..

,

I

1. . . . . .

X Y

f

XxYy . ,

y x ( ).xfy = x U U ( UU), y U U, X - U U() , Y U U () . xOy, ( ),xfy = UU .

. :

( ),xyy = ( ),xFy = ( ).xgy = U

, ,

, () . : UU, UU UU.

U U U : y=const, (x - ),

),10(

.,,arccos,arcsin arcctgxarctgxxx

, , (), U .

U U: ( ) ( )( ) ( ) ( ) .3,sinlog,1 23 xx arctgxfexxfxxxf === U,U U U

: 1.

( ) ,122110 nnnnn axaxaxaxaxP +++++= K - , 0n naaaa ,,,,0 210 K= - (), U U, U nU. U U ( ).

2.

( )mm

mmnn

nn

bxbxbxbaxaxaxaxR ++++

++++=

11

10

11

10

KK

U U. U U

3. , , U U.

4. , , U U.

5. U U ( ). u, u x, x.

( )uFy = ( )xu = . x

( )[ xFy ]= - U U. . U

U B1B, B2B, , xBnB, , > 0 N = N(), N. axnn

=lim . , ax .

( )xfy = .

1. U ( )xfy = b ( by ) U ( ax ), > 0 > 0, ,

0, , Nx >

2. . .

U

,

, ax x . ax , x , .

1. , : ( ) kaxaxaxkax uuuuuu +++=+++ lim...limlim...lim 2121

2. , .

kk uuuuuu lim...limlim...lim 2121 = .

: .limlimlimlim,lim 11111 ucuccuau === .

3. , 0:

vu

vu

limlimlim = 0lim v .

4. )();();( xvvxzzxuu ===

vzu , u v ax x b, z = z(x) ax ( x ) .

5. ax ( x ) y , b .

0y by 0b

6. ) v=v(x), (xuu = ax

( x ) ) ,

()( xvxu

)(lim)(lim xvxuaxax .

7. v , .. , , , .. v < M,

, < M. av =lim

-0, - . { } { } { } 1,,0,,00 U.

x

xsin (U 0x

U) 1sinlim0

= xx

x.

U e.

1. U U n

n

+ 11

U :n U = nlim

n

n

+ 11 . .

: =2, 7182818284...

. x

x

+ 11 , , :

xlim

x

x

+ 11 = .

3. , . . . . . .

U , = )(xfy

. 1. )(xf x , .. U

U, >0, > 0, , )( .

)(xf ax , = )(lim xfax . )(xf , , +)(xf , , - )(xf .

2. ) U (xf U ,

, , ,

0>M.)( Mxf

, U)(xf U .

3. ) U (xfx a ,U , .

4. ) U (xfx , U , ,

0>N

Nx >|| , ) . (xf 1. ( ) bxf

ax=lim b ,

) (xf ax . 3.

, )(xf

( ) = xfaxlim ( ) = xfxlim , .. , .

)(xf

U 1. )(x U

U ax U U x , 0)(lim = xax , 0)(lim = xx . 1. 0)( = x ax ( x )

0, 1=y . 2.

. 3. )(x

)(xzz = ax ( x ), .

C 1. 0lim = , 0lim = , lim 0= . 2. 0lim = , constc = , 0lim =c . 5. UU

)()(

xzx

)(x , 0, .

U

,

, , .

1. / , .. constAconst

ax==

,0/lim ,

, 0/1/lim = Aax , U .U

2. / , .. 0/lim = ( = /lim ), U , ,U U , .U

3. U k- ,U , .. k

limax / .0= Ak

4. / , .. 1/lim = ax , U U ( ~ ).

U = f(x) B0B B0B.

B0B = f(xB0B). =B0B +, . , :

B0 B+ = f(xB0 B+). = f(xB0B+)

f(xB0B). . = f(x) U =UBU0 UBU( UBU0UBU),U B0B

0lim0

= yx

, , 0)]()([lim 00

0=+ xfxxfx .

)()(lim 00

0xfxxf

x=+ .

