:

:

.

1. 充3

2. ..4

3. .

3.1. 充6

3.2.1. . ꅅ8

3.2.2. . ꅅ.10

3.3. .12

4. .

4.1. 充...13

4.2. 充.14

5. .

5.1. 充19

5.2. 充.21

5.3. 24

5.4. .33

6. .

6.1. ..35

6.2. 36

6.3. ..38

6.4. 42

6.5. 充.49

7. 充54

8. ....55

.

, .

, , .

.

1.   .

,

- .

..

. .

, , , . .

, , , . , . .. . , .

, , .

. - , .

, , .

2. .

( - ) . . , .

, . .

. 1638 , f(x). (f(x+h)-f(x))/h=0 h=0, , . , (f(y)-f(x))/(y-x)=0 y=x .

f(x) . x . f(x h)<f(x);f(x) Ph Qh2 <f(x) . h, P Qh <0. h , P . P=0, . . , Q .

, , , . , , , .

.   (1755) .

y=f(x) . . (1789), .

XVIII . . , , .

3. .

3.1. .

f(x), [a,b], . [ , ] [a,b], , .. 󠠠 .

, f(x) ( ), (x0- ,x0+ ), , , .

f(x) < f(x0)( f(x)>f(x0))

, x0 f(x) (), f(x0) () , ( ) . , () , x0.

, ( x=x0)

f(x)<f(x0)( f(x)>f(x0)

, x0 (), .

x0 x1 , , [x0,x1] , , x2 x0 x1 . , . ( ) , , .

, .

(max f(x)) (min f(x)) 0. (sup f(x)) (inf f(x)) [a,b] .

1 , 1 3 , 2 4 . , =, =b.

, . .

, f(x) (a,b) . 0 , , 󠠠 (0- ,0+ ), , , , f (x)=0 . , .

, (.2)

.

, , , , , : .

3.2.1. . .

, , , ; , .

, 0 f(x) , 0 , .

, .

, (- ,+ ) 0 ( , =0) 0 , 0 ( ) . :

I f(x)>0 <0 f(x)<0 >0, . . f(x) 0 . , [0- ,0] f(x) , a [0,0+ ] , f(x) [0- ,0+ ] , . . 0 .

II f(x)<0 <0 f(x)>0 >0 , . . f(x) 0 . , 0 .

III f(x)>0 <0 >0 f(x) 0 , . . 0 , . , ; 0 , f(x)<f(x0), , f(x)>f(x0) 0 .

3 (,,).

, 0 : f(x) <0 , >0, 0 ; f(x) , , , ; , .

, (,b), , , :

a<1<2< <k<k+1< <n<b (3.1)

, , (,1), (1,2), ,(k,k+1), ,(n,b) f(x) , , f(x) ., f(x) , , (k,k+1) , , k k+1, , (3.1).

: f(x) (k,k+1) , ( ) - .

3.2.2. . .

, .

.

3.1: 0 f(x) f(x)<0, 0 , f(x)>0 , 0 .

:

(f(x)-f(x0)

f(x0)=lim-------------

x-x0

f(x)=0.

f(x)

f=lim----------

x-x0

, f(x)<0. (x0-,x0+), f(x)/(x-x0) , . .

f(x)

----------<0 (x0- <x<x0+ )

x-x0

, f(x)>0 , -0<0, >0, f(x)<0, -0>0, >0. , 0 f(x) . , f(x)>0 f(x).

...

( ):

1. f(x) f(x)=0 f(x).

2. , 0 :

-     f(x)>0, 0 ;

-     f(x)<0, 0 .

1 : f(x)=0 , . , .(, y=x3, y=x4, =0, =0, (.4)).

: f(x) , , f(x) .

3.3. .

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

3.2: f:U(x0) R, U(x0) 0, 0 n (n>1).

f(x0)==f (n-1)(x0)=0 f(n)(x0)=0 , n 0 , n , , f(n)(x0)>0 , , f (n)(x0).

:

f(x)-f(x0)=f(n)(x0)(x-x0)n+ (x)(x-x0)n (3.2)

(x) 0 x x0, , . (2)

f(x)-f(x0)=(f(n)(x0)+ (x))(x-x0)n (3.3)

f(n)(x0)=0, (x) 0 x x0, fn(x0), 0. n , 0 (-0)n , , (3.3). , n=2k+1 .

n , (x-x0)n>0 x=x0 ,, 0 f(x)-f(x0), (3.3), f(n)(x0) :

-     f(n)(x0), 0 f(x)>f(x0), . . 0 ;

-     f(n)(x0)>0, f(x)>f(x0) ,. . 0 . ...

