n
1 1
5^Y5
rr
IE 383 3S0 3E) 3» 3» am 3» i©2ffl jeo 3» 3» 3» aro 3» 39D so 360 3© 3fiO 3» 3K a»3e03»
3«^
BO
mi mo
VA
'x'^UITG
IDC
CHUYEN NCaHiiP
M
-van
BO
GAO tjjfa DUNG
(^
csiuiro
!^Zl^IiXa£^irCB3SK
ojc
ISQ^
^»v>;t,i.
•:
wr;;
:;,
i
I'
^^^-i^
Ha-nOl.
-
1978

cua
phop 1$P (2.9) 52
§4
V5
in§t logii phuctog phq? l§p
b^c
cao
thrf hai
do
giSi gSn
dung
phifcmg
trinh
to5n
ti>.
. •
37
C^^ctofa:
haj.
5 M$t
vai
phi:?cmg
phap
ngi
suy
tSng qu5t
svQT T§ng
d|
giai
gin dung phtfcmg trinli

•
• •
^
Gfaifgya;;!^
l?a
t
Ph8n iJng
d^ng
i
1 Mgt
s5
T^ag
d\wig
cho
3191
vai
to^n ti>
cy
thS
61
1.1
Phi?c?n6
trinh
haci thi/c
vo^
bi§n s8 thi;c
61
1^2
H$ phtfc^ng
trinh

ph'.^'ng
phap giai gan
dung
cac
lopi phi?o'nc
trinh
ngt c5ch tong
quit
da
dem iQi
nhieu ket qua v6 cung quan
trgni^.
'2u»
ti^o'nt;
cua
giai
tich
ham khong
nhu^ng
cho phep
dcn
[^Inn
hoe
each
nhin
cho nhieu
phUHo'ng phfip
khac nhau
ma
eon giup ta

l?ng
chiing
(1) Ax
=
0,
trong do A lu toan tu xac
d^nh
trong
ngz
khong gian noo do
va
CO
gia
trt
trong
mgt
khong
i;ian
cung
lo^i.
DS
giai gan dung
phu^c-rig'
trinh (1)
ngi;?o'i
ta dung nhieu
phi?o'ng
phap
kh£c
nhau.

cho
timg
trj'c^ng
hg^
rieng, chang
hgn lop
toon tu that,
loip
toan tu
(to»n dipu
v.^-".
••
Trong
nhiJng
nam gan
dixy nhiou cuSn each
chuyen
khao
da do
cfp
o^t
each
dang
kt toi nh5ng phuxmg phSp
tong
(luat,
Chang
h?n,
c5c
tai

ket
qua thuge linh vi/c
do.
Ban
lu§.n
van nay gom
ba
chu'o'^
- ;'
-
Chu-^ng
1 trinh bay mgt
Q6
cac
phiAD'ng
phap ngi
cuy
tuyen tfnh
dS glM gSn
d'ms
phuv^ng
trinh toan ti (1).
Thi/c
ra
phuang
trinh
(1)
da
'*u^c
nhiou

trong s6 do
phuang
phap
riutcn - '-ingtorovlch
t-uy
di.;vo
su'
ct\mg
i^ng
roi,
nh>mg nhuyc
diem chinh la trong
nhiSu tru?o»ng
hgj) vit;c
xac
^nh toi'n ti>
^AxJ"^
(d^o
ham theo
Preso) thu^'n^i
^gp
kho khan
v-o
ph'^e
t^p.
ITgoai
ra trong
ngt
G6
tr-fo'ng

gia
tr%
Ax^^
va
v^.
th3
khong the op
dyn^j
dug'c
phv?o^
phap
TUu-to'n.
Di;'a
vao khai
niv'ia
ty sax phan
stjy rQng
cho to5n
ti
trong
nhu'ng nhjsi
gan day cac
con^
tA.nh
cua
/"11-13;
22-237
da lan
lirg^t xuSt hii^n nh?aa khle phyc nhu^ng nhu^c
d-icm