. = f () U UBU0U B: 1) B0,B 2) , , )(lim

00xf

xx +)(lim

00xf

xx 3)

)(lim00

xfxx +

= = f (B0B). )(lim00

xfxx

, B0B = f () U.U =B0B U .U

. = f () ,

)0()(lim 000

+=+

xfxfxx

)0()(lim 000

=

xfxfxx

,

1) , = B0 B U )(lim)(lim00 00

xfxfxxxx +

1- U,

)(lim)(lim00 00

xfxfxxxx +2) = ,

= B0B, = B0B U U. , U U.

U

1. = f ()

[a,b], [a,b] =B1B , f ( B1B) f (), , =B2B ,

f (B2B) f (). f (B1B) U U = f () [a,b], f (B2B) U U = f () [a,b].

2. = f () [a,b], , a b =, .

3. = f () [a,b]. f ()= , f (b) =B, , , =, b , f ()= .

4. . , . . . . . .

y ,

, x , x 0, ..

xxfxxf

xy

xx +=

)()(limlim 00

00

U Uf(x) x.

.,),(dxdyyxf

U. ,

=f (x) , . U .

y=f(x) xB0B, ..

x

xxfxy

xx +=

)(limlim 0

00, f(x)

U xUBU0UBU.U [a ,b], U U[a, b].

. f(x) xB0B, U U .

, , , , , .. .

U

1. 0, .. , = const , cy = 0=y .

2. , ..

)(xcfy = , = const , )(xfcy = . 3.

, ..

)()( xvxuy += , )()( xvxuy += . 4.

, ..

,vuy = uvvuy += . 5.

, , , ..

vuy = ,

2vuvvuy= .

U

=f(x), .. , : y=F(u), u =(x) y=F((x)).

y=F(u) u .

. u=(x) x )(xux = , F(u) u

)(uFyu = , y=F((x)) x ,

)()( xuFy xux = , u u=(x).

U

1. , consty = 0'=y . 2. , . nxy = 1' = nnxy

3. xy = , 1'=y . 4. xy = ,

xy

21'= .

5. x

y 1= , 2

1'x

y = . 6. , xy sin= xy cos'= . 7. xy cos= , xy sin' = . 8. , tgxy =

xy 2cos

1'= .

9. , ctgxy =x

y 2sin1' = .

10. , xy arcsin=21

1'x

y

= .

11. yy arccos= , 21

1'x

y

= .

12. , arctgxy =21

1'x

y += .

13. , arcctgxy =21

1'x

y += . 14. , . xay = aay x ln'=15. , xey = xey =' .16. , xy alog= exy alog

1'= .

17. , xy ln=x

y 1'= .

U 1. , )(xCuy = )(' xuCy = . 2. , )()( xvxuy += )()( xvxuy += . 3. , )()( xvxuy = )()()()( xvxuxvxuy += .

4. )()(

xvxuy = ,

)()()()()(' 2 xv

xvxuxvxuy= .

5. ).(),( xuufy == )()( xufy xu = . 6. ,

==

)()(

yxxfy

yx xxf

'1)(' = .

U ,

)(xy)(),( txty == , t Tt

5. . . . .

U

y=f(x) [a,b].

, .lim)(0 x

yxfyx

== , , ,

+= y

xy ,

, .. 0 0. xxyy += .

y , U , x U. xy UU dy.

dy= xy . U

.U. dxxfdy )(= .

, dxdyxf = )( ,

.

U

y=f(x) [a,b]. x. y

, U .U y )(xf .

y . . VIV yy , , VIy

n- f(x) (n-1)- . ) - .

)(ny ( )1()( = nn yy

6. , , .

. 7. .

.

8.

. . , , . . .

, .

.

1. 1) f(x), [a,b], , [a,b] , .. 0)( xf .

2) f(x) [a,b] (a,b), 0)( > xf x(a,b), [a,b].

() , :

2. 1) f(x), [a,b], , [a,b] . 0)( xf

2) f(x) [a,b] (a,b), 0)(

1 (

). f(x) x=xB1B , , .. f (xB1B)=0.