4. .

4.1. .

v=f(x,y,z) D (x0,y0,z0) .

, v=f(x,y,z) (x0,y0,z0) (),

(x0- ,x0+ , y0- ,y0+ ,z0- ,z0+ )

f(x,y,z)<f(x0,y0,z0)

(>)

, , . . , (x0,y0,z0)

f(x,y,z)<f(x0,y0,z0)

(>)

, (x0,y0,z0) (), () .

( ) .

, (x0,y0,z0) ,

, ()

fx(x0,y0,z0), fy(x0,y0,z0) ,fz(x0,y0,z0)

, .

y= y0,z= z0 ; :

v=f(x, y0,z0)

, (x0,y0,z0) ( - c ), , , , (x0- ,x0+ ) x=x0,

f(x, y0,z0)<f(x0,y0,z0)

, ,

fx(x0,y0,z0)=0

, .

, , : ,

fx(x,y,z)=0

fy(x,y,z)=0 (4.2)

fz(x,y,z)=0

, .

4.2. .

, . , ( ) , , .

, v=f(x,y,z) , (x0,y0,z0), , ..

fx(x0,y0,z0)=0,fy(x0,y0,z0)=0 ,fz(x0,y0,z0)=0

, (x0,y0,z0) ,

= f(x,y,z)- f(x0,y0,z0)

,

= { fx x12+fx x22++fx xn2+2fx1x2 x1 x2+ +2fx1x3 x1 x3++2fxn-1xn xn-1 xn}= fxixj xi xj

堠 x= xi-xi0 ;

(x10+0 x1, x20+0 x2,, xn0+0 xn) (0<0<1)

fxixj (x10,x20,,xn0)=aik (i,k=1,2,,n) (4.2)

fxixj (x10+0 x1, x20+0 x2,, xn0+0 xn)= aik+ ik

ik 0 蠠 x1 0,, xn 0 (4.3)

:

= { aik xi xk+ ik xi xk} (4.4)

f : , , x1,, xn. , , .

aik yi yk (aik = aki) (4.5)

y1,,yn (), () , .

, (4.5) (J.J.Sylvester). :

a11 a12 a11 a12 a13

a11>0, a21 a22 , a21 a22 a23 >0,

a31 a32 a33

, , : , ( ).

a11 a12 a11 a12 a13

a11>0, a21 a22 a21 a22 a23 >0

a31 a32 a33

, (x0,y0,z0) , ( 4.5).

.

: , (x0,y0,z0), f(x,y,z) ; , (x0,y0,z0) f(x,y,z), ..

f(x0,y0,z0) f(x0,y0,z0) f(x0,y0,z0)

--------------- =0, ---------------=0, ---------------=0

x y z

x=x0,y=y0,z=z0 :

1)  f(x,y,z) ,

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2

---------------<0 , -------------------------------- - --------------- >0

x2 x2 y2 x y

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2

--------------- -------------------------------- - --------------- --

x2 x2 z2 y z

 

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------- -------------------------------- --

x y x y z2

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------------------------- +

x z y z

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

+ --------------- -------------------------------- --

x z x y y z

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- ------------------------------- >0

x z y2

2)  f(x,y,z) ,

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2

--------------->0 , -------------------------------- - --------------- >0

x2 x2 y2 x y

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2

--------------- -------------------------------- - --------------- --

x2 x2 z2 y z

 

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------- -------------------------------- --

x y x y z2

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------------------------- +

x z y z

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

+ --------------- -------------------------------- --

x z x y y z

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- ------------------------------- >0

x z y2

3)

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0) 2

--------------- -------------------------------- - --------------- --

x2 x2 z2 y z

 

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------- -------------------------------- --

x y x y z2

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- --------------------------------- +

x z y z

2 f(x0,y0,z0) 2 f(x0,y0,z0) 2 f(x0,y0,z0)

+ --------------- -------------------------------- --

x z x y y z

2 f(x0,y0,z0) 2 f(x0,y0,z0)

-- ------------------------------- =0

x z y2

, ( )

4) f(x,y,z) , .

5. .

5.1. .

u=f(x1,x2,,xn) D (x10,x20,,xn0) .

