дифференциалдық геометрия және топология элементтері
312
. . , 2014 Мусин А. Т.
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Transcript of дифференциалдық геометрия және топология элементтері
. .
, 2014
Мусин А. Т.
M
« » -
:
. . – - , ;
. . – , ;
. . – - .
. .
: . – , 2014 – 312 .
ISBN
,
, ,
.
,
, , , -
, ,
.
ISBN
© . ., 2014
©
, 2014
M
514:515.1(075.8)22.15 я 73
79
79
978-601-7427-41-2
978-601-7427-41-2
ƏОЖ 514:515.1(075.8)КБЖ 22.15 я 73
3
VIII,
,-
. – ,
c , -.
- –
.-
,-
. --
, .
– . -
., , -
. –
. -
.-
, ,.
– .
, -
. ( )
. ,
4
..
- –
. –
– . « » -
. ,
. V . -
V
. -.
V .(1288–1344) . V
. (1574), . (1603), . (1658), . (1680), . (1663), .
(1733), . (1766), . (1800), . (1818), . (1825) . V -
, ,.
V. . -
(1792–1856) . 1826 ..
. 1832 . .. ,
. . . . ., -
, -, -
.-
. « »
5
. ,,
. (1323–1382) .- .
,. (1596–1650) .
(1601–1666) . . .-
. .(1826–1866) , -
- ,. ,
,, .
– . ,.
. (1707–1783) . (1838–1922) . . . , . , . , .
.. .
.
. .. . -
, , ,
.. (1854–1918) .
. . . –
., -
,.
, -
6
., -
. -, , ,
, , . 1872 .« » .
. . , --
. ,, -
-. ,
.
.-
.
7
I .
§ 1. - .
.,
.1.1. . t ( = 1, 2, … n)
3RA -.
t, n
- , : = (t1,...,tn) (1.1)
, n - Rn-.
kji ,, , -x, y, z ,
= kzjyix.
(1.1) - 3
x = x(t1,…., tn), y = y(t1,…., tn), (1.2)
z = z(t1,…., tn)- .
-.
- -.
-.
, - )(trr),( vurr - .
:
8
1.2. . - -- -
-.
F(x, y, z) = 0 (1.3) ,
0),,(,0),,(
2
1
zyxFzyxF
(1.4)
.-
.n = 1 (1.2)
x = x(t), y = y(t), z = z(t) (1.5) .
M t ( t) (1.4) -
. (1.2) n = 2 -
:x = x(t1, t2), y= y (t1, t2), z = z (t1, t2)
, t1, t2- (1.3) ..
1.1. - ,, , - ,, .
.:
ltrr 0 (1.6) 0r l , r – . r
- -.
- -. - 321 ,, rrr
:
9
0),,( cbar , (1.7)
13121 ,, rrcrrbra .ar , b c
(1.7) ,cvbuar
cvbuar (1.8) , u v – , r –
.
§2.
-.
,.
1.3. . )(ta – , 0a – , t0 – . t t0 c
0)(lim 00
atatt
(1.9)
, 0a )(ta .-
. 1.2. ,
,0)(lim
0
atatt
(1.10)
.0)(lim 0
0
atatt
(1.11)
. (
2 ) (1- ):
00 )()( ataata.
(1.9) (1.11)- . -),(),( tbta - , m(t),
10
n(t),… t- t0-
0 0 0 0, ,.., , ,..a b m n - ,,...)(lim,)(lim 00
00
btbatatttt (1.12)
,...)(lim,)(lim 0000
ntnmtmtttt
1.3. -- -
. , )(ta )(tb -t t0 c
- t t0 c a0b .
. - :).()( tbta -
.)()( 00 btbata
(1.9) (1.12)-.
. (1.13) 1.4. - , m(t), n(t),…
t- t0- - . t-
t0- ))(),((),()( tbtatatm , )()( tbta -
000000 ,, babaam ..
00000 )()()()()( amtmatatmamtatm
00000 )()()()()( amtmatatmamtatm
. (1.12) -0)( ata 0)( mtm
0tt .
0000 )()()()( btbatabatbta
0)()( 00 batbta
)(lim)(lim)()(lim000
00 tbtabatbtatttttt
),(),( tbta
,...,,...,, 0000 nmba
11
0)()(lim 000
amtatmtt
.)()(lim 000
amtatmtt
,
00000 )(,))(,)((),()(),( btbatbatabatbta .
00000 )(,))(,)((),())(),(( btbatbatabatbta
p q
),())(),(( 00 batbta0)(( ata )(tb +
0a 0)( btb .
(1.12) 0)(( ata 0)( btb -
,.))(),((lim 00
0
batbtatt
p q
qpqpqpqp ,sin
00000 )()()()()( btbatbatabatbta
000 )()()( btbatbata
.)()()( 000 btbatbata
(1.12),
.))()((lim 000
batbtatt
.
qpqpqpqp ),cos(
12
§3. - .-
- 1 --
.1.4. .
)()(lim 00
tatatt (1.14)
, )(ta - 0tt .
1.5. . a -
)()()(lim 0/
0
0
0
tatt
tatatt (1.15)
, )(ta 0tt -- .
)( 0ta - 0tt. )(ta -
21 ttt t ,)(ta t- - . )(ta
,)(ta -
)(ta .
- . -,
.- , -
:1) - -
;2) - ( , -
) )()( tatm - ;3) - -
;4) -
- .
13
-.
)(tr , dtrd
.
1.5. , -,
...)()(....)()(dt
tbddt
tadtbtadtd
(1.16) :
.....)()(.......)()(
0
0
0
0
0
00tt
btbtt
atatt
batbta
1.3- 1.5- . 1.6. - -
)()()()())()(( tatmtatmtatm (1.17) .
0
00
0
0
0
00 )()()()()(tt
atatmatt
mtmtt
amtatm
..
, constcconst,
cdt
tdmctmdtd
dttadta
dtd )()(,)()(
(1.18) 1.7. -
:
dtbdab
dtadba
dtd ,,,
, (1.19)
dtbdab
dtadba
dtd
14
,)(
,)(,)(,)(),(
0
00
0
0
0
00
ttbtb
atbtt
atatt
batbta
0
00
0
0
0
00 )()(
)()()(tt
btbatb
ttata
ttbatbta
. 1.8. 1) t = t(s) s=s0
)( 0stts ; 2) )(trr - -c 00 )( tst )( 0
/ trrt .)(strr - c s = s0
)(tr )(st -,
///sts trstr (1.20)
0
0
0
0
0
0
ssstst
tttrtr
ssstrstr
, 00 stt .-
,. 1.5–1.8 - -
-.
cbacbacbacba ,,,,,,,, (1.21) .
-. :
,,,lim0
uu
rur
uvurvuur
(1.22)
.,,lim0
vv
rvr
rvurvvur
1.9. tr - -
x = x(t), y = y(t), z = z (t) .
15
. . x = x(t), y = y(t), z = z(t) ( ) ,
( -)
ktzjtyitxtr
- ( ).. tr - ( -
) . ( -)
ktrtzjtrtyitrtx ,,,,,
( ). .
§4. -
tr -. t
tr -OMtr
. t t, tt ttr
1 .
trttrMOMOMM 11
1-c 2-c
, tr - 1
.
16
ttrttr
(1.23) 1MM , ,
. t . tr -
0lim0
trttrt ,
1 0t -. , 0t 1 .
-.
t (1.23) tr -, 1 .
1 - – tr MN
(2- ).1 MN
..
1.10. t tr -tr , ,
.
vurr , -ur -
.0vconstv , u
0, vurr .vur ,
L .
ur
L
( «u» «v = const » ).
vr «v » «u = const
» .
17
§ 5. -
TtT0
trr - .
ktzjtyitxtr
. x, y, z – r
kji ,, . 1.9-x = x(t), y = y(t), z = z(t)
. -, :
,!
1!1
1...
...!2
1
*11
2
nnnn ttxn
ttxn
ttxttxtxttx
,!
1!1
1...
...!2
1
**11
2
nnnn ttyn
ttyn
ttyttytytty
,!
1!1
1...
...!2
1
***11
2
nnnn ttzn
ttzn
ttzttztzttz
t*, t**, t*** t tt ,, - .
i -, j , -
k
ktzjtyitxn
tktzjtyitxn
tktzjtyitxtktzjtyitx
ktzjtyitxkttzjttyittx
nnn
nnnn
******
1111
2
!1
!11...
...!2
1
18
...!2
1 2/ ttrttrtrttr
nn
nn tqn
ttrn !
1!1
1... 11
(1.24) .
* ** ***n n nnq x t i y t j z t k . (1.25)
, x(t), y(t), z(t)TtT0
, t-. nq t*,
t**, t*** ,,
nnnn
n Ctztytxq 2***2**2* ][|| (1.26)
n c – [T0, T] ttt .
, (1.24) -n- . -, nq tr - ttt,
t0 n .
§6. -
- tr -, n = 2 -
:
.2
2
2tqttrtrttr
.2
2
2tqttrtrttrr
(1.27) .
trttrr tr -
1 1MM
. ttrMM -
19
tr -
« » (3- ).2
2
21/ tqMM
– . – t.
1MM ,t
1MM t-1MM
.1.6. . ttrMM tr -r tr -
.r - rd .
dttrrd , (1.28) t dt ,
.
dtrdtr
: tr dr -
dt1
.
§7. - .-
.1. . tr -tr tr .
. tr -,
consttrtrtr ,2
.t ,
0, trtr
20
«
t
l
« »
, 2.
3-c
t 0
2
2sin
lim0t
.
tm
t
tm.
.. 4-
1 (1
.
-
.-
tmm
m
0
.
tr.
m
-t 0
sin2m
2sin2
0
).
md
t
-
2n
2
2sin
.
limt
tr
,
ttm
m
4-
1m0
m
t
tm
c
.
.
,
.1
-.
-
21
e - .Oz -
. e -- .
sincos jie (1.29)
(5- ).
sincos,cossin // jiejie (1.30)
2/ ee
,.
: 3. (1.29) e -
e2
.
§8. .
n- , nR-
: nRM1, 2, ..., n .
nee ,....,1 -.
, –
1, 2, ..., n ,)
. ( 1, 2, ..., n),( 1, ..., n) Qxi = yi, i = 1,… n ;
b) , xi ( 1, 2, ..., n).
ee
22
., .
,.
(-
),.
: Rn-, nR, -
. -, ,
.
,,,,
,...., 11 nn
nn ,...,,,...., 11
.,
- .-
., , -
, -.
R , ,C222 Ryx ,
, , b
12
2
2
2
by
ax
.
23
. ,
022 xyarctg
eyx.
,, –
-.
sin,cos ryrxr,
( ).er
,( 1, ..., n) (z1,…, zn)
.-
.xi = xi(z1,…, zn), i, j = 1,…., n, (1.31)
zj = zj(x1,…, xn).
:xi = ai
jzj i, j = 1,…., n (1.32) ( j ).
(1.32) 1, 2, ..., n
z1,…, zn
= (aij) .
= AZ, X = ( 1, 2, ..., n), Z = (z1,…, zn). Z- A = (ai
j)
B= A-1 = (bij).
ki
kj
ji ab i, j, k = 1,…., n
24
,
kiki
ki
,0,1
– , j ., ,
:k
kjj xbzBXZ , .
xi =xi(z1,…, zn ), i = 1,…, n , xi(z1,…,zn)
( ) .
1.7. .nn zzzz
j
i
ji
zxaA
010
1 ,.....,
( z10 ,…, zn
0i(zi
0,…, zn0) = xi
0, = 1, 2,…, n) , (x1
0, …., xn0)
(z1,…, zn).
-
zxJ . -
JzxJ ˆdetdet .
() :
xi = xi(z)
x 0 = i(z10,…, zn
0) 0det0
10
1 ,....., nn zzzzzxJ
, ( 10,…, n
0) z1,…,zn zi=zi(x) 1,…, n
, zi0 =zi(x1
0,...., n0), i = 1,…, n
j
i
ji
xzb
25
, ,
l
k
lk
zxa (1.33)
,
ikk
j
j
i
zx
xz
(j ).n = 1 :
= x(z)
00zzdz
dx
, z0=z(x0) 1dzdx
dxdz
0
z=z(x) (z- x).
-X = AZ, xi = ai
jzj.
dzdx
,
k
iik dz
dxa . 0det A ,
Z = BX, .
:1. . 1 = , 2 = , n = 2.
,sin,cos ryrx ,01 rz .2z
j
i
zxA :
26
cossinsincos
rr
yry
xrx
A
.
0det rzxJ
.r = 0 . r > 0 (
) .20,0r -
, .-
.2. . ,1 rz ,2z zz3 -
1, 2, 3
zxrxrx 321 ,sin,cos. r = 0 , z
« »., :
1000cossin0sincos
rr
A
r = 0 .r > 0 .
20 .3. 321 ,, zzrz
,sinsin,sincos 21 rxrx
.0,20,0,cos3 rrx
:
27
0sincossincoscossinsinsinsinsincoscossincos
rrrrr
A
AJ detsin2rJ
.,0r ,0
.r = 0 ( , – ) = 0, (r, – ) – -
.
5-c 6-c
28
.
§1. . .
,-
..
– .
.2.1. . – , D –
. D ,
,.,
xzxy , (2.1) , OxD
.
trr (2.2) - .
rtfztfytfx 321 ,, (2.3)
. – = y(x) (y(x) –
) M(x,y) -,
y/(x) 0 . = x(y)
.t f/
i(t), (2.3)- t- ( ,
29
t = t(x)) (2.3)- ,(2.1)
y = f2(t(x)), z = f3(t(x)).
ktfjtfitftr 321/ )()(
tr 0 .tr 0 t- -
(2.2) -.
2.2. . ( ) -trr -
. tr 0 , tr = 0 .
( ) tr 0 .
.. tr 0
,. r -
t ., 3t
.3 /2trtrr
00
/
ttr
.00
/
tr
0),,(,0),,(
2
1zyxFzyxF
(2.4) (Fi – ) -
( 0, 0,z0) .
,
zF
yF
zF
yF
22
11
30
7-c
, (2.4) z- . (2.1)
. (2.4)
zFzF
yFyF
xFxF
2
1
2
1
2
1
(2.5) .
kzFj
yFi
xFgradF
(2.6) ,
021 gradFgradF (2.7) .
trr (t = t0) .
00 trtrR
. R – - ,.
«0» , (6- )
rrR (2.8) .
-. -
-
0, rrR (2.9) , R –
N -(7-c ).
31
.1 .
§2.
trr (2.10) , t
, t
trdtrd
(2.11)
.t s
0, tdtdsts
. (2.12) tr
(1.8 )
dtds
dsrd
dtrd
(2.13)
. (2.13)
,222
dtds
dsrd
dtrd
(2.14)
dtds
dtrd
dsrd :
tdtds
.
/(t) dsrd
: .1dsrd
dsrd
(2.15)
.. s .
32
(2.14) 12
dsrd
22
dtrd
dtds
(2.16)
kdtdzj
dtdyi
dtdx
dtrd
2222
dtdz
dtdy
dtdx
dtds
(2.17) ,
2222 dzdydxds (2.18) .
() ,
,. (2.18)
.(2.16) 1(t1) M2(t2) -
dtdtrdds
dtrd
dtds
(2.19)
dttrs2
1
t
t
(2.15) :2.1. - -
– .
§3.
- ,--
.
:
(2.20)
33
1, 2
dsrd
(2.21)
s ,
dsd
dsrd2
2
(2.22) . s
, -.
(2.22) v, k .
:
vkdsd
. (2.23)
1- ,dsd
,v
0v . (2.24) v – -
..
dsdk
dsdv ,
. , -.
(2- ) -. - -
sk
s 0lim
2- 1lim0
mt
sk
s 0lim (2.25)
.k
-
34
§4
(
s
8-c
4.
– v
(,,
.
,
,
2-
k
i
v
.
– 1, 2,
dsdk
.
a
v
v sin
,1
).3
s
.
.
v
v,n
.
(8-c
9
).
.
9-c
.
-
-
(2.26) .
-
--
35
( ,, ) -).
(2.23) . -
, , dsvd
,,
. vdsvd
,dsvd v ,
cadsvd
(2.27)
. 0v (2.24) -
0dsvd
dsdv
. (2.23) (2.27)
0cavkv
0,122 v
+ k = 0, a = – k. æ ( ) ,
:
kdsvd
æ . (2.28)
(2.26) -:
dsvdv
dsd
dsd
.
(2.23) (2.28)
ækvvkdsd
vvv ,0,0
36
.ævdsd
(2.29)
(2.23), (2.28), (2.29) (2.21)
,æ,æ,,dsdk
dsdk
dsd
dsrd
(2.30) .
, { ,, }.
,, k æ ( -
). – k – , –
æ – . (2.29)
dsdæ .
,,
|æ|ss 0
lim (2.31)
. , --
ss (9-c ).
§5.
srr (2.32)
. -.
« » -, ,
.
37
0 – n . s s
0(s0) (s0 + s). 0 -
- (10-c ).
2.3. . sn + 1 , 0n- . ( s-
1- ).2.4. . 0 -
0 -.
d = PM, MM 0 -
,MMnd 0
.MM 0
ssQssrsrsrssrrMM ,62
103
32
00000 .
0030,lim rssQ
s . (2.30)
0000
000
0 , vkdsd
dsrd
dsdr
dsrdr .
33
20000 6
121 sQsvksMM
33
2000 6
121 sQsvksnd
(2.33) 000000
2030
ælim kvkkQs
.
:)
00n
38
, -, d- , .
.b)
0,0 00 vnn (2.34) , (2.33) -
. d-
002
21 vnks
.. . (2.34) n
0 ,0 0 ,
0 .. , -
.-
.)
0,0 00 vnn, 0 -
, .d- (2.33)
00
3
000
3
3
3
æ6
æ66
ksnksQns (2.35)
, .
, (2.35) k0 = 0 æ0 = 0 (
) ..
2.2., . -
t -tr -
t s -:
39
§6
dsrd
.,
6.
,
dsdr
dtrd
dsrd2
2
,
-,
.
10-c
;dtds
vkdsd
v
:
N
R
xx
xX
2
2
2
dsrd
dtrd
v .
N
rN
,, rrr
yy
yY
,
2
2 dtdsr
.
N
r .
0
zz
zZ
,
2
dtd
dsrd
,
0
11-c
,
.
,2
s
2
2
dtrd
-
--
-
(2.37)
(2.38)
(2.39)
--
(2.36)
40
,
(11-c ).-
. -. , , v -
. – -
– (§1, (2.9)); (12-c ).
-. .
-. (2.8)
, (2.9) , - (2.38) .
(2.37) rr -, -
. ,rrrR (2 40)
.
:rrvrv , ,
,rrr (2.41)
,
rrrrR (2.42) .
rrr ,
0,, rrrrR (2 43) .
rr , --
.
41
§7
(2
,
7.
2.25), (2
.,
zzyX
X
,
12-c
2.29)
:
xxX
xzyzx
xxX
)
: s = t.
,:
yyY
zxxyY
zyY
.
.
– l
zzZ
,
yxZ
zzZ
,
,
1
(
xyz
,0z .
13-c
( (
(
.
-
(2.44)
(2.45)
(2.46)
-
-, -
--
))
.
42
,
dsMPd
MPdsMPd
(2.47) (13- ).
,, , P
constMP .
, MP -1-
MPdsMPd
,
dsMPd l
. dsMPd
.
, ,MPdsMPd
,,
. ,
.
lPMPMPMPdsMPd ,,sin
( dsMPd
– , | | –
, lP, – l) .
,MP ,, , (2.30) (2.47)
,vkdsd
43
,æ vkdsvd
(2.48)
vdsd æ
. æ k (2.49)
. (2.49)
.-
. -.
§8.
.s ,
srr.
(2.30)
æ,,,, 23
3
2
2
2
2
kvkdsdk
dsrd
dsrd
dsrdkvk
dsd
dsrd
dsrd
.
2
2
dsrd
dsrdk
,3
3
2
2
2 ,,1æds
rdds
rddsrd
k .
. 2
2
dsrd
dsrd
1dsrd
2
2
dsrdk
(2.51) t ,
(2.50)
44
trr.
(2.16) (2.19) dttrds
, rdsdt 1
.
trr - s - ( – t )
,r
trdsdt
dtrd
dsrd
,22
22
2
2
2
2
2
2
rr
dstdr
dsdt
dtrd
dstd
dtrd
dsrd
33
33
3
3
2
2
3
3
3
3
rrrA
dstdr
dsdt
dtrd
dtrdA
dstd
dtrd
dsrd
( ). (2.50) ,
3r
rrk , (2.52)
2ærrrrr
(2 53)
.
§9. .
(2.30) ,,, ,
,1222 0 (2.54) .
,,,
.
62æ
rkrrr
45
.,
/33 eOe
(14- ).
21,ee/2
/1,ee -
3e /3e , 3e
/3e .