, x.

, , . .

, : f () =0 () B B B.B f

, f () =0 f () , .

2. ( ). f(x) , xB1B ( , xB1B). + , x=xB1 B . +, x=xB1 B .

, .

1. , (a,b), .

2. , (a,b), .

1. (a,b) f`(x) , .. )(xf 0, .

. , , .

, .

3. y=f(x). (a)=0 f f (a) x=a , x=a .

. L , M , M L . 0

.

1. . , =+ )(lim 0 xfax = )(lim 0 xfax , x=a .

x

y yy

x x 00 0

, , .

x=a , ax f(x) , x=a . 2. .

y=f(x) , : =kx+b.

k b: ;)(limxxfk

x = ].)([lim kxxfb x = , .

, :

1) ;

2) ;

3) (, , );

4) (, );

5) ( f (x)=0 f (x)`- );

6) ( f (x)=0 (x) ); f

7) ;

8) ;

9) , .

.

9. . . . . .

: F(x)

() (a, b), F(x) (a, b) ( ) ( )xfxF = dF(x)=(x)dx . ),( bax

() F(x), .

1. F(x) () (a, b), F() + C , C - .

2. F(x) () () (a, b), : () F(x) = C . ),( bax

. F(x) (), ( ) cxF + ()

dxxf )( . F(x) (),

: ( ) ( ) += cxFdxxf .

() (). , , .

1.

( )( ) ( )xfdxxf = . 2.

: ( ) ( )dxxfdxxfd = .

3. :

dF(x) = F(x) + .

( )11

.11

++=+ acaxdxx

aa .

( )0ln.2 += xcxxdx . ( )1;0

ln.3 >+= aacaadxa

xx .

cedxe xx +=.4 . += cxxdx cossin.5 . += cxxdx sincos.6 . += ctgxxdx2cos.7 . += cctgxxdx2sin.8 . + += cxcxxdx arccosarcsin1.9 2 . + +=+ carcctgxcarctgxxdx 21.10 .

caxxax

dx ++= 2222 ln.11 .

++= cxa xaaxa dx ln21.12 22 . 13.

+

+=

cax

cax

xadx

arccos

arcsin

22

+

+=+ c

axarcctg

a

caxarctg

axa

dx1

1

.14 22

1.

: ( ) ( ) = dxxfAdxxAf .

2. :

( ) ( )[ ] ( ) ( )dxxgdxxfdxxgxf +=+ ( ).

3. ( ) ( ) += cxFdxxf , ( ) ( ) ++=+ cbaxFadxbaxf 1 .

, () , (t) (t) t.

( ) ( )[ ] ( )dtttfdxxf = . x = (t) ,

, .

u = u(x) v = v(x)

.

= vduuvudv . .

udv

vdu, .

10. . . .

- :

( ) ( )( )xQxP

xRm

n= (m>n) :

( ) ( ) ( ) ( ) ( ) ...... 22211221 pslkmm qxpxqxpxxxxxaxQ ++++=

:

( )( ) ( ) ( )

( ) ( ) ( ) +++++++++=

ll

kk

m

n

xxB

xxB

xxB

xxA

xxA

xxA

xQxP

22

2

2

2

11

21

2

1

1

...

...

( ) ( ) +++ +++++ ++++ ++ sSS qxpx NMqxpx NxMqxpx NxM 1122112 22112 11 ...( ) ( ) ...... 2222222 22222 11 +++

+++++

+++++ pPP

qxpxqxpxqxpx+ QPQxPQxP

22112121 ,,,,...,...,;,..., NMNMBBBAAA ek .. .

, , .

11. .

1.

, sinx cosx: ( ) dxxxR sin,cos .

t. :

txtg =2

.

sinx, cosx dx t dt;

212;2

tdtdxarctgtx +== .

.12;

11cos;

t12tsin 22

2

2 tdtdx

ttxx +=+

=+= ,

.

2.