t-y
l^i
cao, chung to
ra dgc
bi^^t tign Igl
trong
truVng hg^)
toan tu' A co
dsua haa
phiJc tjip hogic
khong kha vl.
Chi?c^ng
hai trinh
bSy mgt s6 cac
phut)*nc;
phap
ngi
Bvxy
t6ng quat suy rgng cho toon
t&*
PhSn d2u
tien cua
chu'c^ng
nay
la
dUB
TV.
coc khSi nipn
ve
1^

ket
qiTO
c^'ia
ch'.;x»'ni3 I va
cua
coo
toe
gia
khac.
L:5i
phuang
phap 6"
cb^Mn^
T cTuig
nhi,"
& chwng
TT
^eu
dxjt^o
khao
B:OZ
qtia
cac
phSn 5
- ilgi
dung
phittrng
phap
-
S^

IT).
ChL?o'*ng
ba (phan
iJng
dyng)
khao cat
ngt;
oS
v'nx:
^yng
cua cac
pht!t?ng
phnp
o'
di^x^nz
1
va
chi't5»ng
TI cho
m^t
so
to5n ti*?
cy. the;
ITeu
len
thugt
toan, dieu
ki^n
hgi
t\i,

dirg'c rai
roc boo cao
trong
cac buoi sinh
hcpt x6-iai-na
"Cac
phuti*ng ph&p
gl.al gan
dung
^.ihu^i'ng
trinh toan
ti"
vh nhu*ng
ftxg dyng
cu.a
chnng"
cua
to
bO ^5n 'Doan
hgc vinh toan, khoa 'Joan
ir.>?o'ng Bjii
hgc long
hfi'p "^a-ngi
l';?76
va
ia$t oh'&n
da
dug^c
cong b6 trong
/"55-557»

qua trinh hoan thanh ban
lugn v2.n
nay.
"
6
Cn'^^^n^-^;
I
IE
Cg-AT
GAN
DTTITG PHHONG 'H^lTii lOAN
l^T
PJxao
sat
phi?o»ng tri.nh
to5n
t\^
-
(1)
• Ax
^
0,
trong ro A la toan
tu"
phi
IrL^on
anh
x^ ingt inien
loi
^TL

r^ng
^'-]0
toan
tu"
(:^m^5l7)*^
Xot ngt
ham tr\i
t':Vng
Ax, han
nhy chTij^'n
khong gian
d5.n!i
chuan X vao khong gian
dj.nh ehu*n !. , ^,-"
:tiu^T^''*^
Hi hi^u
khong
gi n
cna
riii?ng
toan
tiFVx vaor
la
fX-^
I
J.
Thanh
Igp
khor^-
gian tlch :

(i
"
0,1).
iltWig
ty:,
phsn ti>
cua
B^.^
di^vc
-^rio-i;
dL^'c'l djing
(XQ,
XT_, ,
:5C^„-()»
"^^
^1^^"'
^^ ^
cTl^).
Gia
^ tBn tfi ngu
h^a
tr*a:
ui^ijr'ng
A(x^,
:^:^),
han
nay
chuyen
nhiJng
ph3n

la
ni$t
toan tu
tuySn
tinh.
OJoan
tu'
t-uyen
txnh
A(::^,
oi-j
) thoa man dieu
kipn
(1.1)
AC-,,
x^)
(x.
-2„) =
A::^
-
A^
,
difg'e Qg±
la
ty
sai
ph;'i^ui bye nhSt ei\a
h-an
tru'^u tu'o'ng
Ax l3y

e
X,
(i
=
oT^
joi v(?i c5c phSn
ti:
x^,
(i
=
0,2 )
o5 ^^nh
t^x^ng
X thi
A(x ,
X|,
Xp)
la ngt toan tu
LzonQ tuySn
lanh.
Toan
tl?
so.ng-
tuy-Sn tinh
ACX^,
:20:
, s>^)
thoa nan
*iou
kign ;

t'^
('1.2)
(cau
k^i
•-?,
d-g^c
-coc dyng ^:ji
vS
len
(:<>,
-
^Q^
^ "^^ -^-^^7 ''''^^ •
(1.2)'
A(XQ,
xp
x-)
(x^
-x^J
(:o.
- ::^) =
=
Ax-
-
A:.:^
A(x^,
: )
(x.
-
::.J.

