [1, p].
[ p 1, n]
n
(9.3.1)
-
n s
sj
j
;s
[1, n]
n
-
j 1
,
-
x
, .
k
,
p.
4 .
,
k k 5 .
p.
,
p. 9.3.1 m .
q,
k
m
p q.
343
9.
.
(x )
1 . 9.3.3.
-
( x)
,
x
0
-
.
(x )
2 .
-
( x)
,
x
0
-
.
3 .
(
)
,
.
(x )
, (
( x)
4 .
)
0 ( ( x)
0)
x
, -
,
)
, (
)
.
9.3.3.
n
, (
) ( .
.
9.3.2 (
) -
(
).
344
.
9.3.4
-
n
,
).
, 12
...
1k
21
22
...
2k
...
...
...
...
...
kk
11
det
k1
k2
; k
[1, n] ,
. .
1 .
.
1
k
.
k
,
-
n 1
n 1 n 1 2 i
( x)
.
i 1
2 .
,
n
, .
,
n n:
,
,
-
345
9. n 1 n 1
n 1
( x)
ki
k
2
i
k 1 i 1
kn
k
n
2 n.
nn
k 1
-
( x) ,
(x )
n 1
n 1
,
,
,
,
.
-
,
( x) n 1 k
ki
;
i
k
[1, n 1] ,
i 1 n 1 2 i
( x)
.
i 1
(x ) n 1
n 1 2 i
( x)
2
i 1
in
i
n
nn
2 n
i 1
n 1
( x)
(
2 i
2
in
i
2 in
n
2 n
)
i 1 n 1
n 1
(
2 in
nn
)
2 n
2 i
nn
2 n
,
i 1
i 1
n 1 nn
2 in
nn i 1
;
i
i
in
n
; i [1, n 1] .
346 1
1
0
0
1, n
1
2
0
1
0
2,n
2
, n 1
0
0
0
1
n 1,n
n
0
0
0
0
1
n 1 n
, . 3 .
,
9.1.1
. ,
det
-
11
12
...
1n
21
22
...
2n
...
...
...
...
...
nn
n1
n2
n.
(x )
-
,
(x )
n
.
0
nn
n
nn
,
nn
-
n 2 i
( x)
.
i 1
(x )
,
n, , .
-
, .
347
9.
“
” § 10.3. -
, .
9.3.1.
-
n
, ,
–
.
.
(x )
-
(x )
,
,
,
-
. , ,
det
, 11
12
21
22
...
... ... ... ...
...
k1
k2 11
( 1) k det
21
... k1
1k 2k
... kk 12 22
... k2
... ... ... ...
1k 2k
...
0
kk
. .
k-
k
[1, n] .
348
§ 9.4. -
, 4.4.1.
Oxy {g1 , g 2 } Ax 2
O. 4.4.1
2 Bxy Cy 2
2 Dx
2 Ey
F
0,
E
, ( A
B
C
A, B
A, B, C, D, F C
0 ).
, A, B
-
C
, (
Ax
,
Ax 2
( x, y )
.
9.1.1).
2 Bxy
Cy
-
2
2 Bxy Cy 2
A B
{g1 , g 2 } .
B C 9.2.2
sgn
2
9.3.1
,
– ,
,
rg
rg ( x, y )
–
sgn
-
.
-
, , .
,
,
-
349
9.
D, F
E,
, , E.
A, B, C, D, F
3
Ax 2
( x, y, z )
2 Bxy Cy 2
A B D B C E D E F
2 Dxz 2 Eyz
{g1 , g 2 , g 3 } .
3
,
( x, y,1)
Fz 2
,
0,
,
z 1. z 1
, .
3
-
( x, y , z ) .
S {g1 , g 2 , g 3 } ,
9.4.1.
{g1 , g 2 , g 3 } z 1 ,
S
11
12
13
21
22
23
0
0
.
1
.
Oxy
-
x
11
x
12
y
13
,
y
21
x
22
y
23
,
350
z 1
z
x
11
12
13
x
y
21
22
23
y .
1
0
1,
0
1
1
S det
11
12
21
22
-
0.
.
,
,
z 1
rg sgn
.
rg
,
sgn
.
,
-
-
9.4.1.
Oxy
rg
, rg
, sgn
sgn .
rg
, rg
, sgn
sgn ,
4.4.1,
9.4.1
, ,
, .
351
9.
9.4.1
sgn
rg
1 2
3
4 5
.
x 2 a2
y 2 b2
x 2 a2 x2 a2
y
x 2 a2
y 2 b2
x2
y
a2
b2
7
-
2
y 8
3
1
2
2
3
3
2
2
0
2
2
2
2
1
3
1
2
0
2
0
2
0
2 px
3
1
1
1
a2
2
0
1
1
a2
2
2
1
1
0
1
1
1
1
1
2
1
b2 y2 b2
2
y
6
sgn
rg
2
0
-
9
y
2
-
y
2
, 1 . -
352 .
( x, y, z )
,
z 1
,
-
,
,
0
0
0
1
0 .
0
0 -
S
z 1
,
1
0
1
0
1
0 ,
1
0
1 -
. T
S
g
1 0 1
0 1 0
2
0 1 0
0 0
,
S 1 0 1
0 0
0 1 0
0 0
1 0 1
0 1 0
1 0 1
0 0 . 2
2 . ,
I1
-
,
,
g
,
A C
I2
,
det §4.4.
A B . B C
-
353
9.
3 .
,
, -
.
§ 9.5. n
9.2.1
(x ) {g1 , g 2 ,..., g n }
,
, .
-
,
n 2
(x )
i
i
1
...
2
n 1
n
.
i 1
9.5.1. 1
maxn ( x)
n
(x ) minn ( x) x
,
x
x
n 2
1.
i i 1
. n 2
( x)
i
i
,
i 1 1
2
n
...
n
n
n 2 i
i 1
n 1
i
n 2
2 n
i i 1
2
i
1 i 1
n
i i 1
i
.
354 n 2
1,
i i 1
n
n
2
2 i
i
1
n
x x
1, 0, ..., 0
i
.
i 1
i 1
0, 0, ...,1
T
i
T
,
–
.
.
§ 9.6. , , ,
, .
9.6.1.
{ x1, x 2 ,..., xk } Q ( x1 , x 2 ,..., x k )
k
j Q ( x1 ,..., x j
,
[1, k ] x j ,..., x k )
Q( x1 ,..., x j ,..., x k ) x ,x
Q( x1 ,..., x j ,..., x k )
, , ,
,
-
, k-
.
355
9.
1 . 9.6.1.
k , Fk ( x) ,
-
F1 ( x), F2 ( x), Q ( x1 , x 2 , , x k ) F1 ( x1 ) F2 ( x 2 ) k
Fk ( x k ) , .
2 .
.
3 .
n n
n
, .
-
356
10
§ 10.1. ”, “ .
”, “
”
, ,
-
.
x
10.1.1.
, : 1) 2) 3) 4)
y ( x, y ),
,
-
( x , y ) ( y, x ); ( x, y ) ( x, y ); ( x1 x 2 , y ) ( x1 , y ) ( x 2 , y ); ( x, x ) 0 , ( x, x ) 0 x o, ,
E.
1–4
:
, (
3)
2
( ,
,
1) (
.
4) ,
, .
, -
357
10.
1 . 10.1.1.
§
, 2.2,
.
n
2 .
1 2
x
...
1 2
; y
...
n
n
, n
( x, y )
i
i
,
-
i 1
. 3 .
[ , ] ( x, y )
x ( ) y ( )d . 3
10.1.1.
? .
,
, 10.1.1.
3
-
x
y
E 10.1.2.
1)
(
x
)
x
( x , x) ;
2)
x
y .
358 -
:
...
, -
,
,
, , – .
x, y
E
10.1.1
( x, y)
x y .
– ). .
x, y
E
x 4
0
(x x
y)
2( x, y )
( x, x) 2( x, y ) y
2
2
( y, y )
2
.
,
( x, y )
2
2
x y
E.
10.1.1
y, x 2
y
, 2
0.
.
,
-
10.1.2.
, x
y
.
x, y 10.1.1
E x
).
y
x
y .
359
10. .
–
x
y
2
(x x
y, x 2
y)
( x, x) 2( x, y ) ( y, y )
2 x y
y
2
( x
x
y )2 ,
y
x
y
-
. .
, 10.1.1 (2 ) n
n i
i
i 1
j 1
n
(
n
2 j
2 i)
i
i 1
2 k
;
:
i, i
, i
[1, n] ;
k 1
n
2 j
j 1
n
2 k
;
i, i
, i
[1, n] ,
k 1
10.1.1 (3 ) :
|
x( ) y ( ) d
|
( x( ) y( )) 2 d
x 2 ( )d
x 2 ( )d
y 2 ( )d ;
y 2 ( )d .
-
360 E x
10.1.3.
y
[0, ] , ( x, y ) . x y
cos (
10.1.1)
, E.
E
x
y
-
,
10.1.4.
( x, y ) 0 . ,
.
§ 10.2.
.
E 10.2.1.
{e1 , e2 ,..., en } (ei , e j ) i, j ij
n
,
[1, n]. E
10.2.1
n
. – ). .
1 .
E
n
,
,
{g1, g 2 ,..., g n } . {e1 , e 2 ,..., e n }
.
-
.
361
10.
e1 e2
g2
g1 . e (e1 , e2 )
21 1 ,
,
21
e2
(e1 , e2 )
21
e2
o.
,
o
e2
.
0,
(e1 , g 2
(e1 , g 2 )
–
21
,
e)
21 1
(e1 , e1 )
0;
21
(e1 , g 2 ) . (e1 , e1 )
,
g2
e g2 g1 g2 ,
21 1
21
g1 (
.
7.2.2). 2 .
k 1
,
ek
,
k 1
ek
gk
kj
ej .
j 1
0; j
i [1, k 1] , [1, k 1]
ej )
(ei , g k )
(e k , e i )
,
(e j , ei )
0
k 1
(ei , ek )
(ei , g k
kj
ki
(ei , ei )
0;
j 1
ki
(e i , g k ) ; (ei , ei )
i
[1, k 1] .
ek
, k 1
: ek
gk
kj j 1
ej
o.
o.
-
362
ei , i
[1, k 1]
gi ; i [1, k 1] , g i ; i [1, k ] , , ek o.
. 3 .
gi ; i
-
-
[1, n] , ei ; i [1, n] ,
{e1 , e2 ,..., en } , ek
ek ;k ek
[1, n] .
.
,
,
-
. ,
, .
§ 10.3.
{ f 1 , f 2 ,..., f k }
-
. E 10.3.1.
{ f 1 , f 2 ,..., f k }
-
363
10.
( f1 , f1 )
( f1 , f 2 )
( f1 , f k )
( f 2 , f1 ) ( f 2 , f 2 )
( f2 , fk )
( f k , f1 ) ( f k , f 2 )
( fk , fk )
.
f
E
n
{g1 , g 2 ,..., g n } . n
n
x
i gi
y
j
i 1
gj,
j 1
10.1.1, n
( x, y )
n
n
i gi ,
( i 1
n i
j 1
(gi , g j )
ij
n
jgj) i 1 j 1
i, j
n
j ( gi , g j )
ij
i
j
i 1 j 1
[1, n] –
g
,
-
. ,
, ( .
.
-
1
. 10.1.1), , . 9.1.2)
(
-
:
( x, y )
x
T g
g
y
g
( g1 , g 1 ) 1
2
...
( g1 , g 2 ) ... ( g1 , g n )
( g 2 , g1 ) ( g 2 , g 2 ) ... ( g 2 , g n ) n
...
...
...
...
( g n , g 1 ) ( g n , g 2 ) ... ( g n , g n )
1 2
... n
,
,
364
x x
y
g
g
–
(
{g1 , g 2 ,..., g n } .
y § 2.3.
)
-
,
-
§ 9.2.
,
,
,
e
E ,
,
( x, y )
n
T
x
y
g
i
g
i
.
i 1
g
10.3.1.
det
0.
g
.
10.1.1
,
,
,
{g1 , g 2 ,..., g n }
S )
det S
9.1.1
T
S
g
{g1 , g 2 ,..., g n } (
g
-
S ; det
det
g
g
(det S ) 2 ,
0, ,
sgn ( det
g
.
,
det
det
, .
)
,
1,
e
g
,
0.
-
365
10.
{ f1 , f 2 ,..., f k }
10.3.1.
En
-
, . .
{ f1 , f 2 ,..., f k }
,
-
. 1
,
2
,...,
k
,
,
,
f
1 1
2
f2
...
k
fk
o. fi
1
( f i , f1 )
2
( f i , f 2 ) ...
k
( fi , f k )
i
0
[1, k ] , i
,
-
[1, k ] . (
. § 1.1),
, , 1
,
2
,...,
k
,
,
, (
6.5.2
.
6.5.2).