. 1) ,

sinx = t, cos xdx=dt

( ) xdxxR cossin( ) .dttR

2) ,

cosx = t, sin x dx= dt .

( ) xdxxR sincos

3) tgx,

tgx = t , 21,

tdxarctgtx +==

dt

. 4) , sinx cos x

, dxxxR )cos,(sin

tgx = t. 5) .

. xdxx nm cossin

) , m n ,

xdxx nm cossin

. , n . n=2+1 :

.cos)sin1(sin

coscossincossin2

212

xdxxx

xdxxxxdxxpm

pmpm

===

+

sinx = t, cos xdx=dt. ,

xdxx nm cossin = ( ) ,1 2 dttt pm t.

) , m n . xdxx nm cossin m =2p, n =2q.

:

).2cos1(21cos),2cos1(

21sin 22 xxxx +==

,

+= + .)2cos1()2cos1(2 1cossin 22 dxxxxdxx qpqpqp , ,

cos 2x . , ). .

, .

mn xx ,1. R(x, , ) k 1/m, 1/n, ...

x= ., 1dtktdxt kk =

t t.

2.

++

++ ;...;; nm

dcxbax

dcxbaxxR .

,

. :

,Ntdcxbax =+

+ N m, n,

12. . . . -. . : , , .

, .

y = f(x) [a,b], f(x) >0. , , y = f(x), : x = a x = b.

, [a,b] .

. ,

SBnB, [ ]ba, ( ):

. (1) =

=1

00max

)(limn

kkkx

xfSk

- . kxmax

f(x) [ ]ba, k .

. (1) ,

0max kx

k , f(x) [ ]ba, :

b

a

dxxf )( .

, b . , :

=

=1

00max

)(lim)(n

kkkx

b

a

xfdxxfk

. ,

f(x) a b.

.

,

abdxb

a

= . =a

b

b

a

dxxfdxxf )()( , a=b, . 0)( =aa

dxxf

, , . [ ba, ]

1.

:

= ba

b

a

dxxfAdxxAf )()( .

2. f(x) g(x) [ ]ba, , :

[ ] +=+ ba

b

a

b

a

dxxgdxxfdxxgxf )()()()( .

3. [ ]ba, , ( a, b, ).

+= bc

c

a

b

a

dxxfdxxfdxxf )()()( .

4. 0 )( xf [ ]ba, (a

:

= xa

dttfx )()( ( )bxa . . f(x) [ ]ba, ,

) [ , (x ]ba,)()( xfx = .

-.

. f(x) [ ]ba,

F(x) - f(x) [ ]ba, , :

)()()( aFbFdxxfb

a

= . -. baxFaFbF )()()( = - .

u(x) v(x) . [ ba, ]

= ba

ba

b

a

vduuvudv .

, , .

, f(x)

.

ba

dxxf )(

)(tx = . :

. : 1. f(x) [ ]ba, . 2. )(t )(t [ ] ,

bta )( t ba == )(;)( . 3. [ ])(tf [ ] ,

[ ] =

dtttfdxxf

b

a

)()()( .

1. 0 )( xf [ ]ba, ,

S= . b

a

dxxf )(

2.

[ ] dxxfsb

a += 2)(1 .

3. y = f(x) Ox

[ ]=b

ax dxxfV

2)( .

13. . .

, : b

a

dxxf )(

1. [ ]ba, . 2. f(x) [ ]ba,

( ). ,

.

( )

1. f(x)

+

+=

a

a

dxxfdxxfdxxf )()()( .

, , , , .

, . .

1. [ )+ ,ax :

)()(0 xgxf , :

1. , . +a

dxxg )( +a

dxxf )(

2. , . +

a

dxxf )( +a

dxxg )(

( )

f(x) [ ]ba,

( ), d ,

f(x) . a b .

[ ]ba,

++= bd

d

c

c

a

b

a

dxxfdxxfdxxfdxxf )()()()( .

.

0 , :

=b

a

b

a

dxxfdxxf )()(lim0

.

, .

. f(x) bxa < , = ,

.)(lim)(0 +=

b

a

b

a

dxxfdxxf

.

