vY
t.
vI$N :KEoa ,IrQ.qltl
' /r$r rer " ,.;:
e0r ro.&N' :,r'}e'e'-r..
VIET'NIA}I::"l."
!
i
t
l\
'rEANs nnGs
'Tru s&:
.a
I
1986 ,."
, .+
, sd 152.
!(?
?8 TrIn tlung Bao -- t{} NOi
Drly ndi: 52BZS
sac RA rrAl
I&ag }idn f Sp : Ngug6n C,inb To*s
Phl tdng biilx tQp I I{gO Sqt ?ri'
.$
InONC b$i r,dy. ain ri€p ,u, giO, ihiQu tliem
' v6i bRn dr,r' r'E ung dgn6 aga tii h ngoii
' (co thi rrct r,q,p vJi ii*fr ,il,"nngr vro
hinh lroe phtng.
t. ye thr?c gita 3 rdcro bdr kr.
'['r!.n*y. chring la s€ qui ude b0 g[ d{u mfii
t6n trgn cAe ehfr dnng dB chi vects. Ch&rg bqnp
:.?
a s€ du$c thav b&ng o, sau khi d& n0i.tru0e
a lA mOl vecl<r.
BAy gir), trong rnit phfr,n!, x6t 3 rrectu b{i I{}
.a, li, c.. Ndu rlrring dbng ipluong v6i rhae tbi
vi khing cd gi dl{ng n6i. I\6u c6 tr nh{t mQt
c{p {ecto khOng dbrrg phuqnsr ch$ng trgn n vA
b, thi: Io,--oJ*o *: '
vi veeto cdn lq.i tf c bilu ditn drrgc b&ng.n,gt
tb hop tul6n tlnb cia o;-D rph&n tich c llido,hei'
pbuong s vA D), trtc c6 a, B lA *d thr,rc sso cES
c = *o *$D (il
LLn lugt lly tteh ngoril cta hai. vE v|i a vh, b,l
tr l0 duqe: x
'fr\ -
[c, ol= F[t, ,i. [e, 6] = *[,
rnc a = -'-[o' tj-, p,* :bA
la. bJ [o, bl
H0 thrlc (t) rr& nen
- [4, .]q - [e. a.Jo
, [e. bl
hay ld
t;, alc * to, cla -l- [c, a]6 = o
bt
(2.)
DS IA hS rhue gifta 3 v6cto b6r k, trong m{r
.phfiag. Ta phri f r&ng, tru0ng hqp,;, &,, c:d0ng
phuong vdi nhau tbt h0 thrlc (ir vfln d6r.rg.
2. COpg thirc"cQng cung rrons lrgng gi6c : ,
- "Ta h,,Ef x€t 3 veeto a,. b, c lA nhilng vecto
ilon vi: :
lol =,1b 1 = te,ls l, " -
Diah lj Cfiaelee v& g6c tho ts he tbr]c:
--.-\<.:*.-\r
a.r,* o.&+ b.c *2kr
I IA nqr ed nguy6n, co'7], ;;, Tli-tr
nh&ng 96o lugng giAe (gtic c6 dinlr hu6ug nghla
rQnC' c1 ,h3i tia): Te h&y d{t :
. I
tr
rhi
"r 'll'
Nhin ud tu1ing v0i i, la c6:
' [a, cXbl * [u,.oJ(*, bi + [b, cXo, b)
Pn.u, ir, t:14 ';ects don vi, [a riuge c&ng lhue
srn 'i t l3l = *ip:t . *esp -1" ces* . sinil (4)
'[* do siry ra sIn(* - 9,', trr"(ix t p), cos(z''p),v.r'.