, u=f(x1,x2,,xn) (x10,x20,,xn0) (),

(x10 x10 x20 x20 xn0 xn0 )

f(x1,x2,,xn)<f(x10,x20,,xn0)

(>)

, , . . , 頠 (x10,x20,,xn0)

f(x1,x2,,xn)<f(x10,x20,,xn0)

(>)

, (x10,x20,,xn0) (), () .

( ) .

, (x10,x20,,xn0) ,

, ()

fx1(x10,x20,,xn0) ,, f xn(x10,x20,,xn0)

, .

x2=x20,,xn= xn0 x1 ; x1 :

u=f(x1, x20,,xn0)

, (x10,x20,,xn0) ( - c ), , , , (x10- , x10+ ) x1= x10,

f(x1, x20,,xn0)< f(x10,x20,,xn0)

x1= =x10 , ,

fx1(x10,x20,,xn0)=0

, (x10,x20,,xn0)

.

, , : ,

fx1(x10,x20,,xn0)=0

. (5.1)

f xn(x10,x20,,xn0)=0

, .

: :

d f(x1,x2,,xn)=0

, fx1= fx2== f xn , dx1,dx2,,dxn

f(x1,x2 d,,xn)= fx1 dx1+ fx2 dx2++ f xn dxn=0

: , dx1,dx2,,dxn fx1, fx2,, f xn .

, f(x1,x2,,xn) () , , , . , ( ). , , , .

. , , - (5.1) , , .

, , , ( , , ).

5.2. .

, . , , .

, .

f(x1,x2,,xn) , (x10,x20,,xn0).

= f(x1,x2,,xn)-f(x10,x20,,xn0)

y ,

= { fx x12+fx x22++fx xn2+2fx1x2 x1 x2+ +2fx1x3 x1 x3++2fxn-1xn xn-1 xn}= fxixj xi xj

堠 x= xi-xi0 ;

(x10+0 x1, x20+0 x2,, xn0+0 xn) (0<0<1)

fxixj (x10,x20,,xn0)=aik (i,k=1,2,,n) (5.2)

fxixj (x10+0 x1, x20+0 x2,, xn0+0 xn)= aik+ ik

ik 0 蠠 x1 0,, xn 0 (5.3)

:

= { aik xi xk+ ik xi xk} (5.4)

f : , , x1,, xn. , , .

aik yi yk (aik = aki) (5.5)

y1,,yn (), () , .

, (5.5) , , (J.J.Sylvester). :

a11 a12 a11 a12 a13 a11 a12 a1n

a11>0, a21 a22 , a21 a22 a23 >0,, a21 a22 a2n

a31 a32 a33

an1 an2 ann

, , : , ( ).

. :

,. .

aik xi xk (5.6)

(5.2) () , (x10,x20,, xn0) ().

= x12++ xn2

(x10,x20,,xn0) (x1,x2,,xn). (5.5) 󠠠

xi (i=1,2,,n)

= { aik Ei Ek+ ik Ei Ek} (5.7)

Ei , , (5.7) , (5.7) . ,

Ei=1 (5.8)

m, Ei

aik Ei Ek>m

, Ei , (E1,, En), (5.8) ( ). , , , . . ; , , , ( ).

, (5.3) (5.7) ,, m, . , , (x10,x20,,xn0) , , f(x1,x2,,xn) .

, (5.6) , .

, (5.6) , ,

a11 a12 a11 a12 a13 a11 a12 a1n

a11<0, a21 a22 , a21 a22 a23 <0,,(-1)n a21 a22 a2n

a31 a32 a33

an1 an2 ann

5.3. .

. . . :

1. , .

2. , , .

.

1.,

a11 a12

a21 a22

ࠠ a11 .

2. , - () - , , () .

, n- , n f(x1,x2,,xn).

aik= fxixk .

a11 a12 a1n

(5.9)

an1 an2 ann

(5.9) a21/ a11 .

(5.9) a31/ a11 .

(5.9) an1/ a11 .

