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)à=;) AT = A ;堠)Ҡ=;) AT = A. 18.. : );););). 19.. : ););). 20. , .

21., ,.

22.,蠠 ,蠠  +蠠i( ). 23., ,   = i . , , =i. 24., ,  =(i)1(+i).

25. 2 , ( f, g ) = f ( x ) g ( x )dx., A = 1 d .

26. f(x) (x[0,1]) : f(0)=f(1)=0 : ( f, g ) = f ( x ) g ( x ) ( x ) dx, () . (), Af = x ? , f(0) = 0? ()=1.. 27.,   =*.

28. , , .

29., , .

30. ,   ,   = .

31., .

32.,   ,Ѡ=.

33. , .

34. , ࠖ.

35.頠 ⠠ {f1, f2, , fn} ; {e1, e2, , en} . ,:

)f1=1+2,f2=2; A = 8 1 ;

)f1=2+3,f2=1+3,f3=1+2; A = 1 0 0.

);););). 36.{1,2,3}. , 1 2 37. , :

)1=1+2,2=2;)1=1+2,2=12;

)1= (1+2),2= (12); )1= (51122),2= (121+122).

);););). 38. , :

)1=1+22+23,2=21+223,3=2122+3;

)1= (21+223),2= (1+3),3= (1+42+3);

)1= (1+2+3+4),2= (1+234), 3= (12+34),4= (123+4).

);););). 39. , : )1=1+ie2,Ae2=ie1;)1=1+ie2,Ae2=ie1+2;

)1= (21+(1+2i)e2),Ae2= (5e12(1+2i)2);

)1= (21+(1+2i)e2),Ae2= ((1+2i)e122).

););););). 40.1,2,,nb1,b2, , bn, .?

)1(3,4),2(1,3),b1(5,0),b2(3,1);

)1(2,1),2(1,1),b1(1,2),b2(1,1);

)1(1,2,2),2(1,1,0),3(0,1,1),b1(2,2,1),b2(0,1,1),b3(1,1,0).

););). 41. 1 2.?

););). 42. , :



);););). 43. , :

);););););). 44.),() (). ), .

+ . 45. :

) 1(2/3, 2/3, 1/3), 2(2/3, 1/3, 2/3), 3 (1/3, 2/3, 2/3); ) e2 (1 18, 1 18, 4 18 ),3(2/3,2/3,1/3);) e1 (1 2, i 2, 0), e2 (1 2, i 2, 0),3(0,0,1);)1(1/2,1/2,1/2,1/2),2(1/2,1/2, 1/2,1/2), 3(1/2,1/2,1/2,1/2),4(1/2,1/2,1/2,1/2). 46. , :

) 5 0 ;) 2 1 ;) 1 3 ; (1,2), ); 47. , :

) f3= (1,1,0);)1=0,2=3,3=6,f1= (2,1,2),f2= (1,2,2),f3= f1= (1,0,i),f2=(0,1,0),f3= (1,0,i). 48. , :

) sin cos ;) 1 0 0. 1);)蠠=k,kZ,蠠=0, 1=2=1, = , 1 = 2 = 1, ;)1=1,2=1, f1 = sin 2, cos 2, f 2 = cos 2, sin ;)1=1,f1= (1,1,1);)1=1,f2= 49. , :

,f1= (1,1,1),f2= (1+i 3,1i 3,2),f3= (1i 3,1+i 3,2);)1=1, 2= 2,3= 2,f1= 2 (1,i,0),f2= 2 (1,i, 2 ),f3= 2 (1,i, ). 50.?

););). 51. , 3 , , , . 52., 1.

53. , , 1: A = 9 4 9 8 9 1.

=4/9,=7/9, =4/9. 54., . 55. , .

56.,,:

. 57.,,,, .

58. , () ,( 59.,.

60. , () : 1) ();2)();

3)2=,.

61. , : ; 2 = , .

62. , : ; ; , .

63., ,,,. , .

:((+),(+) =(| 1|,2,3,,n). 64., : = Q1Q, Q ,   ().

65.,: =і1,堠Ѡ, ( ).