{ f 1 , f 2 ,..., f k }
, ,
10.3.1. .
9.3.4. 9.3.4
n
,
366 ,
-
).
det
11
12
...
1k
21
22
...
2k
...
...
...
kk
...
...
k1
k2
; k [1, n] ,
. .
1 .
§ 10.1
,
-
, .
, , , .
2 .
, (
) .
,
n
-
,
{g1 , g 2 ,..., g n } . -
{g1 , g 2 ,..., g k } ; k (
[1, n] . )
10.3.1 ,
-
367
10. 11
det
... ... ... ...
12
21
22
...
...
k1
k2
1k 2k
... kk
( g1 , g1 ) ( g1 , g 2 ) ( g 2 , g1 ) ( g 2 , g 2 ) det ... ... ( g k , g1 ) ( g k , g 2 ) k
... ( g1 , g k ) ... ( g 2 , g k ) ... ... ... ( g k , g k )
0;
[1, n]. .
x 10.3.2.
E
n
{g1 , g 2 ,..., g n } x
1 g
g
b g,
( x , g1 ) g
–
b
,
( x, g2 ) g
...
.
( x, gn ) . n
x
i
gi
gk ,
i 1
k
[1, n] . n i i 1
( gi , g k )
( x, g k ) , k
[1, n] ,
368 . 10.3.1
x
1 g
,
b g.
g
.
{e1 , e2 ,..., en } 10.3.2.
E
-
n n
x
En
e
i i i 1 i
:
( x, ei ) , i
i
( x, ei ) , i
[1, n] .
[1, n]
,
-
x .
(
. § 12.3).
§ 10.4.
Q ,
5.1.4 T
Q
Q
T
,
Q
Q
1
,
Q Q
,
T
E
det Q
1. -
.
369
10.
(
En
)
10.4.1.
. .
{e1 , e2 ,..., en }
En
{e1 , e2 ,..., en }
S
. , T
S
S ,
E
S
S 1
S
S
T
S
e
T
e
S
S . ,
-
T
.
.
E
T
S
S
n T ki
kl
il
k, l
;
[1, n] ,
n
3
i 1
§ 2.9. , 10.4.2.
E
n
,
. .
Aˆ
f g
f
g
f
T g
Aˆ
,
g
T
f g
T g
.
-
370 ,
Aˆ
T g
f
T
Aˆ
g
f g
2 g
f
T g T
Aˆ
Aˆ T g
f
2
f
g
T g
.
g
Aˆ
g
g
f
f
f
g
.
,
f
Eˆ , g
2
,
1,
-
.
.
,
,
,
A
,
10.4.3.
A
Q
R ,
Q –
-
R –
,
,
. . 14
,
A
Q1 R1
Q2 R2 .
A R2 ,
Q1
R1 Q2
,
,
.
14
. .
-
371
10. T
Q2 1
R1
Q1
R2
1
R1
,
–
.
R2
,
1
R1
,
. , .
R2
,
1
R1
, (
T
Q2
-
Q1
1
)
-
. , ,
R1
.
-
R2
,
R2
R1
1
E ,
-
. .
,
10.4.3
A
x
b A – Q ,
R R
x
Q
T
b .
-
372
§ 10.5. E
E2
E1 .
E
x,
E1 . E
10.5.1.
x,
( x, y ) 0
,
y
E1
E E1 .
k 10.5.1.
E1 n k.
E
-
n
.
E
n
E2 –
E1 . E1 {g1 , g 2 ,..., g k } .
x E2
E1
(
.
7.4.1),
( x , gi )
0 ; i [1, n] :
11 1
21
2
...
n1
n
0,
12 1
22
2
...
n2
n
0,
.......................................... 1k
1
2k
2
...
nk
n
0,
373
10. 1i
gi
2i e
...
1
;i
[1, k ]
x
2 e
...
ni
.
n
(
x ), E2 , k {g1 , g 2 ,..., g k } .
,
n k
6.7.1,
,
-
E2 . .
. 10.5.2.
E1 E,
E2 – E1
E2 .
.
x E2
( y, x) y E1
0 ; y E1 . ( x, y ) 0 ; x E2
,
E2 ,
E1
-
E.
.
y x
E 10.5.2.
E , 1 . y 2 . (x
E ;
y, z ) 0
z
E .
-
374
E 10.5.3.
E
k-
-
x E
y –
,
E –
.
.
E
{g1 , g 2 ,..., g k } , k
y
y
E
i
gi .
i 1
(x y
x
y, z )
E ,
0 z
(x i
,i
E
y, g j )
-
0 j
[1, k ] , ,
,
[1, k ] k
(x
i
gi , g j )
0 j
[1, k ]
i 1 k
(gi , g j )
i
( x, g j ) j
[1, k ] .
i 1
(
g1 , g 2 ,..., g k , 10.3.1)
,
)
.
6.4.1 ( .
-
.
{e1 , e2 ,..., ek }
, ,
E x
k
( x, ei )ei .
y i 1
E
375
10.
E4
-
10.5.1.
1
2
2
1
3
4
0, 0
2
E .
E4
E .
.
E
1 .
g1
g2 , {e1 , e2 , e3 , e4 }
-
E ,
, ,
1 2 ; 1 0
g 1e
dim E
2 .
g
2 e
1 2 . 0 1
2,
-
E
10.5.1
2.
g3 g4 ,
,
g3
e
1 1 ; 1 1
g4
e
2 1 , 0 0
376
E ,
-
.
g1 , g 2 , g 3
3 .
g4
-
4
E ,
E
4
{g1 , g 2 , g 3 , g 4 } .
A
E
4
E ˆ Ag 1 g 1 ;
,
,
Aˆ g 2
g2;
Aˆ g 3
Aˆ g 4
o;
o,
{g1 , g 2 , g 3 , g 4 } :
Aˆ
4 .
1 0 0 0
g
0 1 0 0
0 0 0 0
0 0 . 0 0 {e1 , e2 , e3 , e4 }
,
{g1 , g 2 , g 3 , g 4 }
S
1 2 1 0
1 2 0 1
1 1 1 1
2 1 0 0
,
377
10.
Aˆ Aˆ
Aˆ
S
e
1
S
g
S
g
1
Aˆ ,
S
,
,
,
§ 6.8,
Aˆ
e
§ 5.1 ,
e
1 2
1 2
1 1
2 1
1 0
0 1
0 0
0 0
1 2
1 2
1 1
2 1
1
0
1
0
0
0
0
0
1
0
1
0
0
1
1
0
0
0
0
0
0
1
1
0
2 1 11
4
1
1
1
4
8
2
2
1
2
6
5
1
2
5
6
.
-
:
, .
( x, y ) E
,
E
[ , ]
-
n,
-
x ( ) y ( )d ,
– n k
Pn ( )
,
k k 0
x( ) –
E –
E
[ , ] x( )
. § 12.3.
378
§ 10.6. ,
8 ,
. ,
.
A , 10.6.1.
E,
A, ( Ax , y )
x, y
E
-
, ,
-
)
-
( x, A y) . ,
10.6.1.
( x, y ) Aˆ
x ( ) y ( )d
d ( d Aˆ
d . d ,
( Aˆ x, y )
-
dx( ) y ( )d d x( ) y ( )
x( )
dy ( ) d d
379
10.
dy ( ) )d d
x ( )(
( x, Aˆ y ) .
E
{g1 , g 2 ,..., g n }
A
-
A
,
,
A
.
A
A
g
n
g
A
-
,
x
{g1 , g 2 ,..., g n } –
y
1
x
1
2 g
2
y
...
g
n
( Ax , y ) ( Aˆ
x
g
g
)T
n
( x, A y) g
y
g
x
T g
g
Aˆ
y g
En
–
g
,
...
( A B )T x
T
( Aˆ g
B
T g
g
g
,
, (10.6.1)
. T
Aˆ
A ) y
g
T
-
0,
g
x
,
g
y,
, –
, 5.1.2, ,
, -
380
Aˆ
T g
g
Aˆ
g
,
1
Aˆ
O g
g
g
T g
g
{e1 , e2 ,..., en }
,
Aˆ
Aˆ
T
Aˆ e .
e
( x, Aˆ y )
0
x, y
Aˆ
E,
.
10.6.1. .
x, y
E
( x, A y) x
( A y, A y ) Ay
0.
Ay.
0
10.1.1
o .
,
y
,
8.2.2
,
A
-
O.
.
10.6.1.
E
n
.
.
E
n
A ,
A, 1 g
Aˆ
T
A.
g
g
A . A x, y
E
,
A .
A ,
,
381
10.
( Ax , y )
( x, A y)
( Ax , y )
( x , A y) .
( x, ( A
, 10.6.1 A
A
A ) y)
0,
-
O.
.
A 10.6.2.
B,
-
E,
( AB )
B A .
.
x, y (( AB ) x , y ) ,
E
( x , ABy ) ( A x , By ) ( B A x , y ) . ((( Aˆ Bˆ ) Bˆ Aˆ ) x, y ) 0 x , y E
10.6.1 ( AB )
B A
O.
.
(A )
A.
10.6.3. .
x, y
E (( A ) x , y ) ,
(( A ( A ) ) x , y )
10.6.1 A .
( x, A y)
(A )
O.
( Ax , y ) . 0
x, y
E
-
382 10.6.4.
A
E
n
A .
.
1 .
A ,
,
ker A ,
–
A. y ker A ,
,
A y b
Ax , x
-
,
o, En, (b, y )
( Ax , y )
( x, A y)
ker A
2 .
0.
.
,
8.4.3
dim( ker Aˆ ) n rg ( ,
n rg Aˆ 1
Aˆ
T
,
A
rg A ,
)
T
n rg Aˆ
n rg Aˆ .
8.4.1
dim(
n rg Aˆ
10.5.1.
dim( ker Aˆ )
dim( ) ker Aˆ .
,
ker Aˆ .
383
10. : 1)
10.6.4 , 6.7.3 (
Ax
b
),
,
2)
-
b
-
A. y
,
b E
-
m
-
, :
A 10.6.5 -
x
b
-
, ).
A
T
y
o
-
b .
§ 10.7.
R, 10.7.1.
E,
,
x, y E ( Rˆ x, y ) ( x, Rˆ y ) . A 10.7.1.
AA
A A
A , -
A. ,
x, y
E.
A A,
,
,
384
( A Ax , y )
( Ax , Ay )
( x , A Ay ) , . -
.
En
R 10.7.1.
-
, . . 1
Rˆ
10.7.1
g
g
Rˆ
T g
g
{e1 , e2 ,..., e n } Rˆ {e1 , e2 ,..., en } .
S
, T
1
( S
e
S S
T
Rˆ T
Rˆ
T
x
T e
Rˆ x
S )T
e
e
Rˆ
e
T
y
e
.
y
e
x
S 1
S T
Rˆ e ,
e
( Rˆ
e
T
Rˆ
)
S Rˆ
e
( S
T T
( S
:
( Rˆ x, y )
R
e
S ,
10.4,
Rˆ
R
T e
Rˆ
e
e
Rˆ
e
. §
1
T
S
T
T
Rˆ Rˆ
.
S )T
e T
S
e
S
Rˆ
x, y
En
e
x e )T y y
T e
x
e
.
e T e
Rˆ y
e
( x, Rˆ y ).
385
10.
En
-
10.7.1
,
. -
10.7.2.
R
E
n
.
.
:
-
R i,
0. R
8.6.2
.
x Rˆ x Rˆ y
-
y,
,
x
y,
y
x.
:
–
y,
x,
–
( Rˆ x, y ) ( x, Rˆ y )
( x, y )
( y, y ),
( x, y )
( x, x).
R, ( x ,
2
2
y ) 0. 0. .
,
386 10.7.3.
, -
, . .
R Rˆ f1
Rˆ f 2
f
1 1
2
-
f1
f2 ,
f2 –
A
1
2
–
-
. :
f2 ,
–
–
-
f1 , ( Rˆ f1 , f 2 ) ( f1 , Rˆ f 2 )
( Rˆ f1 , f 2 ) ( 1 f 1 , f 2 ), ( f1 , Rˆ f 2 ) ( f1 , 2 f 2 )
( f1 , f 2 ), 2 ( f 1 , f 2 ).
1
R –
,
-
,
( f1 , f 2 )
( 0.
1
2
)( f1 , f 2 )
0,
.
E –
-
10.7.4.
R, E
–
E, E.
E E R.
.
R,
E
x E ,
E : Rx
E . x
E E
–
x
E : (x , x )
0.
–
387
10.
R,
E –
( Rx , x )
R Rˆ x
( x , Rx ) E
x
0.
-
0.
,
E ,
E
R. .
En
R 10.7.1.
,
R. .
R ,
En
,
1.
-
10.7.2
. (8.5.1)
1
e1 .
e1
,
1.
n 1,
.
E1 – En
R.
1
10.7.4 E
e1 , E1.
– n 1
-
–
R. R E
n 1
.