3 4u&ng th&ng, giao dibnr cira &ai dti*rg
t h&nq.
ui b*ring lhiing Ta hfry r6e tlinh rn0t duosg
thrllg blng cAe ir cha dibir a vii phuong u (iec-
t<ri eiia rd. Du0rg thiiag di qua "'L thco phluong {.r'
l* quf.. rieh d-ibrn II sao eho aM dOug phrtong
vdi .rl, tue lA d
['rtri,ui =0 1
huy l*
trinh iir :
[.c * w, W] * o (-q]
Ba dudrg lh$ng {5),. (6) v* (9) l{ d&ng qur
ndu vA. ctri n6u giao eria hai du&ng (5) ve (6).
cho blvi (p), ila* mirn phu<rng trinhr (9). Ei6u
kiQn d6 la :
[[u, B]c'* [8, p]a * [.{, u]u, lVi = I
hav la
Ir,'rllc, wj + [rr, w][4, u] + [w, u][8, i,] * o rth]
li.v lA diBu krEn dbng qui eho 3 duOog ihirg
(5); (,6i vo (9r.
'a
ChuS.l) C6 thb l5-v rlibrrl C liirn g6c e{e ver-
to truQc. Ktri 6y C = {} Bi6u i{iQH (10j trr} n6n :
lo, lYlf{ rrl -i- [1Y, ujlr, u] : o(rt)
tj:0)
2i N€u rn6i dtiong thirng.duqe x{c e{!nlr'i.,dng
? ili&rn^ tuc n€u oho 3 du*ng itnng it&', BB',
CC", thi ta l5v eAtr veeto chi phuur:g la :
u* rNtl - i ** A, u * Blt' : B' * JJ,
I'
\Y * CC'-E ci "* C rbi Ap agog t1i0u kr$n (10)
xDae (Ili'
5. $!ah lj Ceva.
Lio tam g,irc 11.8L'. '1'r0n c{rc c4th (.6 ti}B h6o
daii -BC C.t. ag' tAu lrqrt clro cdc dibnr J'. .8', (.'-
lJEy tim, cIrEu kiQu tiurtg qui pLo 3 iluorlg ih6.ng
:1:L', E}J', CC', :
Ly*rl, a]=o (5)
, .tsqn cft.re nhn r'flng to llf bi*u 0,i ,a -u-*, tan
'luut'b&ng .t/, A I vhi O la g6o di ehqn cljo cAc
vee to ban kii:h cria e6c di&m)r.Co thts odi (;) la,
phuona drinhcria rluong thhog. trcng dO llf li
dtl,at tttg!1 tr6n iIu<rng thing'
s) Grao ein /rai a;bxg tidng
Cho th0m mSt ilrrdng ttring thr? hai. 6i qua B.
iheo phrrtrng u (khAt u)" ma phtrorg lrinh lA
' [a - ,1,t, a,j = u,lu, u] l= o (a)
'fa lriy dat N =' qu * 9u (tl
vtr buQc Jd thoa mirr e& (5) vh {6), sE ilugc
[a-au -Pu,u):0 ,|*
, [B'ctu-Fu.ul =o ll,
{[a,ul +Plu,ui =o
o I [r, ul - afu,,l = o
'Iit ri6, Ia,rrit ra a, F rbi'thay viro (7), ttrrryc
giao rliBin rr)a hai dudng th&ng (5) va (6,! l{ :
Lu, olu * [a. u]u
Iu. ul
(8)
a ilibu-kiQn d.iing rlui cira ba durlng th&ng
i.4i eiro ih€tr el uOng th&ng thu ba v6i phuugrg
llenelairE ('rHv'lr ss 2-1985),
l,'ir\ lu'c\
- *l- ! E * l-.-----t
**i"-*-i 'P*t-.'.*l'
\ .a'; ,i \ r'" /
a.
Tu LI5. ta"su-t ri l
ft
Du t,
r. ' F{
Ntrrr .trong bll i truoc & mue
i-
- 1*c(
t'Ia lqi co:
II-Il. *.i*
ri6i rts diuh lf
td heY det : a
._,*t>.