,

a11 a12 a1n

0 a22- a12 a21/ a11 a2n -a1n an1/ a11

(5.10)

0 an2- a12 an1/ a11 ann- a1n an1/ a11

(5.10), a11, (5.10) a11n-2

1

----------- (5.11)

a11n-2

a11 a22- a12 a21 a11 a23- a13 a21 a11 a2n- a1n a21

a11 a32- a12 a31 a11 a33- a13 a31 a11 a13n- a1n a31

(5.12)

a11 an2- a12 an1 a11 an3- a13 an1 a11 ann- a1n an1

(5.12). (5.12)

a11 a12 a1n-1

a21 a22 a2n-1

(5.13)

an-11 an-12 an-1n-1

(5.12) (5.13) ,

a11 a11 a12

a21 a22

a12 a11 a13

a21 a23

..

a1n-1 a11 a1n

a21 a2n

.

, 1n-1 n. : ; .

a11 a12 a13 a1n

a11 a12 a1n-1

a21 a22 a23 a2n

n-1 .

a11 a12 a13 a1n

a21 a22 a2n-1

a31 a32 a33 a3n

蠠 n-1

a11 a12 a13 a1n

an-1 1 an-1 2 an-1n-1

an1 an2 an3 ann

n-1, . . (5.13),

1

a11n-3 (5.14)

a11 a12 a1 n-2

a21 a22 a2 n-2

..

an-2 1 an-2 2 an-2 n-2

aik .

, n1 , , :

a11 = a1

a11 = a2

a11 = a3

a11 = an .

an

a1n-2 a2n-3 an-1 (5.15) n>2

1.

2 1 5 3

0 4 7 2 1 2*4-1*0 2*7-5*0 2*2-3*0 1 8 14 4

5 6 3 1 22 2*6-5*1 2*3-5*5 2*1-5*3 22 7 19 -13

0 2 1 3 2*2-0*1 2*1-5*0 2*3-3*0 4 2 6

4 7 2

7 19 13 1 4*(-19)-7*7 4*(-13)-2*7 1 -72-49 -52-14

2 3 1 4 4*1-2*7 4*3-2*2 4 -10 8

1 -121 -66 1 -121 -66 1

4 -10 8 2 -5 4 2 (-121*4-66*5)= -121*2-33*5=

= -242 165= -407

2.

3    0 2 1 5

0    4 1 3 6 1 3*4-0*0 3*1-2*0 3*3-0*1 3*6-5*0

1    2 3 5 1 33 3*2-5*0 3*3-5*2 3*5-5*1 3*1-5*5

0    3 4 0 6 3*3-2*0 3*4-2*2 3*0-2*1 3*6-2*5

1 2 3 4 5 3*2-1*0 3*3-1*2 3*4-1*1 3*5-1*5

12 3 9 18 -30 66 -264-108

1 6 1 10 -22 1 69 -105 96-162

33 9 8 -2 8 33*122 66 78 120-108

6 7 11 10

-30 66 -372 30*105-66*69 30*66+69*372

1 69 -105 -66 1 -30*78-66*66 -30*12+66*372

33*122 66 78 12 33*122*(-30)

1 3150-4554 1980+25668 1 -1404 27648

33*122*(-30) -2340-4356 -360+24552 33*122*(-30) 6696 24192

-1404*24192+6696*27648 33965568-182476800-2654208

33*122*(-30) 33*122*30

31311360-182476800 15116544 15116544

33*122*30 33*122 3888

=3888

頠 n, n-1, , 2 a1,a2,,an , ..

sign a11=sign a1

sign a11=sign a2=sign a11 a12

a21 a22

.

a11 a1n

sign a11=sign an=sign ..

an1 ann

. . .

f(x1,x2,,xn). , 0(x10,x20,,xn0). , a1, a2,, an . 0 , a1, a2,, ak k+1 .

0 , a1, a2,, an ,

a1<0, a2>0, a3<0,

, .

, :

1. ,

f

xi i=1,2,3,.,n

2. ik

2f

xi xr

3. 11=1

) 11>0, 2,3,,n , 0

) 11<0, 2,3,,n , 0 .

4. - .3 , 0 .

. ik 蠠 2f/ xi xr ,  ik= ki. , (ik) ik , .

-, , n-1 ,. . 򠠠 n-1 ꠠ n ..

, , .

ik n-1, i=k, i,k=1,2,,n-1.

ik= ik 1 k 1i / 11 (*)

i,k ,

aki= ki 1 i 1k / 11 (**)

(*) (**) , , ik= ki , ik= ki. , , n- , n-1 . n-1 ?

1k (k=1,2,,n-1) n-1 , ..