66.Q,, =Q1Q:

67.Ѡ ,,  =і1:

68.蠠Q , 72., A = 4i 2i 4i. 81=. 73.ࠠ. A : ) 1 1 ;) 0,25 0,15. 74.: A = 24 75.: 76.:

). 77. ? :

) a 2 (1,1, 0) ; 0 2 0 ; ) ) a (1, 0, 0,1) ; 0 0 0 0 ; ) . 78.: ) 0 2 1 ;) 0 1 1 ;) 0 0 1. 79.: 80. {ei} . {fi} :

) f 2 (1, 0, 0) ; 0 2 0 ;) f 2 ( 3,1, 0) ; 0 1 0 ;) f 2 (1, 0, 0) ; 0 3 0.

8) ....:,1975.

9) ....:,1971.

10) ....:,1970.

11) ..,....:,1973.

12) .. . ..:,1969.

13) ... 14) ..,..,.. ..:,2001. 15) .. ..: ܠ

۠Ƞ

Def. G , , z... , ( ),..,yG zG,z=(z=), :

1)xy(yz)=(xy)z;1)xy(yz)=(xy)z;

2) G G=; 2) lG Ge=;

3)G yG xy=;(y=x).3)G yG   ,ࠖ .

(,), . ,.

, , . .,,, , , ,.  , 堖.

蠠堠,.. 4) xy=yx4)xy=y .

Def.G1G,:

1),yG1xyG1;2)G1;3)G1(x)G1. , , . 0 = e, , 2,... ak = e ,.k.

,:

Š;

SBDBD; SAC; SO.

:

SBD SAC SO

SBD SAC SO

SBD SBD E So SAC

SAC SAC So E SBD

SO SO SAC SBD E

, ,, .

2..

Ѡ ,:

1)Š;

2) ;

3) ;

4)S1(1)( );

5)S2(2)();

6)S3(3)().

:

. , S1 S 2 S1. . {, , } (), .

3. . , , b, , d, ... ,,,: f = f ( a ) f ( b ) f ( c ) f ( d ).

f2f1 :f1,f2.

( , ) ( ) !,Sn.

. S4:

,f1f2.

f3f4,f5f6.

: E = A B C ;

A B C A B C A B C A B C A B C

C A B B C A A C A C B A

G1 G2 , 堠fG1G2: G1 G2,蠠 x y,1+21+2. , ࠖ.

G .

Def.1H2G,H3 1 H2 : 3=1 H2={h3 3 =h1 h2,h11, h2H2}. , 1 H2 , H3, , .

G aG, , {}, H⠠G.

(, ):

1.a͠͠ .

2. 1b = b͠ ( b , 1b). b .

3. H , .

4.a.

, , G.

G, ⠠.

Def.G,b,,...蠠 G f G G : f: G G ,a,bGf(a,b)=f(a)f(b). G G.

G G, G ; G, G.

Ұ..

Ұ. f G ͠ G, Ұ.G͠ G . G ͠:G/H.

. . :

1. GL(n) n V.

2. () n.

3. SO(n) .

4. U(n) .

2. 1..

2.?

3. ?

4.,?

5..

6., ?

7. 1 5 3 1 5.

8..

1. 5 , ,렠??

. ) , 5. z1, z2 5. k1, k2 , z1 = 5k1 z2 = 5k2. z1 + z2 = 5k1 + 5k2 = 5(k1 + k2 ), z1 + z2 5,, z1 + z2 5. 5.

) 5? ,⠠,0 05,, = 0 5.

) z 5, . . z = 5k, k. , , z, 5,頠5.

),),),5 .

)頖 5 , (0).

2. 5 ?5.

. : , ,5. .

5 z = 5k, k :

5. n 5, n, 5 n.:

5= 5 = 4.5 , 1, 5, 3, 5 ,51,2,34 .,:

5 = 5.

, n 5 5 =n5=5. , , ,., :

, 5: 5 =5, 1, 5, 3, 5. , .

, 5 n 蠠 n 5, , , 5 .

3. 5.

. G , :

: 5 5 ( m, n = 0,1, 2, 3,4 ) m + n s 5. 5 = 5 , 5 5k, k = 1, 2, 3, 4 .

G (). 4., .

.렠 . Pa a. : Pa Pb Pa +br. ࠠ VP . VP :

, z1 Pz1 z2 Pz2,: z1 + z2 Pz1 Pz2 = Pz1 + z2 ;

z1 + z2 Pz1 + z2.