,
R –
,
388
E n 1,
R
En
10.7.4 ,
, n
x, y x, y
1
E : ( Rx , y )
( x , Ry ) ,
En 1. ,
-
2
e2 .
e2
,
1.
1,
2
,
( e1 , e2 )
n 2, E2 –
0. .
-
{ e1 , e2 } E
n 2
,
3
e3
.
En . .
,
10.7.1,
10.7.2.
-
R n
E . .
§ 8.5. 10.7.3.
, , .
-
389
10. .
10.7.1.
En
R
n
10.7.4.
,
-
. .
R ,
R
,
e
.
,
-
Aˆ
10.7.1 . .
R
,
10.7.5.
Q
D
Q
1
R Q
-
,
Q
T
R Q
. .
R
En ,
Q
-
, , 10.7.1. .
390
A
B En
10.7.2.
Aˆ Bˆ
,
-
Bˆ Aˆ .
.
.
Aˆ a
a Bˆ Aˆ a
,
Bˆ a Bˆ a
a, a ; Aˆ Bˆ a Aˆ a a, ( Aˆ Bˆ Bˆ Aˆ ) a o . ,
, a–
-
, ,
,
n
E , AB
.
BA
O.
.
A ˆa , A
,
B
a.
,
A
.
b
,
Ba
A. AB
-
,
BA Aˆ b
Aˆ Bˆ a
Bˆ Aˆ a
Bˆ a
Bˆ a
b.
A ,
b
. ,a–
.
a
,
, a
b
Ba , B.
b
-
Bˆ a
a.
391
10.
§ 10.8. Q, 10.8.1.
E, ),
(
x, y
(Qx , Qy ) 10.8.1
E
-
( x, y ) . ,
.
,
Qˆ x
(Qˆ x, Qˆ x )
cos
(Qˆ x, Qˆ y ) Qˆ x Qˆ y
( x, x)
x ;
( x, y ) x y
x
–
Qx
cos ;
x, y
y,
–
E,
Qy . Q
10.8.1.
, 1
Q
,
Q . .
10.8.1
x, y
E (Qx , Qy )
( x, y ) ,
,
( x , Q Qy )
( x , y)
( x ,(Q Q E ) y ) 10.6.1
Q Q E
O.
0. ,
392
Q Q E Eˆ Qˆ ,
Qˆ Qˆ Qˆ
O
E.
1
Q .
Q
E.
,
-
Qˆ Qˆ Qˆ
,
QQ
Q Q
,
,
Qˆ Eˆ
8.2.8
.
Qˆ
Qˆ
1
.
10.8.1.
En 10.8.2.
.
.
Q
Q
1
.
Q
10.8.1
Q
1
1
Q
Q
e
e
Q
-
T
Q e.
e
T
Q
e
1
§ 8.3 (4 )
e
,
,
-
Q e.
5.1.4, .
En En
, 10.8.3.
, .
393
10. .
Q
1º.
Qˆ (Qˆ x, Qˆ y) T
x
e
T
Qˆ x
Qˆ
T e
Qˆ
Qˆ y
e
y
e
1
T
Qˆ
e
e
( Qˆ
e
x
e
T e
x e ) T Qˆ 1 e
En
x, y
.
e
Qˆ
-
Qˆ
y
e
e
y
e T
x
e
e
y
( x, y).
e
,
{e1 , e2 ,..., e n } . {e1 , e2 ,..., en }
2º.
,
.
,
S , ,
Qˆ
1 e
S
1
( S T
( S
Qˆ
T
1
Qˆ
e
Qˆ S
e
S ) S
S )T
e
T
1
S T
1
Qˆ ( S e
Qˆ
T e
Qˆ
1 e
S
T T
)
S
( S
T
1
Qˆ
Qˆ e
T e
S
S )T
.
.
A
En
det Aˆ
0
10.8.4 ).
A QR ,
Q
,
R –
.
394 .
1 .
A A ( .
, 10.7.1)
-
Aˆ Aˆ f f, ˆ Aˆ f , f ) ( Aˆ f , Aˆ f ) 0 , (A ˆ Aˆ f , f ) ( f , f ) – (A (f , f ),
.
,
,
f
o, ( Aˆ f , Aˆ f )
(f, f).
0
-
,
Aˆ f
f
o
o Aˆ f
2 .
,
det Aˆ
0f
0.
{e1 , e2 ,..., en } –
,
A A.
-
Aei ; i [1, n] , ( Aˆ Aˆ ei , e j ) i (e i , e j )
( Aˆ ei , Aˆ e j )
,
1
ei
i
Aˆ ei ; i
ij
;i, j
[1, n]
[1, n] .
–
-
i
.
Q {e1 , e2 ,..., en }
3 .
{e1 , e2 ,..., en } ,
,
,
-
Q 1A.
R ,
A QR .
,
,
Rˆ ei
Qˆ 1 Aˆ ei
Qˆ
1 i
ei
i
ei ;i [1, n]
395
10.
ei , i
,
[1, n]
-
R, i
-
,
R
,
e
{e1 , e2 ,..., en }
.
R
10.7.1 4 .
,
.
,
.
R2 ,
A A A QR
A
R Q
A A
,
R Q 1QR
R Q QR
R
R R, 2
A A
R .
,
-
R1
R2
-
,
,
A A
,
R1
R2
R12 ; A A
R22 (
R12
R22
.2 )
, ,
O. .
§ 8.2,
Rˆ12
Rˆ 22
Rˆ12 ( Rˆ 1
Rˆ1 Rˆ 2 Rˆ 2 Rˆ1 Rˆ 22 Rˆ 2 )( Rˆ1 Rˆ 2 ) Oˆ . R1
R1
8.6.8
( R1
R2
R2
R2 )( R1
R2 )
R1
O
, R –
A
,
Qˆ
.
A.
Aˆ Rˆ
1
,
,
R2
O. -
396 .1 .
5.5.2 -
, ,
– , .
A
2 .
10.8.2,
R
,
, -
.
10.8.1.
2
Aˆ e0
A
0
E2 1 . 2
. .
1 .
, A A 0 0 {e1 , e2 }
10.8.2. (
)
Aˆ Aˆ 2 1
Aˆ
e0
0 2
e0
2 0
Aˆ
e0
1 2
Aˆ
T e0
2 2
Aˆ
e0
2 . 3
397
10.
1
1;
4 ; f1
2
2 e0
1
(
; f2
1 e0
,
2
,
-
10.8.2)
{e10 , e20 } {e1 , e2 } :
,
f1
e1
f1
-
2 3 ; e2 1 3
1 3
f2
,
f2
2 3
{e1 , e2 } :
e1
1
Aˆ e1
1 3 ;
1
1 2 2 0
1
e2
2 3
1 2
Aˆ e2 2
1 3
2 3 .
2 3
1 3
398 2 .
2 3
1 3
G
1 3
2 3
2 3
1 3
F 1 3
2 3
( 0 1
0 2
{e1 , e2 }
{e , e } ,
)
{e1 , e2 }
7.5.2,
-
Q
Qˆ
1
F G
e0
.
,
Qˆ {e1 , e2 }
{e1 , e2 } (
)
Gˆ
1
{e10 , e20 }
{e10 , e20 }
{e1 , e2 }
ˆ , Q
Fˆ
e0
e0
,
1 e0
(8.3.1)
1
Gˆ
Gˆ
e0
Gˆ
1
Qˆ
,
e0
{e1 , e2 } .
. (7.4.2), e0
-
F G
G
,
Fˆ
1
.
(
,
),
Qˆ
e0
F
G
1
F
G
T
-
399
10.
1 3
2 3
2 3
1 3
2 2 3
1 3 ,
2 3
1 3
1 3
1 3
2 3
2 2 3 .
Q 1A,
R
3 .
Rˆ
e0
Qˆ
1 e0
2 2 3
Qˆ
e0
1 3
1 3 ,
Aˆ
2 2 3
2 0
1 0 e
Aˆ
Qˆ
e0
4 3
1 2
T e0
Aˆ
e0
2 3 ,
2 3
5 3
,
Aˆ
e0
Qˆ
e0
Rˆ
2 2 3 e0
1 3
4 3
2 3
2 2 3
2 3
5 3
. 1 3
400
11
§ 11.1. b
U
-
ab ,
-
,
-
a
11.1.1. 15
, : 1 . 2 .
ab
ba ;
ab
ab ;
3 .
a1
a2 b
4 .
aa –
a1 b
a2 b ; -
,
a a
0
a
o,
,
. -
, ,
“ :
”. 1
, .
15
3.
401
11.
,
a b
ab
,
ab ,
b a ,
,
-
i:
a
ia ia
( i) 2 a a
(i)(i) a a ia ia ,
i2 a a
a a ,
a a
4
.
ab
b a
-
:
a b
ba
b a
ba
ab ,
,
-
ia ia
ii a a
a a ,
4 .
n
1 . 11.1.1. 1
a
2
...
1
; b
n
2
...
,
i
,
i
; i
[1, n]
n
–
,
, n
ab
i i 1
.
i
,
402 2 .
[ , ]
-
ab
a( )b( )d .
,
,
,
-
.
,
:
a a bb
ab ba .
,
a a bb
ab
2
ab ab
ab b a
a, b U . Un
{g1, g 2 ,..., g n } . : 1 2
ab
1
2
n
g
n
1
2
g1 g1 g 2 g1
g1 g 2 g2 g2
g1 g n g2 gn
g n g1
gn g2
gn gn
1 2
n
n
,
11.
403
–
U n.
g
gi g j
, T g
g
g j gi ,
-
.
A , 11.1.2.
A
T
A,
.
A , A
T
A
E
A A
T
E ,
-
. , .
det ( A
T
A )
,
det A det A
T
det A 2
det A det A
det E
1.
§ 11.2. ,
-
10. , .
-
404
A, 11.2.1.
U,
(
a, b U Aˆ a Aˆ b
),
-
ab . ,
:
n
U ,
.
A , 11.2.2.
U,
a, b U
A, Aˆ a b
a Aˆ b .
A 11.2.1.
-
B,
U, : ( AB )
-
( Aˆ )
B A
Aˆ .
.
.
( Aˆ Bˆ )a b
Bˆ a Aˆ b
a Bˆ Aˆ b
a, b U .
11.2.2,
( AB )
( Aˆ )a b
Aˆ a b
B A .
a Aˆ b .
.
a Aˆ b
a, b U
405
11.
,
-
n
U , A , 11.2.2.
-
n
A
U ,
{g1 , g 2 ,..., g n }
Aˆ
1 g
T
Aˆ
,
g
(10.6.1)
-
.
§ 11.3.
A, 11.3.1.
,
-
(
A= A .
), ,
:
n
U , ,
n
E .
-
: 1 . . 2 .
, ,
-
.
3 . , .
406 4 .
-
U
n
.
A 11.3.2.
,
-
.
.
A
B
-
11.3.1.
,
Aˆ Bˆ
Bˆ Aˆ ,
.
.
.
Bˆ Aˆ a Aˆ Bˆ a ,
,
Aˆ a Bˆ a Aˆ a
Bˆ a
a a, a, ( AB
,
a,
BA) a
a–
o.
,
,
,
, .
AB
BA
O. A
. ,
ˆa , A
a. ,
,
.
B -
407
11.
,
b
Ba
A. AB BA Aˆ Bˆ a Bˆ Aˆ a
,
Aˆ b
Bˆ a
Bˆ a
b.
A
,
,
b
.
a
b Ba Bˆ a B.
a
–
-
b
-
a.
a
.
A 11.3.2 -
Bˆ A
).
Cˆ , .
.
–
f–
-
A,
-
.
,
dim
1. A
,
Cˆ f
Bˆ f
Aˆ f
Cˆ Bˆ ( f )
Bˆ Aˆ f Cˆ Aˆ f
f , Cˆ Bˆ Aˆ f f , Bˆ Cˆ Aˆ f
( f)
11.3.1)
A
f ,
f. Aˆ Bˆ f Aˆ Cˆ f
B (
Bˆ Cˆ ( f )
f
C
, -
f, f,
( f ).
408 ,
( Bˆ Cˆ Cˆ Bˆ )( f )
BC
CB
O
,
o
f
,
B
C
. , ,
-
, . .
11.3.1
:
( a, b)
( Aˆ a, Aˆ b)
:
( a, b)
(a , b )
A: a, b E
ab
Aˆ a Aˆ b
ab
ab
:
A
En
T
A
A: a, b U :
E
A
Un
T
A
E
409
11.
A : ( Aˆ a, b)
(a, Aˆ b)
a, b
A : Aˆ a b
E
En Aˆ
a Aˆ b
a, b U .
Un 1
Aˆ
g
T
Aˆ
g
1 g
Aˆ
:
(a, Aˆ b)
a, b
Aˆ a b
E
:
a Aˆ b
:
A
T
g
( )
( Aˆ a, b)
T
a, b U :
A
A
En
T
A
Un
-
-
-
E
n
U
n
-
-
11.3.1
, .