, , /3e
3e .- 3e
Oe1
, /3e
/1eO
, /1e 1e .
/1e , /
2e , /3e 1e , ,2e 3e .
3,, e /3,, e -
,.
( 1e , ,2e 3e ) ( /1e , /
2e ,/
3e ) ( 1e , ,2e 3e ) ( 3,, e ) ,
/3,, e - , ( /
1e , /2e , /
3e ) .:
– 3e , – ,/
3e .
1e , 2e ,
:cossin,sincos 2121 eeee (2.55)
, 3e /3,e -
:
46
.cossin,sincos 3/
33 eee (2.56) , /
1eO/
1e , /2e :
cossin,sincos /2
/1 ee (2.57)
, , ,sincoscossin 321 eee ,cossincossin 321
/3 eeee
,
,sinsincoscossincossincos 32121/
1 eeeeee,cossincoscossinsinsincos 32121
/2 eeeeee
.
,sinsincossincoscossincossinsincoscos
32
1/
1
eeee
,sincoscoscoscossinsincoscossinsincos
32
1/
2
eeee
cossincossinsin 321/
3 eeee (2.58) .
( 1e , ,2e 3e ) ,,, ( /
1e , /2e , /
3e ) (2.58)
.
§10.
,skk
,.
3,2,1itfx ii (2.59)
.
æ = æ(s)
47
(2.30) . - (2.58) /
1e , /2e , /
3e ,,, 1e , ,2e 3e -
kji ,, , (2.58)
,sinsincossincoscossincossinsincoscos
kji
,sincoscoscoscossinsincoscossinsincos
kjiv
(2.60) cossincossinsin kji
.x, y, z
(2.60) – - , (2.30)-
.sinsin
,cossincoscossin
,cossinsincoscos
dsdzdsdydsdx
(2.61)
(2.54) p a. v
,
æcos
,ctgæsin
,sinsinæ
dsd
kdsd
dsd
(2.62) . , (2.60)- -
,
,sincos,, kkvkkdsd
,cossinsincossinsindsd
dsd
dsd
48
,æcossinsin,æ, kkkkdsvd
dsd
dsd
dsd coscossinsinsincos
,
cossinsincoscossinsin
sincoscossinsincos
kdsd
dsd
kdsd
dsd
(dsd
dsd ).
, -
coscossinsin-cossinsincos
coscosæcossinsincossinsincos
kk
dsd
;ctgæsincossin
cosæsincossin 2
kk
cossinsinsin-æcossinsin
sincossincosk
k
dsd
æcoscossin
sincosæcos
.(2.61), (2.62) ,,,,, zyx -
, -.
, (2.61), (2.62)
000
000,,
,,, zzyyxx
(2.63)
sssszzsyysxx ,,,,,- .
(2.63) -: 000 ,, zyx 0 -
, 000 ,, (2.60)
49
0.
.2.3. 1- ( 1-
) s, 0
- .000000 ,,,,, zyx -
.:
,,,,,, 0000001 zyxP,,,,,, 0001002 zyxP,,,,,, 0001103 zyxP.,,,,, 0010004 zyxP
1, 2, 3 – , 1, 2 -OZ ,
1, 3 = x0; 1 4 –
.-
,. , (2.61), (2.62)
, ( k æ- -).
,. (2.61), (2.62)
0, /0 l l/ ;
l0
/0 .
l l* .(2.61), (2.62) , (
-, ( ) -
(2.30) (2.61), (2.62) ) l/ *,l- (2.61), (2.62) -
, l/ l*
50
/0
*0 TT . (2.61), (2.62)
. l/ l* .2.4. - -
. ,1 ,01 sf sf 2
,sfsfk 21 æ,
- ., (2.61), (2.62) (2.63)
: (2.61) l x, y,
z , (2.62) ,,
1e , ,2e 3e, (2.60)
/1e , /
2e , /3e
.(2.61)
1222
dsdz
dsdy
dsdx
,
22222 rddzdydxds.
ds – l .(2.62) (2.30) -
. (2.60)
/3
/2
/1 ,, evee .
/1e l
.: -
/2e – , sfk 1 – l,
veee /2
/1
/3
, /3e – , æ sf2 –
l .
51
14-c 15-c
§11.
2.5. tyytxx ,: t -0,,: tyxt (t – -
)t
, ,
t (15- ).
,tx ty -0,, tyx
0,, ttytx
- . -0tyx yx
.0yx yx . , yx ,
, xy , – t
. ,
.xy
yx.0yx yx tyytxx ,
,0,,,0,, tyxtyx t
. .
.tyytxx ,
52
. /, / – -,
0) ytyyxtxx. t
02/2/ tytxtytyytxtxx
., - , -
xyyxxyxtyy
yxxyyyxtxx
2/2/2/2/,
. ( 0yxxy).
.
§12.
srr () . -
vk1 -
..
: , - 2- , -
, , -.
1-.
,, .
. ,
,1~ vk
rr
vk
kk
vk
rr 111~
53
16-c .
. ,,
.0k , bsa
b
a
b
a akbkds
kdsr 111~
, ,.
.srr . -
, s < 0 , s, s > 0 , -
..
..
,.
. ,,~ srr
vskvskrr~
(16- ). ,.
§13.
.
(§10) --s
k > 0 æ
54
.k(s) æ(s)
. (2.64) -
.kæ 0 . k = 0
0trtr
. ,. -
. .
2
2,
dsrdr
dsrdr
,rr .1r
0rr
02
2
dsrd
.
21 cscsr ( 21,cc – )
1c.
æ = 0
( )0dsd
.
« »,
, æ 0 . -æ = 0 ( ) .srr 0 -
R0 .0 s . s- s, / ,
55
ssrMOsrOM ,0/
0 -
srssrOMOMs 0
. sr - s
461
21 32 ssrssrssrs
, [n] n- .
.æ,
,
kkvksrvksr
sr
4sæ613
213 32 kvskss
. (2.64)
sk æ(s) ,, s
. 2.4 .
(2.64) srr.
s R0 , , z,
4æ 61
,321
,3
3
2
skz
sky
sx
(2.65)
.ss
3sx (2.66)
56
: 0s, 0s – .
0.
, s s
321 2sky (2.67)
( 0k ). 0
. v , -: v
, ( )., s
4æ 61 3skz (2.68)
s . , 0
. æ.
æ > 0 ; 0k , 0s (2.68) , 0s , (2.68) .
æ > 0 0s s, 0s ,
z . R0 17- .
æ < 0 , 0s (2.68) ,0s – . 0s s
z , , , 0s, , z .
R0 18- .--
« » « » -, « » « » . æ > 0
æ < 0 – (19- ).
57
(2
17
.2.65)
–
æ > 0
1
-c .
z
.
(20
0
9-c
z = 0
,0- ).
æ
21
z
sky
sx
21
z
y
.
æ < 0
0
,3
,3
2s
0
,2kx
18-c
20-c
c .
RR0 -
,
(2.69)
(2.70)
-
58
(2.65) = 0
,3sx
,4æ61 3skz (2.71)
0y
0
,æ61 3
y
kz (2.72)
, – 0
(21- ).
21-c 22-c
(2.65) = 0
0,4æ61,3
21 32 xskzsky
(2.73) , , –
0 1- (22- ).
§14. .
-. (§10)
,
sææ,skk
59
, 0sk , æ(s) -. k
æ ( -) ,
. (§13) k = 0
. k = 0 æ = 0 (2.74)
.
.R0
....246
24
13
2
ttrttr
ttrttrtrttrt
V
(2.75) .
(2.52) æ = 0 0,, rrr ,
rtBrtAr .
rBrBArAr V1
r rBrA
rtBrtAr V11
1
.( tr ) (2.75) -
-
0,, rrR (2.76)
, R0.
tt (2.76) ,
.æ = 0
60
kdsvdvk
dsd
dsrd ,,
(2.77) , const
- . s, k -. k = 0 -
() . -
v. ,
v , - v -.
(23 2 – ). k > 0 ,k , , .
v (§13) 23- - .
k ( ,
§3- )., v ji ,
. (24- )
jivji
cossin,sincos
(2.78)
23-c 24-c
61
s i . (2.78) -(2.77)
,cossincossin jkikjdsdi
dsd
dsdk (2.79)
.s (
* = + const , (2.79) ),
.,
(2.52) .
tyytxx , (2.80)
.
tgdxdy
( – t )
yxarcctg
xyarctg
, (2.81)
dtyxds 2/2/ (2.82) (2.79)
2/32/2/ yx
yxyxk (2.83)
.
§15. .
constkæ (2.84)
..
.
62
(2.84)-,
const 0,æsincosk. (2.30) 2- cos - ,
sin - - ,
,0sincossincos vakdsd
dsd
0sincosdsd
.
constm sincos (2.85) ,, constm
m ., ,
., m
= const ,.cos, constm
s ,
0,vmk. k 0 , m
,sincosm
. ,,
0sinæcos vvk
,0sinæcosk
constkæ
. (2.85)
, (25- ).
63
. mz
,x = x(s), y = y(s)
.kszjsyisxr
. z(s)
cos,, kdsrdk
cosdsdz
.constccsz cos .
kcsjsyisxr cos (2.86) . s – , – -
, x(s) y(s)22 dsrd
, cos,, // yx
1cos22/2/ yx
constyx 22/2/ sin . (2.86/)
., (26- ).
Oz, ,
CMeajiaCM sincos
64
. -,
25-c 26-c
= b .-
CMOCr ,
kbear (2.87)
. e – - ( - , §7), Ozk .
1 b . 2b ,
.. jyixr * -
s* , (2.86/)
2222/2/2* sin dsdsyxds (2.86//). ( – s ).
(2.51)
2*
*2*
ds
rdk
.
65
0,,sin
10,,
,cos,0,00,,cos,,
**
*yx
dsdsyx
dsrd
yxyxdsrd
cossin*
*k
dsrd
dsrd
.
.sinsin 22*
*2*
2*
*2
2
2
dsrd
dsds
dsrd
dsrd
24
2*
sin1 kk
.
,.
, ,
jtitajyixr sincos*
.
dtadtttadsrd 22** sincos.
, (2.86//)-
sinsin
* atss
. (2.86)
kctctgajtitar sincos
. bctga Oz, -
(2.87)
kbtjtitar sincos
.,
66
constconstk æ, – . -
.
§16. .*
.srr
.
vrr *
(27- ).
27-
– .
dsdv
dsvd
dsrd
dsrd *
v , *
,0dsd
= const . *
.*
,
67
cos, *.
dsvkddd *** ,,,sin *** , dsvk
. *|| vv -
.0d* -
, -.
* * -,
sincos*
. *
,* vrr,sincos* (2.88)
,* vvcossin*
. – .
vrr*
dskdsds æ**
dskds æsincos *
.
sinæ
cos1 k
(2.89)
, -
.,
(2.89)
68
.(2.88)- , ,
,--
.
.æ + k + = 0 (2.90)
0, 0, 0 (2.89) .
(2.90) , , (
) .
69
.
§ 1. .
( ).
,
..
. –
.,
, ,
., , ..
,er (3.1) r – - , –
M~ - , e; ,
(3.1) . 0
er 00- 0 (28- ).
e t, (3.1)
tetr (3.2) ,
.(3.2) r (t )
.
70
,,
.3.1. . -
.
, t.
: (t)tettt*
tetr * (3.3) (3.2)-
( – ), t- (3.2) (3.3) (29- ).
28-c 29-c
(3.2) tl.
§2. ,.
.,
,
.
71
«» ,
- ..
3.2. . tlttl (
) tl.
3.3. . tltl
..
,- ,
. ( )
,
. , () - ,
,.
.,
constte 0te .(3.2)
.tl ttl
ttettrtetr ,
(3.4)
.
tteyttrtextr
2
1 ,
( , t t - ).
72
12 rr (3.4) -,
0,,0,
12
12
tterrterr
0,0, 1212 tetterrterr .:
0, tettetextteyttt
.0),(
tettetexytettexytettexttt
(3.5)
0t , 12 rr ,
.lim,0lim00 t
xyxytt
(3.5) t- 0t,
0, tetetet M (3.6) x
tM 0lim – .-
.lim 10tetr MtM (3.7)
tete 0)(te ( -) , (3.6)-
2
,e
eM (3.8)
. tl
ee
eeMM 2,
(3.9)
,
:
73
eetet
tettetettet tt 00
lim1lim
tlMr (3.10)
,eee , (3.11)
er (3.12) (3.8–3.12)- e t - . -
,eer
(3.10) .
§3. .
. ( ) :
321 ,,,z3 ,
,3 ez -
. – ,
,
Mz.
M = 0. , M z
0M (30- ), z M
M .
M = 0 (3.8) 0, 3dzd (3.13)
.
74
30-
2 -
33d ,
332 d (3.14) . -
321 e.
3.4. .-
.
,, 33
122
11 dtddtzd ii
dtddtd 2231
1333
321
212 , (3.15)
zd 3d (3.13)- .
.0,0 23
223
113
03e 03d ,,01
3
01 . (3.16)
,33
22 dtzd
,33
122
11 dtd (3.17)
,12
12 dtddtd 1
313
. , ,0i
jj
i , ji (i, j = 1,2,3) –
31
13
12
21
33
22
11 ;;0
(3.17)- .. .
1) s.
75
dszd { 2(s)}2 + { 2(s)}2=1.
2))()( 3 ttr (3.18)
,
..3
13 dtdds (3.19)
dtdsbap 313
1
21
31
3
31
2
,,,
,
13
12
321
32 ;;;ds
dbds
dbds
dapds
zd (3.20)
.a s p,
b, a ,p=p(s), b=b(s), a=a(s)
.-
. (3.20) -: 1
l(s) ..
r = z + 1 (3.21) .
,.
2zr (3.22) .
0, 1zr (3.23) ,
(3.24)0, 2zr
76
. , -- -
., b, a , ,
., -
.
§4. .
. (3.20)-
;2
dsdb .3dds
,
0lims
b , (3.25)
– , - .
,3 sszr (3.26)
sssszr 3
.,
sss
sssszssz
33
33 ,,
. s3
ss3 -sss 33 .
/33
/33
/
0
,,z
sim
s
. (3.20)
77
ims
p0
(3.27)
,
-. (3.20) ,
cos;sin2222 ap
aap
p (3.28)
. 0= tg . (3.29)
. = 0 ,
dsds
dszda 1 (3.30)
ds1 – ,ds –
.
§5. .
sszr 3
, ssz 3, s( ) .
(3.20)
dsd 33
132 (3.31)
:
,,,, 33
2ds
dds
zdds
zdp
,,,, 23
23
321
dsd
dsd
dsdb (3.32)
78
., 3ds
zda
tltr,
– , – , t – , (3.19)
ddds 3 (3.33)
ez MM (3.34)
22/
,,
ed
edd
e
eM (3.35)
t . (3.32–3.35)
;`
,,,,,,,,2/
//
22233
e
ee
ed
eedd
ed
eededdsdzd M
;,,,,,,3/3
2
33
233
e
eee
ed
edededs
ededeb
/
//3 ,,,,
e
eed
deded
eeddszd MMM
(/
ee )., t
:
e
ea
e
eeeb
e
eep
M/
3/2/
,;
,,;
,, (3.36)
79
2/
,
e
eM .
§ 6. ( ). .
.(2.30) (3.20) , = 0
= 0 -. .
= 0 (3.37) -
. 0 ,ads = ds*
(3.20) -
1*3
1*2
32*1
3*1;;1;ads
dab
dsd
aab
dsd
dszd (3.38)
.((2.30) )
;*, ssrz
;1,21 ka
m (3.39)
æ,32abm ,
13 m.
: ( 0 ), -
. ,.
, ( )
. (3.39)- -,
.
80
, b -. ,
- s.
= 0, = 0 (3.40) ,
0zd, . , (3.40)
.,
.
. (3.2) (3.36, 3.37)
0,,//
ee (3.41) .
, conste , 0/e( 0/ ). (3.39)-
= 0, b = 0, 0 (3.42) ,
-. (3.42)
.
..
(; ).
-, .
.
§7. . .
= 0, p 0 (3.43)
,
81
. (3.20)
;,;*, 21 kabmdspdssrz
æ1,, 3312p
mm (3.44)
.
,
dszd
2.
. -
: 0
0, 3ds
zd
= 0 .
= 0, b = 0 (3.45)
.,
. , (3.44)- k=0,.
,, ,
.
0,tekctr (3.46)
, tjtite sincos – - 7- -
– , kji ,, – , – const. = const (3.46)
( , §15) .
82
.(3.36) (
kct )
;,, / ceekcp ;0,,/ eeeb
.0,;0,/
ekcaekcM
.-
.
§8. . .
-. ,
- .b = 0 (3.47)
. . - – -
. b = 0 (3.20)-
02d ,
2 , 3
( 2 - ) ,2 - .
..
b = const 0 (3.48) .
,,
32 bm.
, (3.20), (3.48)
83
032
dsbd
dsd
dsmd .
.1
,cos2
3 constb
bm
m.
( 3 ) .
84
IV . -.
,,
. n- Vn
neee ,...,, 21
( , , ,)
.
§1 .
4.1. . L x, y, z
B(x + z,y) = B(x, y) + B( z, y); B(x, y + z) = B(x, y) + B(x, z);
B( x, y) = B(x, y); (4.1) B(x, y) = B(x, y)
L-x, y B(x, y)
., L
x y x, y
B(x, y) .4.2. . L x, y
B(x,y) = B(y,x); (B(x,y) = –B(y,x)) (4.2)B(x,y)
( ) .L – n- , ( neee ,...,, 21 )–
i iiiii n,...,2,1i,ey,ex
.
i kikiki
kkkii nkieeByxeyexBB ,...,2,1,,,),(,
,. (4.3)
85
bik = B(ei, ek) (4.4)
B(x,y) = n
kikiki yxb
1, (4.5)
, B(x, y).
n
jjjii ePe
1
/ (4.6)
.x = xi
/ei/ , y = yi
/ei/, = 1,…, n.
B(x,y)n
kikiik
n
kikiki yxbyxbyxB
1,
///
1,, . (4.7)
B(x,y) ( bik bik
/). bik/ -
bik . (4.4)--
.
bik/=B(ei
/,ek
/). (4.8) (4.8)- ei
/, ek/ (4.6)
:n
j
n
lllkjjiki ePePBeeB
1 1
// ,,
B(x, y)
B(ei/,ek
/) =n
1l,jPijPklB(ej,el),
bik/ =
n
lj 1,bjl PijPkl (4. 9)
.. (4.9) , (4.7) .
;1
/n
iijij Px
n
kklkl Py
1
/
(4.7)-
86
n
lj 1,bjlxjyl =
n
lj 1,bjl PijPkl xi
/yk/=
n
ki 1,bik
/xi/yk
/.
(4.9) . , (4.7)- (4.9) , (4.9)- (4.7) .
(4.9) , (4.9)
.B/
B
B/ = P PT
. (4.9)
bik/ =
n
1l,jPij (bjl Pkl) (4.10)
cjk =n
1l
bjl Pkl (4.11)
, (4.10) bik
/ =n
1j
Pijcjk (4.12)
. (4.11)-,
., -
. (4.11)- - (
) . (4.11)
= (4.13) . (4.12)
,/ = . (4.14)
(4.13) (4.14)-/ =
.
87
§2. . ( ). .
4.3. . ( 1, ... , n) ( 1, … , n) ( 1
0, ... , n0)
. ( 1, ... , n) (z1, ... , zn) i (z10, ... ,
zn0) = i
0 x = x(z) , z (z1
0, ... , zn0)
i
(4.15)
( 1, … , n).
(4.15) -.
– -. 1, ... , n
nxf
xffgrad ,...,1 (4.16)
, ixf
f i
. j = jxf
, j = 1, …, n
. x = x(z)z1, ... , zn ,
.
grad f( 1(z), ... , n(z)) = nz
fzf ,...,1
; i = 1, … , n (4.17)
. izf -
i
ji
j
i zx
(4.18)
j
zzj
ii
kkzx
0
i
j
ji zx
xf
zf
88
.
. .1,..., n , 1 = ( 1
1, … , n
1) 2 = ( 12, … , n
2) ( 10, ... , n
0) - .
x = x(z), x(z0) = x0 (z1, ... , zn) ( 1
1, … , n1) ( 1
2, … , n2)
,
. ( ija ) kk zz 0 (k = 1,..., n)
. 1 2
jin
i ijii
211 2121, (4.19)
kjjk
kik
n
i
jij gaa 2121 121, (4.20)
. G = (gij)qk
sjsq
n
i
ik
ijjk aaaag
1 (4.21) .
GG = ATA (4.22)
;ijj
i
azxA ,
– .G gij -
, . y1, ... , yn
zj = zj(y1, ... , yn), j = 1, …, n .
j
iij y
zbB
. 1 , 2 y1, ... , yn
( 11, … , n
1) , ( 12, … n
2)
jij
i a 11ji
ji a 22
zx
89
jij
ijij
i bb 2211 ; (4.23).