11, 12, 13,, 1n-1

,

11

21

31

..

n-1 1

, . , , , .

a11 a12 a1 n-1

a21 a22 a2 n-1

.

an1 an2 an-1 n-1

, 22 ,.. 22, 23, 24,, 2 n-1. , 22,..

22

31

..

n-1 2

, ...

a11 a12 13 a1 n-1

a21 a22 23 a2 n-1

n-1= a31 a32 33 a3 n-1

..

an-1 1 an-1 2 an-1 3 an-1 n-1

..

: . . , ..

a11 a12 13 a1 n-1

a22 23 a2 n-1

n-1= 33 a3 n-1 (5.16)

..

an-1 n-1

, , .

a11= a11 a22- a122

a22= a11 a33- a132 ..

a12= a11 a23- a13 a12 a11 a12 13

23 ..

3.

z=x3+y3-3xy

1.

z z

---- ----

y x

z

---- = 3x2-3y

y

z

---- = 3y2-3x

x

2. ,

3x2-3y=0

3y2-3x=0

(0;0) (1;1).

3.

2z 2z 2z

------- --------- --------

x2 y2 x y

 

2z 2z 2z

------- =6x --------- =6y -------- = -3

x2 y2 x y

 

4. (0;0)

a11=0 a22=0 a12= a21= -3

(1;1)

b11=6 b22=6 a12= a21= -3

5.

a11 a12 0 -3

a21 a22 -3 0

b11 b12 6 -3

b21 b22 -3 6

ꠠ <0 , (0;0) .

ꠠ >0 a11>0, (1;1) , zmin = -1.

4.

w=x2/3+y2/3+z2/3

2 2 2

w`x= ------ w`y= --------- w`z= ----------

3 3 x 3 3 y 3 3 z

x, y, z; ( ) P0(0;0;0). P0 w, (x;y;z) . P0 .

w(P)-w(P0)= x2/3+y2/3+z2/3 P0, , x,y,z . P0 , wmin=w(P0)=0

5.4. .

, . , , .

f G G. G. , , f G, , , ( , , ), , . , G, , , , f G. G, , , f G.

, G , 젠 x=x(t), y=y(t), <t< f(x,y) G f(x(t),y(t)), .

, (. , ).

, , , . , Rn , , , , .

6. .

6.1. .

. , , , .

, Rn x f(x) R ⠠ , Rn x0.

, , . , , , . , , , , 2, , . , ,

(,)=-

+=

, 蠠 (,) , , +=. , R2, . , , : , , =-, (,) (-). . .

6.2. .

G Rn .

yi=fi(x) i=1,2,3,,m (6.1)

x=(x1,x2,,xn). x G , fi i=1,2,3,,m :

E={x: fi(x)=0, i=1,2,3,,m, x G} (6.2)

fi(x)=0, i=1,2,3,,n (6.3)

.

: G y=f0(x) . x(0) E ( ) f0(x) ( ) (6.3) , , .

, f0(x) x(0) , , . , , , .

,

1)   f0,f1,f2,, fm G ;

2)   x(0) f1, f2,, fm , ..

fj j=1,2,,m

xi i=1,2,,n

m- ( ⠠ f1, f2,, fm).

, (6.1) x(0). n- n , 2) , m<n.

2) x(0)

(f1, f2,, fm)

(xi1,xi2,,xim)

. x(0).

(f1, f2,, fm)

(xi1,xi2,,xim) (6.4)

, , (6.3) x(0)=(x1(0),x2(0),,xn(0)) x1,x2,,xm :

x1= 1( x1,x2,,xm)

x2= 2( x1,x2,,xm)

(6.5)

xm= m( x1,x2,,xm)

x1,x2,,xm, (6.5) y=f0(x), .. f0 1,

y= f0( 1( xm+1,,xn),, m( xm+1,,xn), xm+1,,xn)== =0( xm+1,,xn) (6.6)

n-m xm+1,,xn, x(0)=(x1(0),x2(0),,xn(0)) (n-m) Rn-m.

, , (6.3) (6.5) , .

x(0) () g (6.3) , x(0) () (6.6).

x(0) g, :

dg (x(0))=0 (6.7)

, , dxm+1, dxm+2,, dxn. , , , , .. g/ xm+k, k=1,2,,n-m x(0). (6.7) x(0).

, , (6.3) ; , (6.3). , , .

6.3. .

, f0,f1,f2,, fm G.