2. bxa

II

14. . , . , .

.

(, ), , z, z :

z = f (x, y). : z =ln ( )222 + yx .

z. D z. (, ) (, ).

. - MB0B (B0B,B0B) MB0.B - MB0B (B0B,B0B)

2200 )()( yyxx + < ..

B1 B D, , .

B2 B D, D, .

D .

D ( D ), . D , .

z = f (, ) , ..

z = f (,) D.

, . n=2.

. f (, ) B0B, y B0 B ( B0B), >0 B B> 0 , - B0 B( B0B,B BB0B)

),(),(lim 000

0yxfyxf

yyxx

=

f (, ) , .

B1B(B1B, B1B) f (, ), , B1 B .

z =f (, )

B0B (B0B, B0B) . = B0B, . z =f (, B0B). B1B( +, B0B).

BB z = f (B1B) - f ( B0B)= f ( +, B0B) - f (B0B,B0B). z

BB z.

. xzx

0 , z B0B (B0B, B0B).

xz , , ),( 00 yxf x xz ,

xyxf

),( 00 . ,

xz =

xzx

x

0lim =

=x

yxfyxxfx

+

),(),(lim 0000

0= . ),( 00 yxf x=

: , .

B0B(B0B, B0B) B2B(B0B, B0 B+) z ( = B0B, : z= f(B0B, )).

z = f (B2B) - f ( B0B)= f ( B0B ,yB0B+ ) - f (B0B,B0B). y.

yzy

0, , z .

yz , , ),( 00 yxf y yz ,

yyxf

),( 00 .

yz =

yzy

y

0lim = yyxfyyxf

x +

),(),(

lim 00000

= . ),( 00 yxf y

xz

yz

z B0B (B0B, B0B) .

, .

yz ,

L :

tgyz

M =

0.

xz

z =f (, )

=B0B, .. tgxz

M =

0.

15. . . , . .

f (x,).

B0B(B0B ,B0B) . B0B(B0B ,B0B) ( ,)

B B= xB0B+ x, = B0B+ :

z = f () - f (B0B) = f (xB0B+ x, B0B+ ) - f (xB0B, B0B), .. , , , .

z = f (x, ) B0 B(B0B,B0B), , :

z = x + y + , 0 0 . x + y x

y f (x, ) B0 B(B0B,B0B)

dz = x + y. :

d z = xz x +

yz y. x =d y= d.

d z = ),( yxf x d x + ),( yxf y d y ( (, ). , d z = z + z - .

xd yd

.

.

z = f (u, v), u v , , :

u (x, ) v (x, ) z

, x :

z = f [u(x, ), v(x, )].

xz

=xv

vz

xu

uz

+

;

yz

=yv

vz

yu

uz

+

.

1. :

),,,( zyxfw = ).();();( tzztyytxx === , , ),,,( zyxfw =

t: w=w(t),

dtdw =

dtdz

zw

dtdy

yw

dtdx

xw

+

+ -

.

2. , .

),,,,( zyxtfw = ).();();( tzztyytxx ===

dtdw

=dtdz

zf

dtdy

yf

dtdx

xf

tf

+

++

1 -

.

.

z = f (u,v), u =u (x, ), v =v (x, )

.

=dz dvvzdu

uz

+

. ,

, u v . ,

, u v . .

z = f (u,v).

:

),( yxfxz

x= ),( yxf

yz

y= .

, , : . . :

),(22

yxfxz

xxz

xx=

=

; ),(2

yxfxz

yyxz

xy=

=

;

),(2

yxfyz

xxyz

yx=

=

; ),(22

yxfyz

yyz

yy=

=

.

),( yxf yx , ),( yxf xy , .

, .

. ,

xyz

2 = yxz

2 .

=dz dxyxf x ),( + dyyxf y ),( .

.

[ ]dzdz =2d .

=zd 2 2),( dxyxf xx +2 dxdyyxf xy ),( + . 2),( dyyxf yy

,

F(x,y,z)=0, D z= f(x,y), : F(x,y, f(x,y))=0.