( ) -H = (hij) . -
jiij
lkke gh 212121 ),( (4.24)
. (4.23)
)()( 2121lkj
lijik
lkkl bgbh
hkl = bki gij bl
j (4.25).
H = BTGB (4.26) 4.4. . (z1, ... , zn)
(gij)gij = gij(z1, ... , zn), (i, j = 1, …, n) .
(y1, ... , yn ) zj = zj(y1, ..., yn), j =1, …,n ,
gij = gji/(y1, ... , yn) ,
,
j
l
kli
k
ij yzg
yzg /
(4.27) . (4.27) (i k, j l, k i, l j,
gik/ hkl, i
k
yz bk
i, j
l
yz bl
j) (4.25)- .
(4.17) -gij(x) -
jiji
ij xg,
2 )( ( );
= ( 1, …, n). xi = xi(z), (i =1, …,n)
ji
j
i zx
(4.28)
l
j
klk
iklij
xzg
xzgzg )(
/
(4.29)
90
, ,
jiji
ij
jiji
ij gg,,
22 /
. , , - (4.29) , , -
gij .
§3. .
V Wx V
y W A:V W. y = Ax
. e1, …, enx = xkek , ek
Aek = akjej (4.30)
y1, ... , yn
,y =Ax (4.31)
n
kj
n
j
jk
kn
kk
kn
kk
kj
j eaxAexexe1 111
,
jxa kn
k
jk
j ...,,2,1;1
(4.32)
. (4.32) (4.31) -. j
ka(4.32)-
nn
nn
n
n
jk
aaa
aaaaaa
a
...............
...
...
21
222
21
112
11
. e1,…,en.
91
.
..
4.1- (ek) (ek/) -
A*A=PTA*Q (4.33)
A*=QAPT (4.34), P = (Pi
k) (ek) (ek
/) . Pik/
ek/= Pi
k/ei , k = 1, …, n (4.35)
, QT=P-1 .Q = (Qi
j/) n = 3
///
///
///
33
32
31
23
22
21
13
12
11
QQQQQQQQQ
Q (4.36)
,3,2,1,,,,,
/
/
/
/
/
/ jiQPQP ij
ij
ijj
i (4.37)
.. (4.30)-
//~
ii
kkeaAe (4.38)
ei/ (4.35) -.
n
j jj
i
n
i
ikk
n
i ii
kkePaAeePAAe
111////
~;
j
n
i
n
j
n
i
ji
iki
ik ePaAeP
1 1 1//
~
. Aei (4.30) ai
j ejn
jj
n
i
ji
ik
n
jj
n
i
ji
ik epaeap
1 11 1//
~
. (ej) – ,
92
kipaapn
i
ji
ik
n
i
ji
ik
...,,2,1,,~11
// (4.39)
(4.39)- , T, - (4.39)-
PAT= A*TP.
-1
AT=P-1A*TP, a -
A = PTA*Q (4.40) ,
A*= QAPT (4.41) .
ei (4.35) xi
zj
X=P Z (4.42) . (4.42)- -
x1 = P11/ z1+ P1
2/z2 +…+ P1
n/zn,
x2 = P21/z1+ P2
2/z2 +…+ P2
n/zn,
……………………. (4.43) xn = Pn
1/z1+ Pn
2/z2 +…+ Pn
n/zn.
Pij/
(4.42) (4.43) ( ) (z)
j
iij z
xP /
(4.44) . Q
(4.37) (4.44)
ikk
j
j
i
xz
zx
(4.45)
. j
iij x
zQ/
Q -
n = 3
93
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
xz
xz
xz
xz
xz
xz
xz
xz
xz
Q. (4.46)
(4.39) aj
i
k
sj
sj
iik z
xaxza~
(4.47) . , (4.39)
( )
sk
js
ij
ik
lj
js
sk
li
ik
lj
ji
ik PaQaQaPaQPa /
//
/
/
/~~~
j
iijj
iij z
xPxzQ /
/
, ,
(4.47) .A*= QAPT (4.41) .
§4. .
a.
. Ln – n- , e1,…,en – a . e1
/,…,en/ Ln- . e1
/,…,en/
,
nn
nnnn
nn
nn
ePePePe
ePePePe
ePePePe
............................................
,...
,...
22
11/
222
2112
/2
122
111
1/1
(4.48) .
n×n-
94
nnnn
n
n
PPP
PPPPPP
P
...............
...
...
21
22
212
12
11
1
. – /
. ( /), e1
/,…, en/ ,
,det P 0. (4.49)
– Ln- .:
n
ii
in
jj
j exxexx11
., /
/
(4.48)n
jj
jii ePe
1
/ , = 1,…, n (4.48/)
.- ei
/ (4.48)
j
n
j
n
i
iji
n
jj
ji
n
i
i exPePxx1 111
//
. -
n
i
iji
j xPx1
/ , , j = 1,…, n (4.50)
. (4.50) – x1 = P1
11/+ P1
22/ +…+ P1
nn/;
x2 = P21
1/+ P22
2/ +…+ P2n
n/; ……………… (4.50/)
xn = Pn1
1/+ Pn2
2/ +…+ Pnn
n/.(4.50/) 1, ... , n
1/,..., n/ 1/, ..., n/ 1,..., n -
.
95
. -- /
(4.50) X = PTX / (4.51)
.x -
(4.50) -.
,... 1212
111
/1 nnxQxQxQx
,... 2222
121
/2 nn xQxQxQx
............................... (4.52) ,...2
21
1/ nn
nnnn xQxQxQx
jij
i xQx / , ,j = 1,…, n (4.52) .
Q (4.52) :
.
...............
...
...1
21
222
21
112
11
T
nn
nn
n
n
P
QQQ
QQQQQQ
Q
(4.53) (4.51)-
/ = QX (4.54) .
Q (4.53) ;
Q = E, Q = E (4.55) .
(4.55) , a, i
j
.
.,1,0
jijii
j (4.56)
(4.55)
96
ij
kj
ik QP ,
ij
kj
ik PQ
.T – Q j – a -
; Q – , T
j – a .
kn
nk
knk
knk
knk
kk
kk
knk
kk
kk
nn
nn
n
n
nn
nn
n
n
PQPQPQ
PQPQPQPQPQPQ
PPP
PPPPPP
QQQ
QQQQQQ
...............
...
...
...............
...
...
...............
...
...
21
22
21
2
12
11
1
21
222
21
112
11
21
222
21
112
11
=1...00............0...100...01
...............
...
...
21
222
21
112
11
nn
nn
n
n
.
,kj
ik PQ k
.
§5. C .
L -, ( ) -
. ( ) , ( )L .
4.5. . K x, y L1) a(x+y) = a(x)+a(y) , 2) a( x) = a(x)
, ( ) .L ( ) . L- n-
e1,…,en i:
x = x1e1+ …+ xnen .
a(x) = a(x1e1+ …+ xnen) = x1a(e1)+ …+ xna(en) (4.57) .
97
( ) ei ai:
a1=a(e1),…, an=a(en) (4.58) , ai - . (4.58) (4.57)-
( ) 1-
( ) = a1x1+ a2x2…+ anxn (4.59) .
k- k-. k=1 « », k = 2
« » . (4.59) Ln-
.
. Ln-j
jii ePe/
(4.60) e1
/,…,en/ .
ai/ :
( ) = a1/x1/ + a2
/x2/ +…+ an/xn/.
ai/ , ( ) -
ai/=a(ei
/). ei
/ (4.60) ( )
jj
ijj
ijj
ii aPeaPePa/
.j
jii aPa /
. (4.61) (4.61) (4.60) -
. – j
iP . (4.61) -
e1, …, en (4.59) .
kik
i xQx /ij
kj
ik PQ ;
ij
kj
ik QPi
jkj
ik PQ ;
ij
kj
ik QP
98
jj
kj
jk
kj
ik
ji
kikj
ji
ii xaxaxaQPxQaPxaxa
//)(.
jj
ii xaxaxa
//)( ,( )
. 4.2. L
L* .. ( ), b( ) -
.c(x) = a(x) + b(x)
.c(x+y) = a(x+y) + b(x+y) = a(x)+a(y) + b(x)+b(y) =
= [a(x) + b(x)]+[a(y) + b(y)]= c(x)+c(y)
c( x) = a( x) + b( x) = a(x) + b(x) = (a(x) + b(x))= c(x)..
-. c(x) = a(x) .
c(x+y) = a(x+y) = a(x)+ a(y) = c(x) + c(y).
c( x) = a( x) = a(x) = c(x).( ) .
a(x), b(x) L*, a(x) + b(x) L* ,a(x) L*.
(x) ( )L*- . (–1) ( )-
- .L*
. .
§6. ..
L – n- , L* – ( L- ) .L- e1,…,en . L -
e1,…,en { 1, ..., n} .ei(x) ej -
99
ijj
i e (4.62) e1(x),…,en(x) ,
ij – .4.6. . (4.62) L*-
e1(x),…,en(x) L- e1,…,en .ei(x) = ei(xjej) = j i
j ,,
n
n
n
xxxx
xxxxxxxx
1...00.............................................
,0...10,0...01
21
212
211
(4.63)
.(e ) (ej)
jiji
ee ij
ji ,1
,,0),(
(4.64) .
1- . (4.64) e ej
- .2- . e1,…,en ,
ej .e , ej – ,. e ej :
x = 1e1 +...+ nen,x = 1e1+…+ nen. (4.65)
ej ( 1, ... , n) -, e
( 1, ... , n).
(4.65) = ei, = ie (4.66)
.-
(4.66)-ej- , ej- , (4.64)
.
100
jj
iiji
ij xxeexex ),(),( ,jj
iij
iij xxeexex ),(),( .
),(),,( jjjj exxexx . (4.67)
(4.66)-
iij
j eexxeexx ),(,),( (4.68) . (4.68)
. . (4.68)
jjiij
jii eeeeeeee ),(,),( (4.69) .
),(),,( jiijjiij eegeeg (4.70)
(4.69) j
ijijiji egeege , (4.71)
. e ei
(gij) ei e (gij). -
., (4.71)
.(4.71) ek- -
),(),( kjij
ki eegee .
(4.64) (4.70)-
kiki
gg ikjk
ij
,0,,1
. (gij) (gij) -. , (gij) (gij) –
.
§7. .
e ej – , e / ej/ – .
1) e e /
101
e / = pi/ie , e = pi
i/ e / , i, i/ =1,2,…,n (4.72),
2) e e /
e / = qii/e , e = qi/
i e / , i, i/ =1,2,…,n (4.73).
(4.72) ( (4.73) ) (pi/
i) (pii/) ( (qi
i/)(qi/
i) ) – .(pi/
i) (qi/i) - .
(pii/) (qi
i/) .(4.72)- ek- , (4.73)- ek/
, (4.64) ki
ki
ii
ki
ii
ki ppeepee //// ),(),( ;
ik
ik
iik
iiik
i qqeeqee /
/
///
/
// ),(),(. k = i, k/ = i/
),( //i
iii eep ; (4.74)
),( // iii
i eeq . (4.75) (4.74), (4.75)-
. iip / = i
iq / . ( iip / ) ( i
iq / ). , ( )
(4.74) ., -
( ) ( )
, , (4.76)
,.
./ – ei/
. (4.67) . e /-(4.76)-
.ei/- (4.76)-
iip /
iip /
iip /
/iip
iiii epe // /
/
iiii epe
iii
i epe // /
/ii
ii epe
),( // ii exx
iiii
iiii xpepxexx //// ),(),(
),( // ii exx
102
. :
(4.77) , ,
( ) ,
(4.78) ,
( ) .,
. =
.ij
.
ij :
njieee jiijii ,...,1,,,,
:
xi ni nij n2
ij n2
n n2
.
iii xpepxexx ii
ii
i /// ),(),(/
iiii xpx //
iip /
ii xpx ii
//
/iip
xneee ,...,, 21
ii
i exx
n
ii
ijj eeA
1
103
.. –
.
, , (i, k = 1, …, n), (4.79)
1 i n ; 1 j ni j
(4.80)
,
– .
,.
:
, -nl (l = 1, l = 2)
. ( )
.
., , .
nff ,...,1
n
ii
ikk ef
1
n
ik
kii fe
1
ij
n
l
lj
il
n
m
mj
im
11
kikii
k ,0,,1
n
i
ji
ij xx1
/
n
i
ijij
1
/
n
ji
jm
kij
im
k
1,
/
n
ji
jm
ikijkm
1,
/
104
Vn
- : Vn Vn -.
:
.,
, (j = 1, …, n) (4.81)
( ij) .
, (k, m = 1, …, n) (4.82)
.
. ( ij)
(4.82) , (4.81) – – ..
. O -, , , -
. . .
§8. n- .
,-
.4.7. . 1) np+q
,2) f
yAxAyxA )(
n
ii
ijj eeA
1
fn
ji
jm
kij
im
k
1,
/
neee ,...,, 21
njnjnini
Tq
piijjp
q 1;......,11.......;1
1
1............
1
1
nlnlnknk
Tq
pkkll
p
q 1;......,11;.....1
1
1............
/ 1
1
105
,
(4.83) , n- Vn
- , ( -
1- n- )..
( )
.
(V, §5- ) q
. (4.83) ,
0- , - 0- .
10. .1. – –
2. 2- - – - ,
– - , – -
, – -
.3. – – ,
.
(4.84)
1
1
1
1
1
1
11
1
1............ ......
............
....../
j
jl
jl
ki
ki
j
iijj
i i
kkll
q
q
p
pq
p
qp
p
qTT
qp
qp
p
q
iijjT ......
......1
1
00
01
10
11
20
ji
11
lki
lki
n
ji
jl
kij
il
k
1,
/
q
q
p
p
p
q
p
q
jl
jl
ki
ki
ji
iijj
kkll TT ...... 1
1
1
1
1
1
1
1 )(),(
............
............
/
106
, (i) = (i1, …, ip), (j) = (j1, …, jp), i1, …, ip, j1,…, jq 1- n-
. (4.84) -
.
. ,
(4.85) .
(4.85)
(4.84)
.
20. .4.8. .
, S S .
.p
q
iijjT ...
...1
1
p
q
iijjS ...
...1
1
( ) S . S 1 n
qp
neee ,...,, 21p
q
iijjT ......
......1
1
nff ,...,1p
q
kkllT ......
....../ 1
1
q
q
p
p
p
q
p
q
lj
lj
ik
lk
ik
kkll
iijj TT ...... 1
1
1
1
....1
....1
1
1)(),(
/.......
q
q
p
p
p
q
p
q
tl
tl
ks
ks
ts
sstt
kkll TT ...... 1
1
1
1
1
1
1
1)(),(
............
............
/
tj
lj
tl
is
ik
ks ,
q
q
p
p
p
q
lj
lj
ik
lk
ik
kkllT ...... 1
111
....1
....1)(),(
/
.......
............
)(),(
...
.........
......
)(),( )(),(
............
1
111
11
1
1
11
11
11
11
1
1
ts
iijj
tj
tj
is
is
sstt
lk
lj
lj
ik
ik
tl
tl
ks
ks
ts
sstt
p
q
q
q
p
p
p
q
q
q
p
p
q
q
p
p
p
q
TT
T
107
p
q
iijjT ...
...1
1
p
q
iijjS ...
...1
1
.,
:p
q
iijjT ...
.../ 1
1
p
q
iijjS ...
.../ 1
1
S -, 1- n-
k1, …k , l1, …, lq
p
q
iijjT ...
.../ 1
1
p
q
iijjS ...
.../ 1
1 .4.9. .
p
q
iijjU ...
...1
1
p
q
iijjT ...
...1
1
U -.
.4.10. . ,
..
§9. .
.
- , - .
p
q
iijjU ...
...1
1
p
q
iijjT ...
...1
1 + p
q
iijjS ...
...1
1
U = T + S
- .
2. .
km
p
q
iijj
iijj SST ...
.........
11
1
1,
nff ,...,1
p
q
iijjT ...
...1
1
p
q
iijjS ...
...1
1
qp
qp
108
- .
.
-
S . :U = T S. T S S T
.3. ( ).
- , q 2 . (1, ... , q)
; r s (1 r s n)
.
- .
(r s) -.
4. . , (0, 2)- ij. , A = ( ij)
. A-1
. ij .ik
kj = ij.
qp
mk
kpp
mqq
p
q
kppp
mqqq
iijj
iijj
iiiijjjj STU ...
.........
............
1
1
1
1
11
11
kppp
mqqq
iiiijjjjUU ......
......11
11
mqkp
)( ......
1
1
p
q
iijjTT
qp
qrsqsr
.........1
.........1
p
qsr
iijjjjT ...
.........1
1
~ p
qrs
iijjjjT ...
.........1
1
p
qsr
iijjjjTT ...
.........1
1
~~
qp T~
p
q
iijjT ...
...1
1
109
(4.86)
.
(4.87)
. , – ., :
.ij
- ;p
q
p
q
p
q
p
q
iijj
iijsj
sj
iijsj
sj
iijjj TTTaaCa ...
.........
......
......
1
1
1
21
1
21
1
21
ij gij. , Rn-
gij = ij , gij= ij ,
.
-, .
: 1. g(x, y)=<x, y> – .(1, 0)- xi
x = xiei
( ) -.xi= gijxj = <ei, ej>xj = <ei,xjej> = <ei, x>
. xj
-. .
<x, y> = gijxiyj = gij xiyj (4.88).
2. (1, 1)- A=(Aji) Ln
. gij:
Aij = gik Ajk. (4.89)
p
q
p
q
iijji
iijj PaP ...
.........
1
112
1
p
q
p
q
iijj
jii
jjTaC ...
......
...1
111
2
110
A(x,y) = Aijxiyj = xi gik Ajk yj = <x,Ay>
(0, 2)- ., (0, 2)- -
. , - ( )
.( )
., ., (2, 2)-
: 1- – , 2- 3- , 4- – .
....kljiT
. , 2- , 2- . ,
. , 3-..
.....
.kmjilm
klji TgT (4.90)
.
§10. .. , .
-. Sikl… , ,
i k - :Sikl…= Skil…
Sikl… .
S12l…= S21l… , S23l…= S32l…Aikl…
:Aikl… = –Akil…
Aikl… .
A12l…= –A21l… , A23l…= –A32l….
. Aikl… = –Akil… , , A11l… = –A11l…,A11l… = 0.
klijT
111
-. ,
( ) ( ) .
. ( ) ( /) ei/ = k
i/ekTik ( ) , Tik=
Tki , ( /)Tik
/= li/
mk/ Tml = Tki
/.
. SikAik
332313
232212
131211
SSSSSSSSS
Sik
, 00
0
2313
2312
1312
AAAAAA
Aik
. Tik Sik - Aik -
.
Tik = (½) (Tik + Tki) + (½) (Tik – Tki) (4.91) . Sik Sik (½)(Tik + Tki)
, Sik= Ski.Aik Aik (½)(Tik – Tki) ,
Aik = –Aki. .. , .
, Tik
333231
232221
131211
TTTTTTTTT
.Tik ,
332313
322212
312111
TTTTTTTTT
112
(Tik)- Tki .Tki ei/ = k
i/ek
Tki/= l
k/m
i/ Tlm , (i,k,l,m=1,2,3) .
.
.-
..
- ( ) .
.(4.91)- Tik Sik
, Aik
.
.2
6- , 2- .2- ij -
. 2-Cik = AiBk – AkBi
, Cki = AkBi – AiBk = –( AiBk- AkBi) = –Cik;Ai Bi .
113
V .
– – .
,. -
. ,, ,
-
.
§1. . .
-
0),,( zyxF
.F( , , z) -
.
. .5.1. . – , D –
. D
,.
D -,
z=f(x, y) (5.1), f -. (5.1)
. -
),( vurr (5.2)
114
- . – r
= x(u, v), y = y(u, v), z = z(u, v) (5.3) . (
. ., , 6- , §2)x=x(u,v), y=y(u,v)
u, v- u v-
. u,
v- ,),(),(,
),(),(,
),(),(
vuDzyD
vuDzxD
vuDyxD -
, (5.3) – -u, v- ( , z = z(u(x,y),
v(x,y)) , (5.1) ( , z = z(u(x,y), v(x,y)).
,
,
(5.4) ,
(5.4/) .
0vu rr u, v (5.2) -
.5.2. .
),( vurr - .0vu rr
, ),( vurr = 0 .
vuDyxD
vy
uy
vx
ux
,,
kjivur zyx uuuu ),(
kjivur zyx vvvv ),(
zyxzyx
vvv
uuu
0vu rr
115
. (§3) ),( vurr,
.--
. : vu rr = 0 -
, vu rr u,v .
x2 + y2+z2 = 1 ),( vurr,
x = cosu cosv, y = cosu sinv, z = sinu (5.5)
vu rr2
u = 0 , (0, 0, 1)
.
x= sin u, y= cos u sin v, z= cos u cos v , (0, 0, 1)
,
vu rr u=v=0= k 0.