6.1 : x(0) f0 (6.3). f1, f2,, fm , .. , ࠠ 0, 1, 2,, m

0 f0+ 1f1+ 2f2++ mfm=0 (6.8)

: x(0) f0 (6.3) f1, f2,, fm ,

fj j=1,2,,m

xi i=1,2,,n

m, 堠 1,, m ,

f0+ i fj=0 (6.9)

.. f0 ⠠ f1, f2,, fm.

: i=1,2,,n x(0)

f0 fi

xi xi (6.10)

F(x)==f0(x)+ jfj(x) (6.11)

ࠠ 1,, m (6.10), , ࠠ 1,, m .

(6.10) , x(0) f0 (6.3) , , ..

F(x(0))

xi i=1,2,,n (6.12)

, , , . , (6.11) 1,, m, f0, . 1,, m, (6.10) , .. (6.9).

n+m (6.3) (6.8) x1(0),x2(0),,xn(0), 1,, m ( ) , x1(0),x2(0),,xn(0) 1,, m. , (x1(0),x2(0),,xn(0)). , , , .6.5

. : x(0)=(x1(0),x2(0),,xn(0)),

fk(x(0))=0 k=1,2,,n (6.13)

f0, f1, f2,, fm , x(0) .

, f0, f1, f2,, fm , , 蠠 fj/ xi j=1,2,,m,i=1,2,,n m+1. m+1 . , m+1 , ..

(f0, f1, f2,, fm)

(x1,x2,,xm+1) x=x(0) (6.14)

G , 00>0, 0 0<0<00 ,

Q n={x: xi-xi(0) <0,i=1,2,,n}

G , , f0, f1, f2,, fm.

xm+2= x(0)m+2,, xn=xn(0)

x*=(x1,x2,,xm+1)

Q m+1={x*: xi-xi(0) <0,i=1,2,,m+1}

, fj(x1,x2,,xm+1,x(0)m+2,,xn(0)) j=1,2,,m Q m+1. : Q m+1 Rm+1,

y1= f0(x1,x2,,xm+1,x(0)m+2,,xn(0))

y2= f1(x1,x2,,xm+1,x(0)m+2,,xn(0))

(6.15)

ym+1= fm(x1,x2,,xm+1,x(0)m+2,,xn(0))

(6.15) x*(0)=(x1(0),x2(0),,xn(0))

(y1, y2,, ym+1) (f0, f1, f2,, fm)

(x1,x2,,xm+1) x*= x*(0) (x1,x2,,xm+1) x=x(0)

(6.13) (x*(0))=(f0(x(0),0,,0) . ( , , >0 ,

V={y=(y1, y2,, ym+1) : y1- f0(x(0)) < , yj< ,j=2,3,,m}

(.5) , , - Q m+1.

, n,0<n< , (f0(x(0))+n,0,,0), x`*=(x`1,x`2,,x`m+1) x``*=(x``1,x``2,,x``m+1), V`.

(x`*)=(f0(x(0))+n,0,,0)

(x``*)=(f0(x(0))-n,0,,0)

蠠 x`=(x`1,x`2,,x`m+1,x(0)m+2,,xn(0)) x``=(x``1,x``2,,x``m+1,x(0)m+2,,xn(0)), (6.15)

f0(x`)= f0(x(0))+n> f(x(0)) , fk(x`)=0, k=1,2,,n , x` Q n

f0(x``)= f0(x(0))-n> f(x(0)) , fk(x``)=0, k=1,2,,n , x`` Q n

0>0,0<0<0 , , x(0) .

...

. f1, f2,, fm , (6.8) 젠 0=0 堠 0=0 (6.8) . 0 (6.9).

...

5.

u=xyzt x+y+z+t=4c; x>0, y>0, t>0, z>0.

,

=xyzt+ (x+y+z+t)

x =yzt+ =0

y =xzt+ =0

z =yxt+ =0

t =yzx+ =0

yzt=xzt=xyt=xyz

x=y=z=t=c.

6.4. .

(6.10) 0(xm+1,xm+2,,xn), 6.2 (.(6.8)). .

ai1x1++ ainxn=0 i=1,2,,m (6.16)

b1x1++ bnxn=0 (6.17)

C , (6.16) (6.17), (6.16)-(6.17).

: (6.16)-(6.17) (6.16) , (6.17) (6.16).