, . VP.

5. G / H, G , H ,.

. , , H G , , , . G / H , , G H.

, ( H ), . , , . z0 H H z0 ,:

) arg z0 () ) z0.   z0 H , 蠠ࠠ z0 .

, G H .: H r =r0, r .

: H r =r1 H r =r2 = H r =r1 r2.

H = H r =1, H r =r0 H r =1/ r0 .

H r =r0 (r0 +) 頠 G / H.

H r = r0 (r0 +) r0 +,蠠 G / H +, ( ) , G / H +.

.

. ) 1 5, 5 1 ( 1 5 1). , 1 5 . (1,5). ) . 蠠(3,9,7),頖(2,8,6,4).

):

.

) .12 2,34,:

1 2 3 4 5 6 7 8 9 = 5 8 9 2 1 4 3 6 7.

, 12, . .

4. 1.:

);

);

);

);

);

);

);

);

),1; ); ),1蠖1; )1(n). );););););););););););) 2.:

);

);

);

);

);

);

);

);

),3; ); ); ),0. );););););););););););)ࠠ 3.:

);

);

) ;

) .

);););)򠠠 4. :

);

) , ;

);

) l;

);

);

) ;

) ;

) ;

) , : .

);););),l;,Al;);););););) 5. ,:

)*=,*=,>0; )*=,*=,0;

) ;

)렖;

)n ;

),;

),.

9. , ( ),. .

, (41 ); 頠 10.,:

1) ;

2) ;

3) ;

4) 0 .

11. , .

12. , .

13. ,, :

);

)n,n ;

);

)n.

)1;);),堠1 k 1,kn; ) kn ,G 14.

{}().

15. .

ࠠ,ak ,. 16.,:

) , n ( ), 2n ;

) n .

:,ࠖ,b,b1b 17. G2,͠G. ,͠.

18. ,:

) ();

)ࠖ.

19. ͠G.,:

)bG ,1b;

)G ();

) .

20. Gࠠ,͠䭠 , .,͠ GG 蠠.

ࠖ .G={,,2=1b,b,1b},͠={,b},b1ab=a1 :,͠={a,ab},a1H={1,a1b};:, ={a,ba=a1b},Ha1={a1,ab) 21.()G/H,:

a)G;͠ ;

) G ; ;

) G ; ;

)G;͠ ; ) G = Z ; = nZ , ; ) G ;͠.

);),),) , 1, ; ) ;)n 22.,頠 n k (k).

23. G , , ʠ . ,͠G,ʠ. ,G/H.

24. , , > 0, n ,.

25. n 1. :)24;)63;)55;)35. )3;)3;)4;)1 26. Sn,:

)S3;)S4.

) S4 (12) ͠:,,, 27. , n Sn,.

n!/2 28. , 4 6(..18ࠖ ).

29. . :

)(1,4,2)(3,5),;)(1,6,3)(2,5)(4),;)(1,8,2)(3)(4,6,7)(5),; )(1, 5)(2, 8,6, 4)(3, 9, 7),; )(1,2,3,4,5,6,7,8,9),;)(1,4)(2,5)(3,6), 31. :

)(1,5)(2,3,4);)(1,3)(2,5)(4);)(7,5,3,1)(2,4,6)(8)(9);

)(1,2)(3,4)...(2n1,2n);)(1,2,3,,2n1,2n);

)(3,2,1)(6,5,4)...(3n,3n1,2n2).

) 32.:

33. , , , 頠, .

34.:

3 5 4 6 9 7 1 10 8 2, =? )A;)E 35.X =,:

X = 36.,:

) ): X , S. SX = XS X1SX = S. SS=(1,2)(3,4)=X1(l,2)XX1(3,4).젠 , X1(1, 2)ՠ 2, (1,2) 12,X.(3,4). ,XS, S ,, .2蠠(1,2)=(2,1),(3,4)

۠Ƞ

n{1,2,...,n}{1,2,...,n} n.

. {i} {j} , Def Ұ.n.

Ұ. , .

{i}{j}n.: =x1e1+22+...+nen蠠=x1e1+22+...+nen.

(x1,2,...,),(x1,2,..., ). . , ,(.. ). , ( ). ( ,),, 1 { = dim). , . : xi ei = x1e1 + x2e 2 + x3e3 +... + xne n ; ij = 1 + 2 + 3 + + n.

, , :

x = xi e = ( xei ) e ; x = x i ei = ( xe i ) ei. 砠렠,: j=(jei)ei=gjiei,j=(jei)ei=gjiei.gji=(jei);gji=(jei).

Ѡ. ,(gji)(gji).

{i,ei} {i, ei } pii {i} {i }.:

): ii ii i i ii ;

.( p, q ) ,:

1) {i} Vn p+q Ai1i2 i p (1, 2, 3,..., n ).

2) Ai1i2 i p {i } (*) ( Pi i ) ꠠ(ii ), ( Pi i ) . r = p + q .

:𠖠;𠖠 (1,0)(0,1); (2,0); (1,1).

.

1. , .

2. . Ai1i2 i p Be112 er s 3. (,). ,p 0,q 0, ks il : Ai1i12K il Ksi p q kl s. (,q)(𠖠1,q1).

4. 堠堠 im in Ai1i12K imK inK i p .

.

5.堠堠⠠ im in : Ai1i12K imK inK i p . .

{i}{i},..i= ei,ࠠxi=i...

{i}:=ii=(i)i.

{i} {i } :

(pi i) {i} {i }, (pii ) .

: xi = pi ixi; xi = pii xi , . . ,.

,..Ж1=Ҡ蠠Ҡ=Р=.

, .

.r, :

1){i} n r Ai1i2K ir (1,2,, );

2){i} {i }eo:

Ai1K ir = pi11 pi2i2 K pirir Ai1i2K ir, ( ) { }{ }.

. , , , , .

, , ,,. Ai1K ik K imK ir ikim,, ik im : Ai1K ik K imK ir ik im.

.

.

.𠖠, Def :

堠=detP(detP=1).: 1.⠖;

2.⠖;

3.ࠖ;

4.ࠖ.

ikl = (i k)l, i k, l , 3,,:123=231 =312 =1;213 =132 =321=1,ikl.

ikl .. :

ik ( ) ikl ( ) a . r 3r Ti1i2 ir ( x1, x2, x3 ), r.

1. Ti1i2 ir ( r ) r, Ti1i2 K ir ( x1, x2, x3 ) Ti1i2 ir.

2. r r (r1)ࠫ軠, :

3. r r r:

, :

( gradT ) i = i1i2 ir ; ( divTi ) = k i = i ; ( rotTK lK ) i = ikl TK iK.

: : K iiK dV = K iK dSi =K iK ni dS (ࠫ黠); ࠫ: ikl xk dSi = TK iK dli.

12. .

13. ?

14. .

15. , ?

, 16.

??

17. .

18. .

19. ?

20. ?

21. ?

22. 2 ?

23. 2 ?

24. .

25. ?

26. 0?

27. ii n?

28. , ,? Ti1i2i3...ik...in : Ti1i2i3...ik...in.

ࠫ,, 29.

蠫r?

? 30.

r? ? 31.

r?

1., . , {ei }3=1 {ei }3=1 :

ei = (ei, e j )e gij e, e = (e, e )e j g e j , . {ei }1, ( gij ) :

(1) {ei }1 : e j = g ij 1ei, , G :

(2) : e1 = 1 1 + 0 = { 1, 0, 0}, e2 = 1 + 2 1 = { 1,1, 0}, e3 = 0 1 + 2 = { 0, 1,1}.

2.x x = (5, 2, 1). {ei }i3=1 {ei }3=1 i :

e1 = (1,1,1), e2 = (0,1,1), e3 = (0, 0,1), e1 = (1, 0, 0), e 2 = ( 1,1, 0), e3 = (0, 1,1).

x {ei }i3=1 {ei }3=1.

.x:

xi x x i , x x = (8, 3,1) x = (5, 3, 1) .