F(x,y,z) =0 (5.6)
(x0, y0, z0).
.
zF
0 , (5.6) z-
,z = f(x, y) . (5.6)
zF
yF
xF ,,
116
0kzFj
yFi
xFgradF
(grad §1- )
§2. . .
),( vurr -
(u v 0vu rr). 5.1
u, v (5.3) , , z ,
( , ,z) u,v. , (5.2)
u, v. u, v
. – -
., -
. u, v
U, V :U=U(u,v), V=V(u,v). (5.8)
1- ,
U ,V 0),(),(
vuDVUD ,
u=u(U,V), v=v(U,V) (5.9) .
U, V *
.*- U, V u, v
., U, V (5.9) u, v
, ),( vurr = r {u(U,V), v(U,V) } - .
U, V. u, v ,
117
(5.8)- U, V ..
u =u(t), v=v(t) (5.10)
(t – ). (5.2)- ,
rr (u(t),v(t)) = r *(t),
. t r.
,t=u ,
v=v(u) (5.11) .
, ,. (5.11)
v=const, «u»
. v = const«u » . u=const «v »( «u = const ») . ur
vr v = const u = const. u = const v = const
-. ,
.,
. (5.2) ,
. (5.2) -.
,, ,
. ,
118
, ,rr, = cos , y = sin , z = 0.
00rr (0, 0, 0) .=const (31- ).
. (5.5) .
2 + 2 + z2 – 1 = 0
31- 32-
(32- ).(5.5)
cos2u cos2v – cosu cosv + 41
+cos2u sin2u – 41
= 0
cosu (cosu – cosv) = 0 .
u = v
. u = 2.
041
21
,01
22
222
y
zyx
x
119
§3. .
(5.2) (u0, v0) -.
u=u(t), v= (t) (5.12) , u(t), (t) –
, u(t0)=u0, v(t0)=v0 t=t0. (5.12)
:
dtdvr
dtdur
dtrd
vu , (5.13)
t=t0 . (5.13) .
5.1.- .
, ur (t0) vr (t0) (5.12) ( – ).
v=v0 u=u0. (5.13) (5.12)
ur (t0) vr (t0) ,
0,, vu rrrR (5.14)
( r , ur , vr, R -
- ).5.3. .
-.
(33- )., 0vu rr (5.14)
.vu rr .
(u,v)
vu rrrR (5.15)
120
, r , ur , vr.
(5.14) (5.15) . « »
,
, « »
, .
x=x(t), y=y(t), z=z(t) (5.16)
F(x,y,z) =0 (5.17)
33- 34-
,
F(x(t), y(t), z(t)) 0
- .
0dtdz
zF
dtdy
yF
dtdx
xF
(grad F, tr / ) = 0 (5.18)
, tr / – (5.16) . dragF (5.16)
, (5.18) (5.16) .
121
, (5.17) r -
0, gradFrR , (5.19) gradFrR (5.20)
, gradF .(5.13)
.5.4. .
MNm ,. m
, m (u,v),
.
. ur vro
.(5.12) (5.13)
. (5.13) dvrdurrd vu (5.21)
du:dv=f(u,v): g(u,v) (5.22)
dudv
= (u,v) (5.21)
ur f(u,v)+ vr g(u,v) .(5.22) v=v(u) u=u(v)
.,
v = (u,c) (5.23)
u = ~ (v,c) (5.24) ,. (5.22) (5.21)
122
(5.23) (5.24) . dv:du
( (u,v) ).
§4. .
),(),(
VUDvuD 0 (5.25)
(5.9) .
(5.9) .« » -
, ,, u, v, U, V u1, u2, u1*, u2*
.x, y, z
21 ,uu, -
: 21 , uu -
,...,2,1),,( *2*1 iuuuu ii (5.26) *2*1 , uu
,. –
21,uur (5.26)
..
,.
( )- . ,
21
21
uu
uu
rr
rrn (5.27)
123
( ), ,, , .
, ,, -
-.
.
., -
., - .
21,uurr (5.28)
- ( 21 , uu )( 0vu rr ) . -
(5.26)
0det *k
i
uu
(5.29)
, ( ,) .
1 2 . « »
« »
.det,det,
,,,
*
*
**
**
**
2
ij
ijj
iij
j
iijkiuuikui
DDDDuuD
uuD
uurrrrr kii
(5.30)
(5.29) 0*D (5.31)
.
0*D (5.32) .
124
*j
i
u
uj
i
uu
*
,
ikk
i
k
j
j j
i
uu
uu
u
u*
**
2
1 (5.33)
.
kikii
kki
ikik ,1
,,0
– . (5.33)
ik
j
jk
ij
DD2
1*
*
* (5.34)
*
**
*2
1
ik
j
jk
ij DD (5.35)
, j ( )
,j
jj ba j
j ba
.,
22
11 babababa k
kj
j
. , (5.34), (5.35)
*
**
**
* , ik
jk
ij
ik
jk
ij DDDD , (5.36)
.(5.26) 21 , rr 21 ,uu
,.
ii dufuudf ),( 21
*
*ji
ji duDdu . (5.37)
21 , dudu
125
(1.20) ,
*
*
*
*j
iji
j
jii Dru
uur
urr (5.38)
, **jj
urr r
*ju -, . (5.37), (5.38)
jj
iiji
ji rDrduDdu **
**
, (5.39) .
jk
ij
ki
jijk
ki
ii durDDduDrDdur
*
*
*
*
*
* . (5.40)
. -, j
ji
rD *
j- k- ,ji
j duD*
j. (5.36)- ,
kk
jk
kj
ii
durdurdur*
* (5.41)
,kjk
j dudu 1 . (5.37)–(5.41) . (5.37) (5.38)
, (5.40)–(5.41) – (
– ).
iu, -
.5.5. . (5.26)
ppiii
pp jjj
jjjiii aDa ...
...... 21
*...*2
*1
21
**2
*1 (5.42)
126
, p2piiia ...21 -
,**
22
*11
*...*2
*1
21......
p
p
piii
p
ij
ij
ijjjj DDDD . (5.43)
, (4.26) - (5.42)-
jij
i duDdu**
.5.6. . (5.26)
pp
ppjjj
jjjiiiiii aD ...
......... 2121
**2
*1
**2
*1
(5.44)
21 ,uu ,p2
piiia ...21- -
,p
p
p
p
ji
ji
ji
jjjiii
DDDD *2
*2
1*1
21**
2*1
.........
. (5.45)
21,rr.
piiia ...21piiia ...21
-
. , ,.
n -n .
-.
, },{ ...21 piiia }{...21 piii
b ,
ppp iiiiiiiii ba ......... 212121.
( piiia ...21
qiiib ...21) ,
- ,
21 , dudu
127
p
q
q
q
p
p
piii
q
iiijjj
jjjjjj
iiiiiijjj DD ...
.........
...
......21
21
21**
2*1
**2
*1
21
*...*2
*1
**2
*1
(5.46)
pp
q
iiiiiijjj a ......
...2121
21 qjjjb ...21
.
. (5.46)- p
q
iiijjj
......
21
21p
- q .5.7. . ( -
)
- -. ( ).
, 321 iiia21 iib
3213 iiii ac 21 iib (5.47) .
. (5.47) – .
*3
3
3*3
321321
*3
3
2
2
1
121321
*3
3
2*2
*2
2
1*1
*1
12132121
*2
*1
21
*3
*2
*1
321
321*2
*1
*3
*2
*1
*3
ii
iiiii
iiiii
ji
jijj
iiiii
ji
ii
ji
iijj
iiijjiijj
iiiiii
iiiii
iiii
DcDbaDbaDDD
DDbaDbDaba
( (5.42)-(5.45), (5.36) ).
{ 21 ,cc }- ., -
, ., -
,. -
** ii
ii Dcc
128
. , -
jg , p « » -:
I (5.48) (5.41) . -
-, -
.
§5. .
},{ 21 rr},{ 21 dudu
-:
...,2,1;22
11 idurdurdurrd i
i (5.49)
ir -.
- . (5.50)
.)()( 1221 rrrr
.jiij gg (5.51)
(5.50) .idu ,
(5.52)
. idu ,
(5.53) .
piia ...1
p
pii
ii gga ...1
1...
2*2
1*121
2*22
1*11*
2*1
)()()( ii
iiii
iii
iiiii
DDrrDrDrrr
)( jiij rrg
ji
ij dug
jiij
jj dudugduI
129
)(),( 2211 tuutuu (5.54) ,
:
(5.55)
– :
(5.56) . (5.53)
(5.57) .
,,
(5.58)
.
221
222
21
222
21
22
21
2212211
2122211
sincos
)()()(
rrrrrrr
rrrrrrrgggg
, (5.4/)--
(5.59)
. .
((5.55) )
dtdt
dudt
dugdurdurrd
tdztdytdxtdss
t
t
iit
t
ii
ii
t
t
t
t
t
t
ii0
21
210
2
2
1
10
0 0
,
222
),(
))(())(())(()(
2dsI
GgFgEgvuuu 22121121 ;;;
222 2 GdvdvFduEdudsI
02ds
2122211
2212
1211 )(ggggggg
g
0g
ijg
130
, ,.
, 0 - -
. 1- iidurrd
, jj urr ,
= rrd ,
jiij
jiij
jiij
uugdudug
udugrrdrrdcos . ( )
du1:du2= (u1,u2) , du1:du2= (u1,u2)
( )-
22122
1122122
11
221211
22)(cos
ggggggggg
( /)
., , ,1 constu
,2 constu constuuconstuu 2211 , - (34- )
. 1 3 2
212212
212111
,,
,,,
uuruuurr
uuruuurr
1N 2 -.
22 22
1121 durdurrrS
( ,)
S 212121 dudugdudurr
.
131
21dudugS (b)
,.
1-. , (5.53)
.
2121 iiii rra (5.60) .
( ,02211 aa
2112 aa ).
21
2
2
1
12121 ****** iiii
iiiiii rrDDrra .
« »
(5.30) . (5.61)
,
. (§1-
012a ),( ).
.5.8- . –
(5.26)
J . – 1 .
**21a
121*
2*21
2*
1*******
212121
2
2
1
12121 rrDDrrDDrrDDrra iiii
12***21 aDa
2121 *** iiii aDa ** 21 iia
12a
12a
),()(),( 21*
*2*1** uuJDuuJJ p
),()(),( 21**2*1** uuJDuuJJ p
2121 rra
132
. ,.
, 1 .
;
.
*22
21
12
11
21
21
21
21
*2*1
*2*1*
**
**
**
**
DDDDD
bbbDbDDD
bb jj
jj
ii
ii
( )., – 1 . E
, 2
2122211
2221
1211 )(ggggggg
g
. , a , -
.
;
hhhhh
gD2221
1211*
,
iiii aDa **
jjjj bDb **
21
21
21
21
21
21
21
21
*2*2*1*2
*2*1*1*1
*2*2*1*2
*2*1*1*1* ii
iiii
ii
iiii
iiii
DDgDDgDDgDDg
gggg
g
2*2
2*1
1*2
1*1
2221
1211* DD
DDgggg
gD2
2
1
1
2
2
1
1
*22*12
*21*11j
jj
j
ii
ii
DgDgDgDg
** klilki hDg
133
. (5.62)
, g – 2 . g > 0
gDg ** (5.63)
(5.62)- . – 1 .
( , D*D* -
).
(5.61) (5.63)
(5.64)
(5.27) . -.
- – -.
.)()( 1*
2
2*
1
2*
1*
21
2*
1*
12*
1*
1****
iii
iji
ij
iiii
ij
iiiij
iiijiji D
uDrDDrDDr
uDrrr
« » ,. ,
nrrg )()( 211
« » .
(5.65) 0)( irn .
(5.66)
*g 2**
*2*2*2*1
*1*2*1*1 )(1
1
1
1
1
1
1
1 DghDDhDhDhDh
jj
jj
ii
ii
*g 2*)(Dg
g
ng
ag
an 21
*
*2*1*
r ijr
2*
1*21** )()( *
ij
iiiiji
DDnrnr
2112),,( nrijij
134
« ».
jdu
(5.67)
.
(5.68)
..
)(* rrM, (35- )
MPrMPMMPM ** .
n - , MP n
),()*,( nrnPM .
n
jiji du
jiij
ii dududuII
]3[21 2 rdrdr rd
35-
135
].3[21]3[)(
21]3[
)),((21]3[),(
21),( 2*
jiij
jiij
ii
dudududunr
ndurdnrdnPM
. ((5.68)- )
(5.69)
– .
, -, *M
-., *
, du1:du2 . ,
:1) 11 22
212 , , du1 :
du2 ,-
( ).2) ;
II 0 « » ( -) ( – ).
3) – ; – ( – ).
1-
, – 2 ,
gK (5.70)
–. ,
g > 0.
),(2),(*2 nPMnrdII
nPM //*
2** )(D
136
sK
s 0lim (5.71)
.– , –
.
(5.72)
..)( 21
21 dudunn (5.73)
.lim21
210 rr
nnss
(5.74)
1 , ( n , n ) 1.
0,, 21 nnnn ,nnnn 21 , n 1 n 2 r 1 r 2 –
.
(5.75) .
. (5.76)
(5.75)- r 1 r 2- ,
ji rn , kjkijk
ki grr )( . (5.77)
0irn - ,
ijijij rnrn ),()( .
ij kjki g .
s ),( 21 uurr),( 21 uunr
2121 )()()( 21 dudurrrrs constuconstu
n
kkii rn
ki
2112
21
22
1121 rrnn
137
).det()det()det()det(1det 2 kikj
kikj
kiij ggg (5.78)
(5.76), (5.78), (5.74) (5.70) , (5.71) .
ij ji nn
),( 21 uunn -, .
,ji
ij duduIII – , ( n
).i gij
jduj
iji du , g-, (5.62)-
ii
ii
ii
ii
dududugdug
gIV
21
211
.ji
ij duduIV
4- }{ ij
,
ij)(1
1221 jiji ggg
. , 0
(),
V 0 «
» .021122211 ,
..
138
, k1k2 ,
K = k1 k2
( ).
.
.
§6. .
.n – r ( = 1, 2, …, n – 1)
n .r u
,r n n u
.ijb (5.66)
nbrGr ; 1,...,2,1,, n (5.79). r
,n–1, b b nr , -
.G -
. (5.79)- r -
r r = G g (5.80). g = r ·r u -
rrrru
g (5.81)
139
, g = r ·r , g = r ·ru , u
rrrru
g; (5.82)
rrrru
g (5.83)
. (5.82) (5.83) -, (5.81)
u
g
u
g
u
grr
21 (5.84)
. (g ) (g ) , g g =
,
u
g
u
g
u
ggG21 (5.85)
.
(5.86) (5.75) n = 3
. ik
ijb , gij
ijb = kjki g (5.87)
. (5.87)- (gkj) (gkj)
ijb gjl = kjki g gjl = l
kki
jlij
li g
. (5.86)- ,jli ,,
rgbn.
kkii rn
140
ur n
.r = r(u1, u2, … , un) .
(5.84) -gij, ijb. -
,-
, u ijb , gij
, .
§7. .
,. ,
.,
,.
.
.
.u1(t), u2(t) c
, tta . M(t) -
M/(t+dt) a – ad .
, ad .
141
ta -
.ada M ; M
( ). ,
n(36- ).
n ( ada ) -
( ada n), n :
{n ( ada )}na M n a =0 ( n a ),
ada
36-
(n ad ) n. ada
, ( ) ,M ada -
:nadnadaada t .
– , -. tada a -
/ a -aD :
142142
.nadnadaD ( 5.88) /- a aD
ad -.
a.
aD = 0 ./- ad n
, ada -a
. ada- /- a
.ada a (t+dt) - / -
. -/ a
.-
. t ta, 1r 2r
a1(t) a2(t) :a = 2
21
1 rara ,
1r 2r , t.
222
112
2212
111
12
21
1 durduradurdurardardaad
. (5.79) 11r ,12r , 22r
22121
112
12
2111
111
12
21
1 { rGrGdurGrGardardaad
.....}{} 22
2221
122
12
2121
112
22 durGrGdurGrGadu. n ,
da .
143
, da aD -
. r 1 r 2 -
22222
12221
21212
11211
22
22122
12121
21112
11111
11
duaGduaGduaGduaGdar
duaGduaGduaGduaGdaraD
.aD 1r 2r -
. , – -
. Da1, Da2 .
duaGdaDa 111 ;duaGdaDa 222 (5.89)
(5.89)- ,.
a , aD = 0 ,aD - (5.89)-
duaGda 11 ; (5.90) duaGda 22 .
a /
.
1, 2 , G, /
du1, du2
. –
.,
.u
,.
144
a 0 0 0; 1 . , 0
1 , -, 0 1 -. 0
1 a 0.0 1
. a 0
0 1.
..
0 1
u = u (t) (i=1,2, … , n), t0 t t1 (5.91)
, u (t) – .
-
r = r (u1, u2, …, un) (5.92)
- . u (t)(5.92)- r - t- .
(t) 0a .,
.t-
= a (t) (5.93) ,
.u (t) - ,
r (u1, u2, …, un)- t- - -
.0a (t)
a 0 = a (t)ri(u1, u2, …, un) (5.94)
145
. u -t- . t , a 0=const
0 = d ri+ a dri (5.95). dri
.
dri(u1, u2, …, un) = rijduj, (5.96)
ijr ji
n
uuuur ,...,12
.
,rij ri
:
kkijij rGr . (5.97)
kijG ; k
, .
rij = rj
kji
kij GG (5.98)
. ,kijG (5.97)
,
kijG ( ) =
kijG (u1, u2, …, un)
u -kijG -
. (5.97) (5.96)-
dri = Gkij rkduj
. (5.95)
146
0 = d k rk+ Gkij rk aiduj
.k- . rk -
, -; ,
d k + Gkij aiduj = 0
d k = – Gkijaiduj . (5.99)
(Gkij- ai
, duj )..
: (ui) ai
,/(ui + dui) ?
, - ,. , - /- -
ai , -.
dai , (5.99)- ai
, duj -, Gk
ij;- /- , « -
».
10MM ,. (5.99)
10MM,
., .
ui
r (u1, u2, …, un) = uiei , ri = ei, rij= 0 (5.97)-
Gkij =0
. , uGk
ij(u1, u2, …, un) - , (5.97)-
147
rij = 0, ri = constri = ei ,
r = uiei + r0
. - ( -r0 = const – ) ui
.
Gkij
.
§8. .
..
( ,).
-.
kiiT ...1-
1, ..., n
kk iiii Tx
P ......; 11
. ( )
.. //
1 ... kiiT
//1
///1
/ ......; kk iiiiT
xP .
: ?, -
? .(x)- ( /)-
)...( ......
...; 1/1/1
//
//1
//1
/ k
k
k
k
kii
ii
ii
iiii
TDDxx
TP )( ...1 kiiT
x/
xx
148
)...(... /1/1
/1/1/1
...k
kkk
k
ii
iiii
ii
ii
DDx
TDD /1...;x
xPkii
k
k
ii
ii DD /1/1... +
kiiT ...1)...( /
1/1
/k
k
ii
ii DD
x.
//1
/ ...; kiiP /1...;x
xPkii ),,(... /
...; //1
//1/1
xxTSDDk
k
k iiii
ii
.
),,( /...; //
1/ xxTS
kii kiiT ...1)...( /
1/1
/k
k
ii
ii DD
x,
(x) (x/) //1
/ ...; kiiS ,
kiiP ...; 1.
ix :
.
ix :
(x) (x/) ;
, 0//1
/ ...; kiiS . ix « » ( )
T i
, div(T) = i
i
i
xT -
.-
. , (x) (x/) ?
div(T)(x) = div(T)(x/) , ( ) ( /) –
.
.,,)(,,,,
)(
/)(
//
2
)( /
//
/
/
//
/
/
/
xxTRTdivxxTRxTxxTR
xT
x
xxx
xTxx
x
xxTT
xx
xx
TTdiv
xi
i
ikik
i
i
k
ki
ii
i
i
i
k
k
ii
i
i
ii i
i
x
149
/,, xxTR /
/2
i
k
ki
ii
xx
xxxT .
/,, xxTR , ,. /,, xxTR
, R = 0 .
ix.
: -.
«» -
. ( – « ») :
1) Rn- ix.
2) , T - () Ln- ,
T- .. Rn- Ti -
, ( ) – , ( /) – ., 1) 2)
. ( )j
iij x
TT . ( /)-
ijj
j
i
i
j
iij
Tx
xxx
x
TT /
/
/
//
/
. - ,/
/
ijT .