/ c'-.r \
T = l----.- ic
\ZT-I
, lt*tc C*ll.t ,t-TU
A -;
-.---_
, 6 2 ,L E .---
7-u.' l-F l.--T
Llro:, L 1- Ur [s. CG i
-Rr
r Pl ..
, Lj a--, L
.:)
I_P
I
f+ A -"{B
* :;---
l-r
l*d.
F.c-- (l - PtB
i.
U- D
"-"'*'
,tu=C
-t) r' r-p
. A-TB
*. [: C 3----;
I - l*
- Ap rt+ng dibu kigu (ll), ta s€ dirgc
(r - apT)tr, b12 * s
Ntrung ta, a],*0, ehq "n6u
"6pT *'* I (12)
r'
I
i
I
1
'B'(t-at.+
\'
6 Sii to6n 7 /142 crla b6o THVTT rd 5.
r 985.
bE &rir: -- 'ilren c6.e eqnh e &ri' tu gi6o itri c6
r'd)qn ticta S. vb piiis rgtiai. ttr,rng cAc hinh vu6ng.
'IAnr c6c hinh vu0ng d6 tq.o thtrnh m$t tir 6ie€
c6 drsn t-i.ih St Chung minh:
. a) si) u:
. b). Sr. = 25 ktri vrr "ohi irhi'cie dudng cheo cria
' Lu gjae bair diu bir:g nhan v&, vuOlg g6c vai
hhfu.
" G9rii. ehr; tri gi6c I&i ABCD: Ta gpi ,{d,'Jf, Pr
' O ldn lurTt iu 'c6.*drgng ililrn cria A'8, "BC, aD.
D.r, tlai iloqn ,41P ta NQ c0 giao di0ni O nli ta
i '.. bi6t l$ trurrg Cidrn *fia rn6i dualr.'lagQi At,4t,
' '(l-t OitAn iuqlt ll tarn cAc hrnlr vuOug dr,rng tr"en
Ab. BC, Cli. DA vE phia rigodr. Chgu 0 i&nr g$c
cAc r e:lu drBtn.
a) Ta lray tirrh 5.
la c): + -+ *&
"5 : [o,r , uu) )- [on , ncl
: lfr -;z , ;fr1
+
= [c-1 : Dt]7
'-&.+
= [2N.u, ?pryJr
= 1:;lI * N, -v - Pl
:,1[.1{ *I{, tr+ffJ
:s * ri,y, rYj
t
\
*r
la co: a
?si = I*r, .arl * [rr. or] * [cr, .0r] + [ar drl
hay l$
Qhg Icl, riuoer
[u',.u'j 'i- [rr', ro]j * [u,', r'j $ [r,, u]]
_: {irj, jil
lvlat kh;1c n6u ggi rt', B','.C'.
irh eria t, B, C, $ trong plidp
thi:
'l({r' - r') -(.' * 8' "r.r'-.A' = - ty' I
:l(r0' - u') * It' - (j'+-d'* B'.= tfV' I
trong d6.11', &" lA c*.c vectu,11, N da quay goc
t
-.)4 (i)
,D- ii ed.e .vecto
gury g6c -rl2'
(5,
rnc q ;' - :(j'+6,r - - l(c' -r ,.) ;
t2
.1
N-'= j ie, + c't * -J- (r'+d.)
2I
TLr ker qu& (5) ra cd:
:[ir, u'-r'] * t[N- iy'"-' 4'J *
*- -{[rc,rt/,1 -. {lil, f{'L = 4'nf + nt) (6
Vr.s6u d+t l-l1l: * va l.Vl:1thi i
*IM , M'7 = l*l lE := m?, *" h{,iN'J = ll' l? .o lls
d aay. ta.riir& r$ng P x -l[, { = *ff.