: (6.17) (6.16) , ,

b==(b1,,bn) (6.18)

ai ==(ai1,,ain) i=1,2,,m (6.19)

, (6.16) (6.17).

. (aij) (6.16) m0 . , m0<m . m0<m, (6.16) . m-m0 , , m0 . (6.16), (6.17) (6.16) , m0 . , , m0=m .. (aij) (6.16) m .

(6.16) (6.16)-(6.17) . , . (6.16) (6.16)-(6.17), , .. . , .

s , , n , r : s=n-r. , (6.16) (6.16)-(6.17) . (6.16) m , .. (6.19) .

(6.16)-(6.17) m. (.(6.18) (6.19))

b, a1,, am (6.20)

. , b a1,, am.

, (6.20) , ࠠ 0, 1,, m, .

0b+ 1a1++ mam=0 (6.21)

0=0, a1,, am . (6.21) 0, , b a1,, am .

, b (6.19), (6.19) (6.20) m , .. (6.16) (6.16)-(6.17) .

, , b (6.19) :

1a1++ mam=b

, , .

...

, (6.16) (6.16)-(6.17) , (6.16) (6.17) .

...

1 : n Rn, .. n . , (6.16)

(ai,x)=0 i=1,2,,m (6.22)

(6.17)

(b,x)=0 (6.23)

a1,, am (6.18) (6.19) , x=(x1,x2,,xm+1)

a1,, am Rn , . Z=( a1,, am).

(6.22) , Z=( a1,, am) . Rn.

L==Z(a1,, am) Rn.

L=Z( a1,, am), b a1,, am L Rn:b L. , b :b _, , x (b,x)=0,.. (6.22) (6.23). .

2 : , . (6.16) . m. , m, .

a11 a1m

am1 amm (6.24)

(6.16) , n-m (x1,x2,,xn). (6.16). , (x1(0),x2(0),,xn(0)) (6.16). xm+1= x(0) m+1,, xn= xn(0) (6.16) m ( m x1,x2,,xn), (6.24) . x1,x2,,xn, . (x(0)1,x(0)2,,x(0)n). (6.16), x1=x(0)1, x2=x(0)2,, xm=x(0)m .

.

6.2: f0, f1, f2,, fm G Rn, x(0) G

fi(x)=0, i=1,2,3,,n

f1, f2,, fm x(0) m. x(0)=(x(0)1,x(0)2,,x(0)n) f0 f1, f2,, fm , x(0)=(x(0)1,x(0)2,,x(0)n) .

g(x)=g(xm+1,,xn)

, x(0) f0

f0= 1f1+ 2f2++ mfm (6.25)

f1, f2,, fm, ,

F= f0- 1f1- 2f2-- mfm (6.26)

x(0) :

F(x(0))

xi i=1,2,,n (6.27)

(6.25) , (6.26)

F(x(0)) f0 f1 f2 fm

xi xi xi xi xi i=1,2,,m

: f1, f2,, fm x(0) m . , 6.2 ,

(f1, f2,, fm)

(x1,x2,,xm) x(0) (6.28)

(6.3) (6.5) , , xm+1,,xn . x(0) dfi(x(0))=0, i=1,2,,m, dxm+1,,dxn xm+1,,xn (, , ) , ,

fi fi fi fi i=1,2,,m

x1 xm xm+1 xn (6.29)

xm+1,,xn , x1,,xm (6.5). dx=( dx1,,dxm,dxm+1,,dxn) (6.29).

, (6.28) dx1,,dxm dxm+1,,dxn (6.29). 2 , (6.29).

x(0) g(x)=g(xm+1,,xn)

, dg(x(0)). , ,

f0 f0 f0 f0

x1 xm xm+1 xn (6.31)

dxm+1,,dxn , dx1,,dxm (6.5) , (6.29). , (6.29) (6.31). o , (6.31) (6.29) , .. ,

f0= 1f1+ 2f2++ mfm

...

3 : 2 (6.29) Rn, L=Z( f1, f2,, fm) . y T fi , x(0) fi(x)=0 , fi,i=1,2,,m.

, (6.29) , fi(x)=0 ,i=1,2,,m, fi(x)=0 ,i=1,2,,m . , ,.. (6.29), dx (.(6.30)).

2

f0 L=Z( f1, f2,, fm)

f0 T

, f0 dx fi(x)=0 ,i=1,2,,m:

( f0,dx)=0

( (6.31)), .. f0 x(0) . , ꠠ f0, (n-1) 0 , f0(x)= f0(x(0)) . , , f0, 0 , .. Ҡ 0.