/
/
//
/
/
//
/
/
/
/
/
xT
xx
xx
xxT
xx
xxx
xxT
xx
xxT
k
k
i
j
j
i
ik
k
i
jj
j
i
iijj
j
i
iij
;//
/
/
/
/
/
/
/
//
/
/
// 2
kj
i
i
ik
k
jikk
i
jk
j
j
i
i
j xxx
xxT
xT
xx
xT
xx
xx
xx
150
//
/
/
//
/
//
/
/
/
/
/
2
; kj
i
i
ii
kji
kjk
j
iij xx
xxxGGT
xTT
. - ( ) -/
//i
kjG -. -
.Rn- Ti . - Ti-
,
;
.
.~ /////
/
//k
jikji
jiGT
x
TT
– - -.
1- .
..
-/px
ijT jxTi
ijj
j
i
i
ji Txx
xxT ////
//
//
//
/
////
/
//
/
//
//
/
/
/
/
///
/
/
//
///
/
////
2
2
~;~j
j
i
i
ij
kk
jik
jikji
j
j
i
i
ij
k
kk
jiki
k
jkj
j
i
i
jk
i
k
j
j
i
i
ki
k
jj
j
i
i
ji
xx
xx
xxxGGT
x
Txx
xx
xxxT
x
Txx
xT
xx
xx
xx
x
Txx
xx
xxT
xx
xxx
xxT
/
//
~ kjiG
/
//
~ kjiG
/
//k
jiG
/
//
//
//
/
ikk
i
i
i
xx
xx
//
//
//
/ 2
kp
i
i
i
xxx
xx 0/
//
/
//
////
/2
p
p
k
i
ip
i
xx
xx
xxx
151
,/
//k
jiG +/
//
~ kjiG = 0 .
.- (Rn-
),
/
//
/
/
//
/i
kjk
j
iij GT
xTT ;
/
////
/
//k
jikji
ji GTx
TT
- , (1, 1)- -:
ijkT ;i
jj Tx
./
//
/
/
/
//
/
/
/
//
/
///
//
/
//
////
//
/
/
/
/
//
///
//
/
/
///
//
//
22
pkj
ip
ikp
pj
ijk
jk
pi
k
k
j
j
i
i
pj
p
k
q
q
i
k
k
j
j
i
i
p
k
q
q
pj
pi
k
k
j
j
i
i
pj
i
kk
k
j
j
i
ii
kj
GTGTTx
xxx
xx
xx
xx
xxT
xx
xx
xxx
xx
xx
xxT
xx
x
T
xx
xx
xx
xx
xxT
xx
xx
xxx
xx
xxT
- (Rn-
)ijT .
5.2- . M n = R n , (x) – , (x/) – . R n -
p
s
qj
iijqjj
k
s
iq
iqiijj
iijj
iijj s
k
ps
sks
p
k
p
k
pGTGTT
xT
1
.........
1
........
......
...;...
/
//
//1
////1
/
//
////1
//1
//1
//1
/
//1
///1
k
p
iijj
T ......
1
1
. ijk
G (x) (x/)
//
//
//
/
//
2
ji
kkk
kij
kk
jj
ii
kji xx
xDGDDDG (5.100)
. jj
D / /j
j
xx
.
152
. - k
p
iijj
T ......
1
1-
, -- -
. k
p
iijj
T ......
1
1
.. i
jkG
.r = r (x1, x2, …, xn) (5.101)
-
kkijij rGr ; /
/
//// k
kjiji rGr (5.102)
(). (5.101)- xi/-
iiii rDr // (5.103)
, xj/
ijjj
iii
ji
i
ji rDDrxx
xr //////
2
(5.104)
. i k-, ijr - (5.102)-
kijkj
jiiiji
k
ji rGDDrxx
xr //////
2
.
. (5.103)-
//
kkkk rDr
//
//////
2
kkkij
kjj
iii
ji
k
ji rDGDDrxx
xr
153
. (5.102)-(5.100) .
« », Mn = R n ,
.
ixijkG
. R n - –
..
,R n - - .
5.9.- . M n
//
//
//
/
//
2
ji
kkk
kij
kk
jj
ii
kji xx
xDGDDDG
kijG ,
M n -
p
s
q
jii
jqjj
k
s
iq
iqiijj
iijj
iijj s
k
ps
sks
p
k
p
k
pGTGTT
xT
1
.........
1
........
......
...;...
1
1
1
1
1
1
1
1
. M n
: M n = Rn
; kijG
. kijG
!. , , (x) (x/)
.( ) ( )
, M n - . ( ) M n -
.
154
5.10. . ijkG
ikj
ijk
ijk GG
.2- . i
jk .
. ijkG
//
/
//
//
//
2
kj
iii
ijk
kk
jj
ii
ikj xx
xDGDDDG
,/
//i
kjG -, « » j/ k/
ijk
kk
jj
ii
ikj DDD //
//
//
. .5.11. .
( ).
R n - ; --
. M n - -
– M n -.
, -
k . k
kx .
5.3.- . ( ).
(1) ={ k} – .(2)
ijT k
ikj
ij TT ;
.
155
(3) , ( M n - f) ,
:
{ kf} = gradfxf
k
(4) Ti
ikk
ii
k GTxTT
, Ti
ikki
ik GTxTT
.(5)
k PTPTPT kij
ijk
ij
. ijT
P – .. (1)–(4) -
. (5)- . :ijT P -
– . (5) .
j .
jk
iik
jjik
jik GPTGPTPT
xPT
ik
jjk
ijik GPTP
xTPT
x
.jk
ijik
jkk
jiji
kk
iPTPTGP
xPTPGT
xT
(i (i), j (j)) -
.
156
§9. . ,.
§2-. u1=const, u2=const -
r = r (u1, u2). §5- -
,
.5.12. .
.r 1 r 2 u2=const u1=const
,
( r 1, r 2) =0
g12 = 0 (5.105) .
5.4..
. r = r (u,v) .,
du2:du1= (u1,u2) , du2:du1= (u1,u2) (5.106) , ui=const
= 2: 1, = 2: 1
du1:du2= 1: 2, du1:du2= 1: 2 (5.107)
« » .
21212121 :::: , dudududu rdrd
, rd = ridui ,
157
22
11
222
11
1 , rrlrrl (5.108) .
,,
0,, 21 jiji
jj
ii rrrrll
giji j = 0 (5.109)
. = 2: 1 = 2: 1
. ,. .
.( – ),
., ( du2: du1= )
(5.109) 1: 2 ,2: 1 = 1- .
1: 2 ,- ( g
) 2: 1= 2: 1
.(5.105) (5.109)– = 0, = , 2 = 1 = 0,
1 = 2 = 1 .5.13. . 1- 2-
, 1- -,
.5.5.- .
.. (5.107) , 2-
1- -
1lrR (5.110) , r l 1 du1:du2= 1: 2
u1, u2 - . ((3.41) )
158
0,, 2121 ::11 duduldlrd (5.111)
., du1:du2= 1: 2
,i
ii
i rdurrd // (5.112) (5.108)-
ii
ii
ii rdrdrdld 1 (5.113)
,21122121
21121 nrrrrrrlrd j
jii (5.114)
n – . (5.113) (5.114)- (5.111)- ( ( 1 2 – 2 1) 21 rr
)
0, 2121 ::duduii nrd
(5.112)-
0,nrijji
,
iji j = 0 (5.115)
. 1 2 .
2lrR
( du1:du2= 1: 2) (5.115) .
(5.115) .. .
( du1:du2 =1: 2), (5.115)
. = ijdui:duj
1: 2 1: 2
. - , (
).
159
(5.115) – 2= 1=1, 1= 2=0 ,u1=const, u2=const
12=0 (5.116) .
5.14. ..
. ( )
((5.109) (5.115)) ..
5.6.- ..
.,
. g12=0
0)(,0
221211
2211 gg (5.117)
. g11g22 0 = – g11 : g22 . 11=0 ,
. 12 0 ,
+ =2212
22111122g
gg .
C
22
112
2212
22111122 4gg
ggg
022
11
2212
221111222
gg
ggg
. ,
160
.0,02
22222
211111 rrrgrrrg ·
C.
..
.
. ( )
. (5.106)
,. , (5.107)-
a1: 2=du1:du2, 1: 2= u1: u2
. (5.109)- (5.115)-
gijdui uj=0,
ijdui uj=0 (5.117) .
-
.
§10.
F(x, y, z, c) = 0 (5.118) -
. – , F(x,y,z,c). (5.118) x, y, z -
, J( 1, 2)
.(5.118)
F(x,y,z,c+ ) = 0 (5.119)
161
. ( c + J). (5.118) (5.119)
.(5.118) (5.119)
0),,,(),,,(,0),,,(
cczyxFcczyxF
czyxF
(5.120)
. , (5.120)
( ).
0),,,(
,0),,,(
czyxFc
czyxF
(5.121)
. (5.121)
. (5.118) -
. ( ,
).(5.121) ( )
, x, y, z(x, y, z) =0. (5.122)
.
(37- ). x, y, z -
.(5.122) ,
c F(x, y, z, c) =0
= (x,y,z ) , (5.121) -. (5.122)
162
37-
(x,y,z) F(x,y,z,c(x,y,z))=0 (5.123) .
x, y, z
0),,,(2
2
czyxFc
. (5.123)
kz
jy
ix
grad
kzc
cF
zFj
yc
cF
yFi
xc
cF
xF
grad = grad + cF
gradc.
0cF ,
(5.123) (5.118) . //
.5.7.
.
163
(5.118) , + 1c, + 1 + 2. ,
,
0),,,(,0),,,(
,0),,,(
21
1
ccczyxFcczyxF
czyxF
F(x, y, z, c) =0,
,0),,,( - ),,,(
1
1c
czyxFcczyxF
,0
2
1
1
2
2
ccF
cF
( cczyxFccczyxFF 1212 ,,,,,, ,),,,,,, 11 czyxFcczyxFF .
1 2 , (5.118)
(5.118) .
0),,,(
,0),,,(
,0),,,(
2
2czyxF
c
czyxFc
czyxF
(5.124)
().
(5.121) , (5.124) -
(5.121) -. (5.124)
= (x,y,z) ,
164
0),,,,,( ),,(
,0),,,,,(,,
2
1
zyxczyxFc
zyx
zyxczyxFzyx (5.125)
.
...
5.8. -.
. (5.121) , 1
cFgradgradF
.(5.125) 1
*
,grad 1 grad 2
(gradF+ gradccF
) (grad cF
+ 2
2
cF
gradc)
.
cF
=0 , 2
2
cF
=0
, 1*
1 ,
..
= (s). - r
,
F(x, y, z, s ) = (r – (s))2 –a2(s) = 0
165
, s – – . s
,
(r – (s))2 – a2 = 0,
(r – , / ) + aa/ = 0
. /
.
.s
..
– . ,
(r – )2 – a2 = 0, (r – , /) = 0
. = (s),
()
.,
-. ( = const
) (5.124)-
(r – )2 – a2 =0,
(r – , /) =0,
(r – , // ) – / 2 =0
, = (s) (2.30) ,(r – )2 – a2 = 0,
166
(r – , ) = 0,
(r – , ) – 1 = 0
. 1/k
.
s ( s
).,
.y.
F(x, y, z, a, b)=0
,
F(x, y, z, a, b) = 0,
a F(x, y, z, a, b) = 0, (5.126)
b F(x, y, z, a, b) = 0
.
. (5.126)- b
.
167
VI-.
– -.
., ,
( ) -.
§1. .
u v ,-
),(urr.
0vu rr
., constu constv -
),( vuM 1constv ur
, 2 constu -
vr, 3 , vu rr
.
kkiivk
kiiu
vu
ebeeae
ebrear
,
,, 21 (6.1)
)3,2,1,,0,0( kiba . k.
vu,
; :vee
urr i
ivu , .
168
ik
ki
ik
ki bbaa , . (6.2)
,03
322
11
33
22
11 bbbaaa . (6.3)
1, 2 (6.1)
,:
,,,
),(,),(21
21, rr
rrrnrrrg ij
ijijjiji
212221121 ggggrr
,),(,0),(,),( 22212
211 brrgrrgarrg vvvuuu
,),,(),,( 31
211111 aa
abebeaeaea
rrrrr uu
vu
vuuu (6.4)
,),,(),,( 31
211112 ab
abebeaeaea
rrrrr vv
vu
vuuv
.),,(),,( 32
212222 ab
abebeaebeb
rrrrr vv
vu
vuvvvuuv rr
, 2112
32
31 baab (6.5)
.a
012
032
31 ab (6.6)
.vuuv rr (6.7)
3 ., (6.5)- . (6.7)
(6.5)-
169
vu ababab 12
21 , (6.8)
.
:
.,
,,
,,
,,
2313
31212
21321
21
eBeAee
eBeabee
bae
eabeee
bae
ebrear
vu
uv
vu
uv
vu
vu
(6.9)
(6.8) 12
21 , ab
(6.6)
BbAa 32
31 , (6.10)
(6.1)- .,
b, , . (6.7)
ivuiuv ee (6.11)
..
3i
,)()( 2111133 ebaAeAeAeAeA
vee uvvvvuuv
12233 )()( eb
BaeBeBu
ee vuuvvu
AbaBBabA
uu
vv , (6.12)
.1i (6.11)- :
170
.)(
,)(
1221
21332111
eba
abe
abe
abe
eABeabeB
baeAe
baeee
vu
u
u
u
uuv
uvv
v
vvuvuuv
(6.12)- ,
0ABab
ba
u
u
v
v (6.13)
. 2i (6.12), (6.13) -, ( (6.9) -
– ). (6.12) (6.13) -- ,
- , (6.13) -
, (6.12) , , -. (6.12) (6.13)
.
§2. .
u v , (6.1) (6.9) -
. u v
)(,)( ** vvvuuuki
ki bba ,,,
))(( *uurr;****
dudur
dudu
ur
urr uu
))(( *uuee ii
.****dudue
dudu
ue
uee iu
iiiu
.,,, **** ****dvdvee
duduee
dvdvrr
dudurr iviviuiuvvuu (6.14)
171
*u *v:
.,
,,**
2*
1*
*
**
kk
iivkk
iiu
vu
ebeeae
ebrear (6.15)
(6.1), (6.14, 6.15)
**
**
**
**
,
,,
dvdvbb
duduaa
dvdvbb
duduaa
ki
ki
ki
ki
(6.16) .
kkiiuiuu
kk
iiuuu
edudua
dudueeear
eaeedudua
dudurr
**1*
*1**
**
**
,
,,
.0ab , k
iki ba , -
b-:
.,,,32
4
31
3
21
2
21
1 bB
bbI
aA
aaI
abb
bbI
aba
aaI uv (6.17)
-, u, v - ( ,
) . ,.
, -,
.
dvedueeddvrdurrd
iviui
vu , (6.18)
21, ebrear vu , -
21 , bdvadu (6.19)
172
, (6.9), (6.17)- (6.18) (6.19) ((6.16)-
) :
.
,)(
,)(
,
22
411
33
32
412
21
12
31
322
21
11
22
11
eIeIed
eIeIIed
eIeIIed
eerd
(6.20)
321 ,,, eeer - (6.12), (6.13)
(6.9) .
. -,
,:
, -r -
.,
baBA ,,, ,. ,
, (6.12), (6.13) . (6.6)
(6.10)- (6.4)
bBaAbggag
221211
22212
211
,0,,,0,
(6.21)
.baBA ,,, ,
.-
, baBA ,,, -
.
173
,.
6.1- . ijijg , – ()
.
§3. 4321 ,,, IIII ..
-. . ,
( ) -.
. (
) ?)(uvv (6.22)
. -
3erR. , -
(6.22) ( - , §5) 0),,( 33 ederd
.(6.20)
0),,( 22
411
3322
11 eIeIeee
(6.19)-
0)( 34 dudvIIab (6.23)
. 43 II,
constvconstu , ,.
. - – .
174
-. (3.8) constv
322
112
3
3 ),(),(IA
AaA
AeAea
eer
u
uu
,constu
422
3
3 1),(IB
bBe
er
v
vv
., I3 I4
., R1, R2
.
24
13
1,1R
IR
I . (6.24)
I3 I4. H
, , ( ) .
2143
2143
1,11RR
IIKRR
IIH (6.25)
322311 , eRrFeRrF (6.26) .
:,
.,
.//,//)(
33232322
331311131311
eeReReRrFeeReReARaeReRrF
vvvvv
uuuuuu
3erR.
“ ” ( ) -. (3.36) constv
175
;),,(),,(),,(3
11133
3
11133
3
333II
bAv
Aeee
eeAeAeAe
eeeeb u
u
uu
u
uuu
constu
4
23
3
333 ),,(II
eeeeb
v
vvv
.1 2
.
(§9, VI - ). -
. ,
.- (
) .u, v
3e ,.
.6.2 T .
, s.
. ( §5),
abdudvdudvrrS vu
. ),(3 vuer,
dudvABdudvee vu 33
.
176
.43 KIIabAB
S (6.27) .
§4.
,,
.),( 21 uurr
- .)( 12 ufu
nrR (6.28) .
,nRrR
- ,, nRd // .
( nndnrd , )0nRdrd . (6.29)
-nndRnRdrdRd //
(6.29) 0i
ii
i dunRdur (6.30)
, /12 : fdudu . (6.30) 1r2r -
ii
ii
ii
ii
dunrRdurr
dunrRdurr
)()(
,)()(
22
11 (6.31)
.0),( nri
-0),(),( ndrnrd ii
0),(),( jji
jij dunrdunr
177
. ),( nrijij
jij
jij
jji dudunrdunr ),()( (6.32)
. (6.31)
0
,0
22
11i
ii
i
ii
ii
duRdugduRdug
(6.33)
. R-
022
11i
ii
i
ii
ii
dudug
dudug (6.34)
, (6.33)-21 : dudu R1 R2
022222121
12121111
RgRgRgRg
(6.35)
.
0)2( 1212112222112 gRgggR (6.36)
;
2122211
2122211 ,
gggg (6.37)
.(6.36) 3 4
, – :
,1
21 gRRK
.211 121211222211
21 gggg
RRH
(6.38)
1 2, .
(6.34)
.
178
§5. ..
.),( vuM -
.),(/ vvuuM -
]3[21),(),( 2rdrdrvurvvuur (6.39)
. [n] n- -.
(6.9),21 ebdveadudvrdurrd vu
21212 )( ebdvdeadudedbdvedadurddrd (6.40)
.])3[(]2[]2[ 322
2122
12 ebBdvaAdueeevbdeuad
u vt- , /
),(tuu )(tvv.
322
21 ])3[(21])2[(])2[( ebBdvaAduebdveadu (6.41)
, « -» ,
22 bBdvaAdu (6.42) /
( ) -. (6.21)- (6.42) 2
jiij duduII
. u, v2222
43 bKabaIIaAbB (6.43) , dvdu :
,.
179
(6.41) (6.42) [3]
, (6.42) .
, ,
. (6.43) ,
, (6.42) , dvdu : -
.
aAbBdvdu :: (6.42) ,
3 . (6.42) ,
.
. -.
. (6.43)
. , 0du0dv ,
.
.
3 ..
6.1. . )0(K. )0(K
,)0(K
.
180
§6.
-- ,
..
1. ,0,0 43 II (6.17)-0,0 BA . (6.9)
03ue, constv
. constv.
(2.52) , -""u . (6.9)
:
.)(, 21111 ebaaeaeaearear v
uuuuuu
- (6.12) , -
0B
bAa vv
. 1ear uuu , uuu rr // . (2.52)
uuu rr // ,constv ,
– ., -
. 3 – .
03I – .,04I 03I .
; constv –.
043II (6.44) .
181
I3 I4 (6.9)-033 vu ee .
,.
. – .
2. I3 I4.
043 II (6.45) . (6.24)-
constuconstv eI
rFeI
rF |1,|13
423
31
I3 I4.
,
.113
43
321 e
Ire
IrFFF (6.46)
. F -:
.113
33
31 u
uu e
Ie
IeaF
Aa
I 3
1,
13 eAe u
. 1 3
33
1 eI
Fu
u
.
34
1 eI
Fv
v .
- (6.45) - (6.17)-
182
0,0, BAbB
a .
.lnlnlnln bBaA""u
bb
BB
bB uuuu )(ln)(ln (6.47)
. (6.12)
uu baAB
.
aA
bB
uu bbBB
.
bb
BB uu
. (6.48)
(6.47)- , (6.48)-
0ln)(ln)(lnu
uuuu
bBbB
bb
BB
.
0lnva
A
. bB
,0lnlnuu a
AbB
.ln constaAconst
aA
constbB
.
.43 constII
183
.constF (6.49) – ,
(6.49) - -- F .
constAa
IFr
3
1
. (6.45)
(6.49) , (6.45)
– F43
11II
R .
3. 01I ( 02I ). 01I (6.17) (6.12)-
0vv Aa (6.50) . (6.9)- constv
.,0,, 132311 eAeeeAeear uuuu (2.52)
. r -:
,2
,
312
31
31
eaAeaAeAaear
eaAear
uuuuuuu
uuu
0),,(,22
uuuuuuuuu rrreAarr ..
0,3 aA
rrr
ku
uuu (6.51)
, constv – ,.