, :

0=+ xzxx zFxF , .. 0=+ xzx zFF

),,(),,(

zyxFzyxF

zz

xx

= . :

),,(),,(

zyxFzyxF

zz

yy

= . ,

0),,( zyxFz .

16. , . . .

:

u=f(x, y, z).

xu

, yu

, zu

. : lB0B, .

B0B(xB0B,yB0B,zB0B) 0l

r.

B1B(B0B+ x, yB0B+ y, zB0B+ z) 222 zyx ++=

B0 . B B0B B1B u=f(M) u= f(MB1B) f(MB2B).

),,(),,( 000000 zyxfzzyyxxfu +++= , , u = f(M) B0B l .

: lu ; lu ; lu r ; l

ur .

, :

),,(),,(

lim 0000000

zyxfzzyyxxulu +++=

.

.

0lr

:

{ } cos,cos,cos0 =lr .

coscoscoszu

yu

xu

lu

+

+=

.

u=f(x,y,z) { } cos,cos,cos0 =lr rrr

. kji ,, .

u=f(x,y,z). MB0B(xB0B,yB0B,zB0B) B0B , .

. .

u=f(x,y,z) MB0B(xB0B,yB0B,zB0B) . B0B :

zzyxf

yzyxf

xzyxf ),,(;),,(;),,( 000000000 .

B0B, . grad )(Mu 0

grad . ),,( 000 zyxf ,

kzuj

yui

xuMu

rrr+

+=)(grad .

.

,

(2 = ) , ..

. . u=f(x,y,z)

B0B(xB0B,yB0B,zB0B) , . .

.

z= f(x,y), B0B(xB0B,yB0B) .

B0B(xB0B,yB0B)) z = f(), B0 (0< d (MB0 BM) < )

f () 0. .

: B1B B0.

. ( ). z= f(x,y) B0B(xB0B,yB0B) ,

0),( 00 = yxf x ; 0),( 00 = yxf y . ,

, .

(). , . .

. z= f(, ) B0B (B0B ,B0B), B0B

0),( 00 = yxf x ; 0),( 00 = yxf y (.. B0B ). :

Ayxf xx = ),( 00 , Byxf xy = ),( 00 , Cyxf yy = ),( 00 . :

1. = >0, B0B (B0B ,B0B) f(, ) , , < 0 , > 0.

2BAC

2. =

17. . . .

. z

=x+iy, x y , i : iP2P= 1.

y , , z : x = Re z , y = Im z.

1) z = x + iy.

zB1B = xB1B + iyB1B zB2B = xB2B + iyB2B (zB1B = zB2B) ,

xB1B = xB 2B, yB1B = yB2B.

z z ,

z = x + iy, z = x i y.

2) r - - z ,

O. (. 1). r , ,

z :

r = | z |; = Arg z.

r = | r | 22 yx += .

xytg = .

z: ) Arg( Arg +

, , 2.

Arg z : ( ) ( ) 20

: ( ) ninrz nn sincos += . ) . n- z

w (w= n z ), wPnP=z. z w : ( ) sincos irz += ( ).sincos iw += .

( ) ( )( ).102 , sincossincos, ,...,kknr irninzwnnn

=+==+=+=

: = n r -

r, =n

k 2+ (k= ,...2,1,0 ).

.2sin2cos

+++==n

kin

krzw nnk

, z 0 n n, .

18. . . . .

. , , = f(x) .,...,, )(nyyy

:

.0),...,,,,( )( = nyyyyxF = f(x)

, . .

, .

. = f(x), , , .

.0),,( =yyxF

,y = f(x, ). , .

.

= (, ), :

) ;

) = B0B = B0B, = B0B, = (, B0B) .

(, , ) = 0,

. .

. = (, B0B), = (, ), . (, , B0B)=0 .

, . . , .

),()( 21 yfxfdx

dy = (1) , , , . , (, )0)(2 yf :

.)()(

11

2dxxfdy

yf= . (2)

, , , . , ,

+= Cdxxfdyyf )()(1 12 . (2) , (1) .

19. . . .

. f(x, y)

n- ,

f( x, y) = f( x, y). n.

),( yxfdxdy =

, f (x, y) .

x/1= .

.

.

)()( xQyxPdxdy =+ , (3)

) - . )(xP (xQ

: = u (x) v (x) .

, (3).

.

20.

. . . , .

n-

0),...,,,,( )( = nyyyyxF , n- ,

),...,,,,( )1()( = nn yyyyxfy . .

n- ),...,,,( 21 nCCCxy = ,

n , : nCCC ,...,, 21)

; nCCC ,...,, 21)

,00 yy xx == ,00 yy xx = = ... , )1(0)1( 0 = = nxxn yy , nCCC ,...,, 21

),...,,,( 21 nCCCxy = . 0),...,,,( 21 =nCCCx

. ,

, .

nCCC ,...,, 21

,

1.

),(2

2

dxdyxf

dxyd = ,

.

pdxdy = .

dxdp

dxyd =2

2.

,

),( pxfdxdp =

. ,

= (, B1B), pdxdy =

. += 21),( CdxCxpy 2.

),(2

2

dxdyyf

dxyd = ,

.

pdxdy = ,

.

.2

2p

dydp

dxdy

dydp

dxdp

dxyd ===

,

),( pyfdydpp = .

, B1B:

= (, B1B). ,

, .

21. . . . .

. n- , , .. )(,...,, nyyy

)(...)1(1)(

0 xfyayaya nnn =+++ , (4)

) - , , . ,

,

naaa ,...,, 10 (xf00 a

naaa ,...,, 10 )(xf10 =a ( 1,

). ) , , .

(xf

) 0,

(xf .

) 0, (xf0...)1(1

)( =+++ yayay nnn .

1. B1 B B2 B

021 =++ yayay , B1 B+B BB2B .

2. B1 B , B1 B .

. (4) B1 B B2 B [,b], , .. const

yy

2

1 .

.

3. B1 B B2 B (4),

=B1B B1 B+B BB2 BB2B B1B B BB2 B , .

0=++ yqypy , (5) p q .

kxey = , k = const;

.; 2 kxkx ekykey ==

(5), .02 =++ qpkk (6)

(6) (5). , ; kB1B B BkB2B.

qppkqppk =+=42

,42

2

2

2

1 .

: 1. kB1B B BkB2B

; 2. kB1B B BkB2B ; 3. kB1B B BkB2B .

. 1. : kB1B B BkB2B.

xkxk eCeCy 21 21 += .

2. . kB1B= B BkB2.

)( 2121 111 xCCexeCeCy

xkxkxk +=+= .

3. . ,

,, 21 ikik =+= .

4,2/

2pqp == (5)

)sincos( 21 xCxCey

x += , B1B B2 B- .

).(21 xfyayay =++ (7)

: .

- * y

.021 =++ yayay

= y + * (8) (7).

)(xfqyypy =++ , (9) p q - .

, .

1. (9) , ..

( ) xn exPxf )(= , - n . : )(xPn

) k .02 =++ qpk

x

n exQy)(= .

) :

xn exxQy

)(= . )

: x

n exQxy)(2= .

2. xexQxexPxf xx sin)(cos)()( += ,

- , :

)(),( xQxP

) + i , (9)

xexVxexUy xx sin)(cos)( += , - , ;

)(),( xVxU)(),( xQxP

) + i , (9)

]sin)(cos)([ xexVxexUxy xx += . ,

, (9) , ..,

)(),( xQxPxexP x cos)( xexQ x sin)( .

, .

. (, ), , z, z : z = f (,) D.

z =f (, ) 0 (0, 0) . = 0, . z =f (, 0). 1( +(, 0). ( z = f (1) - f ( 0)= f ( +(, 0) - f (0,0). z ( z. . ((0 , z 0 (0, 0). f (x,). 0(0 ,0) . 0(0 ,0) ( ,) ,