• * "V-1
Feu
A ton tgi cac
r^go
han
liun
tye don bgc 1: (thoo
r eae)
thi
cac ty
col
phan xac
d^n^
nho'
coc tich phan 'Urjan tru^i
tii'g'ng,
to
CO 'fc^c lu^j'ng ;
(1.4)
(I
A(x^,
:c.)|U
V'^J
» ^- :.^ .^
(x.
-:c„),
0
^
6
^

th'
chi?no rdnb
5uvc rane» ty
sai phan
b.Jc
k cua ham
tr>ii tuvnjj
Ax
la ham
-^«
<^
-9
-
De
den
gian,
t^:-
day vo sau se ky
hi^^u
:
^01 k ^
''^^^o'
^^•••»
-^1:^
^Jo1 k =
^^^'
^^o'
^^l' '
^1:^
§ 2. Ve mot

2:5
-x° 4=
24_^
-xj_2.
x° :=
x°,
0
<:M.
-^1.
Tu' (2.1) suy ra :
,
(4-^0)-
JL-^(,o_^^).
T-JP
(2.2)
suy ra :
(2.3)
A(xg,
x^, , ^^)
(:^»xj)
(x|-^).,,
(xj - ^^^^) =
:=
(^
" '^
),!
ACT^O
O
ON
^^JO

k
(2.^)
A4-Ax°-T,,^(x°-x°)=-^ o1, k^
(4
-x°y
,
trong do
Aol^ k^
^
A
(x^,
X?
, ,
:^)
(2.4)'
I,,^
= I? ^-'^'
4-1
"-^
^-^^0^
(^
=
k(k-l)
(k-i
M)
i!
L'^t
khac, cung
d^a
vao


(k-D^
(x
-
x*^)
(x
-
-i^)
(x
-
2g_.^)
trong
dc5
k-S
'te°
=
Z
(-1)^
cf.
o
Aoi„(if-l),,
H-
i=1
k-3
o'
'O
+
>•
(-1)^
C^_,

))x-x°||
f
-
11 -
k
(ic)
B§t
dang
thi5c
trong
LQ*
co
nang
tinh ch3-c d}.nh
tinh nhieu
hd»n
la
djnh
lijg'ng.
VSn
do
d-^t
ro
lt>,
co
t^n
t?l cae phSn
tii X ^ u
2J
gS

k-1
-
Axc1^ (k-1)^
n (x
-
xp
0
C
x-O
^
Ta
thSy (1^)
se
dung
khi x
=-
x°
va x -
r^.
(di^a
vao (2.5)
va
each
dgt
6^
* -^^ " ^k ^'^ ^^^'•^* "^-^^ "^"^ '^''"^^
b*ng
vo ph'
vs bang
khSng).

Axo1, (k-l)^
(4
-
X^)^
.
JC
.•
Axo1^ (k-l)^
jT
(.4
-4),
k-2
i=1
(Tlop
trang 12)
-
12 -
to
00
:
(2.6)
Ax -
Ax^
-
To,k (x-x^) = "^—^ Ax^1

(k-^
)^
Cx-x^) .
XSp

ra
t
1-0
„~^
.
o
-x
-
y:
-
'j<5^1:
Ax
long qnat,
neu to dgt
2
xg
:=
c^,
n = 0,1,
Tigp
(±)
.(^^-x^)(^^-:^^^^) ^-
Ao1^ (k-l)^(4-x^)(x^-x|),
k-1

(:4"^k-1^
i ^r
Axo1^ (k-l)Q ; irZ
-:^),


b?ir^-
AOIQ
k^
CO chu^a
Axo1-

(k-1
)_
bar^-
Aol-,
k^.
Tic (5^)
ta se xac
djnh duVc cZ
.
- 15
-
thi ly
lugn
hoan toan
tit^ng
tg'
nhti*
tren to
GO
C6
:
(2.8)
Ax-Ax"-Tn,k
(x-x")-"-

:
(2.9)
:^'^^ - x^^ -
Tn,],
Ax^^
, n
-
0,1,
S\?
hgi ty
ciis
qua
trin^i Igp
(2.S)
^u<^c
the
hiC^n
o^
cac
d^nh
ly
dUol
day :
JtiO
do 1
Gila
su*
toSn
t^jr
A ton tgi

x*|l
o^t^l-
'^
trong
d6 ^ duVc xoc
d^nb
ti^'o'nG
ti;'
nhir ^ tronij
(l.'i-).
i^p
dyng
cone;
thu'c
'lay-lo
co phan
dir
cho toan
tiV
£"1v_7
(2.12)
Ax^
=
A(x*
+
x"
-
x^)
=
=

i=1
l'-
^
COns
tCrng
ve (2.8) (sau khi da thay x
=
x^)
cho (2.13) (chu
y
(2.12)')
ta
nh0n
difn'c
:
(2.14) IK (X X*) - ^
-J^
AJ?
(x^
-
x^)'
II 4
1=^1
- '
k
/
_L«
3^p
}
A

XI
ban
'^au
x°
thoa
niin
cac dieu
ki^in
:
1^/
loan
tu*
AO1Q
ton tai toan
uJ
nijhjch
dao
Aol^^
va
IIAOIOII
4
Boj
2°/
||Ao(k-i)^|i
<.
L^,
(^i
=
1,k-2)
;

V—-^
1=1
k-1
^-^ ,
,i-(v-0
,
= (-1)
iAo^
r L
(-1)
C^^-^
Ao(k-l)Q
i=1
1^
_r>
k-1
. -
V-
r
^r
^ '^r.i
(2.16)
ai,,ic =
C-^r"
Ao1^(I•^Ao1o ./^^-'^^ ^^ Ao(l^-i)
trong do I la toan
tt> den
v^.
The
nhu^ng,

. ^
v-1
^
(l + A0I3
£.
(-1)
-i^^T -'io(i^-^)J)
V'^
-1
^J^
l-(k-1)
.
-1,1 ^
(|(l+Ao1o}_'
(-1)
t.5_^
Ao(k-i)o)
\\
^
-1-
•
-
IG
-
-1 .1
-1
Do do toan
ti nghi
eh
^^^o

lui^^lU ^ ^k '
x"€ ^
v.'l
Vn.
Lhl
do
qu5
trinh
Igp (2.y)
GO
!^;i "cy toi
n-ghl^^n
x"^ cri.a
(1 )
loe
dg hgi ty
d-u<;^c
:-*anh .gi.a
bang
bSt
dang thu'c :
^'^
-
= *
IU
f.
II
'^^-
-
^*

y
=
k-1
/ .
*"7^
^x^
Uv
-
X
^
-I-
i,„
tx
,
X
;,
1=1
^•
trong do
(2.18)
||r^^(x*,
x^)
II 4
-L-
i,^
II x^
_
^f ,
I^,
= cup IIA

-1
*
II
I^ik
II II
\
(^.
^) I)
•
Dya
vao cac dieu
kipn ciia
d^nh
ly va
c5c bSt
dang
thu*c
(2.18),
(2.11),
tii*
bat
dong thu'c
cu5i
cung suy ra dieu phai
c^iihig
minh.
I^u
khdng
gi5
thiet

I^i
f (i = ^)»
trong do lan
cgn liTj ^ duyc
>:ac
dinh
ti^
(2.24).
3°/
II To"'k Ax°|U
>'^,
)i:^.i
-
^_JU
>o'
<i =
^^)'
4^/
Cac hang
s8
Bo,
/^,
L^
th6a man :
k-2
(2.21) 0
^
p^
:=
1

7^
= 2Bo T2 '^ 0
<.
1
vi ^-k
v'''
=_,
;L_____
J^.k
- 2l^-1i,k
3^k-^
k-1
'
2,1,
iF^-1
j^k
(2.24)
u2.^
=
X : [jx
~
x^||:^
Khi do trong
iSn
oan :
k
' "^
^ C ^-Po^o ^
phiJo^ng
trinh (1) sa co

^ -
Po%
'-'
Chfeg
minh
Chu'^
minh rang khi chuyen
ti? ph&n ti5= x*^
song
x
thi
cac dieu
kign
cua dinh ly
v^m
cT>-?g'c
bac toan.
Do
dang
a-^y
ro
rang cac
ph§n tfr
s^,
x\
^
LT/^*
(i
-
o,k-*l).