*b) BAy gidr {inh'Sr /
''Ia L,Iy dat :
,{A ? '$,; IIC : 2u, CD = 2w, ltA -* l|, -
'I'a ggi u r D', Lt), r' lLul l*gl l&' 6.ntr r:&a c&.e vee
!*y h0 rhue (3) eho tai
sr - rlff, Nl * z{.mz + nz)
tt) ho{c
hod.o
- ,Si * s[ac, :y]'+ :{s,8 * ,,r) - 4[M, jvj
: 3.s * t(nr3 ,$ rr2 *- 2m^sin01
la u, u, .r, tr trong pli6p quay rri0t g0c vri0ug Cudi cing tr c0 thb vidt:
theo chiEu im. tue *xll. 'ta th5y ring ; - Sr ; :S .1 f{\m * nis ."r.2ann{i; aia$)l (Z)
,.rn= M * a, Ur ==_L+ ,u',Ct* *lI + t1', _-,Tt h6r qui tie,r, t. -;r..-rrlt lei-ir+o.;;.
:.: . Ot = *ff * r' tl) -Iai to6r. 2,!j
II
l!
s
.l
Nl,dn x|t th6m. Ta hfiy d] .f het du&ng e heo
eira tir,gi{r Al$r61Dr "is c6 {theo.ft} +i tS)i,
P = AtCt: Ct-rt! = -ZU +u,' - &' :
i = -2ff.-l:H-=,Iritf *i.{)
-'4
- q* 8r0t*iDt*pt,:t._2rV*r.._ul =
= -2\, + 2.1r.' = 2(.1{" _ fi)
Cho quay * r/'.:. r"t ilugc
p'= -liM'- ffi = -s.q'* z(-.ry -,S') = p
\'65' ta' e6:
[p. p'l .'{q, q'} * [*p, ql .
,t
lpi' = lci2 = lp, ql
'\'l
M{t kh6.e dtBu kign \ ,
* A A -d
d+..C:=B*De,a
kdo lheo ,*
) sinr{ E sinC
l--:^
i sin B = sin I_)
hay, lr ,
I l.re. ailj lcn r a)
I
) o d b.c
t-*
I lac. a.rl - lo,t. ttt)
t---
t o.tt c.d
c.0
t*e li :
a
e
Kdt luAn Iil: hsi dudng eh4o rlrCi.vA BrDr
bing nhau viL vutrng gtrc l*l nhau. .
7 B{ii tofin I rbi qudc 16 l9SS.
*E 0ril: e ho rr:gt du&ng tr0n tArn & tr6n cqnh
:{# brie n:gt tri'gi6c l&i ABCD vA tigp rLre vdi
p .eqnh cdn l4i Chring ntinh rLng n€u tr? gifc
n$i tidp. ta cd ;,lD * gC * AB,
'Gidt. ?a gpi O l{ tflm cfia vdng trdn <Tfl€}o,
nlm lrdn tB. vA I{y n6 lim g6c cAc vects bAn
klnh cho cdc tlitm I'a dlt:
.rl8 .= g, BC:= &" Cl) ::r c,. OA = d.
14
ca * Io* rre I s IA iI+t)
DiEu ta phii chung minlr lir:
e ri c:&fd (l)
, Dilnr o i,u&i cAe h dBri cdc eg,nh BC, Cn, DA
bing r, eho peJ ,
la, cl = ic, o_1_ [o, e] -_
^_
._i_r*.j--:_--=2r ,2,
bett"
NAU.nel fhd -gi6i kf ni0m 250 ndm ngds sinh
tda J F.*I.agrange,.m$t trong nhttng ahi
toAn hge l0n nh{+ eria th6 ki IS.
,4
.4
e{n cdr vlo (1} v* chfi ,j
[a. Bl = o. ta cti d.n:
( ta.aJ+[D A]
l--T--=
. l oal
eL(
I tc. el + [8. cl
[--, *=
c 'l ?
ta, "al*[a, r.]+tc. ril
bc
le. cl * [4, e] + [c, ol
'c.d
ring .B = l,A,
a
hay li:
r-L &+c*[a -o-lc-.Aa
c b.c e.d
I
rll .r,. --.. ,' - . . -
(t*htlc 1-L
, bc*ed 'b+{i
S* s{nh hai r} sd dlu r,& eu6i {I*1), ta c6!