3. E3 : e1' = (1,1, 0), e2 ' = (1, 0,1), e3' = (0,1,1). {ei }1 {ei '}1 {ei '}1 {ei }1 {ei }1 {ei '}1.

., :

, bii ' bii' . :

(1) , : , bii ' bii' (2),,, {ei '}1 {ei }1, {ei }1 {ei ' }1. 1. bii ' (3), , bii '. {ei ' }1, , {ei }1. :

(4), bii ' :

bii ' bii' , {ei }1 {ei '}1. {ei }1 (2). , {ei }1, , {ei ' }1 :

,:

, e1 = (1, 0, 0), e 2 = (1, 1, 0), e3 = (0, 1, 1).

,,(3) {ei '}1 : 2. 1 {ei }1, {ei }1 :

bii ' bii' : bii ' = 0 1 1 = 1 / 2 0 1 / 2.

{e }1,, 1.

4. (5,2,1). 2, { e1 = ( 1, 1, 1), e2 = ( 0, 1, 1), e3 = ( 0, 0, 1) } { e = ( 1, 0, 0 ), e = ( 1,1, 0 ), e = ( 0, 1,1) } ,,(8,3,1) (5,3,1).

. 3 {ei },{ei } {ei '},{ei '}. :

, ,.. xi ' = bii' xi, ࠖ, x i ' = bii ' xi. , , bii ' bii' . xi bii' , x i bi :

xi ' = (7, 6, 3) x i ' = (3, 2, 1).

5.:

蠠 1 = (1, 0, 0), e2 = (0, 1, 1), e3 = (0, 1, 1) {ei } {ei' }.

. 1. {ei } {ei' } ( Pee' ) {ei } ( Pe s ) {ei' } ( Ps e ' ):

:

6. , 2 2.

. (), Tik . Tik { ei } 1 { e 'i } 1 ' = Ti ' j ' = Pii ' Pjj 'Tij, Pii ' { ei } 1 { e 'i } 1. , { ei } 1 { e 'i } 1 V, Tik : V,V ' = Pii ' Pjj 'V, 2.

7. 2 Tik 2 5 8, Bi (1, 2, 3). Tik Sik Aik . Tik Bi Tik Bi Bk . 2 , , , Tik :

Tik Bi Tik Bi Bk.

1. Tik Bi Ck, :

Tik Bi Bk Ck Bk :

2. 2 ,1 , Tik Bi , .:

8.,:

. ( ):

9., divr, r .

.,:

divr = ij x j = ij ij = ii = 3.

10.,:

. ,:

, a b .

. ,:

12. n S, V.

. ,,ࠖ: , ,, 蠖,. , r ( a n ) dS ( dS ), ( dS ). , dS.

, c a ) dS, c .:

ࠖ:

div( c ) a :

,:

: a ) dS = ( c ). c ,: r ( a ) dS = aV.

4. 1, 2, ..., 1,2, )(1,0),(100,1);)(1,0),(0,3);)(1,3),(1,5); )(1,0,0),(3,1,0),(2,5,1);)(1,1,1),(1,2,3),(1,4,9);

)(1,0, ),(,1,0),(0,,1);)(1,0,i),(1,i,0),(0,1,1+i);

)(1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1);

)(2,1,1,1),(1,2,1,1),(1,1,2,1),(1,1,1,2).

)(1,100),(0,1);)(0, 3 ),(0,3);) 2 (5,1), 2 (3,1);)(1,3,13),(0,1,5),(0,0,1); )(3,3,1), (5,8,3), (1,2,1);)a(1,101,102),(102,1,101),a(101,102,1), a=(1+103)1;) (3i,13i,1+2i), (2+i,1+3i,12i), (12i,2+i,2+i);)(1,1,0,0), (0,1,1,0),(0,0,1,1),(0,0,0,1);) (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0) :

)(1,3,2),(1,1,0);)(1,1,1,1),(1,1,1,0); )(1,1,0,1),(1,0,1,0),(0,0,1,1).

) 6 (1,1,2), 6 (5,1,4);)(0,0,0,1), 3 (1,1,1,3); ) (2,3,2,2), (4,1,3,3); (3,1,3,4) ( f, g ) = f ( t ) g ( t )dt :

) , 1, t, t2 ;

) ,1,t,t2,t3.

) 8 (35t2), 2 t, (13t2);) (35t2), t, (5t7t2), (13t2), (3t5t3) :1(1,0,0,0),2(1,1,0,0),3(1,1,1,0),4(1,1, 1,1)1(1,1,0,0),2(0,1,1,0),3(0,0,1,1),4(0,0,0,1). {i,ei},, iei )=(2,3,5,1);)=(1,2,0,1);)=(3,2,7,0);)=(1,0,1, 0).

)(2,1,4,3),(5,8,6,1);)(1,3,1,0),(3,2,1,1);)(3,1,8,8),(5,9,7,0); )(1,1,2,2),(1,1,1,0) {i}:1(1,1,1),e2(1,2,3),3(1,4,9). ,ei. )(1,0,0);)(1,1,0);)(1,0,1);)(1,1,1).

)(1,1,1),(3, 2, 2 );)(0,1,3),(6,, );)(2,4,10),(4,4,1);)(3,6,14),(1,0, 0) {ei}: 1(1,3,13), 2(0,1,5), 3(0,0,1). , ei. {iei}:

)(1,0,0);)(1,1,0);)(1,0,1);)(1,1,1).

)(1,3,2),(1,0,0);)(1,2,3),(4,1,3);)(1,3,1),(12,5,1);)(1,4,6),(15,6,1) 22) ,2 2.

23) , 23 5.

24) , iklik, ikl 3 , ik 2.

25) ࠠ2Aik.

26) , AiBkClTikl i,k,Cl,Tikl3( 3).

27) 2 Tik : Tik = 4 5 6, B (1,2, ). Tik Sik Aik . : a) Tikk; Tiki; Tikik; ) AikTik;SkAik;AikBi;AikBiBk;)Tikik;Aikik;Sikik.

15 13. ,:

14.:

)iklnpsimnakapbncs;)kmnntzmpsikapazbscz.

15. , r 蠠 a ,:

a) divr ;) rotr ;) grad( ar ) ;) a.

)3;)0;) a ;) a 16. ,:

a)div ( a ) = diva + a grad ;)rot ( a ) = rota ( a grad) ;

(, a, b ).

17.  , r , a b ,:

) ( a r )b ;) ( a r )r ;) ( a r ) ;) r a b.

) ( a b ), ( a b ) ;) 4( a r ), ( a r ) ;)0,2 a ;)0,2 ( a b ) 18. c r a n dS, a c , n S,V.

19. S V,蠠 S :

) ( r ) dV ;)0;)0.

20.,:

44) ,:

) rot(grad) = 0 ;) div(rotA) = 0 ; ) rot(rotA) = grad(divA) A.

). (a b ) ( r , c , (r ) ):

( (r ) , G (r ) H (r ) ):

En :

) ijk kn ;) ijk is jm ;) ijk is jmkn ;

) aij in ;) ij jn ;) ij jnni.

) ijn ;) smk ;) smn ;) anj ;) in ;) En :

) ii ;) ij ij ;) ijk Ai Aj Ak ;) ijk ijk ; )3;)3;)0;) :

) ijk jik ;) ijk jki.) ijk jk ;) Ai B j ji Bm Anmn. )6;)6;)0;) A = (1, 1, 0) B = (4, 3, 2) ,:

) Ci = ijk Aj Bk ;) Ai Bi.) C = (2, 2,1),)7 C = aij aij, i, j = 1,2.86 i, l = 1, 2, 3 )0;)0 a111 = 1 ; a112 = 3 ; a121 = 4 ; a122 = 2 ; a211 = 1; a212 = 5 ; a221 = 2 ; a222 = 2. C = aijk aijk, i, j, k = 1,2.64 a1221 = 4, a1222 = 2, a2111 = 1, a2112 = 0, a2121 = 2, a2122 = 1, a2211 =, a2212 = 1, a2221 = 2, a2222 = 2. C = aijkl aijkl, :

) ( Aijkl + Ajkli + Aklij + Alijk ) xi x j xk xl ;) ( Pijk + Pjki + Pkij ) x x x ;

;) aij t ' s' r ' ami s ' t ' r '.

) 4 Aijkl x i x j x k x l ;) 3 Aijk x i x j x k ;) ij ;)0

22.....:,1975.

23.....:,1971.

24.....:,1970.

25...,....:,1973.

26. .. . ..:,1969.

27.....: 28. ..,..,.. ..:,2001. 29.....:,

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