410
§ 11.4.
. , .
[ , ]
( )
( )
( )
B( ( ), ( ))
,
( ) K( , ) ( ) d d .
x Aˆ x ,
( x) 11.4.1.
A –
x U,
,
-
(
U.
)
Aˆ a
a Aˆ a
11.4.2.
A
a–
.
.1 .
a–
a
(
aa
A
Aˆ a
a Aˆ a
a a
2 .
A
A,
-
,
Aˆ a
aa
.
, ,
.
1) ,
411
11.
Aˆ a a
a Aˆ a
a Aˆ a
, 3 .
Aˆ a a ,
. ,
Eˆ ,
ˆ
E –
,
A
Aa
a
0.
-
,
Aˆ Aˆ a a
a ( Aˆ
Aˆ a )a
( Aˆ a A
Aˆ a ) a a Aa ) 2
(A a
11.4.3.
a Aˆ a
a Aˆ a a
0.
a
Aˆ
a. .
A,
A
-
a
11.4.1.
,
-
,
Aˆ
( Aˆ ) 2 a
a
( Aˆ a ) 2 .
.
A
,
a
.
Aˆ
Aˆ a ,
,
,
A ( Aˆ a (
) ).
412
Aˆ
a ( Aˆ
a
Aˆ a ) 2 a
a ( Aˆ
Aˆ a )( Aˆ
( Aˆ
Aˆ a ) a ( Aˆ
( Aˆ
Aˆ a )a ( Aˆ
,
Aˆ
( Aˆ
a
a
Aˆ a )a Aˆ a )a
0.
11.4.2,
Aˆ a ) 2
a ( Aˆ
Aˆ a ) 2 a
a
(( Aˆ ) 2
2 Aˆ Aˆ a
a ( Aˆ ) 2 a
( Aˆ ) 2 a
Aˆ a )a
( Aˆ a ) 2 )a
2 Aˆ a a Aˆ a
2 Aˆ a Aˆ a
( Aˆ a ) 2
aa
( Aˆ a ) 2 ( Aˆ ) 2 a ( Aˆ a ) 2 .
.
A, 11.4.2.
,
-
, ,
.
.
Aˆ a Aˆ a
a, ( Aˆ ) 2 a
( Aˆ a ) 2
a Aˆ ( Aˆ a) a Aˆ ( a )
a ( Aˆ ) 2 a
a Aˆ a a a
2
a Aˆ a
2
2
a Aˆ a
a a
2
413
11. 2
a a 2
2
0,
a a
1.
a a
2
2
2
a a
a a
2
.
§ 11.5. ,
-
,
A 11.5.1 (c
B,
, -
Aˆ Bˆ
).
a
1 a
4
-
2
Aˆ Bˆ Bˆ Aˆ a .
.
Qˆ
1 .
( Aˆ
Aˆ a )
( Bˆ
Bˆ a ) i (
–
),
Qˆ
( Aˆ
Aˆ a )
( Bˆ
Bˆ a ) i , :
A, A , B , B . a
,
a
a Qˆ Qˆ a 10.8.2,
Qˆ Qˆ –
Qˆ a Qˆ a 1 .)
0
Qˆ . (
.
-
414
Qˆ Qˆ
(( Aˆ
Bˆ a )i)
( Aˆ
Aˆ a ) 2
( Aˆ
a
Aˆ a )
( Bˆ (( Aˆ
3 .
A, A , B , B ,
Q Q
2 .
2
Cˆ
Bˆ a ) ( Bˆ 2
Aˆ a )
Bˆ a ) 2
( Bˆ
Aˆ a )( Bˆ Aˆ a ) 2
Bˆ a )i)(( Aˆ
( Bˆ
a
Bˆ a )( Aˆ
Bˆ a ) 2
( Bˆ
Aˆ a ))i
( Aˆ Bˆ Bˆ Aˆ )i.
( Aˆ Bˆ Bˆ Aˆ ) i ,
,
-
Cˆ .
Qˆ Qˆ :
Qˆ Qˆ a
Aˆ
2
2
a ( Aˆ Bˆ Bˆ Aˆ )ia
Bˆ
a
a
Cˆ a
Bˆ a
Q Q
Aˆ . a
a
, .
-
,
,
( C a )2
.
a
1
A B a
4A B
a
4
0, a
AB BA
2 a
.
415
12.
12
, , ,
, ,
, -
., .
§ 12.1.
, , ,
,
-
.
,
{e1 , e2 ,..., en }
En
(x).
E {e1 , e2 ,..., en } , .
-
n
(x )
416 9.2.1.
,
,
-
E
n
, -
(x). . (
. § 9.2),
-
n n
n
( x)
ki
k
i
x
T g
g
x
g
,
k 1 i 1
-
g
{g1 , g 2 ,..., g n }
{g1 , g 2 ,..., g n } S
g
T g
S .
9.2.1 ( ), ,
. -
, 12.1.1.
,
-
, 16
.
16
, ”.
, “
-
417
12. .
1 .
§ 9.2,
-
(x )
S
ij
T
S
e
e
S ,
–
{e1 , e2 ,..., en }
{e1 , e2 ,..., en } ,
n
ek
sk
es , k
[1, n] ,
e
–
-
s 1
,
(x ) .
S
2 .
(§ 10.4),
S
e
S
1
S 1 e
3 .
T
.
,
S . -
e
(
{e1 , e2 ,..., en } 10.7.1) ˆ ,
{e1 , e2 ,..., en }
( e
S
1 e
8.3.2)
S .
4 .
-
{e1 , e2 ,..., en } –
ˆ.
418 (
ˆ (
.
10.7.1)
,
-
(x ) )
, -
ˆ. .
,
12.1.1
-
10.7.4.
ˆ 12.1.1.
En .
(x )
( x, ˆ x ) ; x
(x )
( x) 12.1.2.
En .
( x, Aˆ x) , ( x, x)
En A,
-
. 12.1.1,
-
.
(x) . ( 12.1.1.
-
(x)
) )
-
A,
.
419
12. .
,
A(
-
12.1.1), n
n 2 ij
( x, Aˆ x ) ( x, x )
( x)
i
j
i
i 1
i 1 n
n
,
2 i
2 i
i 1
, 9.5.1,
i
i 1
, ,
( x)
min
max
.
.
12.1.1
-
. 12.1.1.
E (x )
3
2
1
2
2
1
2
3
2
3
.
.
1 .
{e1 , e2 , e3 }
e1
(x )
B( x , y ) ,
2
1
0
0
0 , e2
1 , e3
0 .
0
0
1
1
2
2
1
3
2
2
3
-
420
1 2
B ( x, y )
(x
y)
( y)
(x
y)
2( 1 2( 1
B ( x, y )
1
2
1
1
2
1
( y )) ( . § 9.2).
2
2
)( 2 1 )( 3
1
2
3
) 2( 3)
3,
2
2
3
1
xe
( x)
2
3
2
1
2
2
)(
3
1 3
3
2
),
3
,
1
y
2
2
e
3
.
3
(x ) 1 1 . 0
,
e
0
1
1
0
1
1
2 .
-
ˆ
E
3
.
-
8.5.2:
1 det
1
1
1
1
3
0
3
2 0.
1 :
2,
1
.
2, 3
1,
,
,
,
,
10.7.1,
421
12.
( x)
2
2
2 2
1
2 3
. 3 .
S –
,
,
ˆ. (8.4.1)
2
.
:
2
1
1
1
0
1
2
1
2
0 .
1
1
2
3
0
, . ,
§ 6.8 (
2
1 1
2
2, ,
-
),
2
3
,
2
3
. ,
3
1 f1
1 . 1 1
2,
-
10.7.2
( ) :
.
, -
422
1
1
1
1
0
1
1
1
2
0 ,
1
1
1
3
0 1
1
1
1
2
1
0 ;
3
0
1
3.
2
-
, .
f1 , .
,
-
1,
1 1 0
.
2
,
2,
f2
0. 1 1 0
,
,
f3
1 . 2
E3
.
, -
,
{ f1 , f 2 , f 3 } .
1
1
{ f1 , f 2 , f 3 }
4 .
,
:
423
12.
1 3 1 , e2 3 1 3
e1
1 2 1 2
e3
0
1 3 1 3 1 3
S
“
1 6 1 . 6 2 6
1 2 1 2
1 6 1 6 2 6
0
{e1 , e2 , e3 }
”
“
”
{e1 , e2 , e3 } ),
-
{e1 , e2 , e3 }
{e1 , e2 , e3 } ,
,
S
1
S
T
, ,
“
“
”. 1
(§ 7.4),
2 3
1 2 3
1
S
1 2 3
1
S
2 3
”
424
1 3 1 2 1 6
1 2 3
1 3 1 2 1 6
1 3
1
0
.
2
2 6
3
, ,
{g1 , g 2 ,..., g n }
-
n
(x ) (
(x ) ,
).
(x ) –
-
,
{g1 , g 2 ,..., g n } ,
(x )
,
. ,
, .17
( x)
17
,
2 1
2B
1
2
C
“
”
2 2
(
), .
A
,
-
425
12. 2
, .
(
.
4.4.1), 2 B cos 2 .
( A C ) sin 2
1
2 1
( x)
2 2
2
( x)
1
2
,
2 sin 2 0, 0 cos 2 , ,
. n
(x ) {g1 , g 2 ,..., g n } ,
(x ) ,
, .
(x )
1 .
-
n
,
{g1 , g 2 ,..., g n } ,
, (
.
9.2.1). ,
-
, .
(x ) . n
2 .
n
( x, y )
k
k
,
k 1
n
.
,
426
{g1 , g 2 ,..., g n } {e1 , e2 ,..., en } , (x ) ,
-
. 3 .
,
{e1 , e2 ,..., en } ,
S ,
(x ) ,
§12.1.1.
(x ) ,
-
E
e
S
-
, T
S
e
S
e T
S
S
1
S
,
T
S
E
S
E .
S
,
(x )
,
(x ) –
. ,
,
S .
,
, 12.1.2.
(x ) (x )
2 1
4
2 1
2 1 16
3
2 1
2
2 2
6
2 2
.
427
12. .
(x )
(x )
1 . .
1 1 1 3
det
9.3.2)
(
4 8 8 6
0 ; det
2
40 0
(x ) –
,
-
(x )
,
-
. 2 .
(x )
. 2 1
(x )
2
1 2
3
2 2
(
1
2
1
,
1
1
2
(x )
2
3 .
2 2
,
2
2
, 2
2 2
1
(x )
1 2 1 2
2 2
2
2
1
)2
-
4
2 1
4 2
1
2
3
2 2 .
2
-
{ 1;
, 2
2}
{e1 , e2 } ,
428
e1
1
;
0
e
0
e2
1
e
.
(x ) 4 e
2 2
2 2
3
.
-
,
4,
2
1 3
2 2 3 f1
1
e
f2
,
e
1 3
2 2 3
{e1, e 2 } . {e1, e2 }
4 .
{e1, e 2 } ,
(x )
2 1
2
(x )
2
2 2 3
5
2 1
1 3
S
.
1 3 ,
2 2 3
4
2 2
,
5
429
12.
2 2 3 1 3
1
2
1 3 2 2 3
1
1
2
1
2
2
2 2 3 1 3
2
1
1
,
2
,
2
.
,
-
, ),
(
-
. ,
-
S
(x ) S
,
T
S
-
E .
-
(x ) S
T
S . 12.1.1 ,
-
,
-
(x ) .
S . T
S ,
S
( S
S
T
)
1
E .
,
,
, ,
-
430
S
T
) 1( S
(( 1
S
T
(
T
(( S
S
)
T
)
)
1 T
)
1
1
)T
S
S S
1
S
1
(
) S . -
,
1
S .
, 1
,
. -
§ 8.5
(
1
) f
(
)
f f
,
o .
f : det
ˆ
ˆ g
0 ,
–
-
(x ) . 12.1.2.
1 1 1 3
-
4 8 , 8 6 -
(x )
431
12.
det (
4 8 8 3
1 1 ) 1 3
4 8
det
0
8 3 3
5 (x ) 5
1
(x ) 2
(x )
,
2 2
1
0.
4,
2 2
4
1
2 2 ,
.
§ 12.2.
E3 e1
2 11 1
2
2
2
23
1
0
0
0 , e2
1 , e3
0 .
0
0
1
12 1
2
2
3
3
{e1 , e2 , e3 } ,
2 2
22
14
1
2
2 24
13 1 3
2
2
2 3
33
34
3
44
0
k
(
0 ).
ik k 1 i 1
E3 . , 2 1 1
2 2
2 1
2
3 3
2 3
2
14
1
§ 12.1.
2
24
2
2
0, .
34
3
44
0,
432 I.
: 8.6.8
1
2
0
3
,
,
det
11
12
13
21
22
23
31
32
33
0.
,
-
, 2 1 1
2 2
2
2
3
3
44
0, :
44
0
1)
sgn(
i
)
sgn(
sgn(
i
)
sgn(
sgn(
1
)
sgn( 2 ) sgn( 3 )
44
) , i 1,2,3 ;
2) 44
) , i 1, 2,3 ;
3)
sgn(
44
);
sgn( 2 ) sgn( 3 ) sgn(
44
);
4)
sgn(
44
1
)
0
5)
sgn( 1 )
sgn(
2)
sgn( 1 ) sgn(
2)
sgn( 3 ) ;
6)
sgn( 3 ) .
433
12.
II.
:
2
2
1 1
2
2
2
34
3
1
0,
2
34
3
0,
0,
44
:
0,
34
2 1 1
2 2
2
2
0,
: 7)
sgn(
1
)
sgn(
1
)
sgn(
2
);
8)
0,
34
sgn(
0,
44
2
);
:
9)
sgn(
i
)
sgn(
sgn(
i
)
sgn(
44
sgn(
1
)
sgn(
2
44
) , i 1,2 ;
10)
) , i 1,2 ;
11)
34
0,
0,
44
);
:
12)
sgn(
i
)
sgn(
i
)
sgn(
2
);
13)
sgn(
2
).
3
0.
434 III.
:
2
2
1 1
24
1
2
2
34
0
2
3
44
3
0.
0, -
:
1
1
24
;
2
2
34
2 24
,
3
34
;
2 34
3
2
24
2 24
O
,
2
2 24
2(
1 1
2 34
)
2
44
3
2 34
2
3 ).
0
: 24
0
0,
34
-
: 14) 24
; 34
0,
:
15)
sgn(
1
)
sgn(
44
);
16)
sgn( 1) 24
17)
34
44
sgn( 0,
44
);
: .
435
12. .1 .
. 2 . , .
§ 12.3. (
[ a, b]
)
f( )
g 0 ( ), g1 ( ), g 2 ( ), ..., g n ( ), ... , (
-
f ( )) [a, b] ,
.
, , .
[ 1,1]
f( ), {g k ( )
k
,k
[0, n]} . ,
n k
n , Pn ( )
k k 0
f( ).
,
-
436
[ 1,1]
, ,
gk ( ) ,
– k
{g k ( )
-
,k
[0, n] }
g k ( ), k
[0, n] }.
n 1
,
{ gk
1
( x, y )
x( ) y ( ) d 1
E.
x( )
y( ) 1
x
y
(x
y, x
y)
y ( )) 2 d
( x( ) 1
x( ) x( )
y( )
y ( ) E),
[ 1,1] . E,
f( )
f
E.
-
n
f
k
gk
E
k 0 n 2
(f
n k
gk , f
k 0 k
.
,k
k
gk ) .
-
k 0
[0, n]
,
2
-
437
12. n 2
(f
n k gk , f
k
k 0
gk )
k 0 n
n
(f, f) 2
n
k ( f , gk )
k
k 0
( g k , g i ),
i
k 0 i 0 2
k
k
[0, n] , n i
(gk , gi )
( f , g k ), k
[0, n] ,
(12.3.1)
i 0 k
2
.
,k
[0, n] ,
, ,
-
,
. 10.3.2, k
f
-
,
{g k
;k
g k ( ), k
f
[ 0, n ]
[0, n] }
,
. 2
n 2
(f, f)
:
n k
( f , gk )
k 0
n k
(
( f , gk )
k 0
i
( g k , g i ))
i 0
n
n
(f, f)
k ( f , gk )
(f, f
k 0
k k 0
, n
f
k k 0
gk ,
g k ).
438
f
,
. -
2
n 2
(f, f
k
gk )
k
,k
[0, n] ,
-
k 0
12.3.1.
,
. ,
{ gk
k
gk ( )
, k
[ 0, n ] } ,
10.2.1, :
e0 ( ) 1 ; e1 ( ) e3 ( )
3
2
; e2 ( )
1 ; 3
dn ( d n
3 ; ... ; en ( ) 5
2
1) n ,
. k
{ gk
,k
[0, n] } ,
, ( ,
,
-
)
{ ek ( )
dk ( d k
2
1) k , k
[0, n] }. ,
,
12.3.1
-
439
12. n i
(e k , e i )
( f , ek ) , k
[0, n]
i 0
: 2
k
( f , ek ) ;k (e k , e k )
[0, n] ,
-
: n 2
(f, f
n
e )
k k
(f, f)
k 0
,
,
(ek , ei )
ki ;
k
{e k , k [0, n] } k , i [0, n] ,
( f , ek ) ; k
k
( f , ek ) 2 . 0 (e k , e k )
[ 0, n ]
2
,
f
2
n 2 k
.
k 0
,
k
,k
[0, n] –
:
1)
f ; 2)
,
-
,
{ek , k
[0, n] } ( . ,
10.3.2). ,
,
. -
.
440 10.7.3,
, ,
, .
Rˆ [ 1,1]
,
18
,
,
-
, .
, 1
d 2x y ( )d 2 1d
( Rˆ x, y ) ,
dx y( ) d
1
dy x( ) d
1
1
dx dy d. 1d d
1
, 1
d2y x( ) 2 d d 1
( x, Rˆ y )
1
dx dy d . d d 1
1
Rˆ dx y( ) d
1
x( ) 1
,
(
dy d
,
1
.
,
,
1
,
)
-
[ 1,1]. 18
10.7.1
,
Aˆ Aˆ .
Aˆ
d d
Rˆ
Aˆ Aˆ
(
-
Aˆ
d d ),
d2 . d 2
441
12.
Rˆ . Rˆ x
-
x d 2x d
x,
2
0,
e
x( ) –
e
,
, -
.
x( 1) (
.
dx d
x (1)
dx d
1
1
3):
e
e
0
sin
0.
: 2 k
k 2;
k
0,1, 2, ... :
xk ( ) k
k
cos k
k
sin k .
,
k
-
k. , ,
[ 1,1] (
,
.
) , .
-
442 : : n? n
2
lim ( f
, 2 k
n
-
,
)
0?
k 0
.
,
E
, (
gk
E f E, ( f , gk )
E, ),
f E , 0; k
-
,
E ,
-
). ,
19
,
19
{g k , k
f
0,1, 2, ...} , , .
E
.
443
. 1.
1
§ 4.4
-
, . .
§
1.1. -
, 4.4.1.
.
1 .
“
”
x2 a2
y2 b2
0
-
{O, e1 , e 2 } . y
2
a
2
. 1.1.1.
a
0
.
3x
2
{O, e1 , e2 } 4 xy y 2 0.
-
444
y) 2
(2 x
x2
0
(
),
y
y
x .
3x .
. 1.1.1.)
1 0, -
1 arctg 2 . 2 2 .
.
“
”
y2
y
0
{O, e1 , e 2 } . 0. b 3 .
. 1.1.1
0
-
1
“
”
x2 a2
y2 b2
0
–
-
{O, e1 , e 2 } . 4 .
“
”
{O, e1 , e 2 }
,
-
445
. 1.
x2 a2
y2 b2
y2
1
a2 .
“
§
”.
1.2. ,
-
. 1.2.1.
x2 a2
y2 b2
1; a
b
0,
.
a2
b2
-
a
. 1.2.2.
.
a
.
0 a
x
-
.
p
b2 a
.
: 1)
–
: | x|
a
| y | b,
446
b a2 a
y 2)
x2 .
L
Ox
Oy, .
x y
x y
L
x y
L
x y
L,
L, .
.
. 1.2.1
( P, Q ) P
-
Q,
–
F1 A
F2 A .
447
. 1.
x y
A . 1.2.1.
,
-
L,
, :
1) r1
| F1 A | a
x ; r2
x;
2) | F1 A | | F2 A | 3)
( A, F1 ) ( A, D1 )
4)
( M , F1 ) ( M , D1 )
5) | F2 B |
2a ; ( A, F2 ) ( A, D2 )
| F2 A | a
M
L;;
F2 B
p,
6)
;
Ox ;
.
.
1 .
(
.
.
. 1.2.1)
(x a )
r1
2
y 2 ; r2
(x a )2
y2 .
,
i 1, 2 :
,
ri
(x a )2
y2
( x a )2
(1
(x a )2 2
)(a 2
x2 )
b2 2 (a a2
x2 )
448
x2
2 xa
a2
2
a2
a2
2 xa
x2
2
|a
0
| x| a
a2
2
x2
x2
2
x| .
1,
a
x 0 ,
-
,
r1 | F1 A | a 2 .
x ; r2 | F2 A | a
2
x.
1 .
3 .
( A, F1 ) ( A, D1 )
a x a x
4 .
;
( A, F2 ) ( A, D2 )
a x a x
4
5 .
.
.
,
| F2 B |
b a2 a
a2
2
b a 1 a
2
b
b a
p.
6 . . 1.2.2. .
A . 1.2.2.
,
x0 y0
,
,
449
. 1.
, A,
x0 x a2
y0 y b2
1.
.
A
y
y0
y ( x0 )( x x0 ) . 2x a2
y ( x0 )
,
2 yy b2
0,
b 2 x0 . a 2 y0
b 2 x0 ( x x0 ) , , a 2 y0
y
y0
x 02 a2
y 02 b2
-
1, x0 x a2
y0 y b2
1.
,
y0
x
0,
a. . 6
. 1.2.1.
A,
450
x0
d2
.
y0
c 0
F2
(
.
3.2.1):
1 x0 ( c) a2
d2
1 x0 a
y 0 (0) b
1 x0 c 1 a2
1
2
r2 , a
a
d1
x02
y 02
a2
b2
F1
. -
c :
0
d1
1 x0 c 1 a2
1 x0 a
a
r1 . a
sin
d2 r2
1 a
,
d1 r1
sin
1 a
. 6
. 1.2.1
. 1.2.1
.
. 1.2.2
.
: ,
2a .
451
. 1.
:
(
) ( (
-
, )
)
.
,
,
: .(
-
.)
,
, A,
r2
(1
a
x
a
(
( cos
a )
a
.
cos
cos ) a (1
1
2
)
p . cos § 4.6.)
.
. 1.2.2
. . 1.2.1.),
a 2.
452
§
1.3. ,
-
. 1.3.1.
x2 a2
y2 b2
1; a
0, b
0,
.
a2
b2
-
a
. 1.3.2.
.
a
.
0 a
x
-
.
b2 a
p .
: 1 .
–
,
| x| a, y
b x2 a
a2 ;
453
. 1.
2 .
L Ox
Oy,
.
x y
x y
L
x y
L
x y
L,
L, .
(
.
. 1.3.1).
y ux v y f (x ) x f ( x) u lim x x
. 1.3.3.
,
v
lim ( f ( x ) u x) . x
y
3 .
lim
, u x
v
lim (
x
b x2 a
b x. a
b b , x 2 a2 ax a b b a2 x) lim ( x 2 x a a
(x 2 a 2 ) x 2 b lim ax x2 a2 x
ab lim x
,
a2
1 x2
a2
x)
0. x
454 .
.
A .1.3.1 .
. 1.3.1.
. 1.3.1
x y
,
-
L,
, :
1 .
r1
| F1 A |
r2
| F2 A | a
x
r1 | F1 A | a
x;
r2 2 .
| r1
x;
a
| F2 A |
a
r2 | 2a
;
(x
x
(x
a) .
a) ;
455
. 1.
3 .
( A, F1 ) ( A, D1 )
( AF2 ) ( A, D2 )
4 .
(M , F1 ) (M , D1 )
5 .
| F2 B | p ;
6 .
;
M,M
L;
.
.
1 .
. -
1.2.1,
(x a )2
r1
y2 ;
(x a )2
r2
y2 , -
.
ri
i 1, 2
(x a )2
y2
(x a )2
(1
x2
2 xa
a2
2
a2
a2
2 xa
x2
2
|a
2
)(a 2
a
x ; r2
x2 )
x2 ) a2
| x| r1 | F1 A |
b2 2 (a a2
(x a )2
2
x2
x2
2
x| .
a
1,
| F2 A |
a
-
x,
456
r1 | F1 A | a 2
x; r2
| F2 A|
x.
3 .
4 5 .
a
.
,
| F2 B |
b a2 a
2
b a a
a2
2
1
6 .
b
b a
p.
, . 1.2.1,
6 . 1.3.1. .
-
x y
a y x 1 1 x y, 2 2 1 1 x y. 2 2 . 1.3.1
-
. 1.2.2
.
: ,
-
2a .
457
. 1.
: (
-
,
-
, (
)
)
. : . ( , .)
A . 1.3.2.
x0
,
y0
-
,
, ,
-
,
x0 x a2
y0 y b2
1.
.
A
y
y0
y ( x0 )( x x0 ) . 2x a2
y ( x0 ) ,
b 2 x0 . a 2 y0 ,
2 yy b2
0,
y
y0
b 2 x0 ( x x0 ) , a 2 y0
458
x02 a2
y 02 b2
1, x0 x a2
y0 y b2
1.
,
y0
x
0, .
, Ox (
.
.
1.3.2). A, ,
r1 a
a
x
( cos
a )
cos
2
a
(1
.
a .
cos )
a(
2
1)
1
p . cos § 4.6.)
. 1.3.2
a.
459
. 1.
§
1.4. ,
-
. 1.4.1.
y2
2 px ;
p 0,
.
p 2 . 1.4.2.
.
0
p 2
x
-
. p
-
. .
. 1.4.1,
,
– .
: 1 .
–
2 .
x 0;
,
L Ox,
x y
L
x y
L, .
460
.
. 1.4.1
3 .
, .
x y
A . 1.4.1.
,
-
L,
, :
1 . r
x
p ; 2
2 .
( A, F ) ( A, D)
1;
3 .
(M , F ) (M , D)
1
4 . | FB |
p;
M,M
L; 5 .
.
461
. 1. .
r
1 .
p 2 ) 2
(x
y2 ,
,
x2
r
p2 4 p , 2
px x 1
2 px
-
2 .
3
.
, | FB |
4 .
p |, 2
|x
2p
p 2
p.
5 .
. 1.4.2. .
,
2
y
ax ,
.
. 1.4.1 . :
-
, (
)
( .
)
462 :
.(
,
, .)
x0
A .
,
y0
.1.4.2
-
,
, , ,
yy 0
p( x x0 ) .
.
A
y
y0
2 yy
y ( x0 )( x x0 ) .
2p, y
y 02
y ( x0 ) y0
p , y0 y0
0.
p ( x x0 ) , y0
,
2 px 0 ,
yy 0
p( x x0 ) .
,
y0 .
0,
x
0.
463
. 1. 5
.1.4.1.
A
y0 p
,
–
p 2 .
x0 y0
y0 ( x0 cos y
2 0
,
p
2
p ) 2 ( x0
py 0 p 2 ) 2
y0 y
2 0
y 02
1 .
0 . .
, , ( 1.4.2). A,
.
.
,
x
p 2
.
y0 p
,
,
p2
.
.1.4.2
464
p 2
(1 cos )
cos
p 2
p
cos .
p p . 1 cos § 4.6.)
465
. 2.
2
4.5.1 , .
-
.
§
2.1. , 4.5.1. ,
, ,
1,
-
,
{O, e1 , e2 , e3 } . , ,
,
x y –
,
0, 0, ,
Oxy .
-
466
{O, e1 , e2 , e3 } . ,
. -
, .
§
2.2. ,
-
. 2.2.1.
x2 a2
y2 b2
z2 c2
1, a
0, b
0, c
0,
.
1 .
–
,
| x | a ; | y | b ; |z | c.
, 2 .
-
: -
;
-
;
. 3 .
, ,
.
,
467
. 2.
z
z0 ,
z0
c,
:
x2
y2
z2 (a 1 02 ) 2 c .(
.
.
§
z2 (b 1 02 ) 2 c z z0 ,
1,
. 2.2.1.)
. 2.2.1
2.3. , . 2.3.1.
-
468
x2 a2
y2 b2
2z , a
0, b
0, .
1 .
–
, , .
.
. 2.3.1
2 .
Oz ;
-
Oxz
Oyz .
z
-
0
469
. 2.
3 .
,
Oz , Oy – Ox z z0 0 ,
, ,
y2
x2
.(
y
,
.
(a 2 z 0 ) 2
-
1,
(b 2 z 0 ) 2 z z0 , .
. 2.3.1.)
2a 2 ( z
y 02 ), 2b 2
,
-
y0
x2
y
y0 ,
x
.
x0
:
y2
x
§
x 02 ), 2a 2
2b 2 ( z x0 .
2.4. ,
-
. 2.4.1.
x2 a2
y2 b2
2z , a
0, b
0, , .
470
1 .
– ,
z –
, -
. (
.
.
.
. 2.4.1
2 .
Oz ;
-
Oxz
Oyz .
3 .
,
Oz ,
,
Ox
,
Oy , –
.2.4.1.) ,
z
z0
0,
-
471
. 2.
y2
x2 (a 2 z 0 ) 2
(b 2 z 0 ) 2 z z0 , z0
.
1,
0
-
:
y2
x2 2z0 ) 2
(a
2 z0 ) 2
(b
1,
z0 .
z ,
x
x0 y
2
x 02 ) 2a 2 ,
2
2b ( z x
x0
.
y
x
2
y0
2
2a ( z y
y 02 ), 2b 2
y0 . ,
-
, , ,
, -
. 4 .
.
472
(
x a
y x )( b a
y ) b
2z ,
,
-
,
x a
y b
x ( a
x a
2 y ) b
x ( a
z
y b
2 y ) b
,
z , ,
,
. ,
, , . ( )
-
.
§
2.5.
\
,
-
. 2.5.1.
x2 a2
y2 b2
z2 c2
1, a
0, b
0, c
0, .
1 .
–
, ,
473
. 2.
z
(
,
).
.
. 2.5.1
2 .
(
.
. 2.5.1)
-
;
; -
.
3 .
,
Oz , Ox
Oy –
, .
, -
. 4 .
.
474
(
x a
z y2 ) 1 2 , c b
z x )( c a
,
, ,
,
x z ) a c x z ( ) a c
y ), b y (1 ), b
(
(1
,
x z ) a c x z ( ) a c (
y ), b y (1 ), b (1
, , ,
.
,
. .
§
2.6. ,
-
. 2.6.1.
x2 a2
y2 b2
z2 c2
1,
a
0, b
0, c
0,
.
1 .
–
, ,
.
x
-
a
475
. 2.
.
. 2.6.1
2 .
: -
;
; -
.
3 .
,
Ox ,
a
,
Oy
, . 2.6.1.)
§
x
Oz , –
.(
2.7. Oxz , Oz ,
,
F ( x, z )
0. .
.
476 ,
-
. 2.7.1.
x2
F(
y 2 , z)
0, .
, . 2.7.1.
,
:
1 .
x2
y2 a2
2 .
z2 1. c2 k 2z2 x2
y2 .
:
.
z2
,
2 px
Ox ,
-
,
Oz
z2
2 p x2
-
,
y2
z4
4 p2 (x 2
y2 ) . ,
. 2.7.1.
z
2
2 px
-
Ox .
.
x0 0 .
,
-
z0 Ox
z0
y2
x
x0 ,
z2
z 02 .
-
477
. 2. 2
, z0
y2
2 px0 , z2
2 px0 .
x0 0 ,
,
z0 ,
,
–
y2
z2
2 px .
478
3
20
, ,
-
{O, g1 , g 2 } .
z
.
1
g1
0
g2
g1
g2 .
0 , 1
-
z z
1 0
0 1 –
:
. 3.1.
z1
1
z2
1
20
( ,
2 2
§ 7.5)
, .
, -
479
. 3.
-
z1 z 2
1
2
1
2
1
2
2 1
.
, . 3.2.
1
{ g1
, g2
0
0 },
1
. 3.1,
–
z
, .
.
1 .
,
-
. 2 . :
z1
z2 z ,
z1 3 .
-
,
z2 z . ,
0
–
,
. 3.2 , -
, .
480 ,
-
:
1 0
z
g1
0 1
g1
g2
(
“
-
g2
”), “
i (
”).
z
z
i, :
z1
z2 z
( (
z1 z 2
1i )
1
i) (
( 1i )(
1
(
2 i)
2
) (
2)
(
1 2)
(
1
2 )i
1
;.
)i ; 2i )
2
(
(
1
2
2 1 )i
1 2
.
, ,
i2
ii
,
i
(0 1i)(0 1i)
0 1
2
1, 0 1
1 0
( 1) 0 i
1.
,
-
i z1 z 2
2
( 1) , (
1
(
1
i )(
2
1
2
1 2
2
) (
i) 1
1 2
2 2
1 1
i
2
1
i
1
2
i2
) i, . 3.1. :
481
. 3.
z1 z2 (
1
1i
2
2i
1
2
1
2 2 2
( ( 1 2) 2 2 1 2 2
2
1
1i )(
2
2i)
2
2 i )(
2
2 i)
(
1 2 )i
2 1 2 2 2 1 2 2
1 2 2
2
i.
z
i:
. 3.3.
1 .
-
Re z .
z 2 .
-
Im z .
z
2
3 .
2
z .
z 4 .
,
cos sin
-
,
2
2
2
2
,
z
arg z
,
z
0.
i
5 .
z
z.
-
482 :
1 .
,
1 , 2 ,
5 , -
. 2 . ,
-
.
,
z, z1, z 2
: 1 .
(z )
2 .
z. z
,
z
z. 2
zz
3 .
2
-
. 4 .
z1
z2
z1
z2 ;
z1 z 2
z1 z 2 .
n
5 .
Pn ( z )
k
zk –
-
,
,
k 0
. n k
,
k k 0
0,
483
. 3. n
0
n
k
k
0
k
.
k
k 0
k 0
-
:
, ,
,
,
.
. 3.1.
z2 1 0 . ,
.
z
i
,
1 0
,
0 . 0
,
1
1 1 0i
0 0 0i
0
0 0
. -
, 2
2
2 , , :
1
0 0
.
-
484 2
2
1 0,
2 ,
0,
0 1
,
0 . 1 0 1
z1
0 1i
i
0 1
z2
0 ( 1)i
i.
. 3.3, ,
-
:
z
i (cos
2
2
(
2
2
2
i)
i sin ) .
,
,
.
g1
{O, g1 , g 2 }
2
1 , 0
-
485
. 3.
z
-
–
-
,
-
,
arg z
,
,
z
(cos
.
,
z
. 3.3,
i sin ) .
.3.1
,
,
:
eiz
-
i
cos z i sin z ,
z
.
486
z
ei
z
,
i
z (cos
i sin )
.
, , 21
z1 z 2 ,
.
-
,
1
e
i
1
2
e
i
2
1
2
e
i(
1
2
)
,
i
0 1i
ii
cos
i
i sin
2
(e 2 )
. 3.2.
i
e
i
e2,
2
2
cos x
.
5.
,
.
cos z
e iz
e 2
iz
;
z
,,
21
.
487
. 3.
ei
x
e
i x
5
2 y
ei
y
1 y
x.
ei
,
i x x
x
5 2 6,
ln(5 2 6 ) ln 2 (5 2 6 ) .
10 0 ,
488
4
§
4.1.
,
,
, ,
,
.,
, ,
,
,
-
. , , ,
. ,
, .
n
, ,
, .
,
,
,
, .
, , , -
489
. 4.
, , !),
(
. -
n
. 4.1.1
,
,
{g1 , g 2 ,..., g n }
-
,
{g1 , g 2 ,..., g n } . . 4.1.1
-
{g1 , g 2 ,..., g n }
n
-
{g1 , g 2 ,..., g n }
x
x
1
g
2
x
g
1
S
x
n
...
j
ji
i
i 1 n
-
f
f
f
g
S
g
g 1
2
...
n
,
n j
i i 1
f (x ) i
f (gi )
ij
g
490 -
Aˆ
Aˆ
g
g
11
Aˆ
12
21
22
...
...
n1
n2
... ... ... ...
1
S
1n
Aˆ
S g
2n
...
,
nn ki n
n kj
n
Aˆ g j
ij g i ; j
mi
jm
j 1m 1
[1, n]
i 1
-
B
-
B( x , y )
B
g 11
12
21
22
...
...
n1
n2
... ... ... ...
...
B( g i , g j ); i, j
B
g
S
,
nn
ki n
ij
T
S
1n 2n
g
[1, n]
n mi
jk j 1m 1
g
g
-
(x )
11
12
21
22
...
...
n1
n2
... ... ... ...
S
1n 2n
... nn
,
T g
S
jm
491
. 4.
ij ij
ji
2
; i, j
ki
[1, n]
n
n mi
jk
jm
j 1m 1
,
S
,
-
n ij
gj
,
ij
gi ;
j
[1, n] ,
-
i 1
T
1
S
ij
,
n
gj
ij
gi ;
j
[1, n] .
i 1
,
:
{g1 , g 2 ,..., g n } {g1 , g 2 ,..., g n } ;
1 .
2 .
S
S
1
,
.
, .
x
,
n
y,
x
1
2
...
T n
y
1
2
...
T n
492
n n, 1
1
2
1
1
2
...
,
{g1 , g 2 ,..., g n } 1
2
...
1
n
2
...
2
n
...
...
...
n
... 1
n
2
n
:
, n
n
k
ki
i
j
jm
i 1 n k
n
x y {g1 , g 2 ,..., g n }
2 .
,
.
m
m 1 n
j
ki
jm
i
m
,
,
,
i 1 m 1
x
y
g
1 T
( S
)
x
y
g
S
1
.
,
-
1
2 . ,
,
-
. , ,
-
, ,
.
F,
, , :
r
-
493
. 4.
Fx Fy Fz
xx
xy
xz
yx
yy
yz
zx
zy
zz
x y , z
D E Dx Dy Dz
:
xx
xy
xz
yx
yy
yz
zx
zy
zz
Ex Ey . Ez
, . ,
-
, . ,
x, y
z.
27
(
,
,“
”).
, : xx
x yx
x zx
x
xy
x yy
x zy
x
xz
xx
x
y
yz
yx
x zz
x
y zx
y
xy
y yy
y zy
y
xy
xz
xx
y
z
z
z
yz
yx
yy
yz
y
z
z
xz
z
zz
zx
zy
zz
y
z
z
z
.
494 n ,
A
,
r
g
nn .
{g1 , g 2 ,..., g n }
-
{g1 , g 2 ,..., g n } . n
A A
S
g
n
1
A
g
S
(
-
n
ki
mi
kj
jm
),
j 1m 1
, ki
n
n kj
l
n
kj
n kj
p
p
l
p 1
jm
T im
pl
j 1m 1 p 1
p
,
A
S
r 1
jm
mi
j 1m 1
l
,
n
n
mi
j 1m 1 n
n
jm
g
, 2 .
1
S
T
A r
S . g
-
,
, .
-
495
. 4.
,
, 1 2
x
n
... n
x.
{g1 , g 2 ,..., g n } n i
,
i 1
,
{g1 , g 2 ,..., g n } ,
, n
n
n
i
ij
i 1
j
.
i 1 j 1
,
, -
: -
-
-
1
2 ,
-
. ,
,
,
S ,
( ),
S
1
S
,
T
( S
1 T
) –
, , .
496
§
4.2. ,
,
,
,
: n
,
(q, p) , q p
(
np
p
q –
n
),
q
jm
(
j1 j2 ... jqi1i2 ...i p
ik
[1, n] ; m [1, q] –
[1, n] ; k
n
: n
n
i1 1 i2 1
ik
[1, n] ; k
n
n
...
j1 j2 ... jqi1i2 ...i p
j1 j1
-
[1, p] – {g1 , g 2 ,..., g n }
),
{g1 , g 2 ,..., g n }
, -
j2 j2
[1, p]
n
...
i p 1 j1 1 j2 1
... jk
jq jq
i1i1
i 2i2
...
i pi p
jq 1
j1 j2 ... jq i1i2 ...i p
[1, p] ; k
,
[1, m] ,
ij
ij
S T
S
1
-
. -
. , ,
.
497
. 4.
1 .
,
q
,
q
p-
p-
p-
q
,
),
j1, j2 ,..., jq
i1, i2 ,..., i p .
n,
1
, . i
,
i
. 4.2.1. i
i
i
[1, n] .
2 .
. ,
, : (
,
), . 2
-
ijk
. 4.2.2.
:
111
,
112
,
121
,
122
,
211
,
212
,
3 .
221
,
222
. -
,
–
.
, , “
” .
-
498 . j k .. i ..l m
. 4.2.3.
.
4 .
,
-
,
. 4.2.4.
.
S, {g1 , g 2 ,..., g n } ,
{g1 , g 2 ,..., g n }
j i
(1,1)
( ),
ji
7.3.2 8.3.1.
, ,
,
-
, :
,
.
,
.
. 1 . . 4.2.5.
( x)
i ij
j
.
499
. 4.
2 . 1 1 1 2 1 1
1 2 2 2
2
1 n 2 n
... ...
2
n
1
n
2
............................................. n 1 1
n 2
2
n n
...
n
n
k i
,
ij
ij
i
k
.
(
-
{g1 , g 2 ,..., g n }
{g1 , g 2 ,..., g n } )
(1,1),
, . 4.2.1.
-
n
(q, p) , q p
n
,
,
-
n j1 j2 ... jq i1i2 ...i p
(
p q
jm ; m [1, q ] – ik ; k
[1, p] –
),
-
{g1 , g 2 ,..., g n } {g1 , g 2 ,..., g n } j1 j2 ... j q i1i2 ...i p
i1 i1
i2 i2
...
: ip ip
j1 j1
j2 j2
...
jq jq
j1 j2 ... jq i1i2 ...i p
.
500
(q
p)
. 4.2.2.
(
-
j1 j2 ... jq i1i2 ...i p .
)
, -
. 4.2.3.
.
.1 .
, , ,
-
. 2 .
, , (0,0). . 4.2.1 -
n
{g1 , g 2 ,..., g n } x
{g1 , g 2 ,..., g n }
(
j
)
( ) (0,1)
j
i
j
(1,0)
f (x )
j i
-
j
i
i j
501
. 4.
(
A
m k
m j
i k
j i
(1,1)
) j i
B( x , y )
(0,2)
(x )
( )
-
km
j k
i m
ji
( )
-
km
j k
i m
ji
m j
i k
ji
(0,2)
ji
( j i
1, i 0, i
j j
m k
j i
(1,1)
) j i
. 4.2.1
.
, –
– , , ,
–
. 4.2.1,
m k
m j
i k
j i
m j
j k
1, i 0,i
j, j.
. 4.2.1 , , . -
502 , m j
,
,
j k
, , j k
m j
.
,
, .
:
(1, 0)
1 .
.
(0,1)
.
2 .
,
, , 2 "
,
".
, 1,
,
-
.
3 .
, –
. ,
,
– .
: 1111
1211
1112
1212
2111
2211
2112
2212
1121
1221
1122
1222
2121
2221
2122
2222
.
503
. 4.
x . 4.2.1.
y
-
4
f ( x, y ) , 1
,
2
,
3
,
4
1
, 2, 3, {g1, g 2 , g 3 , g 4 } 1
f ( x, y )
3
3
2
4
-
4
.
, ,
-
, .
.
1 .
, . . 4 k i
gi
gk
k 1
{g1 , g 2 , g 3 , g 4 } {g1 , g 2 , g 3 , g 4 } .
4
f (gi , g j )
4 k i
f( k 1 4
l j
gk ,
gl )
l 1
4 k i
l j
f (gk , gl ) .
k 1 l 1
,
-
S , . 4.2.1
(0, 2).
-
504
f (gk , gl )
2 .
g1
1 0 ; g2 0 0
g
g
0 1 ; g3 0 0
0 0 ; g4 1 0
g
f ( g1 , g 3 ) 1 ; f ( g 2 , g 4 )
3
g
f (gk , g l )
.
0 0 . 0 1 0
,
0 0 1 0 0 0 0 3 0 0 0 0
.
0 0 0 0
§
4.3. ,
-
. .
(q, p) . 4.3.1.
j1 j2 ... j q i1i2 ...i p
.
(q, p)
j1 j2 ... jq i1i2 ...i p
j1 j2 ... jq i1i2 ...i p
505
. 4. j1 j2 ... jq i1i2 ...i p j1 j 2 ... jq i1i2 ...i p
j1 j2 ... j q i1i2 ...i p
j1 j2 ... j q i1i2 ...i p
,
j1 j2 ... j q i1i2 ...i p
-
.
j i
. 4.3.1.
j i
,
(1,1), ,
-
,
(1,1) j i
j i
j i
(q, p)
j
i ,
j1 j2 ... jq i1i2 ...i p
j1 j 2 ... j q i1i 2 ...i p j1 j2 ... jq i1i2 ...i p
, j1 j2 ... jq i1i2 ...i p
.-
(q, p) nq
. 4.3.3.
p
. -
,
:
-
.
j1 j 2 ... jq i1i2 ...i p
(q, p) .4.3.2.
-
.
(q, p)
j1 j2 ... j q i1i2 ...i p
506 k1k 2 ... k r l1l2 ...ls
(r , s)
(q r , p s )
.
j1 j 2 ... jq k1k 2 ...k r i1i2 ...i p l1l2 ...l s j1 j2 ... j q i1i2 ...i p
k1k 2 ... k r l1l2 ...ls
,
j1 j 2 ... jq k1k 2 ...k r = i1i2 ...i p l1l2 ...l s
j1 j2 ... j q i1i2 ...i p
k1k 2 ... k r l1l2 ...ls
. -
.
n
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. . 4.3.3. ,
x
y
k
y
y
x.
i
-
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i
i
k
,
, .
,
-
.
c . 4.3.1.
a –
( 0,3)
a b, 1 2
3 4 5 6 7 8
( 0,1)
9 10 .
,
b –
507
. 4.
, c
.
(0,4)
,
(
) ijk
a
l
,
ijk
b
9 9 9 9
2 4 6 8
9 9 9 9
l
c
1 3 5 7
10 10 10 10
2 4 6 8
10 10 10 10
9 27 45 63
q 1 jr )
18 36 54 72
10 30 50 70
j1 j2 ... j q i1i2 ...i p
(q, p) . 4.3.4.
–
. ,
1 3 5 7
-
p 1.
,
( , is )
(
, -
(q 1, p 1)
, m ).
(
20 40 . 60 80
j1 j2 ... jq 1 i1i2 ...i p 1
j1 j 2 ... jr ... j q i1i2 ...i s ...i p
jr
is ,
-
j1 j 2 ... jq 1 i1i2 ...i p 1
=
j1 j2 ...m... jq i1i2 ...m...i p .
508 , ,
n
–
m –
,
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(1,1) , . 4.3.3.
j i
,
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-
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,
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.
i
i
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k
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: . 4.3.2.
a –
(1,1)
1 2 3 4 5 6 ; 7 8 9
i j
509
. 4. j
(1,0)
b –
2 3 ; 4
(0,1)
c–
i
2 i j
.
3 4 .
j
i j
i
.
1 .
i j
,
j
–
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i j
j
.
j 1
1
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2
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2
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1
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2
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3
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3
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1
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2
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3
7 2 8 ( 3) 9 4 26.
2 .
,
i j
3
1 2 2 ( 3) 3 4 8,
(0,1)
–
i
6 4 17,
3 i j
j
i
.
i 1
1
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2
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3
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2
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3
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2
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3
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2
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3
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-
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-
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) ,
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,
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3 4
ij k
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i
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6 8
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-
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. 4.3.4.
N=2
N=3
( i1 )
( i1 ,i2 )
( i1 ,i2 ,i3 )
i1
,
1 ( 2! 1 { 3!
i1 ,i2
i1 ,i2 ,i3
i2 ,i1 ,i3
...
...
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),
i3 ,i1 ,i2 i3 ,i2 ,i1
i2 ,i3 ,i1 i1 ,i3 ,i2
}
512 , -
: ,
. (i
j)
.
. 4.3.5.
, ,
N
(
-
), N!
( 1)
, ( k 1 , k 2 ,...,k N )
(k1 , k 2 ,..., k N ) – {1,2,..., N } , ,
.
-
, .
. ,
-
, .
N=1
[ i1 ]
N=2
[ i1 ,i2 ]
N=3
[ i1 ,i2 ,i3 ]
. 4.3.6.
i1
,
1 { 2! 1 { 3!
i1 ,i2
i1 ,i2 ,i3
i2 ,i1 ,i3
...
...
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},
i3 ,i1 ,i2 i3 ,i2 ,i1
i2 ,i3 ,i1 i1 ,i3 ,i2
}
513
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, -
: ,
. [i
j]
.
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,
, ,
.
1 2 3 4 ijk
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.
7 8 ( ij) k
.
1 .
ijk
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jik
i ( jk )
i [ jk ]
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i
j,
(
.
1 3 2 4 5 7
. 4.3.3.)
6 8 ijk
ikj
j
, k,
1 5 3 7 2 6 4 8
,
-
514
, 2 .
( ij) k
1
1
2
2
4
2 3
3
1
2
2
2
5
4
5
4
2
2
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5
6
6
8
2 7
5
13 2
2
2
i ( jk )
7
8
13
8
2
2
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1
1
2
3
4
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5
1
2
2
7
3
7 2 11 2
2
, 5
2
6
2 7
7
8
2 11
2 4
2
i [ jk ]
6
8 2
–
2
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515
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1
2
3
4
2 3
5
3
0
2 3
2
2
7
0
2
2
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2
6
2 7
3
8
2 3
2 4
8
2
§
6
0 0
2
2
4.4.
,
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-
, .
-
. .
E
{g1 , g 2 ,..., g n }
n
x,
, ij
i j
i
ij
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E
n
i
.
( gi , g j )
( gi i , g j )
x
i
,
( x, g j ) . -
516 j
.
j
x
{g1 , g 2 ,..., g n } ,
-
x
-
x i j
ij
.
, .
, (
.
10.3.2).
En
j1 j 2 ... j q i1i2 ...i p
(q, p)
. 4.4.1.
q 1.
,
j 2 ... j q i0i1i2 ...i p
(q 1, p 1)
-
j1 j 2 ... j q i1i2 ...i p
j1 j2 ... jq i0i1i2 ...i p
i0 j1
j1 j2 ... jq i1i2 ...i p
,
-
.