, (6.50) (6.51)-uadus
0v
, , uconstv
.01I
:
184
– (38- ).
02I 01I, constv
constu ,.
021I.
,.
-V- (§10) ( – -
). -.
. -zOu
)(ufz (6.52) .
Oz , (6.52) . xyz
kufveur )()( (6.53)
(39- ). v xOy -, )(ve –
- (§7) - .
38- 39-
185
constv (6.52) , constu
constz u ., ,
0))(,)((),( // veukfverr vu . –
,, 01I . :
)(
,,,)(
,)(,)()(
/
/////
//
efkurr
eueurerkufr
veurkufver
vu
vvuvuu
vu
(6.54)
,,0,1 22212
2/11 uggfg (6.55)
.1
,0,1 2/
/
22122/
//
11f
uf
f
f
(6.55)/
.uu
(6.54), (6.55) -- .
, (6.38)-
22/
///
)1( fuffK (6.56)
, , ,.
2/3/
////
)1()1(
2
2
fuffufH (6.57)
0//// 3
ffuf, 0H (
) .
186
constceecucz
cz
,2 (6.58)
. , Oz
.
(40- ).: (6.52) -
)(zuu)(tuu , )(tzz .
, .(6.52) constu ,
.
§7.
(V , §9) -
., .
(5.117) ., (6.21)-
0,022
vbBdvuaAduvdvbudua
.
AbaB, 43 I , .
§6- 43 I u v ,-
. ,
, ., .
43 I
40-
187
. -.
5.14-.
(§3), 43 I
.43 I ( ) ,
( ), ()
.:
,.
-:
.
.
188
V I . .
§1
S S*
, -.
S S*
, S S* ; S* S. S S*
. ,;-
, .
..
( ),( vur ) -
. ,.
.,
.
§2.
u, v S2
22122
112 2 dvgdudvgdugds (7.1)
, )2.1(igij -vu, .
,.
.
., s
. ,
189
,,)(),( 10 ttttvvtuu (7.2) , s I-
1
0
2
2212
2
11 ),(2),(t
tdt
dtdvg
dtdv
dtduvug
dtduvugs (7.3)
., (7.2)
I .
.,
.
. -u, v .
, Mi Ni ,Ni ui, vi Mi
. ( -
) dvdu, .-
, ds ,dvdu,
(7.1) . -221211 ,, ggg
.ugij , v
,-
. I , -
I ,. , -
, --
, .
190
:,
- .
§3.
-, r -
ijr .
ijr- .
. r -ijr
)2,1,( ji 2- ,21, rr m
:mrGrGr ijijijij 2
21
1 (7.4) 21 , ijij GG , 1r 2r
. i, j ijr. ijr ij
m . (7.4) . i, j
1, 1; 1, 2; 2, 2; . i = 1, j = 2 -, 2112 rr
221
212
121
112 , GGGG .
:
)2,1,,(, kjiGG kji
kij . (7.5)
(7.4) .(7.4) m ( 21, rmrm )
.ijijmr
ij I ij
. (7.4)
191
mrGrGr ijijijij 22
11 (7.6)
. 6- k
ijG )2,1,,( kji.
:)2,1,,(, kjiGrr ijkijk . (7.7)
i, j, k ijkG ,
..,, jikijk GG (7.6) 1r ,
2r .
.
),2,1,(,
222
121
,2
212
111
,1
gGgGG
jigGgGG
ijijij
ijijij (7.8)
(7.8) ji,, (7.8)- 21 , ijij GG
. (7.8)
2212
2111
,212
,111
2
2212
2111
22,2
21,1
1 ;
gggg
GgGg
G
gggggGgG
G ij
ij
ijij
ij
ij (7.9)
. , ijkG ,kijG -
(7.8), kijG ijkG , (7.9)
1 .(7.9)- :
21122122211
2211 , ggggg
gg
2122211
11222122211
12 ,ggg
ggggg
g (7.10)
,
2221
1211
2221
1211 ,gggg
gggg
192
. (7.9)
,,212
,1111
ijijij GgGgG ijijij GgGgG ,222
,1212 (7.11)
. – ijkG , ijg -21,uu
.,, jkkjijji grrgrr ,kiik grr
ku , iu ,ju ,
jki
ijkikj
ijk
kijkji
kij
jkijik
ugrrrr
u
grrrr
u
grrrr
,
,
..
kij
ijk
jki
ijk du
g
du
g
ugrr2 (7.12)
(7.7) , (7.12)-
.21
, kij
ijk
jki
ijk dug
dug
ugG (7.13)
.: ),( 21 uur -
-. (7.13)
,21,
21
211
12,1111
11,1 dugG
dugG
,21,
21
222
22,2122
212
22,1 dugG
dug
dugG
211
112
11,2122
12,2 21,
21
dug
dugG
dugG
193
. 1 2 .
ijkG , (7.13) 1 -. (7.9)- .
kijG I ijg -
– kij
dug
. (7.9) kijG 2
. (7.4) -
, 21 , ijij GG I -,
. (7.4) 1 - ( ) .
§4.
.m 21, rr
)2.1(,0 irm i
. - iu:
),2.1,(,0)( jirmu ij
.0ijij rmrm
ijijrm , ji,
).2.1,(, ijrm ijij (7.14)
2 – mmrr ,, 21
. m im m
194
. 21, rr:
.
,
2221
122
22
11111
rbrbmrbrbm
(7.15)
ijb -
. b- (7.15) 2,1,2
21
1 irbrbrbm iiii (7.16)
.j
ib (7.16)- 1r 2r
.
.
,
222
211
2
122
111
1
rrbrrbrm
rrbrrbrm
iii
iii
(7.14) ijji grr
222
121
2
212
111
1 ,
gbgbb
gbgbb
iii
iii
. 21 , ii bb :
.)(:)(
,
2222
121
2122211112121
21212
1112
122211
2122211
gggggggg
bgggggg
ggb
iiiii
iiiiii
i
2,1,ji
ji gb
. i, j 1 2 -.
§5. .
,,
-. -
.
195
- ,.
, ),( vurr - (V , §1).
. u, v0),( vuf t
, (V , §2) .
0),(),( dvvuduvu, 1
. .
.,
.
),( vurr (7.17) .
),( vuM .
)(uv (7.18)
0),( vuf (7.19)
)(vu (7.20)
)(),( 21 tvtu (7.21)
.21,,,,f
.,
.:
),( vuFdudv
, (7.22)
196
),( vudvdu , (7.23)
0),(),( 21 dvvuduvu . (7.24) (7.24)
00 , vvuu, ),( 00 vu (7.18) (
(7.19), (7.20)) .
« »vu,
., (7.19) ,
vvvufu
*
* ),,( (7.25)
( 0uf
)
. 0vf
),(,
*
*
vufvuu
(7.26)
.constvconstu ** ,
, (1- )
,010),(
),(**
**
**
vf
uf
vv
uv
vu
uu
vuDvuD
2-
.001
),(),( **
vf
vf
ufvuD
vuD
(7.19) 0*u ((7.25) ) 0*v
((7.26) ) .
197
« » – ),( vurr constv
. (VI , §1) (6.1) -
, constv. -: , 3e
, 1econstv , 2e
132 eee1e
. -, . constv
« », -
..
(VI §1- )
kkiivk
kiiu
vu
ebeeae
ebrear
,
,, 21 (7.27)
. ik
ki aa , i
kki bb vuuv rr
32
31
21
12 ,, baababbbaa uv (7.28)
.ivuiuv ee -
(V , §1) -. .
§6. .
. ( (7.17) )
0v. , u -
.
198
v , (7.27)- “ u ”
.:
,~~
~1,~
11 k
kii ea
adued
ae
durd
a (7.29)
)0,(~),0,(~ uaauaa ki
ki . v
.
gg aa
aak
aaduads ~
~,~
~,~
~,~
32
31
21 (7.30)
,
213
312
321
1
,
,,
eedsed
eekdsed
eekdsede
dsrd
g
gg
g
(7.31)
.ggk ,,
,. .
ds ,,
2-
1dsrd .
ggk ,, (7.31) (2.30) -
. ,,
.
1e (7.32) , , (
) ( 1e u- )
199
41-
. , ,,, 32 ee- –
- ( 2e
« » ).,3e
,
(41- ),
,cossin,sincos
2
3
ee (7.33)
32
32
sincos,cossinee
ee (7.34)
. (7.32)- , (2.30) (7.31)-
.321 k
dsdeek
dsed
g (7.35)
(7.34)-)cos(sin 3232 eekeekg
,sinkkg (7.36) .coskv (7.37)
gk -:
, -.
(7.33)-
.cossinsincos2dsd
dsd
dsd
dsd
dsed
200
dsd
dsd
dsed ,,2 -
(2.30), (7.31) (7.36)
dsd
g (7.38)
. – , dsd
.
, - ( , §13) . (7.31)-
]3[)(21)()( 2
321 seeksesrssr g
.
-
]3[)(21)( 2
321 eeke g . (*)
s . (*) 0),,( 21 eerR
].3[21)( 2
213 eke g
)0(M
.
0
3/3
//3
/3
gkk
. ( )
0),,( 31 eerR.
.
201
§7.
-.
),( 21 uurr,
)(),( 22
11 tutu (7.39)
.
321
21 errrrn (7.40)
),(),(),(
),,(
ijjiijij
jiij
nrnrnr
rrg (7.41)
.),( kij rr .
),( kijijk rrG. ( « »
)..jikijk GG (7.42)
(7.31)
gg kedsede
dsede
dsed
21
32
31 ,,,,,
.
neeeedsrde 31321 ,,
3
2
22
2 ),,(,),,(,),(ds
rdnrdkds
nrdndds
nrdgg
(7.43) . (7.39) -
« » , ),( 21 uuf
.dtdtdu
ufdf
i
i (7.44)
202
ii du
udfdf (7.45)
.,
.22
11 du
ufdu
ufdu
uf i
i
.
ii
jiij
ii udrdudurdurdrddrd 22 )()( (7.46)
n ,ji
ij dudunrnrd ),(),( 2.
, (7.41)-IIdudunrd ji
ij),( 2
. V-
. 2dsji
ij dudugIds2
1- , (7.43)-
jiij
jiij
dudugdudu
III
(7.47)
.
.,
dtdt
ddu ii, (7.48)
i (7.39)- .
g
ji
jj
iij
ji
i dudurrrrrnrn
grrrrdurdun
nrdnd),(),(),(),(1),(
),,(21
21
21
21
203
. (7.41)- g
((7.43) )
jijijii dudugg
gIV )(1
212 . (7.49)
, g2ds
..
jiji dudunnndIII ),(2 (7.50)
.nr*
.III (§3) . 4- 1-
jiij
jijiji
g dudugdudugg
gIIV )(1 2112
(7.51)
ii
ii
jj
jj
gdudu
dugdug
dsg 21
212
1 (7.52)
( ).gk (7.42)
(7.46)-
.1
)(])[(1),,(
22
212
1
21
2122
jiij
ii
jiij
ii
kk
kk
kk
ii
jiij
duduGudgduduGudg
dugdugg
rrdurudrdudurg
rdnrd
.1
22
212
1
213 ji
iji
iji
iji
i
kk
kk
g duduGudgduduGudg
dugdug
dsgk (7.53)
204
gk g , -iud 2 ,
(7.39) 2. g 1-
, gk 2-
, , g 1-, gk – 2- .
.
kuueur 212 sin)(cos
, juiuue 211 sincos)(12 uu (V , §2) .
tutu 21 ,,
.0, 221221 ududdtdudu
ijkijij Gg ,, :
,0),(,coscos)(cos),( 21122222/2
1111 rrgtueurrg
,coscos,1),( 22221222112222 tuggggrrg
,1,0,coscos),,(12212
222211111 turrr
g.0,0,cossin,0,cossin 222122112221121 GGttGGttG
(7.47), (7.52), (7.53)- -:
.)cos1(
sin,0,121
t
tkgg
§8.. .
(§5,V ), (§3,V ).
205
. :
? ,.
(7.37) .
2
cosk (7.54)
. (7.47)
222
1
12
2
2
1
11
222
1
12
2
2
1
11
2
2
cos1
gdudug
dudug
dudu
dudu
k (7.55)
. : 1) --
ij ijg ; 2) -21 : dudu -
; 3) -
.,
,.
, ( -
), k.
-.
,
. ,. ,
,
206
. -,
, -.
--
.
.
,,)2
,,0)1
nv
nv
(7.54) -k . . = v (7.56)
( k . .). (7.54) (7.56)-
K = k . ./cos (7.57)
: k . ., k- .
.
(42- ).-
., 21,ee
.},,{ 21 eeM
(43- ). 1e (5.14 ) ,
3e .
11
.k
207
42- 43-
..
-: ,
.. -
dsrdR 1 (7.58)
, (dsrd
–
, s – )
dsdur
dsdurR
2
2
1
11 (7.59)
. (6.1) ( vu, - 21,uu -)
2211 , ebrear
. R 1, 2
dsdubX
dsduaX
2
2
1
1 , (7.60)
. (7.47)
.2221
dsduBb
dsduAa
dsdu
dsdu ji
ij (7.61)
208
(7.60) (7.61)- dsdu
dsdu 21
, ,
1224
213 XIXI (7.62)
. 2-. –
, 1
. 2-.
, 043II , (44- ).
, V (§7) ,
43 II (45- ). ( 0K , 043II )
(46- ).
( ,0K 03I )04I
32
1I
X (4
11I
X )
(47- )., 21,ee
, (7.62) 2-, },{ 1eM },{ 2eM
.
44- 45-
209
46- 47-
V ; §9-
-. 2-
.2- 2- -
( -),
.( )
,k . .- .
(V , §3)- 3I4I
. , (6.4), (6.17) ,
24
13
1,1 RI
RI (7.63)
. 1R 2R , ,
-1
.1
1R 2
1R
(7.61) -. , ,
1 , 2v:
210
.1;12
0210
1 1
2
dudu RR
(7.64) (7.61) 1 , 2
. , rd 1e:
dsedurdur
rderd ),(),(cos 1
22
111
.),( 11
22
11
dsdua
dseduebduea (7.65)
rd - 2e2
dsdub
2sin
2cos . (7.66)
(7.65) (7.66)- (7.61)- (7.64)- ,
22
21 sincos (7.67)
..
(7.54) (7.67) v1 v2,
., v1 v2,
. KII 4321 -, II 4321
.
§9.
- .ggk ,, .
0g (7.68)
211
,.
(7.52) , (6.34)
. , (7.68) -. (6.34) 1-
21 : dudu - ,
( . . , . , ., 1958, , §1)
. V V. , (6.34)
,
02
221
11
222
111
dududugdug
(7.69)
.01du 02du
, .,
- . (§3, V ) -
: -.
V , (§7) ,. (7.31)
. ,
3erR
0,, 33 ds
ededsrd (7.70)
.(7.31)- (7.68)- .
, -;,
(7.70) .
212
2- -.
(7.31) ( ):
,- . -
, 0g
dsrd
dsed 3 (7.71)
.//3 rded (7.71) .
..
()
,.
. 3, er *3
* , err-
.
constee ),( *33 . (7.72)
rde*3 rded //3 ,
0),( 3*3 ede (7.73)
. (7.72)-0),( *
33 ede , 3*3 eed .
(§7) *3
*3 eed .
.// *33
*3 eeed (7.74)
rde3 rde*3
.// *33 eerd (7.75)
(7.74) (7.75)-
rded //*3
213
. ,.
(§1, V )constv constu
. constv , (6.19)-02 (6.20)
11
3311
12
31
321
1111
,
,,
eIedeIed
eIeIederd
. (7.31)
/// 030101 ,, dvgdvdv IkIds
. (§3, V ) 1 3:
1, 3
( ,). , 2 4
..
kbuuevr )( ( - , §7) -
.
,1),(,0),(,),(
,0,,
,,
221222
11
/
/
vvvuuu
vvuvuu
vu
rrgrrgbvrrg
rerevr
erkbevr
0,),,(1
,0),,(1,
2212
11222
122211
gbrrr
g
rrrg
bvgggg
vuuv
vuuu
. (6.34)-
0
)( 22
dug
bdv
dvg
bdubv
214
,
22 vb
dvdu
. , -
Cbvvu )ln( 22
.
§10.
0 (7.76) -
.0 (7.31) -
:
.,,;
;,;,
23
312
21
1 edsedeek
dsedek
dsede
dsrd
dsdk
dsdk
dsd
dsrd
gggg
(7.77)
( ),
; ,2erR « » (§6) . -
gk g -k
.
. -
, -, . ,
- ,.
.
215
7.1- .,
.
.(7.76) (7.47)
0jiij duduII (7.78)
. -, ,
( )
.§9- (V )
0jiij
du1:du2 = 1: 2 « »,
.(7.78)« - » .
du1:du2- (7.78) 2
122211 , gK
((6.38)- ). (7.78) 1- , -
, :– ( <0)
;– ( >0)
;– ( =0)
.-.
.
216
(7.78) constu2 constu1
. ,.0,0 122211
((6.38)- )
ggH 12122
. 012g ,
, .
217
VIII .
, ( -
) ( -) . r -
,,
)(rFr
)(rFF
. )(rF ()(r ) ( )(r
) .
- )(rF,
.
. « » .
§1. .Grad, Div, Rot .
21, rr 21 ,
)()()( 22112211 rFrFrrF (8.1) , )(rF .
. Fc
),()( rFrF (8.2) ),()()( 2121 rFrFrrF (8.3)
n
iii
n
iii rFrF
11)()( (8.4)
.
218
21,, rr (8.2) (8.3) , (8.1)
.
.8.1.- . )(r
, a:
),()( rar . (8.5)
. 321 ,, eee ,3,2,1, iexr i
i . (8.6) ix ie
r .
)()()( ii
ii exexr . (8.7)
)( ie i ,3
1
3
1)(
iii
iii eeea (8.8)
. (8.7)-),()( arxr i
i . (8.9) .
a ,-
. ,, 1, 2, 3 .
ie *ie
:0det,* k
ikkii cece . (8.10)
)cos()( **kiki
ki eeeec (8.11)
. (8.10) ke,
219
*~k
kii ece (8.12)
ikikki
ki ceeeec )()(~ **
. (8.13)
ijijji ee )( , (8.14)
,
.,1,,0
jijiij
ij (8.15)
(8.12)- (8.14)- ,
ijk
kj
ki cc
3
1
~~ (8.16)
, (8.13) ij
k
jk
ik cc
3
1 (8.17)
.a
. *ie
),()( *arr (8.18)
. aa * . (8.8)-*
3
1
** )( ii
i eea (8.19)
. (8.10)- , -
aeeeeeeee
eeccececa
jkkj
jki
ji
kij
i
jik
ki
332211
3
1
3
1
*
)()()()(
)()(
, *a - .a
ii
i
defeeGrada
3
1)( (8.20)
220
. 8.1-),()( rGradr (8.21)
. ,Grad :
rrGradnp
r
)(
,, Grad –
)(r -.
)(r -
:3
1))(,(
iii
defeeDiv , (8.22)
3
1))((
iii
def
eeRot . (8.23)
- (Div) (Rot)
. Div RotGrad - -
. ,
))(,())((3
1
3
1
***jk
i
ji
ki
iii eecceeDiv
.))(())(,(3
1Diveeee
jjjjk
kj
.)( jj eE (8.24)
iijj eaE (8.25)
ija , -
iX ix. ,
221
ii eXr)( , (8.26)
iij
jj
jj
j eaxExexr )()(
.jij
i xaX (8.27)
-ija ,
0det ija (8.28)
-.
0det ija , (8.27)
, . (8.12)
ija ?
*~k
ki
ii
i ecXeX
ik
ki cXX ~* . (8.29) :
kj
jk cxx ~* . (8.30)
(8.27) ki
ki xaX *** (8.31)
. (8.29), (8.30) (8.27)-jk
ji
kji
kkj xcaxca ~~ * (8.32)
. ix ,ik
kj
kj
ik caca ~~*
. i j -
. , i
,~~11
* ik
kkik caca
222
,~~22
* ik
kkik caca
ik
kkik caca ~~
33*
.lll ccc 321
~,~,~ ,,
j
ik
lj
kj
j
lj
kj
ik ccacca ~~~~*
. (8.16) (8.13) ,
ija
ik
jl
kj
il ccaa ~* (8.33)
.
ikc~ k
iik u
uD*
*.
(8.33) kja
., k
ja ,ija
.
ji
ij aa
, .j
iij aa
, ..03
322
11 aaa
)(21)(
21 j
iij
ji
ij
ji aaaaa (8.34)
,
)(21 j
iij
i
jcaaa
223
)(21 j
iij
i
jkaaa
. -
kc . (8.35)
Div Rot ija
.3
1),())(,(
iiiii Eeeedif
.),( 33
22
3
1
11 aaaeae
ik
kii (8.36)
3
1
3
1
3
1))((
iki
ki
iii
iii eeaEeeeRot (8.37)
.)()()(3
1
12
213
31
132
23
321
iki
ki aaeaaeaaeeea
,0
cRot
(8.35)
kRotRot (8.38)
.0Div
cDivDiv (8.39)
Div Rot
. ji
ji a E
~
. ,ija – .
, 321 eeeV.
ii eOA , ( i = 1,2,3).
224
,)()(~
11111*1 i
ii
ii eaeeaOAOA
,)()(~
22222*2 i
ii
ii eaeeaOAOA
ii
iii eaeeaOAOA 33333
*3 )()(
~
( ).
]2[)(),,( 33
22
11
*3
*2
*1
* VaaaVOAOAOAV
[2] – . 2- (, 1- )
.*
33
22
11 V
VVaaaDiv (8.40)
, E~
( – - )
.,
kRot
21
(8.41)
.k
-
ij
ji aa
. (8.37) ii exr -
v :
).()()(
)(21
32
231
13
23
321
12
13
312
21
3212
131
32
axaxeaxaxeaxaxe
exeaeaeaexRotrv ii
ii
k
(8.26), (8.27)-
kv (8.42)
.
225
)(r. -
)(r -
ck .
§2. . grad, div, rot .
)(rF.
- .0rr
0)(lim 000FrF
rr (8.43)
, 0F ( ) )(rF
. )(lim0
0 rFFrr .
)()(lim 00
rFrFrr (8.44)
)(rF 0r .)(rF
( – )
.8.1- .
0)()()(
lim0
rFrF (8.45)
)( , )(rF.
)( .
. ,
0)()()(
lim0
rFrF (8.46)
226
)()( ,FrFrF )()(
FF00
lim)()(lim
.limlim00
FF
(8.45) (8.46)
.0)()(lim0
(8.47)
constt , .
0t , (8.47) .
)()(lim)()(lim000 t
tt (8.48)
( - : )()( tt ) (8.48) ,
0)()(
)()( (8.49)
, -
., . r
)(rFrd
rddF (8.50). ,
ii exr ix
.,
),,()()( 321 xxxfexrF ii
227
ii exxxXrF ),,()( 321
.
3
1i
ii dx
xfdfddF , (8.51)
ki
kk
i
ii
ii dxe
XXedXeXdddF
3
1)(
. (8.52) (8.20)
, (8.21) :),()( rdGradrd . (8.53)
)(r )(rgrad .
)(dGradGradgraddef
. (8.54) (8.53), (8.50), (8.51)-
ii edxrd (8.55)
,3
1iii e
xfgrad (8.56)
. grad.
)(r div rot ()
- Div Rot:
)()( dDivrdivdef
, (8.57)
)()( dRotrrotdef
. (8.58) (8.52), (8.50), (8.26) (8.27)-
k
iik x
X (8.59)
. (8.36), (8.37)-
3
3
2
2
1
1
)(xX
xX
xXrdiv
, (8.60)
228
2
1
1
2
31
3
3
1
23
2
2
3
1)(xX
xXe
xX
xXe
xX
xXerrot
(8.61) .
(8.56), (8.60), (8.61) grad ,div , rot ;
.
, ji k
i
xX ,
Div Rotdiv rot
; ,div rot Div
Rot . , ( ) .
gradconstxxxf ),,( 321 (8.62)
« » .f ( - )
. (8.62) )(,0),,( 321 constcCxxxf (8.63)
, (8.63) )(trr.
,0))(),(),(( 321 Ctxtxtxf (8.64)
.)()( ii etxtr (8.65)
(8.64) -
0dtx
xf i
ii (8.66)
(8.56)-
0,dtrdgrad . (8.67)
229
grad
, grad.
sf
ef
s
def
0lim (8.68)
« ».
exxxff ),,,( 321 – , es ),,( zyxM. ]2[dff (8.51),
(8.56)),( rdgraddf (8.69)
,
egraddsrdgrad
srgrad
ef ,,,lim
ee ,
e:
gradnpef
e (8.71)
§3.
grad rot , div -,
2- grad (div ),div( grad ), rot( grad ), div(rot ), rot(rot )
.
:
.),,(
,
rotdivgrad
(8.72)
(« » )
(8.70)
230
3
1iii
def
xe (8.73)
.
3.1. . « -
»
iii
def
xe (8.74)
, ,
ik
i
i
def
divxXe),( , (8.75)
def
XXXxxx
eee
321
321
321
(8.76)
rotxX
xXe
xX
xXe
xX
xXe 2
1
1
2
31
3
3
1
23
2
2
3
1
. ,3
1
321 ),,(i
ii exxxX
.(8.74–8.76) (8.56, 8.60) (8.61)-
iix
graddiv 2
2
)(),()(
, (8.77) 2)()( divgradrotrot (8.78)
,
ii
idef
exX
2
22
)( (8.79)
,0)(rotdiv (8.80) 0)(gradrot (8.81)
- .
231
3.2. . .3
12
2
0)(i
ix
3
12
2
)(ii
def
x« » .
.(8.80) -
. (8.80) :
0div . ( )
. ,
.rot
, . (8.81)-
grad.
( ,) (8.81)-
0rot. .
3.3. .),,( zyxfu
. f .8.2- . grad u
u
kzuj
yui
xuugrad (8.82)
232
.
),( 0lugradlu
, ,1,// 00 lll
.coscoscos0 kjilll
.1. ( -
) .2. .3.
:222
maxzu
yu
xuugrad
lu
:grad u
.
§4. .
. . . - (1801–1861) . -
-a
1826 . , 1838 . .
.
(1819–1903) 1854 . .
.
( ).
233
.S , S
div - . zyx aaa ,,S
.
TS
dadivda
.TS
n dadivda
,.
..
.. L
( ), LS
:
L Sn
L S
darotdsa
darotrda ,
a L
, arotn – S-, L-
. zyx aaa ,, ,S
.1- . , L
S
.2- .
, arot =0..
234
.
, Y , Y.
, Y.
x Y,
. ,
, .Y
, .
. –
.
§1.
.
:1) – - ;2) ;3)
., -
,. -
.,
. ,.
XY
XY YX \
xfYXf :
21 xx 1xf 2xfYXf :
XfYXf YXf :
xx ,
xx,0, xx xx
xxxx ,,Xxxx ,,
xxxxxx ,,,
xx,
,XXX0
235
– d , M , d .
M .M
, , (9.1) .
Ø , ( (9.1) ).
M
: M, , M = (0,1), N = (0,1) .
, ,M
,.
, – ,
, (48- ).
- M. -
.M -
.
Ø,M -
,.
M .M
XMxx, xx,
NyxNM ,inf, Mx Ny
NM 0, NMNMyx
N0, NM
N
N aa Ma,
xaMa ,inf, Mx
0 x
xx, x ,xO
,MOMx,
Xx,MO
MxOMx ,0, 00, Mx x
x
236
, . -, M , ( -) , -
: , , – .M -
[M] , M -. . M
M , [M] = M, M .
MM , M
. M.
: M -
,.
,,
.,
. , 1 2
.,
. 1
2 ,-
. , ., - (49- ).
-- , -
. - G
. -
, - .
baa , bab , ba,
MM
x
MX \
21
21x 21
x
11,xO 22,xO
1 2
21,xO x 21nR
nRnR Gx0
GrxUr ,:0 0nR
nR
237
48- . 49- .
G,
., Ø G ,
. Ø, , Ø, Ø – , -
( – , Ø – , Ø – , – ).
§2. .
-- G
:IG. , Ø – G .IIG. G
G G (G ).
G (X, G), (X, G) ,
G (X, G). G -
(X, G) . (X, G)
. :I . Ø – .II . -
.
CX CXXX \
XF \
X
238
GG . : 1) G
G ; 2) G; 3) Ø ; 4)
G ,. G
., G
.- G -
,. G
, .1–4
.-
-- – I , II
- -.
[X, ] , -,
G IG, IIG ,(X, G) = [X, ]
. (X, G) [X, ].
- G -.
,
.
. ,
,.
9.1. A . ( M ) -
GG
GG
GX ,
x
239
, M .(M ) Ox ( OM) ,
, .9.2. A .
, Ø , M . - M
-M , [M]x [M]
., M M
, . .
, [Ø] = Ø ., ( « »).
9.1.- . M
[M] = M.
. . , M - ., .
M- ,; -
- , . ,
[M] = M. .[M] = M . M - , - X \ M
. , [M] = MM- , X \ M-
. ,
X \ M – . (M-
, M- M-) .
x x
Ux Vxx
XM OxMx
MM
XXNM ][NM
MX \MXx \
MX \ MXx \M MM MM
MXx \Ux
MXUx \
MM
XM MMMM
MM
240
,.
. . [M] - ,
, M- -, ( - ) M-
. M-, . ,
..
. , -
.
1- .
,.
,-
.2- .
. , ,
b - .3- .
.-
.
Ø, ( )
(b)
XM
Mx x Uxx
My MyUx
x UxMx MM
MM
R
nnGn
11,11
1;1
yxyx, Rba, bxa Rx
ba,cbac ,min cc ,
yxyx, Rba, bxa Rx
ba,
aHCX XH
H HCX H XHCX
HHCC XX
241
.
(c)
(d)
.
§3. , ,
(X, G) – ., a
.H - .
U , H (50- ).
H.
H,
, H
. H
.H
, H. H -
H- ., H
, ,
(9.2)
HCHC XX
HCHC XX
Xa
X aHU a
Hint
bHCV X
b
extH
c
H
XHint extH H
XHextHHint
50- .
242
Ø (9.3)
:
,.
(9.4) .
,- :
. (9.5)
,, (9.6)
, -,
. (9.7) -.
, ,,
.- ,
H intH, extH. , (X G0) -
Ø , ,,
– , Ø, Ø .
X- , H,
, , Ø.51- (§1, 2- ,
4- ) - H , -
.
HHHextHextHH intint
HextCH Xint HextH Xint
HCH X
HHint
HCextH X
XH
XH H X
Hint extHHH
HH int HCextH X H2R
2RHint extH H
243
9.2.- . HintH .
. (52- ).
,, .
intH , §2, 1-
.
Ha int HU a
aU
aUH HU a int
HaaUH
int
int
Hint
51- .
52- . 53- .
§4
, .a H
a H
244
.-
.
(9.8) .
. (9.9) –
,
. (9.10),
– . .9.3.- . .9.4.- . .
. §3- (9.2) (9.3) .
. ..
, .9.5.- .
,
. (9.11) . (9.11)
(9.12) , §2- (d)
(9.13) . (9.8) (9.13)- (9.11)
(9.14) . (9.14)- .
. Ø U . Ø
H HH
extHCHHH Xint
HH
HHHH extH
H
HextHHint
H
BABA
BACBAC XX
BCACBAC XXX
BextAextBAext
BAexta BAU aAU
245
Ø, - -. ,
. (9.15) . Ø Ø (53-
) U .,
Ø, . ,
. (9.16) (9.15) (9.16) (9.14) .
§5.
, ,
.
..
– ,. -
B, B (X, G)
( ) .9.6.- . (X, G)
,U
.. . , U –
. , U
. , , a. , .
. P G ,.
BU a A B
BextAextBAextBextAexta AU BV
a VVUW a
BAW BAexta
BextAext BAext
GX , BBGX ,
BBXa a
aBa UB
aBB
a
aGU B
Ua
aB UBa
a
GP Pa
246
(54- ).
, .
T ,
, .,
,
54- . 55- .
.,
,.
.9.7.- . X
: 1) ; 2) Ø ;
3). X
..
.G G . §2- 1, 3
4 G.
PBaa
BBa
Paa
BP
2E
X
BBBX B
21 , BB 21 BBa 3Ba
213 BBB BB3
B
BB
247
2- , G-G G G -
. (55-
).
. 3-
B-
., G- .
.
. En
. En- ,,
.-
.
§6.
– -.
(X, G) – Y X- -. Y
. Y- G ( ),
(X, G).
, -, -
. , . (X, G) ,
- - .
.
2112 GGG
21 GG
1GBa a 2GBa a BBa
BBa
aBa 21 GGBBB aaa aB
21 GG BBa
GGYG ,
,Y
,Y YH GX ,,Y
3E3E
248
9.8.- . (Y, ) F, Y - -
.. ,
- .(Y, )- . , (Y, ) - .
, - . (Y, )- . , .
, - ..
9.9.- . -, (Y, ) -
.
§7.
Ø U U, ,
, -.
., -
(§2, 2- ).
, X-.
XU V , U = VV = U . -
: -
.X -
, ( ) -. X H , X-
GX ,
F HYGX , HXYFY \\
FF ,Y FYQ \
0GYQ GG 0
0FYF 00 \ GXF FX ,
BB GX ,BYB
XVU VU
XUUXV \
X
,21 GGH
249
Ø, (9.17) Ø, Ø
G G ,.
9.10.- . – . -
.. .
, (9.18)Ø, (9.19)
Ø, Ø (9.20) U V
. (9.21)
. -,
Ø. (9.22) – - ,
. U- ,V- .
, . (9.19)-
Ø,, 1 2.
: Ø, Ø.
.
Ø,
(9.20)- - . – X. 9.10- X-
21 GHGH
1GH 2GH
H XHH X
XVUHHYHU
HU HV
VHUHHUH VH H
VHUHVHUHH X UH VH
VH1
UH2 1 2
VHUHHH21
UH VH
UHUH
a
250
,
. X-
. – b ,
Ø , 9.10- -.
.9.11.- .
.9.11.-T , , -
-. X
X- ., X
..
.
§8.
§2- --
. ,
,. , -
.. -
-.
..
-. §1, 2-
aH bH
ba HH
baba HHHH
251
-.
.. ,
,,
.
.,
.-
( [4], , §6 [6], ).
..
.U , -
U ,,
. .9.12.- .
.. , b
. bU V . -
-U V ,
, Ø. . .
§9
-, .
-
,...,...,1 naaa 0nn
naUan 0n a na
na a
na
na
na
VU
252
. –
- ,.
, -.
.-
, ,. , ,
- ,.
9.13.- . -- -
.. . - -
. ,
-. - , -
-.
.
. .. –
- .MG1 G
G .. MX
U (56- ).- .
GX , G
GX ,G
.GG ,G
)( 22 GaG
a X Xa
x
253253
, -
. G1 G2 ,
.9.14.- . .
. y .MGy Hy y
. MH y Ø , C -: .MC y
, y -
yXMy
HCM
..
-.
9.15.- . n--
. ( -).
– -. .
.En- . .
--
. . , – .
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254
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9.16.- . -:
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255
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., 9.16- ,
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fYV Vf 1
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0xf VfxVfU 10
1
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f
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256
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G YGXf
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f
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257
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258
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259
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260
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21 XX
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262
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nRnW
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263
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264
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1S
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nR
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265
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int - n- -.
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n- -,
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n = 1 , – , – ( ). n = 2 ,
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266
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n,...,1
268
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269
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270
10.4.- . SWW ,...,1 –
r,...,1 SWW ,...,1 -,
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.
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271
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273
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274
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78-. S2 t1 t2
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2. 2 . - 2- .
275
78- . 79- .
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( - - ). a, b, c 2
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. I, II, III, IV -,
I, II, III, IV ,, « »
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(III, IV) (IV, ), ( ) (I, II) (I ,
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(t1, t2) (t2, t3) -, (t1, t3)
- (80- ).
4. 10,10),,(2 yxyxI ( , 0) (1- ,1), (0, ) (1, )
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81- .
276
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277
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W – . W-1) - ;2) W3) W- ( , )...(\ 1 pW
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, W- .§3- S2 2
.
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. « -», , 1
(83- ).p,...,1
.
p2pS -
.S1
, .2 S2
, S2 -
.0)(,2)( 22 MpS p –
1- , -
278
),1(2)2()( (10.7) 2)( (10.8)
.
., . .
10.8.- .
. (10.1)
(10.2) 10.8-.
10.9.- . -,
-.
[8], [9] -.
§6.
. ( m
) ( n) 3- -
, (« ») .
3,3 nm (10.9) .
, k, f. - n
,kne 2 . (10.10)
kmf 2 (10.11).
279
2fke (10.12)
(10.10) (10.11) f -
222mkk
nk
(10.13)
,
21
21111
knm (10.14)
.(10.9) (10.14) , m n-
:;4,3)2;3,3)1 nmnm
3,5)5;3,4)4;5,3)3 nmnmnm(10.10–10.13) , k
f - :
m n k f 3 3 4 6 3 3 4 6 12 8 3 5 12 30 20 4 3 8 12 6 5 3 20 30 12
, 3
. , -, -
.
280
I
1- .1. M0(–1,–1) :
A) (x = 1 + 2t3 , y = t2 – t) B) (x = t ; y = t2 + 1) C) ( = t3 ; y = t +1) D) (x = t3 – 2t ; y = t2 – 2)
2. t = 1 (t) = (t, t2, t3)
A)14
322 kji
B)14
32 kji
C)14
322 kji
D)14
32 kji
3. (t) = (5cost, 5sint, 5t)
A) k = 31
B) k = 21
C) k = 51
D) k = 101
4. L: y2 = x3 + x2 :A) (1, 0) B) (0, 0) C) (1, 1) D) (0, 1)
5. 1(t1) 2(t2)
281
A) L = 2
1
t
t
2 2 2( ) ( ) ( )x t t z t dt
B) L = 2
1
t
t
2 2 2( ) ( ) ( )x t t z t dt
C) L = 2
1
t
t
2 2 2( ) ( ) ( )x t t z t dt
D) L = 2
1
t
t
2 2 2( ) ( ) ( )x t t z t dt
6. 1(t1) 2(t2):
A) L = 2
1
t
t
2 ( ) ( )x t t dt
B) L = 2
1
t
t
2 2 3( ( ) ( ))x t t dt
C) L = 2
1
t
t
2( ) ( )x t t dt
D) L = 2
1
t
t
2 2( ) ( )x t t dt
7. L:3
3
3
2
13;
13
tt
tt
A) (0, 0) B) ) (1, 0) C) D) (1, 1) D) ) (0, 1)
8. r = r( )1(r1= r( 1)) 2(r2=r( 2))
2
1
)()( 22 drrL .
r=sin 1 = 0 2 =
A) 0B) 2C)D) 1
282
9. r = r( ) k
23
22
22
]))(()([
)()())((2)(
rr
rrrrk
. r( ) = 3 .
A) 1B) 1/3C) 0D) 3
10. ( ) ( , , )M t cht sht t t0 = 0 A) (1, 1, 0) B) (0, 1, 0) C) (1, –1, 0) D) (1, 0, 0)
11. 2 + 2 + z2 – 14 = 0 (1, 2, 3) -
A)1
2 = 2
2 =
33z
B)1
1 = 2
3 =
33z
C)1
1 = 2
2 =
33z
D)1
1 = 2
2 =
23z
12. = f(x) 0( 0;y( 0)) -
A) – ( 0) = ( )B) = ( 0)( - 0)C) – ( 0) = ( 0)( - 0)D) – ( 0) = - 0
13. = f(x) 1( 1;y1) 2( 2;y2)
A) L = 2
1
21 ( ) dx
B) L = 2
11 ( ) dx
283
C) L = 2
1
31 ( ) dx
D) L = 2
1(1 + y 2(x))dx
14. r = r( ) k
23
22
22
]))(()([
)()())((2)(
rr
rrrrk
. r( ) = 4 .A) 0B) 4C) 1D) 1/4
15. L (x = t2 , y = 32 t (3 – t2)) - .
A) (1, –1) B) (3, 0) C) (2, 0) D) (1, 1)
16. r = r( ) k
23
22
22
]))(()([
)()())((2)(
rr
rrrrk .
. r( ) = 2 .
A) 2B) 1/2C) 0D) 1
17. y( ) = , z(x) = x2 (1, 1, 1)
A) 6x – 8y – 2z + 3 = 0 B) 7x – 8y – z + 3 = 0 C) 6x – 5y – z + 3 = 0 D) 6x – 8y – z + 3 = 0
18. L(x = t, y = t2 + t + 1) A) (1, 0)
284
B) (1, 3) C) (0, 1) D) (–2, 3)
19. L1(x = t, y= 1 + t2) L2(x = t2, y = t + 1)
A) (1, 3) B) (1, 2) C) (0, 3) D) (1, 0)
20. t = 1 L(t) = (t, t3) kA)
10001k
B) k 0 C)
10006k
D)3 1000
6k
21.3
2
3 43,
23)(
tt
ttt t0 = –2
A) (–1, –1) B) (1, –1) C) (–1, 1) D) (1, 1)
22.2
,3
,4
234 tztytx (x0,y0,z0) -
A) 0
1 =
0
02t
yy = 2
0
0
tzz
B) 0
2 =
0
0
2tyy
= 20
0
tzz
C) 0
1 =
0
0
2tyy
= 20
0
2tzz
D) 0
1 =
0
0
3tyy
= 20
0
tzz .