, x(0) , . Ҡ 0 , .. x(0) , , .

4 : 2 1. , x(0) o , x(0) () , , . 2 x(0) , .. .

6.5. .

, 6.2.

F= f0+ ifi

- (.(6.11)) f0 (6.3). x(0) G (6.3) , .. , (6.10) (6.3). , , , x(0) .

, x G (6.3) ,

f= f(x)-f(x(0))=F(x)-F(x(0))= F (6.32)

, x(0) F, .. F x(0), x(0) f0 .

, (6.32) , f0 , .. , , , , F(x) , . .

x(0)= (x(0)1,x(0)2,,x(0)n) (6.3). (6.6) , .. g(x)=g(xm+1,,xn) , f0(x)= f0(x1,x2,,xn) , x1,x2,,xm xm+1,,xn (6.3) x(0). , f0(x ) fi(x ) ,i=1,2,,m x(0).

( 6.2) , x(0) () f0(x) (6.3) , x(0) () g(x). , , x(0) g(x) , f0(x) (6.3). . :

1)     g(x(0) )

xi i=m+1,,n; (6.33)

2)

2g(x(0) )

d2g(x(0) )= -----------dxidxj (6.34)

xi xj

.

x(0) g(x). , x(0) () f0(x) (6.3). , g(x). , , g(x) , , .

, (6.4) (6.29) , , dx1,,dxm dxm+1,,dxn . (6.29), fi(x) x(0):

d fi(x)=0, i=1,2,,m

(6.3) , :

df=0 (6.35)

f=(f1,f2,,fm)

x(0) F(x). , dF(x(0))=0, .. f0+ ifi=0. 2 , x(0) , ..

dg(x(0))=0 (6.36)

,

d2g(x(0) )= d2F(x(0) ) df=0 (6.37)

蠠 n-m dxm+1,,dxn. (6.37) dx1,,dxm, , (6.35) , (. (6.5))

dxk=d k(x1,x2,,xn-m), k=1,2,,m

(6.6),

f0 (x(0) )

dg(x(0) )= -----------dxj

xj

( ) (6.29), i, F(x) (, i- (6.29) i)., (6.11),

F(x(0))

dg(x(0) )= -------[ f0 (x )+ ifi (x)] dxj = --------- dxj=0

xj x=x0 xj

(6.36) .

(6.37) . g(x) x(0):

2f0(x(0) ) f0(x(0) )

d2g(x(0) )= -----------dxjdxk + ----------- d2xj (6.38)

xj xk xj

, (6.3), .. x(0) :

2f0(x(0) ) f0(x(0) )

d2g(x(0) )= -----------dxjdxk + ----------- d2xj =0 (6.39)

xj xk xj

i=1,2,,n

i (6.39) i, F(x), (6.38) ;

2F(x(0) ) F(x(0) )

d2g(x(0) )= -----------dxjdxk + ----------- d2xj (6.38)

xj xk xj

dxi, i=1,2,,n (6.35). x(0) , , (6.37) .

, d2F(x(0) ) () dxi, i=1,2,,n, , (6.35), dxi, i=1,2,,n , ,  (dxi)2>0 d2F(x(0) ) >0 ( d2F(x(0) ) <0)

x(0) (6.3) (6.11) () dx1,,dxn, , (6.35). (6.36) (6.37) , x(0) g(x) x(0) () dxm+1,,dxn, , , x(0) () , , f0(x) x(0) () (6.3).

.

6.3: x(0) (6.3) (6.11) () dx1,,dxn , (6.29), x(0) () f (6.3).

, (6.11) , (6.37), .. (6.3) ( dxi, i=1,2,,n (6.29)). , () , , , .

.

7..

, , , , , , . , , , . , .

, , . , .

, , .

.

8. .

1..., .. .-.: , 1973.

2... .-.: - , 1960.

3... .-.: ,1966.

4... .-.: , 1981.

5..., .. .-.: , 1984.

6..., .. .-.: , 1981.

7... .-.: , 1981.

8..., .. .-.: , 1990.

9... . .1.-.: , 1978.

10... .-.: , 1994.

11... .-.:, 1986.

12... . .2.-.: , 1968.

13... . .1.-.: , 1969.


 
2012 , , .