D6 l&. rliBu ph&i chuag rninh.
fii niQnn 25O'ndm rag&y cinh
;q
I
Lagrnngc einh nLm 1736 & thfrnh phd Turin
au0c f vl c6 nguln g6e PhAp. N[m lg'tubi Ong
ri!4 l& gito rr 'to{n hqc & trudng phto binh
- o,d
[c -.s. a
i
I
-
Turjn NLm I766 vua Dirc Friedrieh Il (m6t *i
Yu:! Ihudng er{n khoe hsc) ,n6i'oU";r;
Berlin vd,i. philsE loi to ^trr,e;'tOu'leu-, s st *
lo6a figg l6n nh4t chArr.tt ngn "U_g-.U_r, irU
-vuai'vi d4i nhrirr. Ong d.tserlra ebe dEa khi vua
Irriedrir h m{r ( l7s6) ;A- seu d6 ai pr_i_.?' ily
Ong-d& rharn gia vige cir .**t **o io'.i" ct,inf,
suyE.l eicn nr*ng, rr-S thanh g,t; ; j t.,rpog
eao ding\s,6 pham nlrn lTgS ;f 6 iruOng 1reof
khoa nd m t797. dng m{r n[rn l8lg" :
" Lagrunge l{ mot lrong nh&ng ngvdi ila ph6t
trita ptr6p rinh vi rich "phl; b;g? o"J .o ge*g
xly dgng ph6p tinh na"v ,n0t uart, o'tlai'rtre thOng
II;".i:,:::s-:o gii -:6. prruong pni n- *,i* o"[
KIrac nnung rsudi di trrr6e {Neu,ton, l,erbniz,
?.r,"i) 6 ,\6 6;g ru.{r ph6t t* ehu6i 1.a-vtor. Mec
di phuon;; r,hA['nA-v to ra c6 nhi]u haa ,";;';;
::g 1i k.h0flg chu .*i a,:nf **.lo?, i,ii nO, ,o
eua ghu6i fallr,r, "6r e0og ,r,ui, ,lri" Osg rd
gili tich vi eo hoc tt[.thric oAr. ;;; nghirn
ctlu r"Ac btm s6 *Ot ,.6"*h lruu. tuqeg. Cdc hf .
:;:: u,-n".,' l;l n f;i "
r; l*' ; ; s : r, ::i i
-;,:
,',, : i
f(t*h') = f(c) * hf,tr\ + _Au_ /,(r)+ ,..
li do oug rtua ra d&u ii*n. 2
. Tron8 eu6n r'Co hqoc gifii tish D, cOng lrinh e6
.S-i6,,"i ntr{r .eria Lagrangei r{t cl c{e ifrari' ,,,,
tru.6e d{y v6 giAi rlch?a aqe" irirh ;dy thco
mS.[.quaa {i}m th6ng uh{t vA. rfrar Irla, ifre*.
ln_0"*.qua vige Gng. dgng .Oig-.f,i'pit,p'tinh ui
tieh phAn euda aAch dA hEo ah[t crri; nhiBu
ngu1&n lf hhAc n,hau crla co hgc Dav ln *Ot
thdng Igi r-r.rc ro crja gi&i rich ,Ira"",,ir':"^l];i
e{c phu.ong ph6p h6e f,qF t..r*..te;1 T;";
phho m& dtru du6n sach, &ng Aa tf"'U*o vi6t
:,4:e: j I.:oe c0ug.trinh nsy ngLrr!: ra Rhdng.
th{y bdt k} mQt hinh v6 oru, *i.rr; iiay n,a"
phep tlnh 6ai s6 D.
,Lagrange con c6 nh&ng d6ng gr p nh{t diilh
eho vige pb6t rriEn Cq; s6 ,.* -s6" iE,..