285
23. L(t) = (t, t2) k
A)23
2 )41(
4
t
k
B) k 1 C) k 0
D)23
2 )41(
2
t
k
24. 2 + y2 + z2 = 1 A(3
1 ,3
1 ,3
1 )
A) (x = t, y = - t, z = t) B) (x = t, y = t, z = - t) C) (x = t, y = t, z = t) D) (x = - t, y = t, z = t)
25. )(t = (cost, sint, 10t)
A)1011k
B) k 1C)
51k
D)101k
2-
1. (x0,y0,z0)2
,3
,4
234 tztytx
A) 30t (x – x0) + 2 2
0t (y – y0) + t0(z – z0) = 0 B) 3
0t (x – x0) + 20t (y – y0) + 2t0(z – z0) = 0
C) 30t (x – x0) + 2
0t (y – y0) + t0(z – z0) = 0 D) 2 3
0t (x – x0) + 20t (y – y0) + t0(z – z0) = 0
.
23
22
22
]))(()([
)()())((2)(
rr
rrrrk r( )=3
286
A) 0B) 1/3C) 3D) 1
3.2
,3
,4
234 tztytx (x0,y0,z0)
-
A) 0 0 02
0 01 2y y z z
t t
B) 0 0 02
0 01 2 2y y z z
t t
C) 0 0 02
0 01 3y y z z
t t
D) 0 0 02
0 02 2y y z z
t t4. x = t, y = t2, z = t3 t = 1
A) 1 1 11 1 3
z
B) 1 1 11 2 3
z
C) 1 1 11 2 1
z
D) 1 1 11 4 3
z
5. 1(t1) 2(t2)
A)2
1
2 2 2( ) ( ) ( )t
t
L t t z t dt
B)2
1
2 2 2( ) ( ) ( )t
t
L x t t z t dt
C)2
1
2 2 2( ) ( ) ( )t
t
L x t t z t dt
D)2
1
2 2 2( ) ( ) ( )t
t
L x t t z t dt
6. = f(x) 1( 1;y1) 2( 2;y2):
287
A)2
1
21 ( )L dx
B)2
1
1 ( )L dx
C)2
1
31 ( )L dx
D)2
1
21 ( )L dx
7. ( )t = (3cost, 3sint, 3t) :A) k 1/6 B) k 1/2 C) k 1/5 D) k 1/10
8. t0 = 1 L(t) = (t,2t) :A) k 2 B) k 1 C) k 0 D) k 3
9. ( ) ( , , )M t cht sht t - t0 = 0
A) (1, 1, 0) B) (0, 1, 0) C) (1, 0, 0) D) (1, –1, 0)
10. (x0,y0,z0)2
,3
,4
234 tztytx
A)30t (x – x0) + 2
0t (y – y0) + 2t0(z – z0) = 0
B)30t (x – x0) + 2 2
0t (y – y0) + t0(z – z0) = 0
C) 2 30t (x – x0) + 2
0t (y – y0) + t0(z – z0) = 0
D)20t (x – x0) – 2 3
0t (y – y0) + 40t (z – z0) = 0
.
11. L(t) = (t2,t3)
288
A)
23
2 )94(
6
tt
k
B) 32 )94(6
ttk
C) 22 )94(6
ttk
D))94(
62tt
k
12. (1, 1, 1) y( ) = , z(x) = x2
A) 6x – 5y – z + 3 = 0 B) 6x – 8y – z + 3 = 0 C) 7x – 8y – z + 3 = 0 D) 6x – 8y – 2z + 3 = 0
13. L: xy2 = x2 + 2x - 45 :
A) = 1 B) x = 1 C) = 0 D) = 0
14. M0(–1,–1) :A) (x = 1 + 2t3 , y = t2 – t) B) ( = t3 ; y = t +1) C) (x = t3 – 2t ; y = t2 – 2) D) (x = t ; y = t2 + 1)
15. 0 = 1 = 1
A) k 21
B) k 2C) k 0 D) k 1
16. 0( 0;y( 0)) = f(x) A) – ( 0) = - 0
B) – ( 0) = ( 0)( - 0)
289
C) = ( 0)( - 0)D) – ( 0) = ( )
17. 2 + 2 + z2 – 14 = 0 (1,2,3):
A) + 2 + 2z – 14 = 0 B) + 3 + 3z – 14 = 0 C) 2 + 2 + 3z – 14 = 0 D) + 2 + 3z – 14 = 0
18. 0 = 1 = 2
A) k 2
B) k1252
C) k 0 D) k 1
19. (x0,y0,z0) 2,
3,
4
234 tztytx
A) ( 30t +2t0)(x – x0) + (1- 4
0t )(y – y0) –2( 30t + t0)(z – z0) = 0
B) 30t (x – x0) + 2 2
0t (y – y0) + t0(z – z0) = 0 C) 3
0t (x – x0) + 20t (y – y0) + 2t0(z – z0) = 0
D) 2 30t (x – x0) + 2
0t (y – y0) + t0(z – z0) = 0
20. (t) = (cost, sint, 10t) :A) k 1/101 B) k 1/10 C) k 1/5 D) k 1
21. )(t =(cost, 2sint, 3t) :
A) k 1 B) k 1/10
C) kt2cos31
2
D) k 1/5 22. 1(t1 =
2) = t – sint, y = 1 – cost, z = – 4sin
2t
290
A)( 1) 1 2 2
12 22
y z
B)
( 1) 1 2 2212 2
2 2
y z
C)( 1) 1 2 2
1 12y z
D)( 1) 1 2 21 12
2
y z
23. (t) = (cost, 2sint, 0) :
A) k 1B) k 1/5 C) k 1/10
D) kt2cos31
2
24. t0 = 1 L(t) = (t2,t3) :
A)513
k
B)613
k
C) 3
613
k
D) 3
510
k
25. ( )t = (5cost, 5sint, 0) :
A)101k
B) k 1/5 C) k 1/2D) k 1/3
291
3-1. 0 = 1 (x) = 10 + 15
A) k 0 B) k 10C) k 25D) k 15
2. ( )t =)1(
,1
12
2
tt
t- t0 = 1
A) (1/2,1/4)B) (1/3,1/4)C) (1/2,1/9)D) (1/2,1/8)
3. L = {t2, t - 3
3t} t1 = 0 t2 3
A) 4 3B) 3 3C) 2 3D) 4 2
4. L(x = t2 , y = 32 t (3 - t2)) -
A) (1, 1) B) (2, 0) C) (1,–1)D) (3, 0)
5. 1(t1) 2(t2):
A)2
1
322 ))()((t
tdtttxL
B)2
1
)()( 2t
tdtttxL
C)2
1
)()( 22t
tdtttxL
292
D)2
1
32 ))()((t
tdtttxL
6. 0 = 1 = 2
) k 1 ) k 0 ) k 2
D) k1252
7. ( ) ( , , )M t cht sht t - t0 = 0 A) (1, 0, 0)
B) (0, 1, 0) C) (1, –1, 0) D) (1, 1, 0)
8. 2 + y2 + z2 = 1 A (3
1 ,3
1 ,3
1 ) .
A) ( x = t, y = - t, z = t ) B) ( x = - t, y = t, z = t ) C) ( x = t, y = t, z = t ) D) ( x = t, y = t, z = - t )
9. (x0,y0,z0) 2,
3,
4
234 tztytx -
A. 030 03t t
= 40
0
1 tyy = 2
0
0
tzz
B. 030 02t t
= 40
0
1 tyy =
030
0
2 ttzz
C. 030 02t t
= 40
0
21 tyy = 2
0
0
tzz
D. 030 02t t
= 40
0
1 tyy = 2
0
0
2tzz
10. L: y2 = x3 + x2
A) (1, 0) B) (0, 1) C) (1, 1)
293
D) (0, 0) 11. ( )t = (cost, sint,10t)
A) k 1/101 B) k 1/10 C) k 1D) k 1/5
12.___
(t) = (tcos
1 , arc tgt) t0 = 1
A) (1, –1)
B) ( -1, 4
)C) (–1, 1)
D) ( 1, 4 )
13. 1(t1 = 2 ) = t – sint, y = 1 – cost, z = – 4sin2t
A)( 1) 1 2 2
12 22
y z
B)( 1) 1 2 2
2 12y z
C)( 1) 1 2 22
12 22 2
y z
D)( 1) 1 2 2
32 22
y z
14. t = 1 x = t, y = t2, z = t3
A)1
1 = 1
2 =
31z
B)1
1 = 2
2 =
31z
C)3
1 = 2
2 =
31z
294
D)1
4 = 1
2 =
31z
15. 0( (t0),y(t0)) (x(t) = f(t), y(t) = g(t))
A)0
0
( )( )
ty t =
)()(
0
0
txtxx
B)0
0
( )( )
ty t =
)()(
0
0
txtxx
C)0
0
( )( )
ty t =
)()(
0
0
txtxx
D)0
0
( )( )
ty t =
)()(
0
0
txtxx
16. 0 = 1 2y x
A) k 1 B) k 3 C) k 4D) k 2
17. (1, 1, 1) y( ) = , z(x) = x2
A)1
2 = 1
1 =
41z
B)1
2 = 1
3 =
41z
C)1
2 = 1
1 =
31z
D)1
3 = 1
1 =
41z
18. (x0 = x(t0), y0 = y(t0), z0 = z(t0))
2,
3,
4
234 tztytx
A) 0 0 020 02 1
y y z zt t
B) 0 0 020 0 1
y y z zt t
295
C) 0 0 02
0 01 3y y z z
t t
D) 0 0 02
0 01 3y y z z
t t
19. y( ) = , z(x) = x2 (1, 1, 1)
A) 7x – 8y – z + 3 = 0 B) 6x – 5y – z + 3 = 0 C) 6x – 8y – z + 3 = 0 D) 6x – 8y – 2z + 3 = 0
20. L(t) = (t,t3) t = 1
A) k10001
B) k1000
6
C) k3 1000
6
D) k 0 21. (t) = (5cost, 5sint, 0)
A) k 1/3 B) k 1/2C) k 1/5 D) k 1/10
22. L (x = t, y = t2 + t + 1) A) (1, 0) B) (0, 1) C) (1, 3) D) (–2, 3)
23. (t) = (3cost, 3sint, 3t) A) k 1/10 B) k 1/5 C) k 1/2 D) k 1/6
24. L: 3
3
3
2
13,
13
tt
tt
A) (1, 0) B) (0, 1) C) (1, 1)
296
D) (0, 0) 25. t0 = 1 2 4( ) ( , 1, )M t t t t
A)1 1
2 2 4x y z
B)1
1 = 2
2 =
41z
C) 1x y zD) 2 1x y z
4-1. r = r( )
23
22
22
]))(()([
)()())((2)(
rr
rrrr -
. r( ) = 4 :A) 1B) 1/4C) 0D) 4
2. )(t = (3cost, 3sint, 0) c :
A)101k
B)51k
C)31k
D)21k
3. L (x = t2 , y = 32 t (3 - t2)) - :
A) (1,–1)B) (3, 0) C) (1, 1) D) (2, 0)
4. L(t) = (t, t2) :
297
A)
23
2)41(
2
t
k
B) k 0
C)
23
2)41(
4
t
k
D) 1k
5. r = r( )
23
22
22
]))(()([
)()())((2)(
rr
rrrrk
. r( ) = :
A)
21
2
2
)1(
1k
B)
23
2
2
)1(
2k
C)
23
2
2
)1(
k
D)
21
2 )1(
1k
6. t = 1 x = t, y = t2, z = t3
A)14
32 kji
B)14
322 kji
C)14
322 kji
D)14
32 kji
298
7. 0( (t0),y(t0)) x(t) = f(t), y(t) = g(t) :
A) 0
0
( )( )
ty t
= )(
)(
0
0tx
txx
B)0
0
( )( )
ty t =
)()(
0
0
txtxx
C)0
0
( )( )
ty t =
)()(
0
0
txtxx
D)0
0
( )( )
ty t =
)()(
0
0
txtxx
8. ( )t = (cost, 2sint, 3t) :
A) k = 1/5 B) k = 1/10
C) k = t2cos31
2
D) k = 1
9. 1(t1 = 2
) = t – sint, y = 1 – cost, z = – 4sin2t
A)
22
)12
( =
221y =
122z
B)
22
)22
2(
= 21y =
122z
C)( 1)
2 =
221y =
222z
D)( 1)
2 =
221y =
122z
10. t0 = 1 L(t) = (t2,t3) :
299
A)613
k
B) 3
510
k
C)513
k
D) 3
613
k
11. t0 = 1 2 4( ) ( , 1, )M t t t t:
A) 1x y zB) 2 1x y z
C)1 1
2 2 4x y z
D)1
1 = 2
2 = 4
1z
12. (x0 = x(t0), y0 = y(t0), z0 = z(t0))
2,
3,
4
234 tztytx :
A) 0202t
= 0
0t
yy =
10zz
B) 0
1 =
0
0
3tyy
= 20
0
tzz
C) 020t
= 0
0
tyy
= 1
0zz
D) 0
2 =
0
0
2tyy
= 20
0
tzz
13. 0 = 1 = 1
:
A)2
1k
B) k 1 C) 2kD) k 0
300
14. 2 + y2 + z2 = 1 A(3
1 ,3
1 ,3
1 ) .
:A) ( x = - t, y = t, z = t ) B) ( x = t, y = t, z = - t ) C) ( x = t, y = - t, z = t ) D) ( x = t, y = t, z = t )
15. L: y2 = x3 + x2 :A) (0, 0) B) (1, 1) C) (0, 1) D) (1, 0)
16. r = r( )
23
22
22
]))(()([
)()())((2)(
rr
rrrrk
. r = a ( > 0) :
A) k 32
2
)1(2
B) k2
2 2
1 2( 1)
C)
2
32 2
1 2
( 1)k
D)
23
2
2
)1(
2k
17. L (112
2tt
,1
2
tt
) :
A) y = - 1 B) y = 1 C) y = 1/2 D) y = - 1/2
18. t0 = 0 ( ) ( , , )M t cht sht t - :A) (1, 1, 0) B) (1, –1, 0)
301
C) (1, 0, 0) D) (0, 1, 0)
19. (1, 1, 1) y( ) = , z(x) = x2
:
A)1
2 = 1
1 =
31z
B)1
2 = 1
1 =
41z
C)1
2 = 1
3 =
41z
D)1
3 = 1
1 =
41z
20. 0 = 2 2y x :
A) k = 272
B) k = 1 C) k = 13,5 D) k = 0
21.2
,3
,4
234 tztytx
A. 030 03t t
= 40
0
1 tyy = 2
0
0
tzz
B. 030 02t t
= 40
0
1 tyy =
030
0
2 ttzz
C. 030 02t t
= 40
0
21 tyy = 2
0
0
tzz
D. 030 02t t
= 40
0
1 tyy = 2
0
0
2tzz
22. M0(–1,–1) :A) ( = t3 ; y = t +1) B) (x = t3 – 2t ; y = t2 – 2 ) C) (x = t ; y = t2 + 1) D) (x = 1 + 2t3 , y = t2 – t )
302
23. (x0,y0,z0) 2,
3,
4
234 tztytx
:A)
20t (x – x0) – 2 3
0t (y – y0) + 40t (z – z0) = 0
B)30t (x – x0) + 2
0t (y – y0) + 2t0(z – z0) = 0 C) 2 3
0t (x – x0) + 20t (y – y0) + t0(z – z0) = 0
D)30t (x – x0) + 2 2
0t (y – y0) + t0(z – z0) = 0
24. 2 + y2 + z2 = 1 (0, 0, 1) .:
A) z = 1 B) y = 1 C) x + y = 0 D) x = 1
25. (t) = (3cost, 3sint, 3t) :A) k = 1/10 B) k = 1/5 C) k = 1/2 D) k = 1/6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
I D B D B B D A C B D C C A D B B D A B C D A D C A
II C B A B B D A C C D A B D C A B D B A A C B D C B
III A A C D C D A C B D A B C A D D A B C B C A D D B
IV B C B A B A C C A D D C A D A C D C B A B B A A D
303
1. . ., . . . – ., 1976, 2 .2. . . -
. – ., 1975.3. . ., . . . – .: « »,1969.4. . . . – ., 1958. 5. . . . – .,« », 1964. 6. . ., . .
. – .: - , 1980. 7. . . . – ., , 1956. 8. . ., . . .
. – ., 1977. 9. .
. – ., 1964. 10. . . . – .: ,1964.11. . . . – .: , 1948. 12. . ., . ., . . -
: . .1: , -. .: , 2001.
13. . ., . . -. .: « », 1966.
14. . ., . . . – .: « », 1974. 15. . ., . .
.- : - - ,1974.16. . ., . .
.- .: « », 1970. 17. . ., .- ., « », 1966. 18. . .
.- .- ., 1961, 2 .19. . . . , .,1958.20. . ., . , . ., . . -
. — .: - . - , 1978. 21. ., - . . - .: ,1981.22. . ., ., . . -
. . . 3: . — .: -, ,2001.
304
23. . . . — .: -, 1956.
24. . . . — .: , 1974. 25. . . . — .:
, 1971. 26. . . -
. — .; .: , 1948. 27. . . . — .: , 1971. 28. . . . — .: -
, 1952. 29. ., . . — .; .: , 1938. 30. . . — .: , 1953. 31. . . — .: , 1965. 32. .
. - .: , 1964. 33. . .
. - .: , 1985. 34. . . . — .: , 1984. 35. .- . . — .: , 1969. 36. . . — .:
, 1960. 37. . . — .:
, 1960. 38. - . . - .: , 1961. 39. ., . . - .:
, 1967. 40. ., ., . . —
.: , 1971. 41. . . . -
,2007.42. . .
.- ,2007.43. . . . – , 2008.44. . . . – , 2008.45. . .
. – , 2008. – 478 .46. . . . – ,– , 2011. – 367 .47. . . ,
. – . – , 2011. – 422 .
305
48. . . . ( . .):. – : – , 2011. –394 .
49. . . I. ( . .):. – : – , 2011. –376 .
306
§ 1. - ...........................................................7
§2. .................................................9§3. - .
- ..........................12§4. - ............15§ 5. - ....................................17§6. - ..................................................18§7. - .
- ..................................................................................19§8. . ............21
..................................................................28§1. . . ..........28§2. ..............................................................31§3. ..........................................32§4. i a . .........34§5. ...................................................36§6. ..........................................................................39§7.
...............................................................................................41§8. .......................43§9. ............................................................................44§10. ..............46§11. ...............................................51§12. ..........................52§13. .
..........................................................53§14. . ...................................58§15. . .......................................................61§16. . ......................................................................66
........................................................................................................3
307
................................................................69§ 1.
........................................................................................69§2. ,
. .........................................................................70§3. ...........................................73§4. ...................................76§5. ......77§ 6. ( ). ........................................................79§7. . ...................................80§8. . .......................................................................82
IV -
. .........................................................84§1 .......................................................................84§2. . ( ). ...........................87§3. ...........................................................................90§4. ............................93§5. C .........................................................................96§6. . -
.................................................................................98§7. ............100§8. n- ..................................................104§9. ........................107§10. . .
, ................................................................110
V...............................113
§1. . ...........113§2. . .........116§3. ..............................................................119§4. . .......122§5. ......128§6. Гаусс және Вейнгартен формулалары. ..........................................138
308
§7. . .........................................................................140§8. . .................147§9. . ,
. .................................................................................156§10. ..............................................................................160
VI. ......167
§1. .....................................................167
§2. ...............................................170
§3. 4321 ,,, IIII .. ..........................173
§4. ............................176§5. .
. .................................................178§6. ..................................................................180§7. ................................186
V I. .......188
§1 ........................................................................188§2. ...............................................188§3. .................................190§4. ..................................193§5. .
.......................................................194§6. .
......................................................................197§7. ..........201§8. .
. .........................................204§9. .....................................................................210§10. . ............................................................214
VIII..............................................217
§1. . ..................................217
309
§2. . grad, div, rot .
...............................................................................................225§3. .............................................................229§4. .....................................232
........................................................234§1. ........................................234§2. .
.................................................................237§3. , , ...............................................241§4 .........................................................................................243§5. .................................................................................................245§6. .....................................................................................247§7. ...................................................................................248§8. .........................................................250§9 ....................................................................................251§10. ....................................................................254§11. ................................................................................256§12. ................................................................259
.....................................262§1. ................................262§2. ................................................................264§3. .........................................................................267§4. .............273§5.
..................................................................................277§6. ......................................278
I..............................................................................................280
...............................................................303
Басуға 27.06.2014 ж. қол қойылды. Пішімі 60х901/16.Қағазы офсеттік. Қаріп түрі «Тіmes».
Баспа табағы 19,5.Таралымы: Мемлекеттік тапсырыспен –
1000 дана. Тапсырыс № 9205.
Тапсырыс берушінің файлдарынан Қазақстан Республикасы «Полиграфкомбинат» ЖШС-нде басылды.
050002, Алматы қаласы, М. Мақатаев көшесі, 41.