,
-
, .
,
.
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-
E
n
517
. 4.
,
,
-
. ,
,
,
(2, 0) .
S
g
T
S .
g
,
:
1 g
T
( S S
g 1 g
E
E
S )
( S
T
1
ij
g
( S
T
)
1
1 T
) ,
( S
1 T
S
)
1 T
( S
)
S
T
-
S
,
( S
1
S
1 T
)
( S
T
) 1.
,
.
En . 4.4.3.
p 1. i1
j1 j 2 ... j q i1i2 ...i p
(q, p) (q 1, p 1) j1 j 2 ... j q i1i2 ...i p
j0 j1 j2 ... jq i2 ...i p
j0 j1 j2 ... jq i2 ...i p
,
i1 j0
j1 j2 ... j q i1i2 ...i p
.
, -
518
E2 . 4.4.1. ij
2 3 3 5
i jk
3 4 5 7 .
2 5 1 3
ij k
ijk
.
.
1 . ijk
im
m jk
12
2 11
2 3 3 5
12
2 12
2 2 3 1 7,
12
2 21
2 4 3 7
2 5 3 3 19,
.
11
1 11
11
1 12
11
1 21
11
1 22
12
2 22
211
21
1 11
22
2 11
3 3 5 5
212
21
1 12
22
2 12
3 2 5 1 11,
221
21
1 21
22
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47,
222
21
1 22
22
2 22
3 5 5 3
30.
111 112 121 122
,
ijk
21 29 34 47 7
19
11 30
.
21, 29, 34,
519
. 4.
2 . ij
i k
mj
ij
,
mk
–
-
,
-
ij
1
2 3 3 5
11 1
1 11 11
1 21
21
12 1
1 12 11
1 21
22
21 1
2 11 11
2 21
21
22 1
2 12 11
2 21
22
11 2
1 11 12
1 22
21
12 2
1 12 12
1 22
22
21 2
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2 22
21
22 2
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2 22
22
,
5
3
3
2
3 5 4 3
.
3,
3 3 4 2 5 5 7 3
1, 4,
5 ( 3) 7 2 2 5 3 5
1, 5,
2 ( 3) 5 2 1 5 3 ( 3)
4, 4,
1 ( 3) 3 2
3.
ij k
3
1
4
1
5
4
4
3 ,
.
ij ij
i j,
520 1
S
, ,
§
S
T
,
.
4.5. , . , .
, -
. , .
E3 .
ijk
-
,
: ijk
( 1)
ijk
0–
( i , j ,k )
i, j, k
,
(l , m, n) , , {l , m, n} ( . § 6.1).
,
. -
521
. 4. ijk
27
,
,
,
:
1
1. ,
ijk
E3
-
. : lmn
li
mj
nk
l1
ijk
m2
l1
m3
det
( 1)
lmn
( l ,m ,n )
n3 n2
l2
m3
n1
l3
m1
m1
n3
l3
m2
l1
l2
l3
m1
m2
m3
n1
n2
n3
n2 n1
,
l , m, n
,
0–
lmn
l2
,
, 11
12
13
21
22
23
31
32
33
(
)
-
1. ijk
( ,
) ,
, .
522
-
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,
§ 2.4).
. i
(
)
a i
,
-
b.
-
k
9
,
k
, : 1
1
1
2
1
3
2
1
2
2
2
3
3
1
3
2
3
3
.
, : i
k
1
1
1
2
1
3
2
1
2
2
2
3
3
1
3
2
3
3
1 ( 2
1 2 1 2
i
k
k
i
)
1 ( 2
i
k
k
i
)
1
1
1
1
1
2
2
1
1
3
3
1
2
1
1
2
2
2
2
2
2
3
3
2
3
1
1
3
3
2
2
3
3
3
3
3
1
2
0 2
1
1
2
3
1
1
3
2
1
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2
2
1
3
3
1
2
3
3
2
0
3
.
.
523
. 4.
,
,
-
,
. ,
, .
,
,
i
-
k
S ,
, : k
k
ki
i
kj
T ik
j
kj
i
j
ij
i
j
i
i
,
. :
a
(
b,
i
k
E3 ,
) -
i
i
1
1
2
2
3
3. (
.
§ 2.9). ,
( a, b )
ki
i
k
a
. ,
a 0 0
b,
-
b cos b sin 0
,
524
a
–
(a, b)
b.
a b cos ,
-
,
. .
0
1
2
1
1
2
3
1
1
3
2
2
,
1
0 3
2
2
1
3
3
1
2
3
3
2
0
3
,
a
-
,
E3
b
[ a , b ],
,
2
3
3
2
3
1
1
3
1
2
2
1
.
. ,
,
-
(n 1)n , 2
n ,
.
E
, ,
3
E
3
.
525
. 4.
,
,
[ a , b ]i , 1 jk
j
ijk
111 1 121
1 2
131 2
k
.
i 1:
,
k
j
3
112 1
1
122
1
132
3
3
2
2 2
113 1
3
123
2
3
133
3
3
2
3
2
.
,
i
ijk
j
k
3
E .
i
ijk
is
j
s
k
ijk
, jm
kl
m
l
. i,
qi
qi
qi
is
s
qs
s
is
s
q
qi
,
jm
qml
kl ijk
qi
m
jm
l
,
kl ijk
,
-
ijk
.
,
i
iml
m
l
,
.
,
[ a , b ].
,
a
, 3
E ,
b
-
526
0 0
b cos
a
,
0
b sin
a
–
b. a b sin ,
[ a, b] , ,
.
,
,
-
“
”, -
. , .
-
:
1 .
(
(a , b , c )
( a ,[b , c ] )
i
[ b , c ]i
i ijk
j
. § 2.6)
k
ijk
-
i
j
.
k
2 . ( :
. § 2.8)
-
527
. 4.
[ a , [ b , c ] ]i
ijk
[ a , [ b , c ] ]i
ijk
[ b , c ]k
klm
il
jm
j
m
j
ijk
klm
i
,
ijk
klm
j
l
ijk m
im
l
m
(
i
j
klm
l
m
.
jl
m
j
,
il
j
jm
i
im
jl
(a , c )
, [a , [ b , c ] ]
b (a , c )
c (a , b ) .
) i
j
l
(a , b )
m
528
1.
. ,
. 102. .
. .:
3.
.
.:
, 2005.
, 2005. .
4.
. .
5.
.
.: .:
, 1976. , 1979.
. .:
6.
.
, 1983. .
.
.:
, 1952. 7.
.
:
.
.:
,
1976. 8.
.,
.,
. .
, 2001.
.:
-
529
§ 4.1. § 4.2. § 6.3. §
. 4.3. § 12.3. § 5.4.
§ 1.5. § 1.5. § 7.2. § 1.5. § 1.5. § 6.5. § 6.5. § 6.5. § 9.1. § 9.1. § 8.7.
,
§ 1.3. § 2.4, § § 5.2.
. 4.5.
530 (
) § 8.4.
§ 2.5, § 8.7. (
)
§ 8.7. § 2.5. § 2.5. § 2.3. § 2.7. § 2.3. § 2.7. § 5.1. §
. 1.1. § . 2.1.
§ 5.4. § 5.4. § 4.4. § 4.5. § 4.5. § 7.4. § 3.3. § 5.6.
531
§ 2.8, § § 8.7.
. 4.5.
§ 4.5. § 8.2. § 8.3. 23n-
§ 1.1. § 6.1. § § § § 11.4.
§ 9.2. . 1.3. . 1.4. . 1.2.
§ 6.3. § 6.3.
§ 10.1. § 1.1. § 8.2.
§
. 4.2.
§ 9.1.
532 § 9.2. § 8.7. § 1.8. § 7.3. § 8.3. § 7.5. § 7.5. § 8.5. § 8.6. (
§ 9.4. ) § 8.4.
§ 4.4. § 4.5. § 9.2. § 1.1. § 9.2. § 9.2. n § 1.1. § 12.2. § 1.4. § 1.4. § 8.2. § 1.4. § 1.4. § . 3.0. § 1.5. § 7.3.
533 § 4.3. § 4.6. § 4.5. § 9.1. § 8.3. § 8.7. § 10.3. § 1.5. § 7.3. § 5.2. § 1.5. § 1.5. § 9.3, § 10.3.
§ 1.4. § 7.2. § 1.4. § 7.2. § 1.4. § 7.2. § 7.4. § 3.2. § 7.1. § 8.2. § 8.7. § 8.1. § 5.3.
534 § 8.7. § 8.7. § 4.1. § 4.4. § 4.1.
§ 1.1. § 9.1. § 10.3. § 9.2. § 8.3. § 8.4. § 5.3. § 1.8. § 7.3. § 6.8.
k
§ 6.8. § 9.2. § 6.3.
§ 1.2. § 3.3. § 3.2. § 7.5. § 6.6. § 5.3. § 10.1. § 10.1.
535 § 1.1. § 1.4. § 10.1. § 1.7. § 3.2. § 3.2. § 3.3. § 1.1. § 1.3. § 1.2. § 8.2. § 8.7. § 7.1.
§ 8.4. § 5.1. § 7.5. § 5.2. § 8.2. § 5.1. § 8.3. § 1.7. § 6.6, § 6.7. § 6.7. § 6.7. § 4.5. § 6.6. § 5.3. § 5.2, § 8.1. § 5.3. § 8.7. §
. 4.3.
536 § 7.3. 23-
n § § §
§ 1.1. § 1.1. § 6.1. § 6.2. . 4.4. . 1.3. . 1.4. . 1.2.
§ § 10.2. § 5.1, § 10.4. § 2.1, § 10.5. § 2.1. § 2.1. § 10.5. § 5.5. § 10.1. § 1.5. § 10.8. § 1.7. § 1.5, § 10.2. 6.6. § 2.1. § 12.1. § 5.2. § 9.3.
§ 4.4. § 4.5. § 3.3. § 3.1. § 7.4.
537 § 1.8. §
. 2.7. § 4.5. § . 4.4. § 7.4. § 9.6.
§ 9.3. § 4.6. § 4.1. § 4.2. § 1.2. § 6.4. § 1.2. § 1.2. § 5.2. § 9.2, § 12.1. § 9.2, § 12.1. § 4.4. § 12.1 § 5.1. § 5.2. § 8.2. § 8.3. § 8.2. § 8.7. § 1.1. § 1.2. § 8.2. § 8.7. § 7.1.
538 § 7.4. § 3.3. § 3.2.
§ 1.6. § 1.7. § 1.1. § 6.3. 3§ 1.1. § 1.1. § 7.2. § 8.4. § 1.2. § 6.5. § 3.4. § 10.1. § 3.2. § 3.4. § 3.3. § 6.6. § 1.1.
§
§ 10.7. . 4.3.
§
§ 5.4. § 2.4. . 2.4.
539 §
. 1.3. § §
. 2.6. . 2.5.
§ 1.3. n§
§ 6.2.
. 1.4. § 8.6. § 8.6. § 2.2. § 2.6.
§ §
. 1.2. . 2.2. § . 2.3. § 3.3. § 9.3.
§ 2.3. § § 1.1.
. 4.3. § 9.1.
n m
n n § 2.2, §
§ 6.4. § 6.6. . 4.5.
§ 10.1. § 1.1. § 1.6. § 8.3. § 1.2. §
. 4.3. § 2.6. (
)
§ 8.5. § 8.5. § 6.6.
§ 1.4. § 11.5.
540 § 8.7. § 10.6. § 1.1. § 1.2. § 11.4. § 1.1. § 1.1. § 8.2. § 8.7. § 7.4. § 8.6. § 4.6. (
§
) § 8.4.
. 4.2. § §
. 4.4. . 4.5.
§ 8.6. § 10.3. § 9.3. § 6.6. § 6.3. § 6.5. § 10.8. § 6.5. § 7.5. § 6.7, § 10.6. § 8.2. § 3.4. § 1.1. § 5.1. § . 4.3. § 1.4.
541 §
. 3.0.
§ 2.2. § 10.1. § 1.1. § 1.2. § 1.6. § 8.3. . 4.3. § § 11.1. § 11.2. §
. 4.3.
§ 3.3. § 3.1. § 3.2. § 1.6. § 1.6. § 3.5. § 3.5. § 3.5. § 3.1. § 3.1. § 3.1.
§ §
. 1.3. . 1.2.
542 §
. 3.0.
§ 1.8. § 3.3. § 3.2. § 6.7. § 6.7. § 5.2.
§ 8.5. § 8.5.
§ 4.3. § 4.6.
§ 6.6. § 2.1.
§
. 3.0. § 9.5. § 1.1. § 6.3. § 6.8.
543 § 4.4. § 4.5. § 4.5. § 4.5. § 11.2. § 11.3. § 11.3. § 11.4. § 11.4.
§ 8.4.