=
-
19 -
-1
To,k
k-1
2Z
(-1)
1=1
i
,.i
(A
^ 1
^^•o(k-i),
-
A^ci
Kk-i)^);:
=
k-1
''^x4li^,'^-'^
^-1':^o(k-i)^-A(k-i)„i "
+ ^(k-i)Qi
^Kk-i)^
)
11 ^
^ 2 Be
1:2
• o (
k-1
i=1

^j^
=
('I?o,k
%) ton
tgi va
Bo
II
<lc
__
.0,
Dieu
kiOn
1 da kiem tra
xong.
KiSm
tra
di§u kipn 3«
CSn dsnh
gia fl
C,
,]r
Ax^||
V^
n
=
0 va
X = X
,
d'^a
\'ao

!
(2.28)
K;,,
Ax^ii 4
?;^(A,^^
'1.k :o>
o''
"o^
^ (
1,k
^ ^2,k A
)
Po .0
0 .
De dang
chi?ng lainh du^c ??lnn;t nsu /\
la
nghi^
ciia phtJWng
trinii
(2.23) thi
A=
('^'^
^j,
^ ^\^-^
y\""^)
-V
Do do
t'Jf
(2.28) suy ra :

f^
^i^C'o^
k-2
Pi
=
i-(i
-
J;^
C^)
•! .
1,
^1 ^1
" ^
*'l
"^^
^o
- ^^
W-Ou
kign
4 da
dxfg'c ki&a
tra
xon^;.
KiSm
tra dieu
ki?n
2.
De
kl^m
tra

^llx'
-x'^ll + 11 x'^
-x^li^
—^— ''. + ' >
>i
0
Dieu klgn 2 di
dxJg'c
kiem tra
2X>ng.
Th;/c
hifn phep
ch^Jng
minh hoan toan
ti?o'*ng
tg?,
ta co
the
Chiang
minh
di?g'c
cho
tr^.?o»ns hgp tSng
quat khi
chi:Qren ti?
phSn
t^
x^""
sang
phSn tiJ x^. Tif

P?"'
-lo ^^
^o'
j=o
+ +
- 22 -
n-i
•^"1
k-1
,
i
^~
(P
q ^-'- )
•
Khi V
-^ '
t^j?
bSt
dang
thu'c cu6i
cung suy ra bSt dang
tht5»c
(2.25).
TSn
de con
Igi
la
c§n chiJng mirJi x*
Iv^

dgi
li^g'ng gl^l
ngi, do do
t
0
=
lim
Ax^
=
Ax*
n—»
:JO
Binh
ly da
du'o'c ch'?ng
minh*
Gia
siV x*^
thoa man cac dieu kifn cua dinh ly
2^3,
ngoai
ra con thoa man thom :
Bjj.<,B
v^
Vnf
.
k-2
J^—
\ *'" ~i
(i-i>««ao) ^

nghigm
khac
x*^.
Can
c^'5'ng
inin^
rang
x^ trTing vt3l
xT.
TV (2.9) ta suy ra
5
Ax^^^
T^^^
(x^^'-x^) -0
Do do :
(2.59)
It.k^"''
-
%,i,x Ax-
^t
%,k(^''-^^)-
Dypa
vao
(2.59)
va
Ax**
= C (vi
x**
la
n^+nigm

. :,^) J
M|lt
khic,
d\fQ
vao c6ng
th'^c Tay-lo cd phSn diP
cho
toan
ti5 ts oc5
:
(2.41)
Ax-
=
A
(X**
>
X-
-
z^)
=
=
Ax^
.
A'x**
(X-
•-
x^)
.
1^2
(^^*)'

^
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Trích đoạn Chiang ninh djnh ly nay hoan toan tuttog ti^ nhtf phap chúng minh dj.nh l y 2 1 fJKp Mar.^k 1968 (20)

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