-Ore
aa
dlra re airfag phrrr:ug ph{p tActr,e$c ngh:fnr
l!-t.'.:" ];ot nfu"ng trinh dai sd va 64s1, ,g,.
xt cnung. C[.c nghr6n e r]!u e,.ia 6ng vE e*.e phuong
trinh iiq.i s6 c6 bge irln hoo i Uo *,.f ,n,.n
oi,c nhA todn hgt"Abel Galois,,chung minh eAc
,lloo-ng trinh niy kh6ng cd lcri giai ta*g quii
vl phat triln llji tirur 6l nircrm
Ei j Larr
.B.ni l/149 coi .r(il u sd tr6c s6 ctla stl tq
nludrr n. l.r.l: r.i hilu.sd ngugtn tii-oiit kh6ng
u!y,t" o"A i Choing ariah rtteg u6i mpi sd ts
nhlie n la.cd
r(r) * 'rei +... *.?(n) = [n/1J*{n/e] *...
* [n/n] . \
^ r"dt -gtdi: {dr;a,tH.bo NrlugEn llfru phtu$nq, I0CT,
,4.:u;,l::-t],j] gri "{ ri r&p rirr .a ce; cip s15
r.L. u, rrong dO itl d -1.* r. d lt cAe sd lr.r nhrBn:
i (, d ( ,,. v6i m6j_ ";6 di,,;-;i"roor*os *a
i/r) ctp s6 (r" d), vSy tr,i -r,;'*'ia" ruor
nhan cac gia rri rr| 1 dsn fl ra "olu"nia"",rj
ctia d ch inh lA
r(r) -t- r(2) .f" "..* r(n).
[fIr khir A clrirrlr, l,U. ,+.p tllf ci c{c rIp s6
(kd, d) irong do t;5t la i*a" ,g ,u ;,;,r',
1Sl./(*n nn.. r<ri(..n/rl r,,_ylr <'i'< I,/al
(do .k la C6 rr.r otrirn). V6i. *Oi o &'"uinn ,0,
rE rAng ta c6 [n/d] c{p sd (kd, d). Y"Er-l;; _';;
d ltrn luot nhsn e6e gi6 tri tr I'd6n nta c6 s6
phhn tri cta A chinh lA
'$ [ni tl * Lnlz] + tllsl+ ., *,{,/nJ,r
. \'{y t(l)-T 2) .r,.. * f 6r * [n/i] , lnlzJ,*...
* {nln')..
, rY*,<irr rit : CAc ban Ld Dinh Thlng,ttlCT eu6c
hqc, Hu5). fodn eu6o Chitn {A,I'it EHrH fi{
ngi), I:r&n 'l'rong fldnS (11 Ctr tlHsp }{[ ndi I),
Pham .QugnhThu (ncT ptTt{ I_o Qui oon, xna
frang,, 'l'ru.ong ydn pdi (ilT FTTII NguyEn
'frurrg Truc Rirch gie Iir'en Ciang). Dodn anh
t fung (l l, PfTH phnn Boi (;hau; Nghe linh),
fS:rqf"" Quang il*ng tlt,t Pt'r'ir euJng 1'runS
Qui Nhon) ed loi gifri ruong rqi. nhlng""tri bafi
Phuo1g e6 c{ch di6a d4t s6ng nd nh{tI
: D* sd cAc b4n ktr{.: giii ir eo y.,t rryrog ;,tlep
gui ngp.
. * j\fu xC-t e{ q6c rrde nguy$n
nhAn,avE phii v6i I.
i
em tii ph&i
D;B.K
BAi Z/14p. Kttt ut*t dT"de*t dtlnq $) thdp
phan, trang trg&$ e hfrtd thlp ptran.a'ha7fia aa
tigy 1a tht cd t0A0 chtr .s6 t,pn inou J-.,ir.q *,pr
gld trl c ndo {ri hog kl.etng,: