DOAN OUYNH (chu blSn) - THAN OlNH VI$N
THUONG DUC HINH - NGUYEN HUU QUANG
ВА/ТДР
HINH HOL
VI PHAN
l_ I

Phan mot
TOM TAT LI
CHlfONG I
РНЁР TfNH GIAI Tl'CH TRONG KHONG GIAN
OCLIT En VA H1NH HOC VI PHAN CtlA En In = 2, 3)
NGUYEN HUU OUANG
§1. DAO HAM CUA HAM VECTO
Dao h£m ciia ham vecto mot bien A : J -* En, t —♦ X (t) la1
X(t) (afti	и bjju u (t) ЬоЬ<.
NSu X(t> = 2 у>1(1) ej (ei 1A co sd cua X", f' : J — R> tU
x-(t> Ллд
Neu X. X v4 bam s6 p cd dao him tai t e J th)
(X +	(t) = X(t) + '3'(t)
(jpX)’ (t) = y'lt) . X (t) + pit) . ~X ft>
(X . T5r (t> = X <t) . T! (t) + X (t) IB' it).
Vdi u = 3 Vi E3 cd budng thi (X x X)'(t> = Xlt) X Ini
+ X<t) x Tht).
Ki hi&u X'(t) = (X(t)>’ (hoec d^> goi no la <f<?o
hai (nSu cd) cua X
3
И bi«u
E3 al Hutes)
la khdng gian vecW Oclit n chi6u]
X(p) e T U voi mpi p e и gpi J
Vec(U) la tap cac trtfdng vecta X trfib U; X(p) = (pt X<p))
§2. VECTO TltP XUC. TRUONG VECTO.
CUNG THAM SO VA TRUONG VECTO
DOC MOT CUNG THAM SO
2) Khi nao
1.3.	Cho ham vectcJ kha vi : J -* "E". t ►—>X(t) tr§n khoang
J C R va gia sii ‘X(t), X(t) d$c Up tuy&i tmh (dltt) vdi moi t e J.
Chung minh rang dieu kign cfin va dii d£ ^X(t) luon thu$c m6t
. । —mo - hMid
ЛлЬ .. Р : 1 (ttatag tad trong R> - Е". ‘
rtom ГЛо« т *V>) “»» Е”' R "в’с Ыёи Щ Ь4°е ham V“
2.3. Cho bg toa db afin <0. p,, a;)	pbb, Oblft ₽.
. Hay phdc boa bob rfa c4e eng than, sb P  R - E . t - № M[
♦. djnh bdi
Anh х, X -. J - ТВ", t ~X(t) e Tw (E") goi la truing veeto
dot cung thorn sdp : J - E". TraSng vactd X doc/'duoc xfc dinb
Mi ban. vaetd 1 J - g", t —'X(t), X(t> = Q=Ct), l(t)>.


«X)’ =ГХ+ЛХ’
<XxY)* sJfxY+Xxr.
2) />(t) = 0 + cost. в| + sint. e 2 J
3) />(t) s= 0 + cht.Tj + sht.e2.
2.4. TYong mat phing Oclit E2vdi toa d6 D6cac vudng gdc
(x, y), phdc hoa inh cua cac cung tham s6 p : R — E ,
t = (x(t), y(t)), xic djnh bdi (hoic cd phtfong trinhl
2.5. Trong mat phing E2 vdi toa d6 cite (r, y>) hay phac hoa
anh cua cung tham s6 p : J -• E2, <p —*p(>p) — (r(y>), Dtf don
bdi mOt di^m chuydn dOng theo quy luat :
2) ChAn difdng vuOng go'c ha tit tAm 0 cua Ьё true tpa dO DAcac vuOng goc xudng mfit doan thing cd dd dii khOng ddi 2a chuyAn dfing dua trAn hai true tea de (hai milt cua doan thing chuyAn dfing trAn hai true). 2.7. Trong E3, xet m6t true сб djnh Oz, met dtfdng thing Ou JAm vdi Oz mdt go'c khOng ddi : mAt phing (Oz, Ou) quay quanh Oz vdi mOt vAn tAc go'c khOng d& w. XAc dinh quy dao cua diA’m P chay doe Ou : 1)	Vdi met tdc do khOng ddi ; 2)	Cd tdc do ti 10 vdi khoing cAch tit 0 dAn P. 2.8.	MOt trtfdng vecto X trAn mOt tAp md U cua En goi 1A mdt tnfdng хиуёп tAm (tAm 0 G En) nAu V p G U, X(p) cimg phueng ’vdi Op. Goi {E1Li., ЕдМа truong muc tiiu song song ting vdi he toa do afin {0, e^,..., en) cua En.’ X =	<f>' Er Tim diAu kiAn vA cAc hAm s6 p* (trAn U) dA X 1A	2.10. Quy dao t	сй. mW b,t t|ch Мв (t u thM trong mot truitng tO vdi cutmg dp H (col H la mOt truong i.eto trtn mot tap mi U c E>) thda man d.Pu kipB	x (H<¥,, (col B3 da cd huong) tro„g do c la mot hang aO. Khi H la mot trudng vecto song song, hay tim p. (Hudng din : xem hai tap 1.7). §3. DAO HAM CUA HAM SO VA CUA TRUONG VECTO THEO MOT HUONG VA DQC МфТ TRUONG VECTO Dpo ham cila ham sfip : U - В (U la tap md trong ЕЧ d^'h kUh"g Г’ E T₽U <hUd°B “ta' di<m ₽) U h'*U U “p I*’1 d“’C 1	“pW 'S’’(1, + Л|1-«-
mOt truing xuyAn tAm (0). 2.9. Cho trudng vecto хиуёп tAm X (tAm 0) tren mOt tAp md U ейа E3. Xet cung tham sO p : J - U, t —>/»(t) mA m/>” = Xo/> (m 1A hing sO dtiong). 1) Chtfng minh ring /’(J) nira trong mAt phing л qua 0 ; 2) Cho [а, Ь] c J. Chiing minh ring diOn tfch quet bdi bin klnh ?t) « От trong mat phing я la 1) |px /|| dt. Т» dd	. euy ra v$n tdc ciia biAn thidn di$n tich dd (theo t) 1A khOng dAi; CM(;3)( V‘^ ₽huonS trinJivi phSn cia oung tham s6 P trong l9a	1	N«u « = J a>e| va n£u (x1) la tpa do ейа dio’m trong B" ddi udi тцс tiiu afin (0 ; ep.,.,e^) th! “А[”]=Да‘Й1р	W Dpo ham ейа ham p e JV doc Iruong vecto X e Vac (U) i	(dupe kl hidu la X fpl) И him sS X[p] : U - R trong dd X[pj(p) = X(p) [p] vOi moi p e u. N«u X, Y e Vac (И), p, p e !fm U cd edng thde X(p + pl - X[p]+X[p]
4) Khi |[X(p)|| =	(к 1A hing s6 dtfong). Chiing minh ring m0t dUdng thAng qua 0 hay m^t dudng bAc hai nMn 0 Um met tiAu diAm.	(pX) (pj = pX [p]	<2) (X+Y) [pj =X(p]+Y(pJ X[p . p] « XIpJp +pX[p]
Cbo z е Vte (u>.«,« т,и- />: J - v а сипе «»»»« t —/хи irons «<> W •*«, • W thl D<dt°~ d““ е°‘ daa ham ейа Z thee huir.g «р rt kl ЫОи nd Id D„pZ. СЬо X Y E V«c (U). Trudng vecto (dude Id hiOu DXY trtn U duoc Гос dinh Wi <DxY>(p) = Dx(p)Y vdi m«i p e U got И doo ham cua truing vecto У doc tritbng vecto X. Anh X9 Dx : Vec(U) - Vec(U), Y — DXY gpi la dqo ham thu&n bie'n doc X. Vdi X, Y, Z, T G Vec(U), G 5<U) ta cd DX(Z + T)	fe DXZ + DXT DpXZ	=	DXZ Dx.yZ	>DXZ + DYZ Dx(pZ) = X[p]Z + pDxZ XtZ, T]	=	DXZ.T + ZDXT. Ndu Z = £ p'E( ({Bj} j_j „ lb truOng muc tiOu song song	3 3 Xdt toa do eSu (r. p. 8) vb truong tone tidu toa do c<o u3> trong E3 \ inua mat phbng Oazl (0 < p < Й, U _ | < 0 < |). Hay tinh 0, M, U,[pl, U,(8] (i • 1, 2, 3). 3.4.	Cbo X, Y e Vec(U), V 1» mot tip mJ trong E" Chdng minh ring ndu X[p] = Yip] vdi moi p e 1F(U) thi X = Y. 3.5.	U 15 tap mo lidn thong cung trong E". Ching minh ring ham s6 (khO vi) p trdn U 15 mOt ham king khi vb chi khi vdi n,i X e Vac U, X[pl = 0. 3.6.	Cho P : U - В H mOt ham st kh4 vi trdn tap m« U с E", f : R — R la m6t ham s6 khi vi. Chdng minh rang vdi moi tru&ng vecto khi vi X tr6n U ta cd X [fop] = (fop)X[p]. 3.7.	KI hidu {Ep E2, E3} la trtt&ng muc tieu song song tfng vdi hd toa d6 afin {0,	cua E3 vdi toa dd (x, y, z). Cbo cac X = хуЕ, + ezE2 - y2E3 ; Y = yEj + xE2. Tinh DXY, DyX, DX(DXY), Dx(X + xY).
trfen U thi	3.&. (Up U2} la trudng muc tieu toa d$ ctfc trong E2'\{0}. Hay
DXZ - i Xtp‘l B,.	tinh DU; Uj (j = 1, 2) Cung cku hui du cho trudng muc tieu tpa dd cau {U(} (i = 1,
3.1. Xet muc ti/u sfin, <0, ej, fc,, Uj) cua E3 vdi too do (x, y, z). Xet vecto « E T.3 vdi tea dO (1, 2, -1) va didm P(2, 1, 0) E E3, tidu afm (0, 1,, T2, "e3) ейа E3 vdi toa do (£ y.’a). Cbo trudng X - z!E, + zE. - yEs. ИпЬ X[p + VI], Xtp . pl, XiXipll, pXtpl - pX (Pl trong dd p = ay, p = yz.	?, 31 trong E3 \ OZ. 3.9.	U la mdt t&p md Нёп thdng cung trong En. Chdng minh rang trudng vecto (kha vi) Z trSn U la mdt trU&ng vecto king khi va chi khi Dd Z = 0 vdi m<?i <x G TU ho$c khi va chi khi DXZ » 0 vdi moi X G Vec(U). 3.10.	U la tap md trong En. Chiing minh : 1)	Vdi X. Y G Vec(U) cd mdt va chi m^t Z G Vec(U, & Z[y] =X[Y[p]] - Y[X[^]] vdi moi y, G (U) vh Z dd chinh la [X, Y] ([X,Y]= DXY - DYX).
» Ndu X, Y*e Vre(U) 1» nhang tnlimg vecto song song th!
IX, Y) = 0
3)	Ы» f : VM(U> X Vac(U) - Vec(U)
(X Y) —IX, Y]
la R - song tuydn tlnh, pbdn ddi xilng.
4)	Vdi X, Y, Z e Ven <U) thl
[IX, Y], Z] + ICY, Z], X) + IIZ, X], Y] = 0.
3.	11 Cho truing mue Uto (Ц) (i = 1, k... >> cung tham
S6 p : J -* En, t >—*P (t). Chtfng minh :
у 55. e J cl Uj, C| la ham s6 trtn J (j = 1, -n).
2)	Vdi moi tnidng vecto X doc p, bi6t X= 5 Ц thl
3)	Khi {Ujl true chudn thl ma trdn C= [Ci j phAn ddi xdng.
4)	Khi n =3 vh thd Hch cua hinh hop tao bdi edc vecto
<0,(4, U2(t), tT3(t)) khOng phu thufte t thl vft C = 4 Oi = 0.
9(X)(p) = ep (Х(рр.
3. Hfi сйс 1-dang vi phan {0 i} (i =1, ...,n) g9i 14 truing d6i
тцс Иёи d6i ng&u udi trubng muc tieu {UJ nffu (U ) = d\
(= 1 n£u i =i. =0 n£u i * il
d^> (X) = Xfrq	X 6 Vec U.
Mdt dang vi phftn b&c hai ш trftn U la vifec dat tuong dng vdi
T U X I и - к
Ki hi6u ш Q2(U),
Cho X, Y 6 Vec U thl co (X, Y) e difoc xac dinh :
: Wp (X(p), Y(p>).
(kf hiSu la вKd) xac dinh nhu sau :
(0Л <5) (X, Y) = e (X). 6 (Y) - e (Y) . 6 (X) (X, Y e Vec UK
§4. DANG VI PHAN
1. 1- d?ng vi phhn 9 (dang vi phdn hAcl) trtn tip md U trong
E« U vide dht Wang Ung m«i didm p e U vdi mOt Anh xa R-tuyAn
Unh 3 : T U - R, Id hiSu e e Q‘(U).
# : Q1(U)
b : Vec (U) -• Ql(U), X —bX sao cl
(bX) (Y) = X.Y vdi mpi Y 6 Vec U.

dang cau R - khftng gian vecto (vS la _jT(U) - tuySn tlnh)


Chdng minh rang пёи {в-, 62, в3} la trudng d6i muc tifiu сйа
trudng muc ti6u {UJf U2, *9^} tren U thi
En. Chtfng minh :
£ aF
4.5. Trong toa d6 afin (x1, ... xn) cua En phii changjdang thtfc
S p- cbd Л dxJ = £ dx' л dxi keo theo pij= р„ (ij =1, ..., n).
<1
(p, p' (i _ 1.2,3) 6 F(U). Dae biat пёи ,«c la dang th£ tfch chlnh
min^r&ng	D5 VU6ng g°C thU*” tX1’ x2’ хУ|’ Chang
1)	dx' (i = 1, 2, 3) theo dr, d<f>, de ;
2)	(dx* л dyi) (Uk, Up (i, j, к = 1, 2
3)	dx’ Л dx2 Л dx3 theo dr л dp л 
 dz, w = sinydx
.. rr,TT  "	s cac anh X9 ’ №tr«n tu <	14
«n 5(U) V» tu Q2fUl dfn Vec <U> cung bdi - Khi >, = „ la
г	•:«	-s.:
«.л. a , V, ,± Й" “ Mo« ™ «•. о 4=e ~ ™
4) Khi и = R3 \ (0). х« ham г е У<и>, г(р) = ||0р|| : Tim
him s« khi vi f : R+ - • R « div <Pad (fori) = ° <‘™<>ng vooto
X goi la khtmg cd nguOn "e“ div X = Ol'
Tpf(ho»=f.p):TpU—T,(p,V
Tp f (orp) = (tj
ret X. rot X = • db X.
T_f la anh xa tuyg’n tinh
1) Citing minh ring vdi X, Y e Vac (U), g> e УГО) :
rot (X + Y)	= rot X + rot Y.
rot (p X)	= t rot X + (grad p) X X
Tpf (Uj (p)) = i. | f Bj (ftp))
{E:} 1Й. trudng muc ti6u song song trfin V dng vdi toa da
f1 (u1, ... um), F (ul, ... u®).
4.10. Trio E2 \ {0} dOi vdi ha toa do Dd cac vuOng goc (x, y),
f. X = (f. X) (ftp)) = Tp f (X(p))
f. : Vec (U) —* Vec (V)
§5. ANH XA KHA VI
1. Bit, Un go, toa do aliti inh in f: U(md trong B") -
• ftp) if (a1, ... u"), .... Г (u1, ... u"), f khA vi khi
Anh xa khd vi vdi cac dang vi phftn
к = 0, p e J(V) thi Г f = y».f, f e J(U)
к > 0 (Г ш)р (oq ..... a,,) = w((p) (T/Cotjh..., Tpf(ock)).
Tinh chdt
P (d<p)	= d(pof)
(gof)"	= Po g*
P (ш Л я)	= f* ш л
Pd	= doP
Vdi	'
I Тю-
Ung vdi {e). Chiing minh
1) N«u /> :J-U, t —
trfin f(U).
2) Cung cku hoi trkn doi vdi f : R3 * R3 ;
2)	V —• R = E1 Ik mdt ham s3 trfen tap md V C En
(UjV) —»(x = u + v, у = cosu, z = v sinu).
1) Hky bifiu difen (theo du, dv vk tich ngoki cua chiing) knh bdi Г
{(uo, v)| ve R) (u0, v0 la nhtfng hkng s6) ;
2) Tim tkp cac difem (u,v) 6 R2 tai do f la mdt dim. Hdi thu
5.6. Cho knh xa (khk vi) f : U(C Em) - V(C E“) (U,V- tap md
X G Vec (U), X G Vec (V) goi Ik f - tuong thich nfeu f. (X(p>i =
= X(f(p)) vdi moi p G U. Chiing minh :
1)	X, X Ik f - tuong thich *=*X[y>of] = X[^]of vdi moi f G f(V)
2)	X, Y G Vec(U) f - tiiong thich theo thtf ttf vdi X, Y G Vec (Vi
th! [X.Y] f - tuong thich vdi [X, Y].
3)	NffuJ lh mC-t vi phdi tfc U 1ёп V thi X G Vec (U) f - tuong
thich vdi X G Vec (V) khi vk chi khi f.X = X.
tnidng vecto	qua y- ,
5.4. 1) Xet knh xa

vko U xkc dinh bdi :
Ik ham sfl trfen U, bi<t :
1 i(Fo0	1 a(Fol) . ,	<>(Fof) .
v‘“f ' S'	”of * 7 ТГ-' ” of = “ST
dtup = cu(. л a>k
3. Vi phdi dAng cu. Vi phdi f : U -
- T((p)V b&o t6n tlch vO hudng.
TRONG MOT TRUONG MUC TI6U TRI/C CHUiN
va ki hifeu dp 6 Q’(U) xdc djnh bdi :
dp (U,) « 0 ; dp (U2)

Gig ей dfi vdi trudng muc ti6u song song (ЕД trgn U
b) DU, « dp. U2 ; DU2 « - dp. U,
6.2. Xet trudng muc ti6u toa dd tru {Up U2, U3}.
DUj = dp.U2, DU2 = -dp.U, ; DU3 = 0, (r,p, z) la

trong d<5 C-1 la ma trfin nghjch dAo сйа ma trftn C = [Cjj.
2. Phuong trinh cau triic. Goi {£'} la trudng ddi muc tidu
• --i.
(2)
c£u trail U = E3 \ mat phAng x+Oz>
xdc djnh bdi
U, = U, ; U2 - rU2.
Tim trudng d6i myc tieu tuong ting {01, ^1- Vi6t d^ng hen kSt
□) сйа E2 trong trudng myc tifiu do. Vidt cac phuong trinh c4u
. . ... ... .........................  .g.»Uu	HJ.. U\l.
6.5. Cho trudng muc tiSu {U,} (i =
dang liftn k<t cua En trong trudng muc tifiu dd.
Hudng d&n : vdi mfli p E E" - {0}, Tpf : l£n —• "Ё” bifn vecto

[Су] true giao
Trtf&ng ddi muc tifeu (в *} tuong Ung vdi cAc dang Ii6n kfit w 1 ейа
trong tru&ng muc ti6u {f, Uf} (i, j = 1,2, 3).

ph lie va gia tri phiic).
CHlfONG II
DUONG TRONG En (n = 2, 3)
bifin da’i vi phOi bao giac cua R2 la mat bi&j d6i d6ng dang
(tile fix) = az t b hay a, b e C vi a # 0).
6.7. Goi f la biAn ddi nghjch dio tAm 0, phuong ti'ch к ейа
E" \ {0} : Of(p) = к - jgrp vdi mpi p 6 EB \ {0} (k 14 mdt s6

1,1. Xet h6 toa dd Dd cac vudng gdc Oxy trong mat phang Ez.
V» cung Г xac dinh bdi t	= «Л У<‘»- s* Г cd tidp
tuydn tai moi didm ; ki hidu M = P <t). P la hint chidu thing gdc
1.5. Xdt toa dd cue (r, ?) trong E2 \ {0} ( r > 0). Cho ham sd
1) Cung dd cd diSm kl di khdng ?
2) Ki hi£u {Up U2} la trudng mile ti&u toa dd cue trong E2 \ {0}
3) Tim Г ma tidp Anh tai moi did'm khdng ddi ;

r(u) = (D Chu, Dshu) (ddi vdi he
Vifft ON = b(p) U2(M), OT = a(p) . U2(M) (a, b la ham sd trtn
J) va gpi в (p>) la dd Ion cua gdc giQa vdi /’’(,?)• (Хеш Г cd
hudng). Chtfng minh b(p) = r’(y>) va khi r’(p) * 0 thi
a(p) =-7^; ^-7w-
3) Tim ata sd r(p) dd : S (p) khdng ddl ; alp) khdng ddi ;
hip) khdng ddi.
1.6. Tim cung chlnh quy trong E3 xdc dinh bdi tham PS hda
p ; t — pit). Bidt phuong trlnh tidp tuydn tai mdi didm t cua nd
§2. DO DAI CUNG VA THAM SO HOA TV NHltN
CUA MOT cung CHlNH out
JlkwIH*
s-1 giOa hai gi
) trong khoang [0, tj.
khoi <p. Khi </> = 1, goi G lb trong tAm ейа cung doan (d6ng chfit) Г.
S3. TlCH PHAN DQC Мфт CUNG DOAN
Z(p) = Ope E3
2xz = a2 giOra hai mat phAng у
«1
<и _ xd? ,.yfe trong toa dfi Оё cAc vudng gdc Oxy.
Ti'nh Jw ; Г xac dinh bdi t G [0, 2л] •—• P(t),
/>(t) = (cost, Sint) G E2 \ {0}. Тй do suy ra w khdng phAi 14 m6t
§4 CUNG SONG CH1NH QUY THONG E’.
□0 CONG, DO XO&N. MUC TlSU FR£n£
quy n£u	d6c lAp tuygn tlnh X />”(t) * 0).
Phuong trlnh mfit phAng mftt ti6p tai digm song chlnh
(R-A AP”) » 0

2) GiA sfi trudng vecto Z tr6n
If II’
{T, N, B} 14 trudng muc tifiu trite chutfn doe Г gpi 14 tnidng
Cong thde Frene :
duy nhat X doc Г thoa man dteu kifin
Jcoap(t)dt, z(t) = bt.
Chtfng minh rang khi <p (tj
3) TYung phAp tuytfn tao met gdc khOng ddi vfli mOt phuong сб
djnh (0 cAu niy them didu kiAn : d$ xoAn khAc 0 tai moi diO’m).
nhau mOt gdc khOng ddi 6.
5) (T, T”, T’") = 0 (trong dd T 1A trtf&ng vecto tidp xtic don
4.8. Cho hai cung Bong chinh quy ph An bifit Г vi у trong E3
ic djnh theo thd ttf bdi cAc tham s6 hoa p ; J -* E3, t •—♦ /’(t)
ip: I -* E3, v—piv).
Xdt vi phOi Я : J - I, t
= (acht, asht, at) ;
./>(t) - (e>, e"',	;
-./W = «I, tat, t!) ;
= (eos’t, sin’t, eos2t) ;
-/41} = (tcost, tsint, al).
i cSc dtam ейа cung t —pit}
dO’ do cong cd giA tri ctfc tirfu.
Cung cAu hoi dd d6i vdi cung :
t•—»/>(t) = (a(t - Bint)a(l-cost), 4acos-).

A(t) d£ cAc phfip tuydn chinh
§5. DINH Ll CO BAN CUA Ll THUYET OUONQ TRONG E>
Phuong trinh tv hhm Phuong trinh к = k(a) »AT = rls> trong
dd a - k(s) (k(«) > 0) vi a - r(a) li hai him ad khi vl tap 0.

cung song cbtah quy dinh hudng trong B3 (cd hudng) vol do cone
5.1. Kh6ng dung dinh И ca bin cila И thuyet dtfdng, hay chiing
Cdng thvrc Frene . kN, ™
k(t> -
((X’(t))2 + (y’(t)№ .
Phuong trinh tu ham : к = k(s>. NghiSm ейа phmmg trinh
tp him trtn My : x(s) = Jcosp(s) ds ; y(s) = jsitv(s) ds trong dd
n (s) (noi chung s khOng phai la tham
s) . ?(s) <c 14 hing sd).

§6. CUNG PHANG.
16a2 (a la hing s6).

S.S. Til» Me cung that» M thoa m»n pbuong trinh U Rsi&3t = a ;	3) R = nt i « • - at^ (R 14 bin Huh chinh khdc, »14 d« Mi cung). S 3 Xat toa (» CMC (r, f) trong B2 \ <W vi. cung chinh quy dinh hudag xic dint Mi t /40 - (t « Kt),	= t) (trong d6 t _ r(t) Id'mot him s6 cho trudc, r(t) > 0 vdi moi t). 6.4. Tim cung tdc Ь6 cua cAc cung sau day :	6 g Tim cung than khat eta parabol *(t) = t, у W - Л*- 6.9. Cho cung chinh quy Г trong E’. Cung chinh quy у gp, Ц mot cung tue bg Ma r ndu cd tham ad Ma W nhi«n a — ria) Cha Г v» tham aS hda a - P(s) cia у ma tigp tuygn cua у tai a la mat phap tuyfti Ma Г tai s (vdi moi s). Chdng minh ring khi do />w ’rW + k5jS(s) + “TW ds)J trong dd N, В theo thd tu la truOng vecto phap tuygn chinh It triing phap tuydn cua Г ; k, r theo thd tu la dp cong va dh join
1)	x(t)	=	aflntg	g	+ cos t] 2)	x(t)	=	a(t -sint)	;	y(t)		a(l	-cost)	(XyclOit) ; 3)	x(t)	=	t,	y(t)	=	at2	(Parabol) ; 4)	x(t)	=	acht,	y(t)	=	bsht	(Hypebol). bdi tham s6 hoa s •—•/’(s) + (c- s) ^‘(s) (h&ng s6 c lufin khdc s) 2) Chtfng minh ring khi у cho bdi tham s6 hoa t >—►/’I(t) thi cd X(t) th* cho bdi tham s6 hoa t —r^t) =/>, (t) -	 J l|p'1(t)|[ dt. S)	Tim cac cung than khai cua cung tron.	Khi Г lh m6t cung phang m& d& cong к kb6ng dat otic tn (k'(si * 0 vdi moi s). Chdng minh rang moi cung tdc b6 cua nd U cung dinh 6c (t6ng quit) va trong s6 dd cd quy dao сй.с tftm chinh khtic cua § 7C CUNG HlNH HOC VA DA TAP МфТ СН1ЁО Tieu chu£n nh£n biet da tap mpt chifeu trong E2 va E3. Trong E2. тар con у (С E-) = {(x,y)| F(x,y) » 0} 1ft da tap con m6t chifiu khi vh chi khi thoa man diSu ki6n ma tr6n	^7!
6.6. Chiing minh ring cung than khai cua cung xoin 6c Logarit r = ca* (trong tpa dp cue) cung la mdt cung xoin 6c logarit bang 6.7. Tim gid tri cda a d6 cung than khai cua cung xo£n loga :	cd hang mdt tai moi ШЙт p(x,y) e у ((x,y) Id t<?a d$ afin cua dirfm Trong E3 Tap con у (trong E3) = {(x,y,z) | P(x.y.z) » 0, G(x,y,z) = 0} la da tap con m$t chi«u khi vh chi khi thde mt» 3S
trong Б3)-
: u(tap mcr trong E2) -

Tip cic khOng di«m nd» him sO F <Wc Up can difm <x,y) e U
ihda min F(x,y> - 0) gpi I4 (phjng) xac dinh bdi phumg
trinh вп F(xj) = 0 <I>.
quy. Hay phac hoa у (du&ng Viviani). Vjgt phifong trinh mat phAng
7.4. ViSt phtfong trinh in (trong toa dd DAcAc vudng gdc trong
E1 2) xac djnh tip cAc diAm M e E2 mi tich cic khoing cAch dAn
PhAc hoa cAc tap di£m dd, nd thay ddi thA nio khi a biffn thifi
7.5. Cho die’m P thay ddi tr6n mdt dd&ng trdn dudng kinh OA
► /J(c)
1) F (x(c), ylc), c) = 0
F’e (x(c), y(c), c) - 0.
2) F\ (xfc), yfc), c) vi F“y (x(c), y(c), c) khdng dfing th&i trift tiftu.
3) x’(c)» y’(c) khdng ddng thdi trift tiAu.
dd DAcAc vuOng gdc) xAc dinh tip cac di€m M nim trAn dtfdng
thing OP mi OM = PQ. Tim cAc difm kl di cua nd. Hiy phAc
trong E2 vi mfit difm О сб djnh thudc dudng trdn dd. Vift phtfong
= P’1 Mp)) li mdt da tap nrft chiAu trong E2
hop dfic bifit khi i

АР}.
 0 khdng ddi}.
Q(x,y)dy
d(F(x,y))
f'8

F(x,y,A) = X2 - 2лу 2xy =


la phtfPng trinh vi ph&n cua hp du&ng thi P(x,y)dy - Q(x,y)dx
xoSn cua da tap mdt chidu song chtah quy khdng d)nh hddng
(trong E3 da cd hudng).
2) Hoi khi ddi birdng cua mdt cung djnh hudng Г trong E2 3 (E2
d& cd budrig) thi cac trudng vecto T, N dpc cung dd, ham dd cong
(cd dSu)k cua cung dd they ddi thd nao ’ Chiing minh ring n«u
tong mS tham ud hda p : J- E2, t ~ (t) cua cung (khdng dinh
hudngl Г. not dang vi phCn ; ru, - k(t) || /М dt e Q4J) (t~ k(t)
Hai minh tham s6 x -. U- En, r : U - En goi la tuong duong
ndu cd vi phdi Л : U -* U dd r = тоЛ. Dd U q„an hd tuong d„ong.
м л	d say *,(IC B0i 14 m“ ; 2 duoc goi и
1.1. ViSt tham sd ho'a (hay phuong trinh tham sd) cua cac :
CHVONG III
MAT TRONG E3
§1. mAnh tham SO'
1.2. Viet phuong trinh tham s6 cua mat tron xoay true Oz
1) du&ng dfiy xich (x = a ch- , у = 0, z = u) (a > 0) (mat
catdn&it);
2) du&ng truy tich (x = asinu, у = 0, z = a (Intg + cosu))
(a > 0) (mat giA c£u, cung goi la mat loa ken) ;
3) du&ng trim khdng cAt Oz (x
(0 < b < a) (mat xuyen).

аби moi di£m cua nd 1й di6m chinh quy. 2- phang di qua di6m
Ьё toa do Ddcac vudng gdc Ox, y, z ейа E3) :

r/u0- vo‘ vdi khdng gian vecto chi phuong (r’u(u0, v0), r’u(uo, v0))
g9i la mftt phing tifip xiic hay ti6p di6n cua r tai (u0, vo).
3) r(u,v)

: у0 + bcosusinv,
(u.v) * <0, 0) ;
, uv+l
2) Khi у la dudng thAng cat trite giao Д (hai chuyAn dong tren
cd tOc do khOng buOc tl 16 nhau), luc nay ta dapc mat cOnCiit diing.
(u , v0) la mdt di£m khOng ki di cua nd. Ki hifiu л la tiep di«n
cua mAnh r tai di£m (uQ, vQ) (nhu v^y theo dinh nghia thi л
2- phAng di qua did'm r(uo, v0) mA cd phuong la khOng gian vecto
2 chiAu( r’u(uo, vo), r’v(ti0, v0)).
1)	Chiing minh rAng phuong cua л cung dupe xAc dinh bdi
T(uo. v0Y
2)	Chtfng minh rAng я cung dupe tao bdi cac ti£p tuySn tai tQ
t—»v(t) 1A hai ham sO xac dinh tr6n mOt khoAng nao dd chda tp.
u(t0) = u0, v(t0) = v0, (u((t0»2 + (v’(to))2 # 0.
1.7.	Chdng minh rAng mAnh tham s6 r : U —• E3 xAc dinh
trAn tAp md liAn thOng cung U ейа R2 mA mpi phAp tuy&i ейа nd
r(U) nlm trAn mat mat cAu tAm 0.
(u, v) — r(u.v) =
E3 va u ~3(u),
trade trong E3.
tinh ti€n.
3)	Chiing minh rAng parabfilfiit (eliptic hay hypebalic la mat
tinh tKh).
mot mdnhjnat kd trong E3 (p : J —. E3 la eung ehinh quy A
hhm vecto A : J —* B3 thda man dieu kidn A'ui * 3 vdi mpi ue Jl
1) Chiing minh ring life (u,v) Id mdt Mm ki di cila i khi vk
chi khi hai vecto ? (u) + v^’(u) vi A(u) phu thudc tuySn tinh
2) Cid sir r khdng cd did'm ki di. Chiing minh ring cdc titfp
didn tai moi didm cua .during sinh thing u = u„ trilng nhau khi
vi chi khi ha vecto ^(u0), X(u0), A’(u0) phu thudc tuydn tinh. Mu
mAt khA triAn.
3> Chang minh ring ngu (X(u), J(u), h thuS<_ |u^„ dnll
vdi moi u e .T th! u
I 4) СЫНЕ mink ring mat tru, mat ndn. mat tide tuyfn la nhdng
mat kha tridn.
I 1.9. Mat mat M trong E3 cd tham sd hoa r nbu <S bhi 8. Chtlng
minh ring ndu (Alni, A’lull d?c lap toy«h tinh va <A(u),
X'lul.^lu)} phu thuftc tuydn tinh vdi moi u e J thi cd day nhat
cac ham s6 f va g trtn J M cho ?<u) = f(u)3<u) + glu) A'lul vdi
moi a e J Chdng minh ring ndu Г - g’ » 0 thl mat кё da cho la
mat ndn. con nSu f - g- к hong tridt t»u tai u nko thl mat кё da
cho la mfit tiep tuy£n.
1.10. Cho minh trong E3. x&c dinh bdi (u,v)
„ Ne“ (x'........x"> Л c4c tt* da afin trong E" va rln. ,) = („,
x3(u, v),...x»(u, v)) thl nd! r Ik tham »6 hda kidu d6 thj.
Vdi U md c U thl riffle S rung la mat minh hlnh hoc ma rl -
la mdt tham s6 hda cua nd.
014 sit <x’.......x"> 14 toa alia trong E", S la minh hlnh hoc
vdi tham s6 hda r : U — E" vi r(u,v) = (x4u,v)...., >•(„,.)) Khi.
dd d6i vdi moi po e HU> cd minh hlnh hoc r<U) Э p„ (U md c U)
thda Ohan mat tham sd hda kidu dd thi Wong duong vdi tham ad
Tap con khdng rdng S cua E" gpi te mdt da tap 2 chifiu (hay
Gho tpa do afin (x1, .... xn) trong En thi tfipnon khdng rdng Sc En

 кл) ( к nguyen tiiy y) cua R2.
§2. MANH H1NH HQC VA DA TAP HAI СН1Ёи
TRONG ЕП
dim ding phdi ten inh r : U (md trong R2) —♦ En ; r goi te m6t
s6 ч> : W —* R khi vi, cd hang 1, mi W Л S
2.1. MOt 2- phang trong E" cd ph hi ft mdt da tap 2 chiOu trong
En khdng ? Cd phhi la mdt minh hinh hoc trong En khdng ?

1) Didm p thudc mat mdc р'Ча) ft mat didm ki di cua mat
khi va chi khi grad p(p) = 0.
и e Tp E3 thuhc Tp (p- Ча» khi va chi khi «[pl = 0.
2.3.	Tim cac didm ki di ейа mat xac djnh bdi phuong trinh &n
2)	(x + y) ‘ - ar - a v 0 (a la hhng sd) ;
3)	p fx.y.t) « 0. trong dd p la mdt ham da Hide Ь4с hai cua
ha hidn sd x.y.i (lien he vdi difm ki di ейа mdt sign mat Me hai
afin <xem giao trinh ‘Dai sd tuydn tinh vh hinh hoc’., Tim mdi
lien he giua ti«p dien tai didm khdng ki di cua mat xae dinh bdi
phuong trinh in p (x, y, il = 0 dd vdi khai niOm sign tidp dien
eda mdt sign m$t hac hai afin.
2.4.	Chung minh rang Snh xa san day ft tham sd hda cua mdt
nftnh hinh hoc trong E3 (vdi toa dd Ddcdc vudng gdc x, у i) vi
hay шб tA manh hinh hoc dd :
(trong toa dd DecAe vuOng gdc (x, у. a) cua E3).
tigp cua mdt cung song chinh quy trong E3 vdi mat ti<p tuytfn cua
2.7.	Cho da tap hai сЫёи S trong E3 khdng cAt dudng thing
mot chidu hay khong cat no'. Chiing minh ring пёи phSp tuyen cua
§3. РНЁР TfNH VI PHAN ТРЁМ DA TAP HAI CHl£U
TRONG En
2.5. Cung hoi nhtf d bai toan 2.4, nhung vdi anh xa
R2 -* E3
Gid sd S la da tap 2 chieu trong En, j : S -
vi (Idp Ck) пёи jof khi vi (Idp c*).
Anh xa g : S —* W (W md trong Еш) goi la khi vi (Idp 'Ск)(,|
neu vdi moi tham s6 hda dia phuong r : U —• S cua S, Anh x? gor d
kha vi ddp Ck) ; gor ddpc goi la biSu thtic tpa dd cua g trong r.
► s2 goi 14 khi vi ndu h lidn tuc vi vdi

Cung cd thfe' ndi ddh vi phdi vi vi ph6i dia phuong gitfa cic da
tap 2 chiAu trong En.
trfin S. Khi = p thi (t0) dupe goi la vecto tiep хйс vdi
Cd thA ndi vA Anh xa Tpf : TpS, — TpS2 khi cho Anh xa khi
vi f : Sj —»S2. Cung kl hiiu Tpf bdi f,p.
Khi f : Sj —» S2 14 vi phdi v4 X 14 mdt trudng vecto ti£p хйс
trAn Sj thi f.X 14 trudng vecto tiAp хйс tren S2, xAc dinh bdi
f.p(X(p)) = (f.X)(flp)) ddi vdi mpi p 6 Sr
N6u г : U —» S, (u,v)»—♦r(u,v), 14 tham s6 hoa dia phuong ейа
S thi dAt
khi dd (Ru, Rv) 14 mdt trudng muc tiAu tiAp хйс trAn r(U). Th cd
Gii suf X 14 mdt trudng vecto tidp хйс trAn S, p 14 mdt h4m sd
trtn S. Dinh nghia him X[p] trAn S nhu sau : vdi p G S, xet
cung tham sd t~/>(t) trAn S mi pttj = p, A<t0) = X<^t0)) v4
<Ut X[FKp) 14 dao him tai to ейа him sd t (/Xt)).
/>(t) G r(U) vdi mpi t cd thA’ viAt dupe dudi dang /’(t) = r(u(t),
v(t)) (t —»u(t), v(t) 14 hai h4m sd (khi vi)).
2) Hiy khai tridn p theo R„oA I^o p.
3.2.	XAt Anh xa r : R2 — E3, (u,v) —*(x(u,v), y(u,vJ. z(u,v)),
•y(u,v) = (a + b cos 2ли) sin 2nv,	Ф
z(u,v) = b sin 2ли
(a v4 b 14 cie hAng so, a > Ь > 0, (а, У, 1) 1» <*« a« D4c4c vutof
goc ейа E3).
1) Chiing minh ring г la m6t dim. 2) Chtfng minh rtng r(R2) IS met da tap 2 chiSu T2 trong E3, phAng chtfa dudng tr&n nhung khOng cAt no (m$t xiiyAn). Vift phuong trinh Sn cua mat xuy₽n do’.	X« »*t eiu S xdc dinh Ы1 phuong trinh in a2 + y2. ,1 . * p (R > 0) vi. mat tru C xie dinh bdi phuong trinh r t r . ,1 ,, A Vdi mil didm M ейа C. gpi KM) 14 gU° ейа S vdi пйа duong tU v OM (gde 0). Tim 4nl> сйа 4nh « f Ph4i ch4”g ' 14 vi PW 4n dnh ? Ndu dung, hay add djnh f. U2, f.U}. trong dd U,, U} |4 hap Idn С ейа ede trudng veetd U2, U3 trong trudng mVc tidu t,, dd tru {U,, Uj, U3) trtn E3 \ OZ.
3.3. Xet he toa de DScSc vuOng go'c Oxyz trong E3 va mat cAu S thin 0, ban kinh R > 0.	3.6. Cho hd toa dQ Ddcdc vudng gdc Oxyt trong E1. X« nut ciu tarn 0 bin kinh R > Owa tham ad hda dia phuong rta nd r : U = <u,v) e R2 | 0 <u <2x, -ur/2 <» <л/2) - E3,
1/ Chtfng minh ring hai trudng vecto Up U2 cua trudng muc tieu toa dfl cSu (Up U,,.U3} trtn E3 \ OZ khi thu hep trtn S \0Z hai trudng vecto Sy 1Й Uj va U,.	(u,v) —• (Rcosucosv, Rsinucosv, Rsinv). 1) Тгёп E3 cho d?ng vi phftn 0 = yzdx + zxdy + xydz. Tim biSu thiic 6 | r(u) = p|d(uor-1) + pjdCvoT"1), <pl li him s6 trtn r(U). 2) Tinh d0 vS tlm bteu thiic d6 | r(U) = ^>.d (uor”1) A d(vor'1).
2» XSt cic ham.kinh d6 f, vi dp tt trtn S \ {(x,o,z)| * > 0}, coi la e.ic ham trtn tip md do cua da tap S\OZ, hSy tinh UJyi], U,[y>], t’j м u,[ei.	trong d<5 f IS him s6 trtn r(U). 3) Trtn E3 \{0> cho dang vi phAn xdy a dt + ydz A dx + 2dx A dy.
3> Хй ЙС ham toa а» x,y,r, eoi |s C4C him s6 tr6n	hSy dob U,M, U,(y),	Udx], V.ly), tiy.j.	*	(X2 + У2 + a2)»2 Tim bife’u thiic ц |	- y».d(uor’1) л d(vor" ’), trong d<5 №
3.4. xa hi <10 Dtale vuOng gdf 0>:yI trong SJ v4 ma[	, * rtm 0. boo (J„b П , 0 iV,, l=2 l._, ,6 ,r,Idug mM t..u ия  * eSu trtn E-> . OZ. Ш hisu eve Me (0.0,8, ейа S la p, rti vdi .»«. qe S.(pl xel r<q) la giao ci. a„e„g thJ„g „д p	уЯ pMng I + К - о (ti6p d,e„ eM s w	pam (ад	• .h. dune ph ep chifu dia cSu f W Slip) ls„	phing dd Ch minh f IS vi phdi vA hSy tinh fBUp ftU2.	hAtp s6 trtn r(U). 5a. мфт s6 tinh chAt tOpO cua DA TAP HAI CHIEU TRONG En 5 С X C E'. T goi la rod trong X nfti nd U giao ct*X «Л
3.S. Cho h$ tpa d<) D&ac vudng gdc Oxyz ейа E3. 64	mdt tap md trong Ea у И| ,4 d(Sllg tn)ngx ngu x j lM0| X 7 ddng khi v4 ehi khi mol day d.«m p,, p,, e J md P, - P e X thi p 6 y. x goi la ii4B .hong cung ndu vol 


§5. РНЁР TfNH TfCH PHAN TR^N MAT	К 14 m«t miAn	•=<> hue”S v6i bd trt” da 2 s trong E" ho4c К 14 da tap 2 chiAu compAc cd huong vdi W	I E", e 14 mdt dang vi phAn bac 2 (khA vi trtn mdt lAn c4a mi-	x
S la da tap 2 chiAu trong En. Mien compac vdi bo trAn S 1A tap con К C S thoa mftn cac diAu kiAn : К compAc ; biAn <?K cua К la da tap 1 chiAu khA vi ting khuc ; Ж chia lAn cAn m6i diAm chinh quy cua minh thAnh hai phia. Ndi К cd hudng nAu dA cho mdt hudng trAn mdt lAn cAn md cua К trong S. NAu К la mdt mien compAc (hay miAn compac co hudng) vdi bd trAn S thi co' mdt ho httu han miAn conipAc vdi bd C; trong R2 va mdt ho anh	•, xs kha vi rs : Cj — S sao cho Ur, (С,) «К, r, thu hep lAn Cj = Cj \ dGf la vi phdi (theo thii tu bao tdn hudng) lAn mdt tAp	| mo cua S va r; (С;) Л г- (СЛ = 0 neu i * j (luc nAy ndi К dupe lat bdi hp r- :C, - S). NAu trong dinh nghia da tap 2 chieu S trong En. thay cho ddi hoi mdi tham s6 hda di’a phuong r : U -» S la Anh xa tft tAp’ md U trong R2, ta ddi hoi U la tap md trong nufa phAng {(u,v)|v > 0}. thi dupe khdi mem da tap 2 chieu S vdi bd,. bd aS la tAp cAc didm p 6 S ma cd tham s6 hda dia phUdng r vda ndi dA’ r(u.O) = p. Cho К 1A rniAn compAc vdi bd trAn da tap 2 chiAu S trong En hoac К la da tap 2 chiAu compAc vdi bef trong En, <p : К —» R 1A ham s6 liAn tuc. Lat К bdi hp г : C —* K, (u,v) —. r( (u.v). Ta dinh nghia tich phAn ciia y> trAn К la bie’u thiic : Jy» (cung viet Jy> dS) = 2 jy> (г-(и,, v)) ^Сг(Г|)’и. (г})\>) dudv. BiAu thiic J1 (tiic JdS ) dupe goi la diAn tich cua miAn K.	К trong S). LAt К bdi ho r, : С, - К <r, thu hep l«n C, 14 и phK ’ bAo tAn hudng lAn Anh), (u V) -> <(u, v). Dinh nghia tich phtu cua n trtn К 14 biAu thiic Jp = Л Jy>, iu.v) dudv, trong dd , 14 hAm аб sao cho f t du Adv = r,>. Ro rhng khi ddi hudng cua К ;	thl Jp ddi dSu. DSi vdi dang di«n tich chinh tAc trAn К thl jpo la diAn tich cua K. Cho К 1A miAn compac co' hvtdng vdi bd trAn da tap 2 chieu S En, в la dang vi phan bac inpt khA vi trAn K. The thi xa’y ra hd thiic (gpi la edng thiic xtoc): Cho V la miAn compac vdi bd trong E3, и la dang vi phan bac hai knA vi trtn V. Khi dd cd hd thde (gpi 14 ebng thde Gaoxo - OxtrAgraxki): JcLk = J . Cho К la mien compae cd hixdng vdi bd trAn da t?p 2 chiAu S trong E-’, hoac К la da tap 2 chiAu compdc cd hudng vdi bd trong
Ndi riAng, khi C la mdt miAn compac vdi bd trong R2, f : C-* R la mot ham sd khA vi vA К la do thi cua f thi diAn tich cua К la	 j * rj + f* dxdy.	EJ, ho4c К la bo AV cua miAn compAc vdi bd V trong E! Goi n 1» trudng vecto phap tuydn dun vj trtn K. Cho X la mpt trudng vecW hta tuc trtn K. Khi dd ham X.n la hAm sd HSn tpc trtn K. Tidt Ph4n J(X.n) dS dupe gpi 14 thbng lupng cua trudng vecto X qua К
U thi dido tich cua r(C) Id
2) Vift rt tich phftn dd khi bifft:
cua E3\ nita ph4ng Ox+z.
Khi p thay dSi thi H
§6. ANH XA VAIGACTEN (WEINGARTEN)
6.4.	Xet mat S xac dinh bdi tham sd hda ki&'u dd thi
Chiing minh ring vecto <x e TpS (p = r(u, v), pe = aRu(p) +bRv(p))
xac dinh m$t phuong chinh cua S tai p khi va chi khi
E(u,v) F(u,v) G(u,v)
J L(u,v) M(u,v) N(u,v)|
S6 k(oc) _	(«X G TpS, <x * 0) goi Ik de cong phap dang cua
S theo phuong x4c dinh bdi «.
6.1.	Tinh da cong Gaoxo cua mat parabfilftit
6.2.	Tinh cong Gaoxo vi d<J cong trung blnh cua:
6.6.	Xac dinh phuong chinh tai mdi difem cua:
b) Mat tron xoay trong E3.

6-8. Chiing minh rAng di£m p thufic mat S trong E3 la mdt
vtii cAc h§ аб ейа dang co bAn I, tiic 1A
L(u,v)
ЕЙ = К? 'GM •
2) H(p)2 - K(p), trong do' H va К theo thii tu la dd cong trung
6.9. Tim cac diAm гбп ейа : 1) elipxdit ^ron xoay ; 2) parabdldit
•ron xoay ; 3) parabfilfiit eliptic ; 4) elipxdit (khdng tron xoay) ;
1) Chdng minh rAng ndu X, Y IA hai trudng vecto tifip хйс v* 8
vA h(X) la trudng vecto p e S —*hp (X(p)) thi h(X). Y = DXY. 0.
2) Chiing minh ring nAu X, Y 1A hai trudng vecto tifp хйс vAi
S thi [X, Y] cung 1A trudng vecto ti£p хйс vdi S.
6.12. 1) Z 1A mdt trudng vecto phAp tuyAn (khac 0) хАс ОД
hudng cua da tap hai сЫёи S trong E3, {X, Y} IA m$t trudng mue
К vA dd cong trung blnh H cua
------—7 ' тДй x D Z) . (X xY),
(X x Y)2 ||z|p X »
H = --------------  JL (DXZ x Y + X x DyZ). (XxY).
2(X x Y)2 H x	Y
°g(P) 1 - n(p). (Goi nd la Anh xa Gaoxo. Nd khA vi).
tpa dd DAcac vudng gdc, lAy Z = (gradF) |s
hay tinh К va H, ling dung : tinh сц thA' khi F(x, y, z) = ax2
d) Parabdlfiit eliptic,
e) Calfinfiit (хеш bai 1.2),
Ы Mil dinh de diing (xern bai 1.4).
6.13.	S la da tap hai chiAu cd hudng trong E3. Vdi p 6 S и
dat IIIp (oc, fi) = hp (oc). hpQS) (thA thi П1р 1A dang song tuyAn tint
ddi xiing trtn TpS, goi nd 1A dang cd bAn thd ba cua S t»i pt
Chiing minh rAng IIIp - 2H(p) IIp + K(p)Ip = 0.
6.14.	Cho mat tron xoay S xdc dinh bdi tham зб hda sau diy
trong h$ toa dd DAcAc vudng gdc : (u, v) —*r(u, v) = (y>(u > cosv,
^(u)sinv. v-(u)) vdi + v.< = 1 Chung minn
ddi vdi dd cong Gaoxo ейа S:
6.15.	Mat trong E3 mA d$ cong trung binh triAt
diAm g9i la mat tdi tiAu.
I 3) N£u {ос, fl} 14 co sd true chuSn cua TpS thl
2лгН(д) = = J k(cos0.oc + sirf.fl) d0.
6.17. Gii svt К U mfit mten compic vdi bd trdn da tap hai chi£u
S trong EI * 3 dinh hudng bdi trudng vecto phap tuydh don vi n, gid
nd V In mdt ham sd Ш vi trtn S va gid si vdi g > 0 dd bd, tap
ode didin {p + »>(p)n(p) | p e S) ciillg Um tMnll m?t da „р hai
Si bd trtn da tap St (coi S„ - S, Ko = K).
Ki Men SiKj) la difin tich eda Kj. Cbdng minb
S(Kp = S<K) - 2tJyH + 0(E), trong dd po la
§7, NHUNG DUONG DANG CHU V
TR6N MAT S TRONG Ё»
D(nqd) .
n./> song song doc P. Rd rang nfeu p trfec dia thi || f’|| la hfem hfeng.
Mdt dudng trfen S goi Id mdt duong tifen trac dia cua S nfeu nd
cd tham sd hda dd trd thfenh cung trfec dja cua S. Vfey, cung chinh
7.1. Xet mfenh hinh.hpc S trong E3 xac dinh bdi tham s6 hda
la hp dudng trfen S хйс dinh bdi phuong trinh vi phfen dd.
hypebdlic.
3) Tim диу dao trUc giao cua ho dudng toa do v bing hing <
(u> v) ^_r(u, v) = (ucosv, usinv, и
trong toa as Dtaic vnOng gdc (x, У, 2) ейа E3.
mot dudng у dudi mdt gdc khOng ddi. Chiing minh ring nfeu ?
dudng chinh khiic cua S thi nd cung la dudng chinh кЬйс ейа S
2)	Ling dung cdu 1, dd tim dudng chinh khiic cua mat trim xoay.
7.	4. 1) Cho ham vecto r*: (u, v, w) e W (tap md trong R3) -♦
•—»rTu, v, w) £ S3. Chiing minh ring nfeu "Fu’ . ?v’ =7V’ . 7W' =
-x.’ •	= ° ^’uv •	=^’vu-  Лг Л,И л = °-
2) Cho ba hp mat trong E3 хйс dinh theo thii tu bdi cac phuong
trinh £n ^](x, y, z) - Cj = 0, y>2(x, y, z) - % =®» у, z) - Cj = G
ma PjQ, - PjQj, * 0. Chiing minh rang tai Ifen can moi dife’m
P = v) cd minh hlnh hoc S C S (S Э p) va mdt tham s6 hda
7.2. Xet mfenh hlnh hoc S trong E3 хйс dinh bdi tham sd hda
b) Goi E, F, G la cac hfe s6 cua dang I trong tham sd hda dd.
Chiing minh rang trfen S hp dudng хйс djnh bdi Pdu + Qdv = о
true giao vdi ho dudng хйс dinh bdi (QE - PF)du + (QF - PG)dv = 0.
hang sd хйс dinh mat cua mdi ho). Gid s£t y~ 2j~ 1
dinh bdi ax2 + by2 + cz2 - 1 = 0 (abc # 0) trong tpa do Dfecic
vudng gdc (x, y, z) bang each ghep chiing vio mdt hp^cia ba hP
mat thoa man eftu 2).
7.5. Chiing minh r£ng mat кё S trong E3 cd dd cong Geoxd К
tifeu Frtnfi {T, N, B} doc y. Gi4 sii у 14 mdt dudng ti$m can сйа S. 1) h(T) = ± TN (T 14 dd xodn cua y); 2) Thi moi p S y, d$ cong Gaoxo сйа S tai p 14 K(p) = -T(p)2.	Ы v x S£).(a4W = 0 ,4 tn,dBE теяо phsp tuyfn Лт „ cua S). 7.12. f> J -. S, s —14 tham rt hda tu uhlta cua met cans dinh hudng у trtn da tap hai chiSa S tang B3 cd hudng xac
7.7. у la mdt cung hlnh hoc trtn mat S dinh hudng bdi trudng	.	dinh bdi trudng vecto phap tuyft. don vj n. Xdt trudng muc tie» true chain (T, Y, Z) doc у nhu saa ; T 14 trudng vecto tigp айс don vi doc y, Z = noft Y = Z X T. (Truimg mac ti4u nay duoc goi
1) у la dudng chinh khuc va la dudng tiOm c4n cua S khi va chi khi у nkm trong mat phang tiOp xuc vdi S doc y.	‘	la trudng muc triu Dacbu).
2) у 14 dudng chinh khiic va la dudng tien trie dia cua S khi vk chi khi у nam trong mat plring true gia^vdi S doc y.	ЧГ =5Y +?"z’ £ = -V	.1 = -v - T& trong
3) у la dudng trim can va 14 dudng tien trie dia cua S khi va 7.8. Hai mat S vajS trong E3 tiep xuc doc mdt dudng y, ChUng minh rang neu у la mdt dudng tifin trie dia cua S thi no cung la mdt dudng tiOn trie dia сйа S. 7.9. Hai mat S va S trong E3 cat true giao doc dudng y. ChUng minh rang пёй у 14 mdt dudng trim rin cua S thi no' cung 14 mOt dudng tien trie dia cua S.	dd к,, = k(T) (k 14 dd cong phap dang сйа S) cung goi 14 dd cong phap dang cua cung p, Tg = h(T). Y goi 14 dd xo4n trie dia сйа cung p, kg goi 14 dd cong trie dia cua cung p. 2) Тй dd suy ra : p 14 cung trie dja сйа S khi v4 chi khi dd cong trie dia сйа nd b4ng khdng ; P la dudng trim c4n сйа S khi v4 chi khi dd cong phap dang сйа nd b4ng khdng ; p la mdt dudng chinh khUc сйа S khi v4 chi khi dd xoin trac dia cua nd 3) ChUng minh rang vd gi4 tri tuydbddi, dd cong trac dia сйа P tai p = p(s) b4ng dd cong tai p cua hlnh chteu thing gdc cua
7.10. у 14 mdt dudng song chinh quy trong E3. ChUng minh:	P(J) Idn tidp difin сйа S tai p.
1) у la mdt dudng trim c4n сйа mat кё tao bdi cac phdp tuyOn	4) Tim edng thUc ti'nh dd cong trie dia сйа cung s — r(u(s), 1	v(s) trong tham зб hda dia phuong (u, v) •—• r(u, v) cua S.
2) у 14 mdt dudng tiSn trie dia сйа mat кё tao bdi c4c trimg phfip tuy£n сйа у. 7.11. ChUng minh ring cung tham s6 chinh quy t •—»/>(t) trtn mat S trong E3 хйс djnh mdt dudng trin trie dia сйа S khi v4 chi	5)	Tinh dd cong trie dja сйа: a)	Kinh tuydn сйа mat trtn xoay, b)	Dinh 6c trtn и •—»(acosu, xsinu, bu) (a > 0, b * 0) trtn mfit try (u, v) —Часот!» asinu, v);

6) H6i khi S la m6t mat phang E2 С E3 thi cdc khdi niSm de>
trfin S tro thftnh cac khii ntem gi ciia du&ng phing?
$8. CAC PHUONG TRiNH CO BAN CUA
Lf THUYET MAT TRONG E’ VA UNG DUNG
S la mdt mat trong E3 dinh hudng bdi trudng vecto phap tuyfti
da»2 = - Ц Л a>l3.
Th cung cd : hp(<x) = w|(oc).U,(p> + <u|(«).U2(p),
dw| = K.e’ Л02 (K la dd cong Gaoxo).
muc tidu chinh). Goi k, la dd cong chinh cua S Ung vdi phuong
chinh xic djnh bdi U, (I = 1, 2). Khi dd a,3 - к.в1, ш3 - к^,
Goi {U-p. U2, U3) la trudng muc ti6u trite chu£n doc V tuong thich
vdi S nSu U3 = n]v; {d1, 92, 03} la trudng ddi muc tifeu cua {Up
(kj - Jij)®|(Up.

Mat com рас thi cd chtfa diS’m p0 ma tai dd K(pQ)
ай, = <»? Ы U2(p) + <»3 Ы.п(р),
D«U, » WU/p) + <о3 (»).п(р),	-
D«U3 = WU;(p) + cvj Ыи2(р),
vdi moi oc G TpS, p 6 V; Tit dd ta edcu}. = —ct»^.
Cac phuong trinh co b4n cua li thuyft mat (Ung vdi {Up U,, U,}
tr6n V c Sh
kj(po) > МРсЛ kj CVC ($a phuong), kj dat eye tfe’u (dia
phuong), thi khi dd K(p0) « 0.
S la da tap hai chidu comp 4c, liSn thdng, cd hudng E3, vdi dd
cong Gaoxo К hang, th£ thi S la mat eSu ban kinh -^= (Dinh 11
8.1. p : J . s, s	la tham s6 hda ttf nhifin cua m«t
duhag trtn mat S trong E3 ; S diroc djnh hudng bdi trudng »«U0
ы dd ™dis' de “‘° ‘rtC "
phSp dang ete /> (xew bii to«n 7.12).
ш tb\ zч ddi “u сйа
rwi<t-’>'2'3’-4'^lieokaciaStrong
va dang II cua S trong tham s6 hda r ; w3 a>3 14 nhfing dang lien
ket trfin S ling vdi trudng muc tiOu tuong thich vdi S 14
thich vdi da tap hai chieu cd hudng S trong E3. Chiing minh rang
си^Лв1 - Цлб2 = ЗНб’лв2, trong dd 01, 02, w^, ш2 la cac dang ddi
и Tinh de>. as2, a-4 <4 Mth“ ИвЬ d’vi * d- c“
vdi da tap hai chiiu cd hudng S trong E3 ; 0 la mOt diem thuOc E3.
1) Xdt cac him sd y>‘ : S -* R, V’Kp) = (5P . U/p) (i » 1, 2, 3).
1) Chiing minh ring {Up U2, U3} 14 mOt trudng muc tifeu chinh
dp1 = 91 + w2y>2 +
dp2 = в2 +	+ cu|jo3
trong do 61, 02, tu2, la cac dang ddi ngiu va cac dang lien kdt
trSn S trong trudng muc tifiu da cho:
8.4. (u, v) •—»r(u, v) 14 tham s6 hda ciia m^t S trong
c4c dudng tpa dO 14 dudng chinh khiic. Chiing minh ring
sau : vdi её G TpS, 0(tx) = бр.(« X T?3(p)), fiM = e(hp(«)).
dfl = 2(1+р3Н)6,лв2,
ф/ = 2(H + у>3Ю0’ л в2.
hudng trong E3 тй do cong Gaoxo К duong v4 do cong trung binh
9.9. Chiing minh гЫ cic « eong chinh ей. da ЭД. ha, chMu lihn thdng. cd hudng S trong E3 14 nhhng ham King th S 14 m . Ы phhn № thong ейа m»t phang. mat cdu hay m»t tru tron xoay.	Anh xa Vaigdcten h, h ciia S v4 S, tUC 14 f. (h(«>) . h vdi mol vecto tWp хйс « ейа S. 9.1. Hoi anh xa dang cu ейа hai mat trong E3 сб bdo и. л.ж cong chinh, do cong trung hinh. dudng chinh khuc, dudng ti»„ c4n, didm rSn khdng ?	£
59. ANH XA DANG CU VA TUONG DUONG DOI HlNH	9.2. Chiing minh ring dd cong trac die cua dudng trtn I trong В3 (xem hai 7.12 va hai S.li bat bidn qua anh-ха dingc.®
Sj. Sa 14 cac da tap hai chiSu trong EJ ; anh xa (khi vi) f . S, -» Sa	1 ml 1'a mot anh xa ding cu ndu vdi moi diem p G Sp ТГ-TS,-» T„P,S; bio tdn tlch vd hudng vk goi la mOt vi phSt ding e J nfu nd vita 14 mOt vi phOi via 14 mdt anh xa d4ng cu. Anh xa ding cu nd cac tinh chSt : la trti, 14 vi phdi ding cu dia phuong, bSo tdn gdc giOa c4c phuong tidp хйс, Ъ4о tdn do dai cua cung trfen da tap. N6u f la vi ph6i dang cu thi : Г1 la vi phfii dang cu, f bio t6n Tlch c&c anh xa ding cu la anh xa ding cU. Tlch cac vi phfti ding cu li vi ph6i ding cU. Anh xa (khi vi) f : Sj - S2 14 ding cu khi vi chi khi vdi moi	' p e S, cd tham sd hda dia phuong r : U -• Sp r(U) Э p mi for :	. U -* S, la mdt tham sd hda dia phuong cua S2 va cic bi£u thtfc S, trong tham sd hda for trung nhau. f : S -» S la mdt inh xa ding cu ейа cic da tap hai chiiu trong E3 thi f bio tdn dd cong Gaoxo, tdc li K(p) = K(f(p)). Vdi moi p e S (Djnh 11 (egregium) cua Gaoxd). f : S -* S li mdt vi phfii dang cU bio tdn hudng. Khi dd cd ddi hinh F cua E3 dd F|s = f khi vi chi khi f giao hoan duoc vdi cac	cua cdc mat trong E3. TU dd suy ra ring khai nidm dukng trie e dia trtn mat trong E3 cung bat bi On qua dnh X. dd. 9.3. Chdng minh rAng khdng cd hai mat nao trong cic mat sat t trong E3 thia nhan mdt vi phdi dang cu Hr mdt lan c4n md cj, m?t dia'm ейа mdt mat lan mdt tap mcl ейа mat kia : mat c4u mat tru trtn xoay, mat dinh 6c ddng, parabdloit eliptic.	' c 9.4. Anh xa f ; S, — Sj giOa cdc mat trong E3 goi 14 mdt Mob I xa bio giac nSu cd ham sd duong p : S, - R sao cho vdi mpl a /3 e TpSj dSu cd Tpf(«).	(vS| m?j p 6 s, Chiing minh : 1) Moi vi phdi bdo gidc Vi ding di5n (xem bdi S.7) phii li vi phdi ding cu (ta dd suy ra khdng cd bln d(S (phlng) nlo cue m« , m.& trdn qua dat vita bio tdn gdc vua bio tdn diln tick). 2) Ndu f 14 anh xa Gaoxd ttt mat lien thong cung S. vie tell Ш s	2 trons E3 <xem bii 610) «М r bio giac khi va Ы klu S, nlm trtn mat mat clu hole mat tdi tidu. S.5. Vdi mdi t, 0 к t < |, xet anh X, r, : R2 — E3 , «mne’r “St<ShuC0SV' shu ’in’. v) -kaintl-chuain v. chu ««.»> <'«ngtpad0 D4c4c vudng gdc ейа E3>. 1) Hdi ro Cd U4n quan gi dgn mat dinh de ddng vi r, cd W* <1 4“an gi dg„ catendit?	SO
‘ 0 thl гг1(У) md trong Up rpi(V) md trong
2) Но cdc Г{ (Uj) phu todn bd M.
3) N6u goi m6i V С M mi r“'(V) md trong Uj vdi moi i e I Id
cac t&p md (trong M) V, W sao cho p G V, q e W. V П W
(Tinh chfit Haoxodorf).
Anh xa f : M - M gida cdc da tap goi Id lidn tuc ndu nghjch inh
cua moi tap md trong M Id tap md trong M. Cac khai nidm vd
hai chi du trong En.
Anh xa f : M — M goi Id khd vi (Idp Ck, к phdi khdng Idn han
Idp kha vi сйа M va M) пёипб lifen tuc va vdi moi tham sd hda
dia phuong rj : Uj -* M, - M md f(r. (Uj)) C r/Uj) til
rj^ofori Id kha vi Idp Ck
CHUONG IV
DA TAP RIMAN HAI CHIEU
II. DA TAP m CHIEU VA РНЁР TlNH GlAl TlCH ТРЁМ NO
0) la tap hpp M (phfin tut
p thi dfit cx[y>] la sd хйс dinh nhu sau (goi nd Id dao ham cua >p
theo oc) : 15y « = fl, />, t0J ma /»(!) С V rfii dat <x(p] la dao ham
ki hiAu X[y?], Anh xa tifp xuc Tpf, dang vi phAn trAn da tap... gidng
nhu d chuong HI.
> M la vi ph6i thi v6i mpi trudng vecto (tiSp xuc) nhan
(f.XXftp)) = (Tpf)(X(p)) vdi moi p 6 M.
Neu r : U( C Rra) —* M la mot tham
г. <E,1 con dupe ki hiAu bdi RJ va (RJ) (j = j, 2, ... , m) mpt
trudng muc tieu nhan trAn r(U).
Vide dinh hudng M la vide dinh hudng R - khdng gian vecto
T M vdi mpi рем, sao cho vdi a,Si p„ e M cd tham sd hda dja
phuong r : U - M, r<UI Э p0, ma vdi moi p e r(U), hudng da
chpn cua T_M xac rfinh hAi p
4) M'gpi 14 H«” thdng "eu t,p co" khOng rfngt v4a ”>« vj.
ddng trong M phAi 14 Man bd M : M goi 14 lien thdng cung nJu <1
vdi moi p в M vi mpi q e M cd anh xa lidn tuc p : I = (0.1) _ g.
sao cho p (0) = p, P m = 4 Chiing minh ring ddi vdi da tap u
cbSt lion thane tuong duong vdi tinh chSt lidn thdnv <<
trAn da tap nhin M thi cd mdt va chi mdt trudng vecto nhin trtt
sd nhan p trAn M.
DC.YJW = X [Y [ to]] - Y [XMJ.
2) r : U —» M 14 mdt tham sd ho'a dia phuong cua da tap hai n
chidu M, (u, v)—*r (u, v). Chiing minh rAng cAc trudng vecto nu.
R,, trAn r(U) thda mhn [Ru. Rv] = 0.
thich vdi trudng vecto_X_e Vec(M) ndu (f.XKftpl! = Xtf.pll.
Chiing minh rang nSu X, Y e Vec (M) f - tuong thich theo the
tu vdi X, Y e Vec (Ml, thi PC, Y] (хеш bAl toon 1.21 f-tuong thich c
vdi [X, YJ.
minh rAng vdi mpi trudng vecto (nhAn) X. Y trAn M. ta cd
d d<X, Y) = XW(Y)J - YtetXlJ - eUX, Y]>.
1.5. XAt cung tham sd trtn da tap m chieu M. tdc 14 Anh »
(khA vi) /> : I M, t (t)> j |4 khoang mi. trong R. Trdtoj
vecto X doc p la vide dat tuong ting mdi t e 1 vecto UAp x«c tt
Г т’ЛУ'”11' Ndi X kM Vi E1 nSu C“ kho4ng mS J 3
J c L dA vo, mdt ham sd .e khi vi trAn mpt tAp md chda P Mm
—X(t) И kha vi tai to. Ndi X khA vi ndu nd khA vi t»i mpi t„ 61 «
W°cVm v^vwt1™4"8 m"c ti4u ”h4n tren t,p m“ cki"R
X(t) = 2 pi(t) ц W1))
Ъа s6 thtfc khdng ddng thdi trifit tifiu x&c dinh sai khAc thtia s6 chung
( T f(<x), T f<jS)> =	P>-
1) f bio tin hudng Me 1* vOi moi p в M, T„f bio tin hoong,
„4 di«u nay thi ini c6 ugh,a la T f biSn muc ttitt thui.> trong T M
think muc tiio thuin trong TI(r)M. vi bio gme («=n> hit tip 2.2).
21 T.IW. W) = Jgp) < V
...	.	- m I. . —„„ ЛХ j va .1-IS cac anh xa
ndi trong bai t$p 2.1 d6i v6i M va M-
2 4 Cho hai da tap Riman (M, < , > ) (M, < , >)• Chtfng
minh rang trtn M x M cd efiu true Riman duy nhatJS' vd. moi
tp. p j e M x M, khdng gian vecto Oclit T{p -j (M x M) ddng eSu
ImTpjp e ImT^ip
du,' = ке'ле2 goi li do cong Gaoxo cua tm,	anh x.
, MO tin di cong Gaoxo. S li da tap hai chiiu trong E> vg
trie Riman cim sinh til tich vi hudng thl docong Gaoxo,
i day vi i chuong III la trung nhau. Cho M la Up mi trot
vii toa di D&ic vuing gdc (x, у) ; (p : M - R+ la hint И
duong ; (M,	14 da tap Riman cd can 14 c«u trtc J
chinh tic trtn M; thi thl:
2 chiiu (k li hing si duong). Hiy so sanh cac di dhi cua ctag
mdt cung, cac dien tich cua citng mOt miin Icompic vdi bd), iff
tich vd hudng ciia hai vecto ti£p xiic vdi S2 tai mdt did'm la tich
vd hudng tu nhien trong thl S2 trd thanh mot da tap Riman
hai chieu, ki hi6u (S2, can). Khi dd trtn RP2 cd cau true Riman
я : (S2, can) -* (RP2, can)
§3.	DANG Ll£N КЁТ VA Оф CONG GAOXO CUA DA TAP
RIMAN HAI CHI&U
Cho da lap Riman hai chiiu (M, < , > ) ; (U,, U.) li trudng
muc tiiu true chain trtn tip mS V cua M ; e‘(U ) = "(J, j = 1,21 ,
dang vi phan bit mot trio V thoa min de' = -„1Л9:, dd- =
= n«jAd' (quy Ute= -.1) go, d„g Ип ka сйа	}
(Up U,1 Ham si chin К duy nhit trtn M sao cho vdi moi tntbng
muc Mu true chain iU„ U2) trtn tip md V tuy у d, M d«„ J
3.2.	M = l(x, y) e R2 | x2 + y2 < 4a2 ) (a li king si duongij
Tinh do cong Gaoxo cua da tap Riman hai chiiu , —у, vdi
3.3.	Trong E3 vdi toa di Dicic vuing go'c X, y, z xdt mil
S tint tai diSm С = (0, 0, II, hinh kinh bang 1 Goi N » (0.
la cue hie cua S. Xdt phep chidu dia ciu f : S\(N> - Oxy
chiSu N). Ching minh ring f la vi phii din- eV ti, <S)(N>,
lin (Oxy, trong dd y>(x, y) = 1 + (x2 + y2) ; 4. J
3.4.	Cho ba him si nhfa E, F, G trtn mit tip md U С Й?fl
E > 0, G > 0, EG-F2 > О (tai moi diSm cua U). Chtlng minh <M
cd thi trang hi cho U ciu trtc Riman <, >di' E F C 14 ode h*'j
-a dang co bin 1 cua (U. <,» trong tham ad hda chinh <й ‘
r = Id,, : U - U.
, to3,;8’ U dl,™E trdn d°n ’i Vong E2. Vdi mdi didm 1
, е-f') e S' x S', xdt tham ad hda din phuong j
4. DAO HAM THUAN Bl£N VA CHUY^N DOI SONG SONG
TR&N DA TAP RIMAN HAI CHitU
«п M, via X|v = y-’Uj + y>2u2 (y»‘ la ham s6 trfen V) va dat
2XKt' = <Z(»l)(p) +	(Z)(p)j U,(p) + Щ*2! (P) +
>‘<p) wf (Z)(p))U2(p).
-*/’(t) trong H, /’(t)
»bpg veew doc p, kl hiSu zdc dinh nhu san : vdi t„ e I, l«y
«hng muc tidu true ehudn (Up U2} trong ISn can ейа vi£t
It) .	(p(t)) + pi (t)U2(p<t)) vh dat
"S4'”’ = (л" <tv)+«’!<‘o4V,,<‘o»V1Wto)) +
(r<V))Uj(rty).
Riman chinh t£c).
ss. CUNG trAc dia tr£n da tap riman hai chiEu
song song doc p. Khi dd t •—•||/’’(t)|| 1A him hAng.
hda tu rhien p : J - M,_s —*p (s) (trie = 1) dat T =p 1 va
vdi m6i s 15y N(s) 6 Tf (s)M sao cho {T(s), N(s)} la co sd true
chuan thuin cua T^s’)M. Khi dd'^ = kgN. Ham s8 kg (doc
Nfu via f <»> = T(s) = cos pls) U,(/>(«i) + sin As) U2(A»)).
As)) ad кг -	(?•).
Cung srtn (M, (,)) S°i Ik mot ucu m. n.,_
than, so boa ® <rd -Ь“Ь CU"S ,raC d,a-
Cung chinh quy xac dinh bdi t (t> trtn (M, (,» li 4ui,
НАЛ trie din khi vi chi khi do cong trie d.a k, cua nd tti« (j
hay khi va chi khi (/, PH »bu«c *4* tinh (vdi moi Л
Anh xa ding co bio tdn hudng giUa cic da tap Riman hai ch
co' hudng cd cac tinh chAt sau : bAo tdn d<5 cong trac dia cua a
dinh hudng, bien cung trac dia thAnh cung trie dia, du&ng j
tiAu true chuin {Up U,}. ViAt P' = V1 (UjO/») +	. (U2op) ‘
dy1 ,	®
do’ la cung trac dia trAn M khi vA chi khi + V2.cu| (/’’? = 0,tap
+ y1 tuf (?’) = 0. (Nhu vfiy day la cAc phuong trinh ейа cun|^un
trAc dja p).	«
Cho a e TpM thi co' cung trAc dia p : J -* M vdi 0 6 J. p’tQ' = «
J IA t£p con thuc su ейа J?. Cung p do' goi la niSt cung trie dj>
no d6u xac dinh trAn toan Ьб R.
Than, sd hda r : U M, (и, V) (u, v> trtn (M, (, » g»i *
than: so hda ClerO ndu F = 0. chn E va G chi phu IhuOc и 1
5.1. Cho cung trac dja i —fUl) trtn da tap Riman hai
(M, (,)> (voi A * 0) va trudng vecto X doc cung dd.
iixi? cthdng 17ii^iizrAngiin®u ?song song d9C p
efiJL hi	~HX(td bam hAng vA gdc giUe f
cung hing doc A
5 2 Tinh g* hslenhmi ейа chu vi Um giac trie dia (hop bdi
L cung tide trie dia) trSn mat ейи bin kinh R trong E3 (vdi eSu
nle Riman chinh the tren mat cSu dd).
' 5.3. X<5t da tap Riman (R2, vdi f»(x, y) = 1 +	4
Phan hai
GIAI HOAC TRA LOT
Chiing minh :
CHUONG I
РНЁР TINH GIAI TfCH TRONG KHONG GIAN
OCLIT En VA H1NH HQC VI PHAN CUA En (n = 2,3)
St. DAO HAM CUA HAM VECTO
pt ft), pfm
1.1. 2X3 ; 2j . A” ; (A. A') :
»-|»'жи ’м|'вд
Ipc^lP -	+ <A(t)2  (A°<4) ’
... tt
||T(t)|| = |К(Ч|! vdi moi t e J U A0(t) - о vm n* .
cd phuong khdng ddi.
Ctek / CIA rt X A', X" P<«, dgt Sit) =	
j5 : J - t5 U ham vecto ЙА vi va thoa man:
S(t> . Ait) = N’ <t) • X(t) = X W  A”it) (li-
ner tun Uy dao bam hai »t cua cAc ding tWc ill ta <
R’iti. Ait) + Xit).X(t) =0 suy ra X.A =0 (21
Vi II = 1 пёп N’.fi =0 (4) vacac vecto A, 1,_N ddc lap
tuvfn tinh пёп til (2), (3), (4) suy ra №(t) = 0 hoic N(t) khdng
dii vdi moi t. Vfty A (t) ludn thudc mot khGng gian vecto con hai
chi4u cua "23 (bii vudng gdc vdi <$>)•
Nguoc lai n£u H (t) luOn thudc mdt khdng gian vecto con hai
chidu cua "23 bii vudng gdc vdi khdng gian con sinh bdi vecto N
khdng ddi cua E3 (trong dd ||Я|| = 1), ta cd :
"*r*	0 (5). Ti£p tuc Ifiyjao ham ci hai v£ cua (5) ta cd:
XX’ = 0 nhung N' = 0 пёп
= 0 (6). Tuong tu ta cd:

г,Л Л.А-:А-А. A-A.A—.-"J-	«
Suv n В z B1 = ®’khi v4 chl khi A' s? %0, 4p ‘l’n«
bai loan 1 2a) 5 X 'S' = tuong duong vdi diZu kiAn В cd phuon J
kh6ng°doi Khi dd A ludn thudc mdt khdng gian con hai chiSu co.
E’ ibu vudng gdc vdi ( В )>.
2> Vdi n > 3. Trong E". ddi vdi cu sd («,}
Ait) = S >:> 5* , tu dd suy ra X X, X” pttt khi Va chi kM I
x"i it) + fit) x’’(t) + git)x'(t) = о ill li = 1.., n)
(trong dd f, g la cac ham sS). Moi phuong trinh cua hd (1) (Л
ta cd nghiem cua he (1) li |
V (2) trong do T, b la сйс vecto theo thill
bf) ; X b d.l.t.t.
1.4. 1) Him so л : J -» R, t •—» A (t) = J ||X’(t)|| dt drtng biftl
VI || A (t>|| > 0 vdi moi t G J nin nd co ham ngtfdc 1
Л vA Л 1 d4u khA vi, Л'(1> =11 X*<ttlL (Г1) (s) = -1—
z'al"l(»)
Vay л la mot vi phdi.
2)3:1->,9^ (s) = Ior»(8)
3’(t) = asin p. r<t) HAft) + i) (2)
«*с _ -
В" (s) - (А Х|^Х А I, = •»’ <»>
1.5.	?«> ft) = ? ft) (к 14 s0 tv nhikn bit ki)
Я«*Ч ft) = г ft + | )(к = 0, ?» = 7)
Я»г)(й = Е(1+л)
H«‘3)(t) = 7ft + ^).
1.6.	((? X ф X !?•) = (?' X ?) X S+(? X <$) X 5 =
=<?x^xi’.-чТЕ.+ 3.ф. 3 + (У . 3 + 3.») .3-
о. ?+<?.». 4’
. К? X?) X <1х 3)]' = -K'S X 3) . $’ + ® X 3)3+
hr х 3). 3)3 + кЗ X ®.?+й х 5 Л+1!х J).?.
§ - - ((3 X 3) . 3) . 3’ + [(3 х 3) . 3) . 3'.
1.7.	1) Нёс к = О кокс 7=7 nhung cung phuong vci 3 ta
Л 3 ft) - 7(vecto king).
NSu 7 = 0, 3(t) = к X 3 ft) * 3 suyra
3.3-0 = ||X ft)|| = const (gii si = a)
3ft) . к = (3 ft) . £) = 0 «=3 ft) . T = const.
Chon muc tiku trac chugn {0, 7p t2, 73) sao cko 7 =«^
" = const « 0 (vi 7 la vecto kkkng d6i cho trade). lh cd thg vift
3ft) = «sin у . £ Uft)) + acos p. T3 (1)
ЙК 2“:“A”kh6ng d,!i vi r T ““"e i6‘-
T x 3 ft) = « asin p . 7 [A ft) + 2]. (да
So sinh (2) vi (3) ta cd A" (t) = «=. Aft) = «1 + 0 0 14 king s«i.
V4y 3(1) = asinp . f [ «I + 0] + a cos p . 7
Chon /3 = о ta cd
| ~^(t) = asin <p, 7 (oct) + acos-y.^~|‘	(4).
2) Khi 7 * 7 Ap dung c&u 1) va chon muc tigu toa da nhu trtn,


asiny?.7(oct)
+ acosp.T3. Suy ra
A(t) = В + sin <p . £*(oct - ^ ) + a cos p . t e"3
d day la vecto hang, a la hang so duong, « la hang sd
1.8. D ^2 e’' (a*(bt) " (bt + f)] + =
S2. VECTO TlEP Хйс, TRUONG VECTO. CUNG THAM SO
VA TRUONG VECTO DQC MOT CUNG THAM SO

2.2. 1) Th co th£ vi£t
kite (!) la fa-u hypeholic
3) Mdt nhanh hypebol x2 - y2 = 1 (nhanh Ьёп phAi true tung. x >0)
2.4. 1) Hypebol	= 1.	2) Hypebol 3-3 « 1. kh
a b	a b" 3 ,u
3) Elip = 1.
2.5. 1) Vbng trbn co' phuong trinh (x - a)2 + у2 =a2. Я Г
2) Dudng thing x = a.	3) Dudng thing у = b. . £
4,Е11”Й + Й=1-	5) НурЛй £-£.*. h
6) Parabol y2 = 4x + 4.
2.6. 1) aHJiA sii difi’m M cd toa d6 (x, y) ddi vdi muc йёи «пл
chu&n (0 ; i , j} nAm trdn vdng trbn tam I bdn kfnh a l&n khfing t:
truqt trdn true Ox, di<m xudt phat la 0. Goi t 1A gdc PD4 (? »
hlnh chi£u ейа I 1ёп Ox), ta cd (h. 2)	1 I
OM = OP+g + f& = atr+аУ+аГСу - t) (7(t) =	£
= cos t. , + sint . jT OM = aft - sint)i*+ + a (1 - costlf . ]
Quy dao la du&ngjXy-cl6-it, 14 Jah cua cung tham ad I
t —> P(t) = =fafr-sinft оП-сааШ

phang xOz. Dife'm P(x, y, z)
chuySn dOng xufit phdt tit 0. Gdc
j s ° jj IlHt) x /Hi) H
2.8. Di8u kite d« trudng vecto X = S E, xuyen t4m О la
r = mt. Th co phtfdng trinh
2.9. 1) D»t At) = O^t). VI m/>" = Xoft X 14 tnltmg „
Л> (0) Bta A’ «?ng tuyfn vdi ~hoic ~p\ x p= f v6i mQi ,
- NSu ~P (t) x vdi moi t пёп p (J) nAm trong m<|
N«U £ = (Г, ~P va ? c0ng tuy&j, ap dung kft qud bAi toan
x?tt, + Atjlhllhti)^'^^ '—I


IWd hifu Stt) =	|| Л”) x F (») || it
= 5^ * II = |lWI khta« d<ii-
3) Cung tham s6 p : J —* E2 - {0}, t —» p^t).
• £fy>) пёп ta сб
Suy ra (p x p) = X (khfing ddi).
= (r" - r »>*2) Ж&0 + (2r’ f’ + r io") Stip +
X о p = (fo />) Op (theo gii thift) ndn suy

~к.(Тр = 0. Та со hai trdbng hop:
a) NSu А = thl />(t) chay trfin dttdng thing qua O.
b) Niu ~K. * 0 thl p{t) nlm trong mat phing qua O, vu6ng go'c
Viy khi * 1Г thi I? ₽ "0 vi ta ctfc
к.ф_И = м,
T£t do suy ra
(||. cosy, - l)r -	= 0, (г, <f>) li toa d$ cdc trong m»t
phang do' d6i vdi g6c 0, true OB. Viy trong Ьё toa dfl do
EE
т U phuong trinh т,е «оа « «#с (г, р) ейа mW cbn!c пкй» О
- кы к > о (thiI > 0), Me dd ьифс ||3|| > к “ <* =W с“”г
,1» trtn mOt nhinh ейа hypebol пЬ4п О 14m tieu didm.

nSm trtn mM nhanh hypebol.
- NSi c •= 0 hay 3 = 0, t —P(t) 14 cung thing
- NSu 3 « <T v4 C * 0 chon hi tpa de Dfcac vuOng gdc
(О, 7p ~ег, 73) aao cho e"3 then phuong ейа 3 (- 3or(t) = <x
vdi mol t), 4p dung b4i tap 1.7 ta thu dupe kft qu4:
Pit— fft> “ 0 + 3? +	- I) + cacosyj t^ 4 day
7 14 ham vecto hing, a • |P (t)|.| (khdng ddi), p la go’c tan bdi
§3. ОАО HAM CUA HAM SO VA CUA TRUDNG VECTO
THEO МфТ HUONG VA D1?C МфТ TRUONG VECTO
1) Vdi p = a= + y2 - 23, - _ (J, 2, -u, p)
^--Ip-^L-4 = 2-<I
’',w=1?1“'5ip=ix4+2x2+('
2) Vdi v = e* siny + cosz ta cd •

Др dung ЬЫ 3.6 la со-
ци D[ = - ,inp . UjtplEj + rasp U,[p] . Ej = OB, + OE3 = 0.
Trang tu DuU2 - 0 ; D0V, — ~U2 1 Du,U2 = "7Ui
V»v |pU|U, = РцУг-0 ; РиЛд7и2 i Рц,ц2 = -fo|(1)
D6i vdi t<?a dd c&u (<p, 0, r)
TiRJng tu nhii cSch tlnh d trfen, ta co:
Du Uj = U, [- sinp] Ej + UJcos p] E2 = - созр Uj [ц>] Er
-Л^и.И Ej =	- ^E2
DuU2 = “^Ei_5^E2
Buu, = 0
Sau khi biSu thi Ep B2, E3 theo Up Dj, U3 nhu Л сДс ket qajj
3.9.	LSy mOt trudng muc tieu song song <EP E.,.. EJ trdn U
vd viet 2 = 2 pi Ej , f' e JV thi do V lien thong cung, Z M
mot trudng vecto hang khi vd chi khi moi p' Й ham hdng. Nhung
vdi moi a e TU va moi X s veo U, ta cd
D„ Z = 2 « [pi] В, = 0 «=*« [pi] = 0 (i = 1,2........Я) A
TJn Z - 2 x [p'] E, = 0 «=X [p1] =. 0 (i = l,2,...,n>.	"
Viy chi cdn cSn Ung dung chu у d muc 3.1 chuong 1 cudn Hlnh
hoc vi phan cua Doan Quynh vh bfci tfip 3.5.
3.10.	li Theo dinh nghia. trudng vecto [X,.Y] = DVY - DJC. !
Ta can chtfng minh
(x, Y] (p) = X [Y [pD - Y ( X tpn. (1)	1
Thgc v»y, „е„ x = 2 x. Ejj Y _ 2 у1 E_ (4	,
X (Y tril - Y [X (pH . ПЙЕ; и -i V в, M Bj tri +
• 2xlYi(Bi[^W -^W
Nhung BJBjtri] - BjtBitri) = 0 Y* ®XY) W =
= J x’ в. Mb. tri, ® Yx)tri = 2 y1 в, txJj B; tri>
ij-1	U->
n«n suy ra dude (1). Ngoai ra, chu у ring trudng vecto xac djnh
bdi tic d6ng fen moi <p 6 JV theo v6 phdi cua (1) la duy nhfit
(xem bai tdp 3.4).
3.11.	1) Th co' Uj(t) = (ftt), Д (t) Ci = 1,.-, n) T. Id cac hdm
Dll	-»	-
-^(t) = <J> (0, u\(t); Ifyt) 6 T/,(t) En, U.(t) la co sd- ейа
T^t)En =» —phdn tlch duoc m6t each duy nhSt theo { Uj(t)},
viet ftyt) . t Cl (t) Bj®, q : J -. R, t _ Ci(t)
2)	Gid sut X id trudng vecto doc p, X = zt р'(Ц),
X(t) =£ ri4(t), Suy ra
DX <	A DU, • |
—(t) =| 2 ri'tt). U|(4 +2. ri(t) -Jfiftj. Лг dung (i>
S(t) = t	+ t Cj(t) .U;(t)
5|(t) = J + J pi Cj) Ui(t). v»y
I = 2 У + £ ri c;j u,	j (»
3)	tu,) true giao « U| . Uj =
d	DU, DU. «
df^  UP = ЧГ • ui +u< • IT = f + c>4'
«^(CKdti + C^ = Ci4-Ci-0
=* ma tran cf| phdn ddi xting.
4)	(15,(0, l52(t), Uj(t) = hdng s6 * 0 «
« (U,(0, U2(t), U3(O’ = 0
Suy ra (CJ + C* + C|) (tf/t), ГТ2(ОД(О = 0
= i ci > о
§4. DANG VI PHAN
4.1. Ыу X e «с <U), f1 —•	e K
Cbo r ejo, MCA
аа^да-xifopl = <fo p> x ад №0 w з.е;
= Га u>) d <p [XI.
4.2. Th da bift (xem vf du d 3.1.2 giao trinh)
dr (Up = U, [r] « 1, dr (U2) = U2 [r] = 0
d / (Up =	[f₽] = 0, d ^(U2) = U2 [p] = -
Ttfong tit nhu tren (bai 4.2) ta cd trtfdng d6i muc ti6u cua
trucmg {Up U2, U3} la {91, в2, 03}, trong do
E
2) D6i vm loa do Ли e, r) trong BJ\ (0) U cd trudng
«К*	<4. IU,. U;. f,;. rue ,r,nt..  . ,го.-ь.. .ja,
U6u tUOTg ling te1, 9:, e2 3! 11:


tiftu сйа nd. GiA 8Й Uj = С| Ej thi dx' = J Cj » (chi cfin tAc
d6ng cA hai v6 vao Uk).
Ap dung ktft quA trtn vAo trudng hop 2 3 1A tnfdng muc
tiCu dug vdi toa do DAcAc vu6ng gdc, <U,} 1A trtfdng muc tiAu toa
(1)
dx3 = rcoad de + sinS dr
2) Л cd (dx' A dxi) (Ut, U,) ,= 2 Cj, Ц if A «’ (Uk, U,) •
dx' Л dx» (U„ Up - sin e, (dx' л di») (U2, U3)
dx' д dx' (U3, Up = cos в
dx2 л dx3 (Uj, U2) = cos 6 cosy»,
dx2 A dx3 (U2, U3) = -sinp
dx2 л dx3 (U^ Up = -sinfi cos/»
dx1 л dx3 (Ult U2) = -sin/» cosS,
dxi л dx3 (U2, U3) = -c°s^
dx1 л dx3 (U3, Uj) = sin 6 sin/».
л dx2 л dx3) (Up U,, U,) (p) = thA tich hinh hAp tao bdi:
UJnV n.fni •	- 1
dx' л dx» (U„ U,)  1, Ы ,, a,» (U„ up . «
dx' Л dx»(Uj, U3) a 0
dx2 л dx»(U„ Up - 0, dx» л dx» (U2, Up - cos F
dx» Л dxS'tU,. Up - sin v
dx* л dx» (U„ Up . 0, dx' л dx»(U2, Up - - sinF
dx' Л dx» (U„ Up = cosp
(dx' л dx» л dx») (U„ U2, Up = 1.
ddu dugc bids th| /1 ®	S' л $ Л 0 », trong dd
Py,» = / <u.' ui' UP' <e'h .in14 «»“*»» d® “C' tlSu 4“* ’al
trudng muc tidu {Up.
Ntu n = 3 (trong Е») thi fl  V л V e
Xdt X e Vec (U)
• X etf (U), - X = ., Theo dinh nghia
(• X) <Y, Z) =0 (Y. Z) = д(Х, Y, Z)
= (v (S' л d» л S’) (X.Y.Z).
1И
£ ex>
VWdivX = -td(«X)] = 2.—
Chdng Id div X khang phu tbu6c via p.
С П1)<Пг,и3)»/‘<В1-и2.иР=*
(. и,«Up и2) = с V,XVP и3) = о ‘ <• FP^ иР-
v»y	•V, - f 62 л
Tvongtd	-V»#»»1
• и3 = ? в1 Л в2.
v«y -i / и, =1 true2 Л e3> + р2®3 л Л + ч№ л 9’> (1)
div (X + Y) . = • [d M-X + Y)1 = • [d<- X) + d(- Y)J
= V[d( • X>] + d(* Y = divX + div Y.
vdi X Y e Vec (U)
div (p X) = *W (*F xb = *£d
= *[? d (* X)] + • [ (d F X * X))
= p [* d ( • X)] + * [* &p (X)]
V£y | div (g> X) = y> div X + grad p . x j vdi moi p G УП. (3)
Th cung co th6’ thu dupe k£t qui tr6n b&ng edeh dua vao (2).
2) Trude h6t hay bi£u thi div X ddi vdi toa dd afin (x1, x2, x3)
trong E3. Gii вй X =	.X’ Uj va dang thd tich p = k/x0, trong
do к la hang sfi, p0 = 0' л 02 л 03, {Uj} va {0 ') la trudng muc

• X = к (X1 fl2 A fl3 + X2 03 A 01 + X3 6’
dcx> = к 1 ® w1 л e2 л s3).

Th cd for : p —• f(r(p))
grad (for) = (for). U3
* grad (for) = (for) (* U3) = (for) 01 л в2
= (for) r2cos6dy> л de
d[ • grad (for)] = ~(f’or) r2 cos 0 dp л d0
= [(f’or)r2 + 2r(for)J cos 0 drAdijP л dd 4
div (gra(for)) = *d[ *grad (for)] = ~ [(f’or)r2 + 2r(for)J i
119
| ciJa phuong trinh
I T(t) + *f(t)
b) G14 sd f : R+ —* R, t —» f('
(for)X khdng co' ngudn (div(for)X = <
[(for) r] (• U3) <

[(for) r3 cos e + 3r2 cosd(for)] dr Л d((> A dfl
(for)r + 3(for).
(u>! A cm2)(U,V)	•	[•(«, Л ш2), U, V]
= (• (»! Л Ш 2) .(U x V) (4).
So sanh (3) vdi (4) ta cd (2). Bay gib ta tinh Rot (.
Rot (p X) = *db(p X) = • d(F bX)
= » [dp л bX] + pd (bX)
= *[dp л bX + *p db X].
Ap dung (2) ta cd.
• [dp A bx] = ( d p) x (#bX) = grad p x X
* pdb X = p (* dbX) = pdiv X.

4.9. 1) Vdi X, Y G Vec (U), V 6 jfU ta cd
Rot(X + Y) = » db (X + Y) = • d(bX + bY) =• (dbX) + *(dbY)
[Rot (X+Y) = Rot X + Rot~Y| (1)
M chiing minh Rot (pX) = f> Rot X + grad? x X ta hay chiing
• (w, Л w j) = (#Wj) x (#a>2) (2) vdi сир ш2 G П1 (U).
= w J (U).cm2(V) - e)j(V) . CM2(U)
= (#to! . U) (#<m2.V) - (#0»! . V)(#cm2 . U)
« (#<Mj x #<m2).(U x V). (3)
§5. АМН ХА KHA VI
vgy V I, = (/> °Mj)’ ft) = p ft)
b) F : v—. R, p’e yv #(V с R”), vai moi« e Tp у o, .
p la cung tbam stf trAn V, p : t —• P ft) (p ft) « p) |
тр p ы = <y> орут
Tp ptvj = «р и E <y> (p)).
Vg,VW = «W|p ^|.w
5.2. a) Anh bdi f cAa
1) ftp { (u,vo)| u e R) ft aba true Ox* nSu v„ = 0 va 1A paraM
x = ~ - v? ngu v * 0;
4v2o
2) Tap { (uo, v)| v e R) la n<la true Ox- ntfu uo = 0 vl la
parabol x = - ^ + u2 ngu Uo w o;
3) Tgp ((u,v) e R2| u + v = hAag аб e ) 1A nia true Oy- afc:
= 0 vA 1A parabol у = —~/2 * nAu e « ft;
2 4) TAP <(u,v) e R2| 1 « V « 2 ) 1A тип gidi h?n bdi hai parabol'
y* = 4(x + UvAy2= 16(x-+4)(kA eAbai parabol do.) |
t ~/> ft).
A»h w tiSp xuc cua p lai t, T, /> : T, J —. T4l)U.
TV хеш tab xa d«ng nMt I<U IrSa J 14 cung tbam sb trta J :
1<и : J —• J, t —• t rt veclo tifip xilc cua cung IdJ la —
122	dt
«,'	cua 1 ,a +	"4° thu h?p cua f Un
doa iah	1 Ли h-₽ 116 kbtog ph4i 14 vi ₽Mi vl kh«”»
517^’2 r^ie;.y=u) u w 4пь ” 4
'• Ш = <те“-'-,)£^-хЛ + л
Vй/	<*	Л бу
5.5. 1) Trvdc het hay zft tnltag hop I5„g qtl4t „„ M „
Phdi Г : U (C K- ) _ R-, p _f(p)i p	«
R",’f(p) od toa de <ж>...ж»)	t„ng R„ Ch0 Ш	e0, wn)
(tojOf) dfJ + ...
f . _ afi
d““
Trong trudng hop f : R2 —» R3



= (xoMtaoO - (yoO . (xof) d(yof) + (zofidlzoO
= (u+v) d(u+v) - vsinu cosu d(cosu) + (vsinuj d(vsinu)



Pdz + F(yzdz) л f*dx + f-(e« dx) л fdy

2) ТЫг lai ом? tbde Г1и1 ’ d°r
Vdi и * xdx - yxdy + zdz> c°
f j, = - ,(ГЛ) Л (My) = -«»•“ '“2“ d“ л *'
fat = (u + v + v sinu cosu + v2 sinu cosu) du +
+ (u + v + v shAO dv.
Ddi vdi cac dang khdc ta cung cd k6t qui Pod = doP.
5.8. 1) X e Vec (U) f - tuong thick vdi X € Vec(U)
«f, (X (p) = X (ftp)), Vp-G U
«f. (X(p)) [p] = x (f (p)) [pl, Vp e f(V), Vp g v.
«X (p) [pot] = (X [p]) tf (P)), Vp G v, V p G J(V)
«(X ipof]) (p) = (x (pj) (f (p), V p e v, v v g 5(V)
<=»x [,pofl = x Ep] of, v p g /(V).
2) Lfiy <P G 5(V)
[X,Y] [pof] = X[Y [[pofl]’ - Y[X[[pof]]
= X [Y [p] of] - Y(X [pl of] ; X [pl. Y [p] G 7(V).
[X.Y] [pof] = (X [Y [p]]) of - (Y [ X (pJDof
[X.Y] [pofl = ([X, Y1 [pl) of, V p G 7(V).
vay [X, Y] 1Й f - tuong thick vdi [X, Y]
3) Kki f la vi pkdi U —* V
f. : Vec (U) — Vec (V)
X = f X Khi vi chi khi X(f(p)) - (f.X)^(f(p»t j
V p G u'ma (f. X) (f(p)) = f. <X(p)) пёп X = f.X khi va chi 1
X f - tuong thich vdi X.
5.7. Theo giA thift gradF = Ц ndn cd thd vigt I
dF = 2 p’ O' « 0'1 j = i 2 3 № trU&n8 d6i mvc tidu cua trudng
PdF=S (^'of)r (O' =(y’1o0f*(rcos6 dp)-Kv>2oOP‘(rdg)-Hu3onf*^J

Mat khac ta cd
Г dF = d(Fon =	+ de + dr (2)
So sanh (1) vi (2) ta thu duoc
§«. OANG LlhN КЁТ VA PHUONG TRlNH CAU TRilC CUA б"
G mOt Tr^ONG muc Tl£u TRUC chuAn

DU, = sin® dp U2 - cos» dp U3
DU2 = - sine dp U, - dd U3
CAch 2 : Dilng cfing thdc DU, = 5 tyi U, cung ckc baj tap 3.2,
6.4. Ik № bids thi w', в2} then {e1, e2)
f1 <u,) * 1. e2 <u,) = о
e1 <u2) = о, e2<u2> = i.
They U, = U,, U2 = r’TTj tn cd
6 1 <U,i = I ;<> (r Uj) . г ?i(U2) - 0
S2 <U,) = 0;S2(rU,) = гё2Ш2) - 1.
Vay say ra
I.*-?-*-
Dd tint ma tr»n cac dang lien kdt, ta ip dung cdng thiic
DU, - 1 3f Ur
Л cd DU, = DU, = dp U2 = 1. d p . u2
DU2 - D (r. Uj) = dr. U2 + r DU2
6.6. U, = £ cjUj.DU, = 1 dCjUj + t O.DUj
ОЦ - i aquj +1 cji ^U, = t (dCf+t ctfju,'» 
M»t khac
HI
du, = J 4 Uj = 1 (2 ii of) 
So Blob (1) v« <2> ta aS
acf + § q «f = 2 Ц Cf
dC + W. C = CS.
|w - C-' dC + C-1 WC|	(3)
lien k£t d6i vdi muc ti6u {U,} va {Ц}.
Gid sS № = J BJ, d* <4>, U, = 2 Cf . Uf
S (Ц) = 5 в), ekS cj, Up = dj
= 5 Bi Cf Sk (U , = £ Bj. Cf й
K.P - 1	K,p - 1
Vfty suy ra
вб phdc thl f IS m?t dim, bSo idn hudng, bio giSc khi vi chi khi
f IS met hSm s6 giAi tich va Г = 0 (cSng thiic Cauchy - Riemann).
ta cd Tp floc) = (fo/>)’ (t)
vay Tpf : TpE" = gb --- ТЕ" = g"
“~l№
Г a." = C = _jl_j [u"„ (u', dx - u”ydy) -
v”n 111 + dy)]
r«2' = -Г «,2 = —1— [ -u-^dx + U’xdy) + v'n (-u'x dx +
+ «’,dy)I. .	‘ У
6.7. 1) Cung tham ad p ; t />(«, trang dd p(tJ = p,
= «. Qua duh xa f :.t ~ f(p(t)) l4c dinh ba.
<гЛр («р» =
«.ity (f (p)) =
<f.U3) (f (p)) ..
[r(p)P <^1<р)' 2(ГУр)-Д(р)) 63(p)=-J— Q
tf(P)l2 = ~ №)]2 U3(P>'

Nhung nt(p) = U|(Kp,) (i =	2 3) do _k_ c r№)/ -
ndn ta dupe	(r(p)]2 ЕД
CHUONG II
BUdNG TRONG E"
§ 1. CUNG TRONG En
Т = °ША = -Ж.зц?га

Cteh 2 Ap dung cdng thiic
D<r,U|) = 2 (f.Uj)
t* cung di d£n kft qui trtn.
Anb cua cung them sd ndi trSn goi IA dudng truy tick.
1.2. fit) = (1, -	r*(u) = (Dsbo, Dchu).
Tai didm chdng M = At) = r(u) ft = Dchu, — = Dahu) u <
?(t). 7(u) = 0, cac cung tham s6 p vl r true giao.
1.3. Th chon diem c6 dinh dd lam gdc cila hi true toa da Ddeft
^”3 3»c Oxy trong E2, 0/>(t) ctag tuygn vdi A’(t) aft
ОАО cd phUdng khdng ddi vdi Vt e J. Cung chlnh quy cd 4nh J
dudng thing qua O. Trong trudng hop phap tuydn luOn qua 0, tft
°.'*0 suy Xa ^>11 = a (h4ng 561 “’’S chl”b
quy co inh li vong tron tim O.
r(t) trong khoing [0, U. Th co
(h • 9). Dudng bin goi 13
dudng Astrdit, no co' phuang
2) Dudng Xy-cl6-it
2.6 Chuang I). Cac di£m ling v6i t = 2k (к Ik эб nguyfin) 1Й cdc
di£m lui.
3) Chi& Id Dd-cdc (h.10)

!
1.5.	1) Cd did’m kl di nSu fim duoc gid tri <f> thoa man h$ phuong trinh
r’(p)cosy> - r(y>) sinjp
««ag lai g<fe toa do О (Ung vOi t =
0), cdt true hoanh tai
OAip) = OM = (r (<p) cosy», r(y>) siny,)
(/Xy,)) = (cosy,, siny>)
^2	= (-sinp. cosy,)
7^* (y,) = (r'(y,) cosy, - r(y,) siny,, r’(y,) siny, + r(y,). cosy,)
p ip) - ,Чг>. О, VW + r<n>> V2 W>
ОТ = ОМ + MT - ОМ +х <₽)- Р W (1)
®Т eOng tuyfn vdi р, Я la him sd ddi vdi f ).
Thay ОТ - •<?> tt2 </><₽))
OM = rip). C, (P(pi)
?'W = I’M П; V><F>> + r<»>) tf2 W)
r : f - r(|0) = ± -Л- v4
goi la xo£n hyperbolic, (h.15).
N6u b =s const, ta cd cung
r : p - a(p - 5Ро) vh gpi 14 хойп
a<y) C, <«F» = rip) U, w> + >. (p) r'ip). V, VW +
+ Mp) . r(r) U, (Pip)) = (r <₽) +Л (fl T'(f) O1 (pip)) +
+ i (p) r (p). tr2 VW-
t* nd <ь!з)
1.6. Th kf hi6u bT. (t) =
=(a,ft). hj(t), cj(t) i =.1,2).
Theo gii thjft
™, X K,) (t) » J Vt. Tifp
tuyffn di qua P(t) n£n he
Hlnh is
3)	NSu e . const ta cd cung г (jo) = ke*<
* ‘	h‘“* *>• C“”* “	Logarit (L14).
• (1) p.(t).^=-d,«
i Л/t). ач = -d2(t>
dung vdi moi te J.
VI N.(t) . /^’(t) = Nn(t) . ^’(t) = 0 пёп bdn phuong trinh sau
toe a (khOng doi) пёю
Bieu th} qua ha Вёсйс vuOng go'c Oxyz ta cd:
(x (t), y(t), z(t))
M с^цуёп ddng doc theo dtl&ng sinh la z’ (t).
2) Trong tnldng hop v&n toe ейа
M doe OL ti 1ё vdi doan dudng OM, ta
P = к sin6) (h.17).
1.10 1) Anh ейа cung p nAm trfin
2) No' nam tren mat cAu cd bdn
kinh R = 2, tam О (goc toa d6), mat
true cua mat tru trung vdi Oz. Goi M la chfit di£m сЬиуё’п dOng,
M(x, y, z). x, y, z la cac ham sd ddi vdi t (xem nhif th&i gian). Vi
| Nf(t) . AO
<2> NW  Я‘) =
Th cd = c (cosnt). Suy ra z(t) = kec‘, к la h£ng s6. V£y
•—»p(t) thoa mSn cac digu ki£n da cho thi cdc ham so
x(t), y(t), z(t) phai thoa man 4 phtfong trinh (2). NgUOc
dfing th&i trifit tteu tai t nao thi p :
(Dudng Viviani, хеш hinh уё 22).
й+ Й

« 2 DO OAI CUNG VA THAM SO HOA TV NH1£N
CUA MOT CUNG CH(NH OUT
2.1.	1) а И ыеи м «	««<=*• “"S d<*°
4)	J2a|sin ||at.
71 d’  4>.b| = S>
5)	Lp = apnsintj - lnsint0J (0 < tQ, tj S
1ИР’-*,!+ъг
*2-^=^=^
=	[WW]* 2 + [(«Л(О-(ртЯ'(О]2 + [го/ТО]2 .di.
2) D6i vdi tpa dd cAu (50, в, r) : tuong tu cAu 1), d dAy ta cd
V-j = (rof) cos (g. p)
2.4. 1) 8<2a;
2) 10;
3) s « J\l2 achtdt = a <2 sht
4) Cd th£ tham so hda cung dd dtfdi dang
tfng vdi tic gifi tn | va 9a cila у la cac gid tri a vi 3a cua x
•‘•[Д2Й^'Ь=9а-
§ 3. TfCH PHAN doc мфт cung
3.1. 1) Gia Бй phdp chuydn tpa dd afin trong En
= S Aj-jA + Bj phu hop vdi phdp chuydn toa do afin da cho (1).
VAy dinh nghla vA G khfing phu thuOc vAo toa d6 afin dA chon.
- sh7
2t, 2t
(Ч - к.) + в-5» (8h V -sh~r)
2(sh-^ - sh—)

| ||Aff||(OX+Afft) it
OG = --------------------- = OA + - AB (trung diem cua
JHJIdt	doan AB)
Г & chu vi ейа tain giac ABC, Г dtfOc xem nhtf inh cua cung tham
s6 />: [0,3] - En
1A + AB.t t G (0,1)
В + BC (t - 1), t e [1,2]
C + CA (t - 2), t G [2,3]
suy ra trong tam G:
,Л; = а.ой +ь.Ой+с.<Я; (M, Ni p the0 thlJ 14 tnlng d|fi<
cua cie canh ВС, CA, AB). Nhu v»y G chink la tint ti cu ci»
dlnh A, (i

(G trung vOi I).
Г 1» met cung trim Um O, ban kinh R.
P : t - РЮ = 0 + R. c(t), t S it,, I,]
06 =	---! Om . ||^(t)||.dt
Nhung ?Ы + T(/3)

«V dii Сй* C’°h BC’ “ Сйа Um e,'4c>- H°»e
VSy®
3) N£u p : [to, tp — E3, t —»(acost, asint, bt) = (- asint, acost, b),	= ^az + bz TYpng tarn G cua cung doan do* cd toa dd	Goi у U cung doan khep kin trong U khi vi thug khuc (gqp cua hai cung Г, vi Tfi Ta cd tbp chm w tham * Ьба la r: [to, t2] -» U trong do
/afsintj - sint ) a(cost - cortj) b	
Didu ki€n cfin va du dd’ G nkm trdn true Oz Ik xG = yG = 0 Suy ra t] = to + 2k л (к la sd nguyen * 0). 3.2. Trudng vecto Z : p •—*Z(p) = op Zop : t '—►(acost, asint, bt).	Ta co J, u = J ш' = J a + J a,	0 ,	* VI Гр Г2 la cac cung doan bfit ki ndi p, q trong U пёп cung
Luu so cua Z doc cung Г (xkc dinh bdi p) la	NgUdc lai ndu ddi vdi bat kl cung doan у khep kin dinh hudng.
/ (Zq^ ,/>Tt) dt = /lAdt -	- tj).	khk vi tUng khiic di qua p, q trong U ma J tu = 0, ta cd thd chia Г] di tif p ddn q va Г2 di tit q ddn p.
3.3 a) Gia sut Гj va Г2 Ik hai cung doan bfit kl, dinh hudng, Л	tjAu./W = p,/^) = q- Л : luo« uiJ - U> W = P’	= я- Theo gik thidt	hoac J и = J o» (Гр 17 lb hai cung doan dinh hudng di tU p ddn q trong U). Didu dd cd nghia la tich phan сйа ш doc cung doan djnh hudng kha vi ting khuc trong U chi phy thudc vao diem p vk didm q
/“  f° i "4	(diem dku vk dik’m cuoi). 2) Ndu ш = d >p, <p 6 ^U,
Ho^c cd thd vidt: J<u+J<v = 0(rli cung doan nguoe hudng cua Г,).	Г Ik cung doan bfit ki di qua p, q, djnh hudng, khi И tfcng khuc trong U, dupe xkc dinh bdi tham sd hda P  lto, t,] - U ; W “ P« = q‘ 153

. j[d/ = jtaw =
M,P  It , t,l - К Лага sd lie» tue).
V», гЛ	-ням> -да»»
L, - ,№-р(р). Tick ph»» »*» cW Pbu thud» V4O dita d«u p
. didm cu6i q ma khOng phu thudc v4o cung do»» Г qua P. q
(thou Wt quA da cluing minh A cAu 1) thi Jra = 0,(7 la eung o»n
VSy lim HP^^W = Iim j ?(P + ta). 5Г du '
t-X)	,^00
= J (lim?(p +t3) . Sfdu = J?(p) . «du
= t (p). « = top (<x).
Td dd : ham F cd dao ham theo moi phitong «p [F] = wp(«),
hen nda (p, Kpl — a-p (cu) lifen tuc пёп suy ra F khA vi vA I
dF'= to	(2)
V*y « kh6»g pMi 14 mOt dang chlnh xac, vi у 14 mdt cung doau
khep kin.
khfing phu thudc vao cung doan Г qua chung. Th хеш q la mfit
di^m сб dinh cho trUoc va dit L = F(p) (1) (hoSc kl hifiu
Jw = F(p)) F la ham s6 xac dinh trfen U. Сйп chdng minh dF =
JzdT- / (ZqP) . f> . dt - / f Z,[X'(t)..... x" (*)^Ж dt I
(z, (x1,.la thanh ph&n cua z d6i vdi trtfdng muc tieu song j
song cua toa do De cdc vufing gdc)
Jzar = J Z z, [х1(4),....х“(ч] dxl = J mop
•' p + toe (?(u) = tot). Sau khi dat # w = Z ta cd
= J?(p + tu«) . to?du.
ft « Q4U), и =-2 Zj dx'),
hoac | Z ds* = | tu.
Ap dung kft quA cua bAi toan trSn (3.3)) ta cd:
| Z ds = jf ш = 0 khi vA chi khi ш = dp.
Khi dd

(bZ 6 Q:(U), xem 4.1.3 trong gido trinh Hinh

Oxyz ейа E3) trung vdi Д. Тй dteu kiftn Z(p) vdi moi p e U lufin
vufing goc vdi Д (ttfc Oz) vk || z (p) || chi phu thufic vAo khoAng
Z(P) = f(r) | Ej + f(r) E2
f(r) dx

trong U).
Z - (for) Uj (f : R* -♦ R) г la cung doan kin duoc xac dinh
bdi f : [a, b] —U,
156
t ~p (t) trong do p (a) - p (b) Vjy
/Zd? = )(foro/>)(U10/V dt = Jtfor^dtrqP).
CM у ring (U,o/>).f = el^j . dr(n =
Vay ^ZdT = F(rof)(h) - P(roP)(a) .0 й
(trong do F u met nguyen Mm oli f).
СоеЛ 3 _Сйпв ohor, toa de nh« trtn, Z = (fo r) U, t. hay tin.
F . R — К di grad (For) = (for) U, «#d(For> = (tori U,
ho4c d(For) = (for) dr «=> F’ = f.
‘ Tom lai Z = grad (For) khi vi. oh! khi F 1* mW ngoyan ham
/>:t0,2al — Bdt-Jla+ooaO.Jaimj
A(t) = |(.-ии, eoat) ; ||(Г(< = |
F4P(t) = |-((1 + cost/-rsitrt = asih^
(yi t e [0, 2,r]).
fp /(pop) (t) . dt = a27sin|dt = 4a2.
3.6. Th cd ||f (t) || _ <3 et. Goi (x0, yG, ao) 14 toa de ейа
Cung d6ng chst da cho. Th cd;
CUNG SONG CHlNH OUT TRONG Es.
OO CONG VA DO XOAN CUA NO
4.1 1) p : t —o + a 7 <0 + bt 5 suy и
'Г(1> ” ^at+p (?г(‘+'г)+bI)
S(t) = -fft) ; ff(t) =	(-> f(t +f) +-Ч-
Suy ra tiffp tuy&i cua cung P tai t:
R ; I — R«) - p (t) + (a ? (t + I) + ьС) Л-
-p . t _/ед = m +
At) . 0 + aZft) + btr + ^=7» * f) + «*) 
= „ + af(t) +^S-(t + j) + (Ы+«).Й
lb _ la I
= vi-" - - ' “ "
p <tl = о + yl <t + toi + (bt + «) T
trong dd r = ija2 + b-', cos to = ~ sin t0 = t  -j
Sau khi ki hi6u <x = ex - bt0 ta cd
P(t) = (o+«I?) + «eft + t»)+b(t + tjlb
Thue hien ddi than! so t —. u = t + to,
thi p = plO„,
Д(п) = o' + ; Flul + buf (o. = 0 +
Vay Anh cua (Г) la mot dinh de trtn
4.2. Gia si cung Г cd tham sd hda tv nhi«n
r : s —> r (s) e E3.
1)	Mat phang mjt tiffp сйа Г vudng gdc vdi m(>t phaong сбфЬ
uong duong vdi diSu kien vecto trung phip tuySn В cua Г li
king, suy га В' = 0 hose T = 0 tai mpi didm cua cung. V»y Г» ‘
uung phang (xera chung minhh chi tidt 6 hai tip 5.1).
1) N6u D = (N, N’, N”) =
Th ki hi$u N(t) = (a (t). b(t). c(t)) la vecto phap tuygn cua mat
> 0 v6i moi
»P (t) =((x (t), y(t), z(t)) lh tham s6 hda cua
Гад, T5 (t) = - aw

(goi la
trudng Ddcbu).
h& Cramer, ta tinh’dupc x(t), y(t), z(t) mdt cdch duy nhfit theo
D, vi "e(t) khdng phai la ham (vecto) hing.
2)	NSu D = 0. khi do' N(t) ludn song song vdi mdt mat phang
сб dinh (xem bai todn 1.3 chuong I). Suy ra mat phang mat ti£p
cua Г lu6n song song vfli m6t dudng thang cho tnldc, Г 14 mdt
cung phing (xem bai tap 4.2 chtfong П).
a3[^t), ?(t), 4*(t)):
*G?t), ?(t), ^(t)]2
Suy га |!?П ' ' W е
vay a (ТоЛ) = 1 s> 7 а
2, Ngcac lai. neu do twin T = - thing sS) thi I
af (S) = N- Is) X ? <!> = [| (rN X Blj(s)
T(s) = [? (B * ^)]«-
Vay	 '	;	0 a-a f X  ds. И
4.6. Theo gii thiet ta cd
p'it) = (asinp(t), acosys (t), b)
/’”’(t) = (ay>”. cosy?(t) - a <p’2. sinp(t), - ap".siny> (t) - ap’2 созуЦцЫ
jP’d> x /’"(t) = (ab^’. siw(t), ab y>’cosp(t), - a2y>’(t) jfl
x p-ltil] =	(a2 + b2)
l?'(t), p"W, p-ttp = - a2bp’2
IM! = а2 + Ъ2, r’ . 0
dinh 6c tron.
4.7. Ttuoc het ta chhng minh 1), 2), 3) ttfang dtiong. ТлЛ’Ю
g'4 si a ]a vecto kh6ng dft 1ц8п ]in> vS. tuvSn с4а И.3Г
met gdc khong doi e.
= “s£) 'khong doi) 7= = 0
аоГк “ ^.= 0 01 * 0) -7. » = 0 «phap tuyin chink 1*
s song vol mot mat phang сб dinh — 'B.'a = ± sin« I
1»
(thing phAp tuyfin ludn lAm vdi vecto cho trade "a m$t gdc khdng ddi).
GiA su "S ludn lAm vdi "a (vecto don vi cho trade) mdt gdc khdng
ddi. (TLT) = О «-Т '^.‘a = 0 (T * 0)
7 = 0 =» "f ludn lam vdi "a mdt gdc khdng ddi. VAy 1), 2).
Bay gid ta chdng minh 1) vA 4) tuong duong:
GiA 8Й ~a la vecto khdng ddi ludn lam vdi ? mdt gdc khdng ddi в,
ta suy ra T.^’ = 0 ttfc 1A
(2k'T + kr’)B. Tich hdn
k3 {kf [T, N, В] - тк’ (T, N, BJ)

dinh de.
1^ = cotgfl (khdng ddi). I

Nguoc lai, ndu ludn khdng ddi doe Г ta sA tim dupe vecto
mdt gdc khdng ddi.
Xet trudng vecto X doc Г
(vi (rj = 0). VAy X 1A trudng vecto song song docT (xAc djnh
bdi a khdng ddi).
D6 dAng thft lai T ludn lAm vdi X mdt gdc khdng ddi в (mA
 k’N + kTB.
R = foA : J - E3, s ~ R(s) = p[A (a)].
=r(s) + <x(s).5(s) (1) ddng then thda mAn R’(s).B(s) = R"(s).B(s) = 0. (2)
(« ; J —► R, «•—»<x(s) 1A hAm sd khA vi Idp c\ к > 2, В la trudng
xoAn сйа Г). Cung Г 1A cung phAng ndn у cung la cung phAng (y
vA Г sai khAc nhau bdi mdt phep tlnh tidn theo vecto «.Bl.
didm tugng i3ng thl
R'(s) = A(s).T (3).
R"(s) = Г. T +AkN.
R’ x R” = A2kB (4).
164
<{T, N, B) la «tag t»“ Fr5”e c™8 r>-
те (4) suy ra chc trimg pbap tuyta cua Г va у song song tai
cap digm tuong ling vh th dd, cac phap tuySn chinh cua Г vh у
► E3,
-r(s), {T, N, В} 1A tnibng muc tiAu FrAnA doc
Theo dinh nghia cung song chinh quy у liAn hop BActrAng v6i
Dua vAo (2) (vdi chu у 1A a’
r’(s) + a(s).N + a(s).N’
7: J —. E3, s t-,7(s) = r(s) + «(s).S (1)
(a la ham sS bicn sfi s, khh vi bae 1, a nd! chung khdng phai la
d (r(s), 7(s)J

didm) chiia N(s). Тй do suy ra :
(?(s), ? (s), N(s)J = 0 Vs (
I?(s),P’ <s), N(s)] = at(l - ka)' - ar'(l - ka)
boac In |1 - ka| = Ln |b t| (b 1A bAng аб).

(BX
3) DAt 9 1A gdc tao bdi hai tiAp tuyAn cua Г vA у tai didm toon
dng, ?(s) 1A vecto tiAp tuyAn cua y, T(s) 1A vecto tidp tuyAn do
tuong ling tao think mdt gdc khfing d6i (0).
4) Cung ti (2) ta cd: sing =

2t
(1 + 2t2)2 ;
S5. DINH Ll CO BAN CUA U THUY£t DUONG TRONG Е»
25sintcost
GiA sft cung
 9a2c2t2 + 9b2c2t4)
2) t = 21or (k Ik cac s6 nguyen).
гСа^2 + 9а2сЧ2 4- Э^сЧ4)
(a2 + 4Ь^
r(s) = 0 + ]Г(±
(khOng phu thu^c vao t).
Suy ra (a2 - a2b2l) + (4b2 - A9a2c2)t2 + (9c2 - 9b2c2A)t
?(s) = + rWs)) (2)
?’(s) = + r(«>(s)).«,’(s). Suy ra
= (a,2bt,3ct2), p" = (0, 2b, 6ct)
= (0, 0, 6c), pl = a2 + 4b2t2 + 9
3abtfa2 + 4b2t2 + 9AW


trade (ta cd th< chon trpe Oz) mdt gdc khdng ddi 6 (cd thd gi*
thidt 6 nhon) va cotgtf =	. Cung Г cd tham s6 hda ttf nhidn:
r : s »-*r(s). Ta cd
7(s) = sine. Г[р(8)] + cosS.^. (4)
Suy ra 7’(s) = saae^’f^(s)).₽’(s),
v& |[?’ (s)||2 = sin2^. p’2(s) = a2 (a2 chinh Ik blnh phtfdng cua
II? X Ml « a2 .« (A2(c2 + 1).
L8y c = 0, a (a) = f cos (arctg^ds.
y(e) = J sin (arctg^)ds. Sau khi d*t t  arclg ^ . - = tgl
_tacdxls.t),|ln14^-‘ ;	JL..
const, vay Г la cung dinh 6c. Phuong trinh


Anh cua Г nAm trfen mat ndn trdn xoay co' phuong trinh
hai vecto $t) va ^(t). Th co
гГсГ+Р) ’	.c(a2+b2)
“ * ’ |Н1.И+Л1 “ t|a2+b2r|c2(a2+b2)+ai2
3) Cung p . t = (x ft). у ft»
I x (t) = a(2t + sin 2t)
I у (t) = a<2 - cos 2t) . (Xj-elOit).
6.2. 1) xft) =-----Ц-. У <2> = -act«*  <P*Mbol>
56. CUNG PHANG (luc cung trong E2)
6.1. 1) |k(a) da • J^77da = arctK"+c-
2) x(t> ж | a" (cost + sinti; y(t) - | a1 (cosl-sint)
(xoAn 6c logarit)
3) x(t) - aft.sint + cost) ; y(t)' » arfaint - t coati
| Xft) = aft + sin t) = aft + Я - sinft +я)) -
| Yft) = - a(l - cos t) = a(l - cos ft + л) - 2a
Xft) = - 4a2 t3
Yft) = 3a2t2 +^~ (h.20)
6.5. 1) Cung Г xdc dinh bdi tham
? : a ^S) =/>(s)+(e-s)X(s) (1)
P Is) = ? (s). Ih cd
x’2 (t) + у’2 (t) = a2 cotg2 t ;
x’(t) y"(t) - x”(t) y’(t) = + a2 cotg2t.

Й=)"
V5y p(s) = fa) _ (c _ s)TfsJ = fa) -
P <t> = (R (cost
6.6. Chuy^n sang toa d6 D6cac vudng gdc
tuydn cua Г tai digm tdong Ung (s):
? (s) = Дз) +_ N(s) y 14 Mc bf сйа r
2) D6i tham s6 Л : ,	(*!*.« dai cuntl
Suy ra : p' Qp) = ca^tlna.coy — sinp, Inasinp +cosp]
c-ln(l + fiI+4))
Vecto tidp tuyffn cua / la
x' N +« Я’ +(J" В+ДТ5'
Ap dung (1) trong bai toan (6.5) ta cd cung thin khai cua cung
	V = »-' Ср» = < <x. У) s и I iP (X, У) - a = 01
• ТЙ (2) ta suy ra a = - (к * 0) (4) Тй (3) suy ra:	Di6u Иво cdn va du di у la mW Ла W mW ehidu It « *F (liy) . — (xy) khOng ddng thdi trial lieu tai moi diflu	I
,1Г "+7 i+< hoic	U-y = T «* (arccotg ’ = T. 1 + («) Suy ra = cotgfjlds +.c) с la hAng s6 V = ^costgj rds+c) Vfiy	Pte, y) g / ; d dfty ?(x>	F <x» y) “ a- Di€u ki®n dd cflng tuo®8 duong vdi diAu kien grad	* 0 tai cAc difi'm p £ y. у 1A cung phAng cd phuong tr’inh <p (x, y) - a = 0. Tidp tuy?n cua nd tai p(z, у) e У 1A Ч>\ (X - x) + F’y <Y - y) = 0. V*y trudng vecto phap tuy6h don vi la (tran y). 7.2. Th ki hifeu у la tap cac dig’m p(x, ]x|), x e R (d6 thi cua hhm sd у = |x|. Tad lAn q£n U cda diSxn 0(0, 0), у П U khOng vi - phfii vdi mdt khoAng nao trong R cho пёп (у nU) khdng phii 1A
p (s) = ris) +^N(s)+^cotg(Jtds + c)B(s)|	7J. Th kf hi6u	F(x, y, z) = x2 + y2 + г1 - a2. Ц) G(x, y, z) = x2 + y2 - ax.	(2) ‘ Thi difi’m A (a, 0, 0)ла trhn sau day hang 1-
Khi Г la cung phAng, 7 = 0 vdi moi s. Tfi (3) ta suy ra /3 ос’ - а p’ = 0 hofic	In а = Inqd (c la hang s6). Г la cung phAng thi В khdng d6i vdi moi s) mdt gdc khdng dd’i в ' cotg 6 =	= - hAng s6. V$y tiic Ьё сйа Г khi do la cung dinh 6c.	[5 J i] -[t z:] Tii cdc did’m trdn у khac A(a, 0, 0) ma tr$n (3) hang	| hai пёп cd lan cAn trong у la mdt cung chinh quy.	О Tip у - {A} la Anh cua	L\
§ 7. CUNG HlNH HQC VA DA TAP MOT CHlfcU	;	hai cung tham s6 :	^>*7^	'	J
7.1. GiA sit (p) = y> (x, y) = a 6 R	X 3 1 P (t) = (acos2 t, acost j . sint, asint)	22
T>i dtem В (О, 0, a) (Ung vdi t = |) cua cung p dd cong bing
Mat phAng mat tifip cua у tai В cd phuang trlnh X = 0 (mat
phing yOz) (h.22).
Difu ki$n |MFj . |MF2| = a2 (ta ki hi$u |MFj li khoAng
^(x + b+y* . )|(x -b)2 + y2 = a2
Kx + b)2 + y2] . [(x — b)2 + y2] = a4
(x2 + y2 - 2ax)2 = 4b2 (x2 + y2)
N$u a > b tap di£m khdng cd di£m kl dj, li mdt dudng khdp
b Tip didm cd mdt didm kl dj (didm O), khdng phAi
Khi b >a dudng cong cd didm lui tai didm I (2a - 2b, 0).
Khi b < a dudng cong cd didm t\t cAt tai 0.
Khi b - a dudng cong cd di*m lid t»i 0, khi dd dudng cong
gpi 1A 6c sen Pascal.
1.1. Chon Ы toa d« Otole ™Ong gdc trong &. Oxy. W»c °*,
dudng thing d cd phuong trinh a = 2a <n > 0>. Mix, y)
r, duoc xac dinh bdi phuong trinh in
cd diem th cat tai A (a, 0) vh nhhn doing thing a = 2a lam tidm
a(l ± siop)
khdng phii la da tap mdt chifiu trong E2
(x2 + y2) (x - 2a)2 - b2 y2 = 0

У cd 4 didm lui dh (a, 0), (0, a), (-a, 0), (0, -a) . (h.4) V
7.9. GiA suf cung tham s6 p : t —»p (t) = (x (t), y(t)) тйг
dudng у. Th cd F [x(t), y(t)] = 0.
Suy ra F’x x’(t) + F’y y’ (t) = 0 (1) ; F\, F’y khdng dfing th#
bAng 0. Ik cd th£ vi£t ’
Tft (1), tiep tuc Ifiy dao ham theo t ta cd:
k(t) = ^,(t) y,,(t)-x»(t).y>(i) ,
Г7/	X’2(t)+y’2(t)
Dua vao (2) va (3) ta tinh dupe dd cong к tai didm khhng M
AB la tiep tuy&i tai mfii di£m M (khfing phAi dife’m ki dj) cua tAp
diem у (dudng Astrdit).


k(x, y) = Fy ° I
<Fi +
W
-	~ an* „.ou cua cam giac (S > 0)
dd- 4h4ng V" “C tr’C 0X1 °V tam =б “*" >“
Kijong ddi s co phuong trinh :
® = - 4рс[х - +Р«ф - бГ * рс> - 4р2с <1 ♦ А » 0. (2)
те (2) ta аиу га у + 2сх = 0 ; е = - £ (3)
ТЬау (3) vao (1) ta thu dupe '
1) Dudng chuS’n сйа parabol (х
khac phia vdi dinh ddi vdi tifeu didm, cd ban kinh bdng — .

Quy dao dupe x&c dinh bdi p : t p (t) md (x (t) = x(^(t),
y(t) = у ( />(t)) ld chfit didm trong kh6i (m > 0) thi luc tdc ddng
?(t) = m ^"(t) (dinh lu&t Neuton). d day = - mg j*( Г la vecta
sd, t6c dd ban ddu).
Suy га ОЛЧ = -gj*y + v„r»t+iT	®
P <l) • (a (t)  v0 eoa «.t, у <t> - ata «  t -	> Л»У«п
5 phuang trlnh dpng £n:
Р(х, у) dx + Q(x, у) dy = О
(b&ng cAch dSt
P(x, y) = F’x (x, y, A (x, y))
Q(x, y) = F'y (x, y, A (x, y)).
TiSp tuySn cua dudng cong cua he tai p cd vecto pb4p tuygn
Suo, y„> = (P<xv yj, Q(x„, yj).
. CHVONG III
MAT TRONG E3


у (2xdx + у dy) = О

hp Elip cd phuong trinh:

7.15. 1) D6i hudng ciia mdt cung djnh hudng trong E3 (cd
hudng) thi Tdfi hudng, N khdng ddi htfdpg, В d6i htfdng. Dd cong
k(s) khdng d<5i, dd xoin T (s) khdng ddi (vi = - TN).
b) Ddi hudng cda m$t cung phAng djnh hudng thi T ddi hudng,
d6i hudng. do cong k(s) dtfi dfiu vi = kN.
Vfiy minh nay cd inh li ellpxdft.
5) Тй gii thiS. di tdi x2 + y2 = z, tdc li
f! + 2- = 2z, vdi (x, y, z) * (0, 0, z). Viy minh nay co inh li
mit parabileft eliptfc trim xoay true oz loai trd dii’m (0, 0, 0).
6) Anh cua minh niy li hyperbdldft m$t ting
J + d _ — = i loai trd hai dudng thing sinh
x = f(u) , у = g(u), z = h(u).


- Trudng hop V 1* dudng thing cit true giao Д thl у cd phuong <
trlnh:
vi do dd phuong trlnh сйа mit dinh 6c dtfng cd dang: x = и coev, I
2) Trudng hop у la dudng thing cit trpe giao Д (vi hai ehuytfn
ddng khdng bit budc ti Id vdi nhau). Luc niy phuong tqnh сйа у li
Do dd phuong trlnh tham s6 cua mit cdndft ddng si li
x = и cosv, у = и sinv, z = y(v).
1.5.	Phuong trinh mit tinh ti6h: r(u, v) = 0 + 1 (u) + E(v). CJ
и cAc gia trj Up u2 ki,6nS d6i ta lin luot dupe cic dudng tpa dd
. rUj(v) = 0 + A(U1) + B(v) = A, + B(v)
.' ruj(v) = 0 + A(U2) + B(v) = A2 + B(v)
He 1Ыс niy ebung to ring khi tinh ti&> 4nb cua dudng ru thM
vect0 AjAj thi dupe inh cua dudng ru .
2)	Gii зй hai dudng cho trudc li
mu (m li hing sd).
> (u, V) = 0 + 1(A(U) + B(vJ) = o + |a(u) + |b(v) nghia la
3)	N^u {0: f j”k} 1й muc ti6u true chu£n trong E3 vdi tga dO
Beede vufing gdc x, y. z thi phuong trinh mat parabfilfiit (eliptic
~ ± y = 2z (p.q > 0), hoac dudi dang tham s6
Vfiy mat parabdldit la mat tinh tifn.
4)	Ta da bi6t (theo bai toan 4) mat dinh 6c diing co phuong
C.«y	(III.,,;
2) Vdi a ?„(u„, r„) + b ?v(%,	. -5, xet

(trong do u chinh la khodng cach/dai sd" tit did'm r(u, v) dgn true
Ox).
Вб phftn U = <(u, v) I - c < u < с, V e R } -* r (U) (c > 0
(y>, yi) »—•	— c cog1*0 Ф , v —	I® nipt v* P^6’
ir^MF, V) « 0 + j(C^>) + a? £) + |(C^V) + ayV)  (Ndn luu у
ring r +1 (C7(lp) + v4 * _0 +1 (C7 (Vr) + av,]5 x4c
1.7. Vi U lien thfing cung nSn vdi bfit kl hai didm p, q e U
dSu cd cung tham s« p-. 10, 1) - U, /40) - p, />(1) > q T>. ci
( 0(V)2) ’ = 2.0(r,/5 . (Tjy = О vl (TjrfW l»m0t veetu ti«p xue
vdi mdnh r (U) va О(г^) <t) la veetu chi phuong cua phap tuydn
cua nd tai cung didm irt.₽i(t). V4p khc.dng each ... .......
khflng d6i, ||Or(/>(0))]| -	> a TO dd, lai do tinh Hftn
thfing cung ейа U, ta cd |j6r(u, v)]| = a vdi moi di£m (u, v) G U.
Vay r(U) n&m trfen mat c£u tam 0, bin kinh a.
1.8. VI r(u, v) = /Xu) + v. X(u), (1)
Cho пйп ndu l$y didjn сб djnh 0 G E3 va ki hi$u
7(u, v) = Or(u, v)‘ ДиТ = О/ХиГthi ta cd:
"?(u, v) = Au) + v. l(u).
7U<U1V> =	+ v- (0)>
1)	do, theo dinh nghia didm И d,
Vo) la dife'm ki di cua r khi vi chi khi ^(u0) + v0 о 0
phu thufic tuytfn tinh.
2)	Doc theo dudng thing sinh u = u0 (ndi tit li dudng sinh
uo), vecto phap tuy€n ciia mat cd dang
^<u„. V) « ?,(«„ V> X ?, <“o.	,
= (7(u_) + V. A’(UO) x A(u0)
= ^(u0) x + v- x ^uo^-
Ю hifeu hang tit thd nhit 6 vg phii cua (2) la ex va hang tvf
thii hai la J. Vi ^nay la nhUng vecto hang cho пёп mii hang
hai trudng hop:
a) Hai hang tut dd khdng cdng tuy&i. Luc nay Mu0, v) thay
ddi phuong khi di£m thay ddi trfin dudng sinh uo (vl luc niy v
3)	Liic niy t cd phuong khdng ddi nin ta cd mat trjj,
mat non, mat tidfp tuy&j li mat khi tria’n. Ngu<?c lai, mat khi
i.e. Via r(u, v) = Au) + v.l(u) - />,(и) + (V - v>(u»jt(u),
 F(u)S(u) = (f(u) + p’(u))l(u) + (g(u) + ^(uWlXu).
Khi Г - g' khdng tri$t tiSu tai u nho, cung chpn
V - - g thi ?,(u) = (Ku) - g'(„))l(u) » 7 vg ,
Cta кё da ch°' ><“ " =
^(и))А(и) mA O’ //^
ddi khi difi'm chay trfen dudng thing sinh u0.
Difiu к>ёп hai hang tti dd cOn'g tuyffn cung Ih diiu к!ёп d£
(? X 7) X (Ж X I) = X* 1). ЦЯ - «? x I).!')!
=•-<?, X Я) X
Nhu vgy, sv eOng tu.yfn cua ?(uQ) x Xu0) va X’(uo) x 7(u )
Wong duong vdi su phu thuOc tuy£n tlnh cua 7^(uo), "X(uo), ^'(u0).
r + 1» ”«» khi H hieu n'„(u, v)
+	>i?_
Tit dd, nffu (u, v) 14 digm khs ц d
•'«tW(«,vllihai mstphbgsu^.;;/
Nhu v^y;

196
gradyj(p) = t . Ej(p) (1)

Trong trudng hop nay
gradpip) = 0	= -~^|p = ^|p =
Рц x ?v = (R + 1)2-cqA'.Z*(u) + (R + IJ^sinv cosv £ Dp do chi khi
R + 1 = 0 thi r moi co' cac digm ki di, chinh la toAn thg cac diem
?g(F->)i
ax1 If
- Vdi | + 2fcr < v < -y + 2br (k nguydn tuy y) (2), fcUc
cosv < 0 thi tfip cac diem r(u,v) la mat cAu tarn О ban kinh (В- 1|
2) Cach 1: Do (1) cho пёп bigt grady>(p) cung tUc la bigt cac
choc thing hai digm thing hang vdi hai digm da no'i d tren khi
khi R - I = 0 thi r mdi co cAc digm ki di, 1A toAn thg cAc digm
(u, v) e R2 mi v thoa man (2).
bang khdng khi p khdng phAi digm ki di cua mat vA luc nay chdng
la cac toa dd cua mdt vecto chi phuong cho phap tuygn cua mat
§2. МАНН HlNH HOC VA OA TAP 2 CHIEU TRONG En
ос e Т Е3, PhAi chdng minh rang
2.1. xa 2 phing E2 С E". Cbpn <0; 5J,..,%> <n>uc titu afin)
®>»g En sao cho О e E2 vh "e,. e’, thuSc Т52. Xet anh xa r: R2 - E2,
lu, »t _.(u. v, О......О). R6 rang r la dim ddng phdi Ida E2. Vay
E2 1* ropt minh hlnh hoc irong E", va do dd cung lit m6t da tap
2 chiAu trong En.
«« € Tp(|p"4(a))- W (1) d,ing
CAch 3: Vdi vectc ti6p xuc <xp cua y?-1(a) th! (graty) (p)- «p
*<lf(p).'«p =	- XpW = 0 do <p 14 him hing trtn цГЧа
2.3. 1) Via gon (ж2 +У2)2 - Зг - 1
^ = 2x(x2+y2>
= 2y(x2 +y«
* =.-6x = 0,

21

0. Th£ nhilng, dife'm
Ис hai хйс djnh bdi difiu kifin:
F\z(x(u, v), y(u, v), z(u, v), c(u, v)) khOng d6ng thdi trifit tifiu.

Minh hinh hoc trong bai niy dUdb bifi’u thi bdi:
у - у = 2t (diy li mit parabdldit hypebdlic).
dinh mdt mat cia ho. Gii svt F khi vi Idp Ck (k » 2) trfin mfit
no'i din cac cung hinh hpc trfin Г. Goi p : t *—•/’(t) li m$t tham
sd ho'a cda mdt cung hinh hgc у nhu th& Ki hifiu К li giao сйа Д
dift’m (u, v) e U sao cho vdi mdi difi’m (u, v) 6 U„ m£t cua'ho ting
d vudng go'c vdi Д vi dudng thing К /4t) пёп d vudng gdc vdi
diem khdng ki di vi ti£p difin cia no' tai do' trung vdi tifip difin
C° £ (W)2) » 2 • W) . w
dd Г li ci dudng trdn tim I bin klnh a.
э РНЁР TINH VI PHAN TRfiN DA TAP HAI CHlEU
TRONG E
s I D w f AM vi «6» w m vi, lai Vi r(U) duong nh.=n la
minh hint hoc nhn anh a, p = r-pP khi vi. DI nhihn cd th« b.«u
thi P(t) = <u(t), v(tl) vdi u(t), v(t) kha vi. Vay ta cd
- LAP phOT”g trinh fc сйа ”4'4° T2' Тв M U> my »
2 + 2 + z2 = a2 + b2 + 2ab cos2nu = a2 + b2 + Л
(do a
VAy du<?c:


2) VI f(t) = r(u(t), v(t)),
r(u(t), v(t)).
u-
v’(t)
= ^or> l(u,v) W T^u’4(U.v)-
d r’„ = Ruo r, r’v = RyO r), tdc la:.
X(t) = u’(t).(RBo r)(u, v) + v’(t).(Rvo r)(u, v).
3.2. 1) Tinh r’„, r’v. De dang kife’m tra thSy r&ng r’u,

2) Chiing minh r(R2) la хиуёп T2. That vAy, cho dudng trim
trinh la


xoay) thi phuong trinh thu duoc ейа хиуёп trung vdi cac phuong
trinh xAc djnh r(R2) da cho trong d6 toAn. Ki hi$u hfi phuong trinh
ttfc la T2 la m6t da tap.
Cung cd tW cluing minh T2 U da tap bAng cich nhan хй Л1
2)	Nflu (£	0)14 foadfl cfiu,y> = £1	,0 = fl| thl
- ~	_ JS\<p.p‘}’	ls\<p.p‘>
do d§ tinh Uj, U2 tac ddng vao £ va в trong mflt bii toin <5 §3,
chtfong I. ta suy ra U,[p] =	, UJfl = 0, U2Iy>] = 0,
U2w = |.
3)	Vi x = Rcosycos# ndn Uj[x] = Uj [Rcosy>cos0] =
= R(U] (cosy»] cos0 + cos^.Uj [cos0] =
= R(- siny» Uj [y>] cosS - cosy» sin® Uj[0] = - siny. Cung nhu v§y,
U][y] = cosy», Uj[z] = 0, U2 [x] =? - cosy» sine, U2[y] =
3.4.	1) S = {(x, y, z) e R3 | x2 +
E2 ={(X, Y) e R2}, л = «X, y,' z) I z =
Xit tham sfl hda r2 : E2 - л, (X, Y) 
- R} С E3, p = (0, 0, R) e S.
~(X, Y; - R), cua n. Vi
+f(q) = л П (dudng thing nfli p, q), q <
f(q) = (X, Y, - R) thi cd A dfl’ A pf(q) = pq tUc
J в у e	m& x2 + У2 + z2 = R2’ n6n 8иУ ra
' ° fXup	- *cW
Г»: x - S\{p}, (X, Y, - K> “(x, y, z) xdc djnh Mi x = ЛХ,
У-И..-Ra-W.tzongddA^^y^.
Vay f la song anh v& f,
Г1 lien tuc, tiic f la dflng phfli. Vdi
tham sfl hda dja phuong r,: (u, v) —»(x(u, v), y(u, v), z(u, v)) e
S thi cic ham sfl (u, v) ►—x(u, v), y(u, v), z(u, v) la khi vi, do dd
tpofcr,: (u, v)
—»(X(u, v), Y(u, v)) li khi vi, viy f khi vi. Vi
« rang Г1 о r2 khi vi nfln rf1 о Г1 о
r, khi vi; viy Г1 khi vi.
[Cdcb khdm_xdt F: E3\(p} - B’XJp), , „F(q>, ,ao ebo
pF(q) - 4R2 j|jgrjp (nghich dio Um q, phuong tlcb 4R2). Chdng
minh dupe F 14 vi phM vi dj thdy F biSn S\(p) thhnb л tir dd
F|sw = rttviphW
2)Й“^ = (	• ^^•““7^7.2.)
e S (Zo * ± R) la vi tuyfln, /^(t) = Uj(f(t)). Tft dd
t —*f о P(t) = (X(t), Y(t) - R), trong dd
X<t) = S^’^ -7^.
. Y(t) ’ R^HS1-ZS ' ,iBq'fet-zg
(r2'ofo/>)’(t) = (X’(t), Y’(t)) =	0/1(1».
trong dd {Vj, v2} 14 trudng muc tiflu toa dfl cUc trong E3\{0}.
VSy H hi4u V, = (г,). VP Vj = (r2). V2 thi f.U, = - _2^ rl V,
b) t —»//(t) = (x(t) = Rcosy»o.cos— , y(t) = RsinpQ.cos— ,
z(t) = Rsin e S	la kinh tuyfln, //’(t) - U,(j<(t))
t — (f о д) (t) = (X(t), Y(t), - R),
cos-	cos-
X(t) = 2Rcosy»o .---------- , Y(t) = 2Rsiiy»o.--
1-sin^
л - rv V-, v,) 1Й trudng muc tifiu tpa do c£u trong ]
vZ “thu h?p Vp V; tr5n svp'p’’thi *'
truang vecto tie₽ 1йс сйа SV<P’ p,) v4 ta du’c
R v	*	4 I
№'7FnSF7v''
-	b) t ~v(t) = U„. y„, « e c <XO = Re°st%. У = Rsiny0, u
fR2coaw R2sir™ fl
-	u3 Wt». Vgy t -<fo)(t) .	, __5, _L.|
3.6. 1) Til r: (u, v) _ (X о r(tt> v) = RCMUCOeV|
yor(u, v) = Rsinu cox,,, 2 0 r(Ui v) = Rejnv) suy ta
Г (dlIlr(U)) = " R’ioucosvdu - Rcosuxinvdv,
Г tdyL(U)) = R coeucosvdu - R sinusinvdv,
r’ (‘‘‘IdU)) = R“svdv - ’
Mwkhfc, theo Uigt; e = ytdx +	+
^^:Zu^+(2or,^:'
i =	+ .in2UCOx,(M.2v . „o4MvJ.

Vay, dat u о i
2)	dd = d(yx) л dx + d(zx) л dy + d(xy) л dz
 (ydz + zdy) л dx + (zdx + xdz) Л dy + (xdy + ydx) л dz = 0
Do dd de = 0.
se la hyp jria t&t cA c&c tAp md trong S! (kidu nhtf U(x) n£i trftn),
псгЫя 1A Г 4V) md tronc S..
trong Sp .	.	.
Khi dd S2\A md trong S2.
(xor)d(yor) л d(zor)+(yor)d(zor) л d(xor)+(zor)d(xor) л d(yor)
((xor)*+(yor)! + (tor)%
S,\r>(A) = Г'(82\А);

2) Cdu trA Idi 1A: khi dd f lien tyc. ChAng han bay gid ta chiing
minh cho trudng hyp tinh "md", nghia la ta её chiing minh: nfiu
lidn tyc.
§4. MfiT S6 TiNH CHAT ТбРб cua DA tap 2 CHlEu
THONG En
1) - Chiing minh n£u f lien tyc, V md trong S2 thi r*(V) md
md trong Sj nfin r>(W) md trong Sr Vay, chinh r*(W) drfng vai
s; m» I g U(X> V» f(U(i)) c W; l»i vi W с V nta KUW) С V,
* do dd U(x) . f-1 (ftuu))) C r*(V).
md trong S thi sd suy m SA
. 10 S lion thftng cung.

 Gil s4*nguoc lai ring moi Up cbila C, md^trong
kMc rdng.vdi SB-
Vi S U da tap 2 Chidu, -do Un tai than. U d>a
, U - S eda S (tie li mdt dim ddng phdi l«n inh) tv huh trim
V c R= vko S sao cho a £ r(U). V. didm r-(C) cda U nd. dupe
vdi moi diem trong U ndn C ndi duoc vdi mpi didm D er(Ul П bB
M 0 do gii thidt phin chdng). C ndi vdi D chlng han hot inh xa
Ilin tuc f- (0, 1] - S mi «0) = c, «» = D, C ndi vdi A gli =«
thi D ndi dugc vdi A bdi anh xa lien tuc h: [0, 1] S хйс dinh
- Chdng minh SA ddng trong S. Lfiy day {A(} C SA mi hfii tu
v6 mdt di£m C G S. Th chi cin chdng minh C G SA la dtfoc (dua
theo mdt di6u kifin ейп va du d£ mdt tap la tap ddng trong mdt
Mdt mat, vi S la da tap 2 chi6u cho пёп cd the' chon duoc tham
s6 hda dia phuong r: U - S cua S, vdi U la hinh trdn md trong
R2 vaC G r(Ur VI r la d6ng phdi va trong U di&n r-1(c) noi dupe
vdi moi digm khac, пёп trong r(U) digm C cung ndi dupe vdi moi
digm khAc.
Mat khac, vi {A,) Mi i«.( С ч C e r(U), cho nan trong r<U)
Nhtt v»y c niS d“’":di6m A|*' Ь1' V6" C° nei dU0C **A
BoTc»fi^''i)iA’w‘!1‘CeSA
. , Tit dinh nghla .« lien thing any ra mdt tap con ® CUa
khhng gia” OcIit lk t,P П6П th<>"g khi Vk Ch‘ khi "6 kb(l”g ,h*
14 hop * 2 kheng r6ng’ Vto dd"g ’** тЙ tr0"gX v4 «
giao rdng.
Biy gid gid 8* X 18 rai?t tS₽ liSn th£,ng trong E"' T1 <*d»g
minh fC®> 114» thOng trong B™.
Щ4 sd ngUcc lai rang fl®) 14 Up khdng lien thdng cua £•
Thd th! cd 2 Up con khdng rdng A, В cua f(X>, vita ddng vita mo
trong f® mi «) = AUB, А П В = 0.
Dodd® = Г1 (ГСП) = Г1 (A U В) = Г‘(А1 U fl(B). Vi A
В vita ddng vita md trong ft®) cho пёп theo bai Up 1 thi ГЦА),
Г’(В) cung via ddng vita md trong®. Mat khac, vi A vk В d«u
. Mat khic, vi A va В d«u khong rdng пёп f 'l Al. Г ‘(Bl cung din
khdng rdng; va vi А П В - 0 пёп ГЧА) Л Г'(В) = f '<А П Б|
= 0. Nhttng didu via thu dude chdng td X khdng lidn tMng
dteu nAy m4u thuin vdi gid thidt.
v4y fl®) la Up con lien- thdng trong E‘n.
4.4. Lfiy bfit Id 2 digm A, В e Lay A e Г'ГАМ'ё
(В). Thg thi.A, В G X. VI X lign thOng cung пёп co the
A vdi B& B Ьд* dU&ng Khi do cbfnh P = f 0 P dtfdng nii
Mdt mat phAng hit ki P cd tfnh li«n thdng cung bdi •'
pb® А, В e P thi dd ding chi ra mdt Anh xa lidn №
nXMc4a'a2,diSm	m*' «» S Ki hi«u lAB
H	d g trtn 1<in «Ao dinh bdi A. B' cua S fft dhng
211
xSy dung duoc 4nh la lien tuc f: [0, 1] - S sao cbo f
f(l) = В <v4 «t0, 1]) = (AB)). Vay S lien thOng cung
- Cho bit kl 2 di«m
E. r.l- . "
- Cho b£t ki 2 dtem
A '	_ !<
5\{р}, mA bien
lifen thOng cung.
[AC) U (BC). Do do mat хиуёп X
3 chfnh la x, y, z). Hi£n nhifin grad y>(q)	0 khi
, шЬ11 S khdng lid» tbdng dung Xet cac di«n
- Xpb»"”6 Р^Р “Р"6 W "h“ trfnd5taXhbhXX'U0C
rlne * Mt rtu. phang. tn. trtn «"•? dfV '"”h hl,Ong
4. e. 1) KI Mau hai die'm bi bp di сйа mat ciu S 1» A vk B. Xdt
M o ®J S> 'J (A) U <B>.
Di«m A (va diem В cung vay) khbng cd Un can md nao nam
tron ven trong tap M. Do do M khdng phii la mbt tap md trong b
“do do- S khi da bd di bai didm A va В thi khdng phai 1» tap
done trong E3. vay, S khi da bd di hai diem A vk В khdng phi.
: : V :	a:»; kn:  ::> &
ki hai didnl P, Q trtn nd ta cd the dd dang xSy dung duoc anh xa
litn too f: [0, 1] - SiiA. B), sao cho Я[0,1) C Si (А, В), KO) = P,
< 0. Nhu viy (theo dinh li Bdnxand - Cdsi tha
. VI g xdc djnh trtn S vi g (X, y. z) = z cho nan (li chungu
z„> - f(to) e S mi tpa do thd ha rta Bj
 y, z) e S ddu cd dac diin»' 1A |zI » 1. V4y S khdng lien thing Mg
TrAn day la mdt vi du thuSc loai ma d« toan yfiu cku.
2) Ki hieu S = < (x, y, z) e E3 |
Tap S nay khdng phai la tap bi chan cua E3. Do dd S khdng

Cap mat phAng song song xdc djnh chAng han bdi phuong trlnh
z? - 4 = 0 cung la mflt vi dp don giin thubc loai dd
4.10 Gii set {E„ E2, Ej) la trudng mpc ti«u song song uag Л.
ha toa do Ddcdc vuSng gdc da cho. Ddi vdi t)iam sd hda du phucaf
t : U - S, (u, v) _r(U| v), сйа S ta phan tich:
2	>> Chi can chdng minh
J- Л	A dx)|r(uj . A tWlJfl^.
Ф	LtD-r-Ro^^thiXW^u ..I). |
V))~ST- <«.v) + H(»fu.v)) Tit dd:
n,	(dy	л	dx)	1^ (RU,‘R.) - “1	№	- <¥,).
x,	(dx	л	dx)	|rtu) (R„. \) - »2	W	• W.
.	X,	(dx	Л	dy)	1^ (R„. Ry)‘- ">3	(Pg2	- Pg1)-
Cfing ba Ьё thtfc пйу se thu dtfoc: vG trii li vG trii ейа (1) con
vG phAi thi bing
° •<R"* B»> - l^x^n  R««к. - IIе» x к« II =
• Wx^iy.
Mat kbfc	R) = ^Gr(R„, R,) d (%r‘) Л d (v0 rb (R„ IQ =
=SGr(RL. K) • V2y (1> du’c eh,i”B minh-
'> л%и, +^1Л'4‘>1^-+^(а1Л.ад1ки> = "'<2)
- %rf(nt^^W'(R»’R») =K^W ^IdU)®"'14’’ '
° Hm^II w  R“ x = <K“ X IV •I”1' Nh“°E ’*
г вр и =2f (6 day hidu I1, X2, x3 Ifa lupt 14 (x, y, a),
eko xfcUR,, x R.) = (Pg3 - Pg2) *+ (Pg1 " W * + '
' ₽Л fc ’ <dy A dl) <R“’	+ Wl * “ <R“'	+
♦ ® (dx л dy) <R„, R,) '
Viy (2) chi<?c chiing minh. .
§5. РНЁР TfNH TiCH PHAN TRfiN MAT
cfiu cd do Xdn « (radian) thi di$n tich hai "mUi* nAy )A
cung suy ra di$n tfch mat cAu IA 4Кхл. M6i hai "miii" cSu dinh
A, dinh B, dinh C dOu chtfa cA tam giAc cAu ABC vA tan g4c eiu

,5.4. 1) К xAc djnh bdi x = cosu, у = sinu, z = v (vdi 0 < u а 2л,
0 a v a 1). Ap dung bAi 5.2, cd diAn tich cua К bAng 2л.
2) a) XAt m(>t trong cAc hinh try cd dd cao 1/1 vA xAt mOt trong
cAc tam giac cAn duqc ndi tdi trong dd toAn, chAng han tam gUc
cAn DBC (DB = DC) vdi 'dudng cao DH. Goi 0 la tAm ciia dAy
hinh try Ay mA cd chtfa В, C. Gpi N 1A hinh chiAu cua D l«n dAy
 dd. D4 dAng tinh ra di$n tich tam giAc DBC 1A
v»y suy ra bifu thtfc cin tlm cua AO, m):
Ы That vgy, tu biSu thdc сйа AO, n>) any ra Othi m Л dinh'
llm A 0, m) > lim (2n> aln i . 21ain:	> >
=) Chi cln ай dung diiu quen Ы«: |im — . 1.
nghu“dfenBH Уйа 8141 ch0 ““y rin» “lth6"'tl’* d,’b
сй* "♦* “*"h l‘ gi« han dian Ucb cdc ma. da
219
di£n "nOi tiSp’ minh do gifing nhtf dinh nghia d6.dai cung m6t
J llF'2+F’2F(u)du
JSf'2 +<i’! (u) du
S=Jf(u) i|r'2(u) +v'2(u) du Jdv =2xJ f(u) t| p'2(u) •
p(u) > 0). Suy ra d6 dai cung doan nay li
(1)
ZirJ f(u) i|f'2+f'2 (u)
Jj/2*»"’2 (a) du

• toa dd afin trong E’ thl. goi trong tdm cua Г 1A digm G cd toa dd
(2)
 -У -	-----JllZW II • xW))df
' Jllr^llat * ' ’ .
Ddi vdi cung doan trong ЬЫ toan dang xet thl:
x' </№) = Flu), i2 (/> (u)) = 0, i’№> = *<")i *
oi c£c tpa dd;
x1 = Sa> x'i + Ы.	(1)
Jx>dS = J (S^xl^FdSutlAiJxMS+VjFdS,

(ZAjJx’ifdS+bl f,rdS) -
PdS 4
Cte be tide (1) V» (2) ebdng to G khOng ph“ ‘Ь““ ”
S = Jt|EG - F2 (u,v) du dv = J du dv.	®
Til djnb ngbia trong tarn d ciu a, trong tim cua C cd toa de la
= J ududv	ududv (xby ra Mu bing the nWt
- d«i'biXs6 cua tick phin ba Mp). Me lb
V = 2л J u du dv. (6)
(5) vA (6) chiing td ring S . 1 - V : dAy 1A diAu cAn cd.
3)	Ap dung cAu 2) KAt quA: V  2 X2 ab2.
t -	5.7. 1) Trong E2 cho h$ tpa dd DA cAc vudng gdc Oxy. XAt tham
~ s6 hda r : R2 -• E2, (u, v) —• r(u, v) « (x » u, у = v). Suy ra
 E(u, v) = 1, F(u, v) = 0, G (u, v) = 1. Do dd, theo bAi 5.2.1) khi
W". goi C la mifin compAc vdi bo trong R2 sao cho r(C) trung vfli mi4n
compAc vdi bd К dA cho trong E2 thi К cd di$n tich
S = J ^EG-F2 (u, vjdudv =. Jdudv	(1).
Jr' Gpi f lb phdp afin trong E2. M - (x, y) —f (Ml = (x\ y’>
j x’ « ax + by + p	(j).
~	| У’ = ex + dy + q
 - R° r4ng r = for Mng 1A< m«t tham .6 hda eila E;
MC) =. f(K) vi , : (u, v) (u- v) = (au + bv + cu + dv .
V Chone E(u v) = a2+^ f(u v)_ab + cJ_ g(ua)rb, +d;
j; Ma [<K)^’V> = ~ЬС)2 D° dd' C<i°g the° b4i 5 2 11 d'‘° 1Й
s - J<EG-F2 (U.v)dudv . |ad-bej . Jdudv (3)
' ,4 Л. Щ	”M₽ аП" **• E’ 14 И ph<" d4ng M" ““
. Phep d4”gaf,„ cua E2 ( fad - bc| - d« Г = 1'
2) (Chdng minh thfing qua cAc tinh tofin binh thuOng)
.trtf&ng vecto Пёп tuc trfin К va do dd cd th£ noi dfih thdn
va tu do suy ra (bang tinh toan gifin don) ring
56. ANH XA VAIGACTEN
224
2) Trudng vecto X chi xac dinh^trfen E^\ {0} (theo gifi thifit),
V thi khdng chtfa difim 0 (cung theo gifi thifit), do do' cd the
0) la tham sd hda cua mat parabfilSit (ehptic dng vdi dfiu
cong, hypebfilic tfng vdi dfiu trit). Tit dd tinh ra
Ket qua: kor = 0, Hor = (k la dd cong, t la dd xoan cua

ttf пЫёп сйа у, />(s0) = p; k(sQ) la dd cong сйа у tai s0, N(s0) la
Monid):
k(s0).N(s0) nV4so)) = к V’’.(s0))	(1)
Xet tiet didn phang у cua S bdi mat phang л ch da A va tiet
goi в la gdc gifla я'чА л thi tit (1) suy. ra
(2)
Th£ thi (2) cho ta 
Do Oxy la mat phlng иёр xuc vdi S tai О пёп y>(0, 0) = 0, (y>’x)0
, ttai gfc 0). VI e, »4c dinh hai phuong chinh ей. S,
0 (do gid ‘ЫЙ) * ° k(e” “ П<В'’ " L° °	”’Л ‘“°’8
it - (/'A 
Vi «„>» - Mo = II(<r'«)o’ (r’A> “ П <el’ ’2’ = ho<«P e,
- 6j e2 = 0. Tft dd khai tritfn Thylo ham s6 <p d Iftn cin difo (o,
0) 6 R2 ta ditoc:
,(x, У) = y,(0, 0) + <y>’,)0 (x - 0) + ((₽',,)„ (у - 0) + | (р"Д ,
x(x - 0)2 +	- 0>2 + (»»"„)„ <« - 0).(y - 0) + OV2, .
js,*2 + ЛЛ + o^-
- 6.5. (aRJ + bR°) la phuong chinh «3 к ,
hp(aR£ + bR°) = k(aR° + bR°) (R*. R° tinh tai p) ~3 к d$ (liy
’ aL° + bM° =. к (aE° + bF°)
L aM° + b№ = к (aF° + bG°)
(L°, MQ) ... tinh tai (u, v))
^|aL°+bM° aE°+bF°|
|aM°+b№ aF°+bG°
b2 -ab a21
" V M° № ='0.
|в° ?o Go
6-6. Sic dung bM 6,5. K«t qui:
1) Dung tham ad hda (u, v) —»r(u, v) = 0 + ucosv.i*+ uanv j -
+а».Гсйа mat dinh 6c ddng thl phuong chinh tai p r(u, «> c«*
"o’ 14 ± <u2+a2 Ru(p) + Rv<p>.
2) Phuong сйа kinh tuyfti vi phuang сйа vi luyft
S.7. Sd dung cSc phuong trlnh dMg chlnh tlc M cd iham ss M, kiSu dd thi, rti tinh dd cong Gaoxu vS suy ra dac tinh сйа сйс diem trOn mat. Ket qu4:	..уЦу-л) _	N(u,v)	» B(u, v) F(u, V) “ G(u, »)
- Тгёп elipxOit moi digm d£u la digm eliptic, - Тгёп hyperbOlOit mdt tfing moi digm d£u la digm hypebolic, - ТУёп hyperbOlOit hai tgng moi digm deu la digm eliptic - Тгёп parabOlOit eliptic moi digm d£u la dig’m eliptic - Тгёп parabOlOit hypebolic m<?i digm dgu la digm hypebolic, - Тгёп сёс mat tru bac hai va cac mat non bac hai cac digm dgu la cac digm parabolic.	2) p la digm гбп «к, = kj	- kj)2 = 0 ,,».+i^)2 _ ^2_h!. K 6.9. 1) Hai digm nay la giao cua elipxOit dd vdi true trdn xpay. 3) ParabSlOit	= 2Z (p Л q > 0) ch bai dijm rfu:
Cd the dan tdi cdc kgt luan trOn bang each duOi d4y. Tigt dign phAng (co th'g gidi han lai d cac tigt dign chtfa phap tuygn cua mat) cua mat la dudng bgc hai: dudng bac hai khOng suy bign thi co do cong Шс khOng, dudng bgc hai suy bign thanh dudng thang (кёр) thi cd dO cong bang khong. Vay theo cOng thUc Ole: К > 0 пёи mat khdng phai la mat кё bgc hai; ngu qua p E S cd dung hai dudng thang hoan toan thuoc mat bac hai thi K(p) < 0 (hypebOlOit mOt t£ng, yen ngua); ngu qua p cd ddng mOt dudng thang hoan toan thuOc mat bac hai hay nhigu hon hai dudng thing nam hoan toan trtn mat bac hai thi K(p) = 0 (ndn, try, (trong dd cd mat phang)). 6.8. 1) Theo bai 6.5, p la digm гбп khi va chi khi moi * = aRu<p) + bR^Xp) dgu хёс dinh phuong chinh cua S tai p, hay	j	(o,±^, ^£). ’	€ 4)	Тгёп elipxoff^; + ^2 +	(a > b > c) cd digm гбп: 5)	Тгёп hypebOlOit hai tang	= -1 cd Ьбп diem '«"= (O.*b^^.±c^^)- 6.	Ю. i) xat t —»/4t) 6 S (cung tren S), p(t) = p, /’’(t) = и e Tps
lb*	-ab	S I E(u,v) F(u,v) G(u,v) = 0 vdi V a 6 R, V h e R, |L(u.v) M(u,v)	N(u,v).|	thi Tjg(3 = (gJ9)’	= <no/>)’(t) = - K*p(<x). v?y T^g =	d day ta ddng nhSt TpS vdi Tg(p) 2 (khdng gian vecto hai chifiu trong E3 true giao vOi ntp)).
«с 1» khi vi chi khi |F(u.v) C(u,v)|	|G(u,v) B(u,v)| _ |B(u,v) F(u v) I _ N(u,v)| " |n<u,v)’ L(u,’)|	|U“-’> 228	2) a) (u, v) ►—»r(u, v) = (acosv, asinv, u) la tham s6 hoa cua mat tru trim xoay. Th« thi ?„ = (0, 0, 1), ?, - (- asinv. acosv. 0), do dd n = - cosv.i; - sittv.^ V»y »«u ki hi«u toa d« cua g.p'
idn xlch dao).
e) Mat cAu khOng кё’ mOt cap diAm xuyen tAm ddi.
f) Nita mat cfiu khOng kA cdc, dtfqe Ifiy vO s6 lAn.
О nen 0,= X[Y.n]
Mat khac, ((DxY).n)(p) = Dx(p)Y.n(p),	(2)
(h(X).Y)(p) = (h(X))(p).Y(p) = hp(X(p)).Y(p)) = - Dx(p)n.Y(p) (3)
Cac Ьё thiic (1), (2) (3) chtfng to diAu cfin-cbtfng minh.
2) IX, Y].n = (DXY - DyX).n = DxY.n - DyX.n. Dung k<t quA
cAu 1) va ch А у rAng h(X).Y = h(Y).X se suy ra [X, Y].n = 0, ttfc
6.12.	1) Lfiy co so {«, p] cua TpS; giA sue hp(<x) = a« + ЬД
V - о + ф thi Kip) = ad - he, H(p) = l(a + d). Do dd
hpta> x hp(0) = Kjp)« x 0	(1)
ЬрЫ X f + hjOT x « = 2H« x f	(2)
Ap dung (1) u dugc D,, x D)t = И x y, Mc j4
ЧАЫн)=kxxy'
(hd>z+ y'[h].zJ 'kx>
i Nbto vd hudng bai * c"a thiic nhy vdi X x Y < khi ohin
сЬЙ У hai Ito =*	tMc <a X Б> . tc x S =
- (fTcWB) t» tbu <,u^c cSng thdc cSn thi6t tinb К
Mot each tuong ta, B\1 dung <2) se ebdng minh duoc aing th^
cto thift df tinh H.
2) z = <F\, TV FJ Is (gih thift r; |s » 0),
X = <F’Z, 0, - F’P |s .
‘-.(0, F-2, - F’y) |s, D/ = F^gZ - Fyj^Z >
Ft  (FW > F,y  F7V “ Fy (Fxz  Fy2 ’ Fa>- Thay vho tong
FjCF'J+F'J + F^E (r’t'F”>a	2Fi'FfF» +
+ F?F"t)	- (р^-р;Г!Р„-
Fz Fyz + Fx Fy FZ2^ (d vg phAi la thu hep cua biAu thiic do' ten S),
“ * У xrat(i‘ z „ - 2F’ F F'2)
<-APhail,thuhepc,abi6.uth^d6^ У П z)
к'!^ГаЪ=“2 + Ьу2 + Сг2-1(а’Ь-С*0’‘Ы
|S (H* 1“4: vfli elipxoit vi hypebOIOit hai
e M К lain duong, con vdi hypebdldit mdt tfag thi К luOn to»;
We bitt, dag dung cho m,t cSu (Um о) bin Hnb r u dd din
nhSn l«i dupe К - i, H = | .
hp(«)  bpW - 2H(p) . hp(«)j5 + K(p)<x.^ . о	(1)
Cick 1 Tu ddng cSu bp cd da thdc dte (rang U Л2 - 2H(pM + K(p)
ndn dp dung dinh II Cayley - Hamilton cho
hj - 2H(p)hp +K(p) Id = 0, tile 14 hj<«) - 2H(p)hp(«) + K(p)« = 0
(rfi mpi «) <*). rfi vdi moi Д chd у ring hj Ы.0 = hp<hp<o<»^ «
= hp<«> hp W nhn 16y tich va hudng hai vg.cia (•> vM p thi dune
vecto d6i vdi nd, ta nhgn thSy (1) ding.
S £(«,> - (к, -	+ mk, = f (k, + kj) =	, ^EG-F2 <1 - 2г(Ног)(рог> + 0(£)) (VI binh phuong hai vS he thiic niy её suy ra he thde d
3) Gii ВЙ («, fl), (ep e2) cung dinh hudng thuhn. Dit - t e,. »> thi cosS« + sinS^S = costs + pje, + sin(S + pjej. Vay klcosS . ex + sinS . 0) - ktcostS + p0)e1 + sintS + pjej = - k, cos2» + yo) + kj sin2(S + fj, tS dd:	S J Gr(ft)’„. (г,)',) du dv =
jf it (costs + (^Je, + sintS + ₽0>e2 d® = kj J cosz(S + po)dfi +	= E J (1 - 2c(p о г)(Н о г» Gr(r’,)u , <r,)’v dudv + 0 (£1 I 4
+ iQsuAS + vjde = (k, + kj).	StKj) - SOD - 2c J	+ 0 <£).
6.17. Ki hieu f: S —* Sr p —•₽ + e . ptpl.nlp). Lat К bdi hp I,: C, — S. Thd thi K, dupe Kt bdi ho J" = for, : C, —. Sf Difiu sap sila trinh Ьйу dupe thUc hiSn ddi vdi chi sd i hdt ki cua hp, Jo dd и ве tain bd qua chi sd i. Tk dinh nghia cua f suy ra: r In, v> = rtu, v) + s(p or(u, v)).(nor(u, v)>, f'u = ё*и + ctp о r)’u.6Tо r) + c(por).(nbrt’u. = ?, +	» r)'v.<n о r) + s(por),Xnor)’,.	'	§7. NHLTNG DUONG DANG CHO t TRtN MAT S TRONG E’ 7.1. 1) Phuong trinh vi phan cua hp dudng toa do u la dv = 0. Phuang trinh vi phan cua ho duhng toa dS v la du = 0. 2)	PjCa, v)du + Qjtu, v)dv = 0	(1) ?2<“. v)du + q2(u, v)dv = 0	(2)
В = E - 2s . (porii + fi [ (pnryl + (f,Or)2	, F = F - 2£.(port.M + E2 [<por)’u . (por)’v + (por)2(nor)’, . (S>rj’2j,	i G • G - 2c. (port .N + g2 [(par).; + ((ет)2 ^M).g, BG - F: * (EG - F2)(l - dctHor) (per) + 0 (fl);	W- p2Q,	0	(3): Th goi Pltli, v) = С,	(4) Г Mnl	?‘У) = С2	<5> vi"phta (2)t! Th?thi ш	irinh * phan (1) vi ph“'"lg triDb ‘rt lai duoc (1) va 1k T" P " tO4n PM" hai ”f	“ dune (2), do dd.	У ₽Ь4П ‘°an ph4n hai v( c“a <5) “ trd lal
	MS'
Nhu rty didu Men (3) trd thinb didu Men:
X4t bfit ki (u
Iftn cftn uo cua dife'm (u
„ P2)<U, v) = (F1(u), y>2«)
du = dV1(u, v) _ V)do + ^(U1 v)dv =
= ₽,<«. v)du + Q,(u, »)dv.
Nhs Vly difci kidn dj = 0 tuong duong vdi dido Men P,(u, v) du +-
+ Q|(u, v)dv =_0. The mi phuong trinh du = 0 xic dinh bo de duimg
•* dO v ed. S; do dd P,(u, v)du + Q/u, v)dv = 0 <*“S
0« v« bp duimg Ц» d« tbd hai vdn dd eung bobn wan WOng W-
- P . (F Д, + G p2) + Q . (E + E = 0
hay (QE - PF) Д, + (QF - PG) ft = 0.
Nhifng cung do nhan xet da neu d trtn,_h? thtfc cufi cung n4y
chilng to vecto /3 tiep xue vdi hp dudng xac dinh bdi (QE - PF)du +
(QF - PG) dv = 0. Vay di du khang dinh trong d6 bAi 1A dung din.

v)—r(u, V)iP<»)+’ dei
= С (C: hAng B« ЙУ #).
KSt quA: - if <^(u»4« + C (C la bAng a« thy y).
e) Dang tham -6 hda (u v) ~r(u. ’> - PM »?.
mat tiAp tuydn sao cbo Ilf „II - 1 KSt quA. u + v -
aS toy У).
d) Dung tham sS ho. (u, .) -r(«, v) - 0 + V g (u> + bn к dSt
vei dinh Sc ddng KSt quA: v = С (С 1A hA.g s6 tuy /).
vdi (no)’ vh do dd song song vol к, ™	«« «w b
’ КЫ no p khdng song aong Bdi n о fl ti <no fl.l'n 0 fl’ . ,
vA Cn о fl.(*n О PY = 0 Buy nt <-n О PY
lai vSn cd f song song vdi (no />) tt £n o />>; do dd (о о fl swg
song vdi p' tdc у 1A chinh kbit tx«n S.
2) MAt phlng c4t mat trim xoay doc kinh tuySn hogc vi tuyft
dudi gdc khdng dSi. Moi dudng trdn mat phdng dSu 1A dudng chinh
m so hoa iu, v.i —»
2vu t d(Si vdi parabOlSit hypebOlic. Kft qui: ho
C$ng (1) vdi (3) r6i trie di (2) s6 thu dupe
(3).
3) КЛ qui: 2u
chinh khuc trfin S nfin (no/’)’ song song vdi /’’• lai v6n cd nop true
> C khdng d6i. Suy ra
(2)
2) KI hiau у: R3 -* E3 la Anh xa hign (x, yf zl e R3 thanh diim
 Pl • Рз)
VI----у " * ° nfin A la vi phdi die phuong. Do dd cd thd
. xdt nhSng vi phdi idn Anh cd dang r= i«A-l|Q:n- S3 I» mS>
tap md thlch hop cua R3 lan W = g. о Г1 (Я) mb trong E3. И>
rang khi w = C3 (hAng sd) ta dupe bp phgn trong W ейа mW mil
cua bp thtl ba (vdi v = C, va и = C, cung tuong IM). mA (u,v) —
-s r<”’v’c’'’<u’ v‘ Cj) e °3’14 Лвв“ 86 hda c““ nd

trong ho thii nhat) la cdc dudng chinh khuc. D6i vdi cAc mat khac
cong cua у thi ha thtfc trtn trd thAnh k.N.i
3) Trong E3 vdi tpa dft DficAc (x, y, z) da cho, ba ho mat xAc
dinh bdi cac phuong trinh fin.
thda man cAc diAu kifen d cau 2). Xem mat bac hai S da cho, xac
dinh bdi:
ax2 + by2 + cz2 - 1 = 0 (abc # 0), (1)
1» met wit cia bo thd ba, ling vdi Л, = 0. Luc n4y aS the’ chdng
minh ring I, va Л, khac nhau va khic khhng Til dd, ip dung
c»u 2) thi khi ki hieu Л, bdi u, Л, bdi v, hai bp dudng chinh khdc
trtn S duoc xac dinh Idn luot bdi hai hd phuong trinh sau:
(1)
I U>
7.5. Dung tham ad boa (u, v) •—r (u, v) = />(u) + v 1 (u) cho
mat kd S. Tinh ra he aS N cua dang II la 0, tir s6 cua ha ad M
cu. dang ПИ (? x 1) H. TU dd:
- h (T) =	= ±	± TN.

. h(N) = aT + bN. Suy ra h(N).T = a. Mat khdc, theo cAu 1)
h(T).N = ± T. Lai do h doi xiing пёп tit do' suy ra a = ± T.
h(T) = ± rN, -h(N) = ± tT + bN, ttfc la ma trtn efia h troi
{T, N} ia
7.7. LAy tham so hda tu пЫёп cua у la s -т* /(s) thi diAu kifen
no chmh khuc la (no/*’ ii p'. dieu kien nd Hem can la (no/’l’-f = 0.
didu кйп nd tidn trie dia 14	// (nqp). Dimg ba di8u kien edn
vi dd nby va ding edng tbdcJ^nL=abX^Jl-xa-duoc nhing
di6u cAn chdng minh.
7.8. у 1A dudng tien trtc dia trtn S пёп co' tham sd hoa t —* />(t)
сйа у dA — Song song vdi tru&ng vecto phAp tuyfin don vj n cua
S doc y. VIS ti8p Xie vdi 5 doc у nto dpej. tn»ng vecto pimp
tuyto don vj 1 cua -5 song song vdi n. Suy » so»8 »»8
n. V»y у li dudng tida trie dia cua '5.
Vi у dudng tiSm rtn trtn S пёп (no pYHN (хеш Idi giii bai
MJ), tile too /VII 5j- - W “i »’“ - Oi *> dd Z'’. (no pl • 0.
GU si cung tham s6 chinh quy p-. t —/>(t) xdc dinh cung
' ь „’uv r cua S. VI у chinh quy ndn cd tham s6 hda tu nhis„
"L rto), vd Ido nhy cd thd vift a = l.(t) vd p = ro Л, TW thi
П dd suy ra dang thtic (/>’ X j) . (no/>) = 0 tuong ducng vdi
2^ x (nori = 0..
Mat khftc, f>”.p = 0 nfin /’".((no p) x p'
0. V&y no/’///’”, tiic
у la dudng йёп trtc dja сйа S.
I*ftiung dAng thiic cu6i la didu kifen cAn va du d£ r xAc dinh mdt
cung trtc dja cua S, con sU kiAn tham s6 hda ttf nhifen г (cua 7'1 xac
dinh cung trtc dia cua S la digu ki£n cin va du dA' у IA dudng tiAn
; 7.12. 1) Da bift (0 chuong 1, § 6) khi bi€u thi dao ham (doc
theo cung) сйа cac trudng vecto trong mdt trudng muc trtu true
chudn theo trudng muc ti6u dd thi ma trtn cua su brtu thi la ma
trtn ph An d6i xiing, do dd d day cd thA vift
va do dd h(T). т
= pY + qZ
! - pT + rZ

2) Tuong Ы nhu trtn, mit И too Mi ole treng phlp tuyln cua
r Ifc dinh bdi (s, v) _ rls, v) = />(s) + v.3 (S), Goi n Id trudng
voeto don-vj phlp tuyln cua mat niy thi cd no p =	~ N
dang 1N Р "П°° P' WC Г 14 Ufa tri' dta сйа “** H
га
Nhan Vd hudng hd thiic th* ba Vdi Y dupe — Y = -r hay
r = _ Р(ЭД _
 ds h(T)Y = rg. Goi {T, N, B} la trudmg muc tiAu
CungZn r CV h“eng he tln,C ,hd nMt (1> Z *
Д И zS Рг4пё' ГЙ	Mnni*. ta duoc
2I = UY + k.X.5I = -kT + rZ^
k„T - T, Y, (2)
4) Тй cAu 1) suy ra
kg =	. Y - 5J((»qP) X T) = (T, 5^ , T>0/»j ttlch Ып tap).
5) a) kg = 0; b) kg = 0; c) k, = -^L
3) Xet mat tru £ vdi dtfdng chufi’n у va dudng sinh thing song

y. Goi {T, N, B} la trudng muc tieu Fi-ёпё dpc у ; tai />(so) - p e у
thi Y(s0) la vecto phap tuygn don vi cua у tai p va la hoac sai khac
cong phap dang cua S theo phuong T(sQ) bing' k(s0) 15 (so).V (s0)
Cfing vi S la mat phing пёп = kN. Suy ra kg
va bang ± kj (so) Y
Monie cho у C 2); '
Nkuag	+ k„Z nSn k<s„) U (sQ) . T (s0) =
.-wMy = v=o>-v’y Iwl = Iswl-
- Cach giii khdc. DAt к = n (p) = "n <p (s0) thi co' tham s6
hda s —V3, (s) = p (s) + у (s) T mk/», (sD) =p (sQ) = p va Д . It =0
(til dd у (sp) = 0 = y’(s0)); Д(в) = f (s) + y* (s) T пёп (sQ) | =
= <1 +y’(so)2 = 1 va Ky dao ham ding thdc dd theo s thi tai
s„ ta dooc k, (.„) T5, (,„) = к (s0) T5 (So ) + F”(So) t, », (so) 14
vecto phdp tuydh chinh don vi cua у tai s0, tdc ± ^(s^); v&y, ISy
tick v6 hudng vdi vecto t(s0) tbi ± k,(s0) = к (s„) N <soj.T?(s0)
^7 =	+ И» so v4 fc*c11 v0 fating v6* V(SO) nhu
trtn cUy, suy ra ke(s0) « ± kj(s0).

8.2.	Goi u la ham kinh do, v la nam vi uu
]a toa dd ейи trtn E3\x*oz thi td dinh nghia, 5p|s = u. «Is :
= a (ham hing) va da cd trong chuong I, § 6
DUi -=	- sin0 dsp U2 + cosfl dp U3
DU2 = sinfl dp Uj	+ de U3
DU3 = - cosfl dp Uj - dfl U2
пбп сйс dang Пёп кй cua S\x+oz trong truong muc ueu uep *««.
true chufin Ujs, U2|s trtn S\x+oz vdi hudng xac dinh bdi U3|s la
to> = - sinvdu, wj =cosvdu, coj =dv (thu hep trtn S eda cac dang
trong (1)).
{rcos 0, dp, rd 0, dr} la trudng d6i muc tiSu cua {Up U2, U3}, do dd:
1) dfl1 = - и’ л 02 = asinvdu л dv, d 02 = — ш2 л в1 = О,
Vay khi kl hi6u u = uo
vor"1 thi 01
3)	Cd t, (KU, v)) = 1P'(U) £(V> + V’ ('
T52 (r(u. v)l = ? (« +f)
T53 (Ku, »)) = - V’(u) К») + (’’(»>
cho пёп
. ftv) = p’(u) Q1(r(u,v)> - 4,'<u) “в3 (Ku, v))
Г(»+|) = Ъ2 (Ku, v))
f = v’ <u) X, (r(u, v)) + y>’(u) 13, (Ku, v)).
та dd duh DJTJ^rfu.v)), D^U|(r(u,v)) (i = 1, 2, 3) rdi khai
2)	Di thfl tMy л e> +из Л d fl. -
8.3.	1) Kinh tuydn va vi tuydn tren mat Iron xoay 14 nhflng
dudng chinh khic Tfl dd suy ra {Ur U2, U3> 14 trudng roue tiSu
chinh trtn mat trdn xoay S.
2) Vift Ku, v) = 0 + jp(u) 7 (v) + v>(u) T (y> > 0, y>’2 + ^’2 = 1)
thi r„(u, V) = F’(u) £(V) +V’(u) X PJ = 1,
Vay
"?= (p or l)dv, = ((р'ог“1)(р”ог‘1)-(р”ог"1)(р,ог~,)Ми ,
4)	Tu esc kft qu4 trtn, dung h„ tr)nh Мп c41 |(
tbuyft mat suy M: d el = 0, d =. (y'or') du л di,
Щ = ((y'm-l)(v,-or-l)(v,..or-1) dJ л d7

8.4.	MOt mat, ti. phuong trinh the ba trong djnh nghia cic
dang lifen kft cu’k suy ra
и -“’-ihi
tdelhm^or)-^.
Nhu vfty:
= toi(Ruor) = tui((r*^)OT) = «1((Г*Е1М = (’’•"iJfEi)- П>
Th lai cd
 >>-4r •(2)
(1)	vi (2) chiing to
a«j (Ei) = (»•«№)•
u^fT^or) = -	= 0 (vi 2 ho dudng toa dd la chinh khuc (theo
gift thifit) пёп M = 0). Nhu v£y:
0 - o?(B,or) «	. e3((r.BJ)or) = (r*«’)(Ej),
TUc 14 (r‘«|) (E2) = 0;	(4)
th lai cd duj (Ej) = 0.	(5)
(4) v4 (5) ehiing to:
248
Hoan toan tuong tu nhu tren. Th se chiing minh duoc
VI h (Ui) = (Up.Ui +	(Uj).U2
h (U2) = (U,).Uf, + (U2).Uj,
(«’(UJ «’(Ujj
[«’(U,) »’(Uj)J
Do dd 2 H = »’(UJ + «l(Uj).

2H 01 л в2 (Ц, U2) = coffUj) + (D
Mat khAc, di ding tinh dU$c
(«I л - <4 A S2) ( u„ Uj) = »?(Up+n^Uj. (2)
fl) va (2) chdng to: '
2H 6‘ л в2 (U,, U2) = M A -»3 л (UV U2>'

= /4t0) vs ЛУ- D1<P>- ™ л! <<d *’1)<и1))1р) 
, (V, trW) - M’ fr’W " S <р’^<0)) "
=	I, (the°dinh ngbia c,i* ’’1)
M(St. - *Pi<rm)\,
(Trong do' At) la viet gon ейа
. ?<y.T5,(p) + (И»).	|,o
= T3, (p). Т3,(р) +wy.
- i+^cy-atytfXU-O’
Nhung ta co':
^<и,)-‘в1<о>>+*Ч<и.)+’^(и'’
hay dr'tD,) =	+ P1"’ +*A“?(U>)' <Z)
Tuong tu:
к £ (2) va (3) Chang td dp1 = e' +	+ш?р’
SJ Cd tha chung minh ding thilc ейп Ipi mOt each tuong t(|.
2) Cd thS chiing 160 = - f2e' + ip1»2. Тй dd d92 = - d»!2 л S'
л 62 _„3 л 9l (theo bai 8.5), dupe de = 2<1 + F’H) 8' л в2
Тй dinh nghia сйа ц, д(ос) = 0(hp(oc)), suy ra
д = - ^2W3 + ^lw3 Do dd d/4 = d(-w^2 + w^1)
» co’ A 02 +0» Л <y| +2^«’ A to’ = 2H0’ A 02 + 2p’K0‘ Л 02 =
= 2(H + y>’K) 01 A 02.
j Stei 8'7' к; + k2 = 2H = hAng sd 2a пёп к, = 2a - k; Vi da tap S
| dang xdt 1A compAc пёп kp k2 dat cue trf trtn S. GiA sii k, d#t cue
(V = D^U, =
* »?(«У)  typ) ♦•») ®3>

Do dd (1) trd thAnh:
« dF>)(U,)(p) = 1 +^to)(«,}(p'(tJ)U,(p) +ш’<Л(1о)).П3(р))
- <e‘(V,»(p> + .яуи1(р) +ш’(А(1о)Иуи3(р)
“ W'cupxp) + «/{(ЛДрНрНр) +“i<Ui(P))p3(P)
- W'CU,) + +yyj(u1)Xp), v P e s,
EJ-	w«o Libman, S la mil c4u ban kinh rA,
IM
|	8.8. 1, Truong hOp 1, - m9i digm u	I ij , 0
» Л mat la mat w Phan lien thang ей. mat phing, 1 - k, • «
S -HlT'J4 тф‘ PhS° li6n tbe”g сйа c4u (Xem Doan Qoynh.
je Hlnh boo vi Phan-. NXBGD 1989, trang 257 - 2591
2) TWSng hop I) # ka Dung muc tteu chinh (U,, Ua> Ung
U,[k,) = (I2 - к,) . пДОц suy ra n,2 = 0 v4
,060 = d»' = -Ц Л <y| = V A V2 = л »2. Wc К = 0.
VI w5 = tW] = 0 пёп DoUj = «*(«) U2(p) + a»3 (<x) n(p) = 0, tiic
Uj la trudng vecto song song: chtfng to cung thing qua moi diem
cila mit vdi phuong Uj nam trdn mat. Vay mat la mat tru. Giao
cua mat trv nhy vdi mat phing thing gdc vdi dudng thing sinh
minh k^p) » kgf(p)) vdi mpi p e y. Sau dd chiing minh ring nffu
у li dudng trie dja trtn S thi у la dudng trie dia tren S
Z = n|^ la trudng muc ti6u Dicbu doc у = f(y) (co thg coi f la vi
phdi vi vfin d£ dang xet la cue ba) vi f ding ci/. Khi dd
kg = w|(T) vi kg = w2 (f^T) (xem baj 8.1). Nhung do f ding cu,
f*a>J = (xem chUng minh djnh И Gaoxo trong: Bohn Quynh.
Hinh hoc vi phin. NXBGD 1989, trang 269). V$y kg = wfCf.T) =
=. Гш2(Т) = W2(T) = kg.
Vay trtn mi€n mi tai dfiy xac djnh dupe trudng muc tifiu
§S. ANH XA DANG CU VA TUONG DUONG DOI HlNH
(u, v) •—» r2(u, v)
(u, v) —» r3(u, v)
chtfng minh.
2p 2q

(cosu, sinu.v) va mat phing cho bdi tham s6 hda
Ki£m tra thSy f la anh xa dang ctf, nhung khdng bAo t6n cac
cd ng la mat trong E3), f<y) = у la cung tren mat S. Ki hifeu kg li



Suy ra = 1. VAy f la vi phdi dang cU.
rd rang la vecto hang (khi (u, v) сб dinh), tdc
К t -* n,(/>(t)) la trudng vecto song song doc p.
-».• - Tinh ra K,(P(t) = -------7: d&y la bAng s6 W*i (u, v) сб din
n (chu)3
yAy t -* Kt(/>(t)) la ham hang (khi (u, v) сб dinh)
9.6. GiA sit f giao hoan duoc vdi Anh xa vaigacten. Th6 thl
ilp(«, 0) = hp(«).£ = f.(hp(<x)) f.QS) =
; ' = h,(p)	. f.QS) = H,(p)(f.««), f.yS)).
cfiu>, mat tdi tiAu (H = 0).

► P(u) — (shu, 0, 0) quay

2) Tinh ra Ц = -Np Et = Gt = (chu)2, F, = 0. Do dd dd cong
trung binh H, = 0; VAy mAt. S, t6i t£u.
Tl,(p)<f,(«),	= np(<x, 0), tile 14
h,(p)a.<o<)). f.^> = hp(«).^ = f.(bp<<x)). f,<|S))
ring nSu cd f : 3 - s la vi phdi ding ы bio tdn hudng tu mat
b 14n mat eJu S thi cd ddi hlnh сйа E’ biSn S thhnb S.
 VI f 14 vi ph« пёп s c(ing comp4<. 1Ио (h (Bhtf g) Vi ,
c4u S^td* hi° S “°® Ga°X° Ь‘”В “ C°”g G*0,° C“* m4‘
cau S, tuc bAng 1/R2.
•S vD40Sdathu d‘nh “ Libma"’ 8 14 m4t c4“ “n kinh R. V»y ci
° va S ddu 1A mat <>£..	__u_t_  , . „
CHUONG IV
DA TAP RIMAN HAI CHlfeu
§1. OA TAP M CHItU VA PHEP TlNH
GlAl TlCH THEN NO
1.1. 1) GiA sil V md trong M theo nghia d6i vdi m6t ho than»
ho {tj (Uj)} (i e I) la phu md пёп cd ho con {rk(Uk)}(k € к С I)
pbu V = V n r(U). Vdi mfi к e K, tap u’ = r-1 (V n r„(Uk))
mJ trong uk va anh xa rk th VR 16n Wk = r_|(V П rk(Uk)) C U
14 vi phdi. Suy ra Wk md trong U. Vay, tap r“*(V) = UkK Wk md
1.2. 1) Dua theo chuong I, § 3, bai 3.9, Ifiy tham $6 hda dja
phuong r: (u, v) e U -* r(u, v) e M, kl hi€u X, Y e Vec(U) sao
cho r.X = X|^u)t r. Y = Y^, thi bufic r.IIX, YD = IX, Y]|r(U,
va each xac dinh dd cua [X, Y] kh6ng phu thudc r.
2) Vi	=0 пёп tCr cku 1) suy ra [Ry, RJ = 0.
1.3.	Dua vko M thtfc (f.X) (f(p)) = Tpf(X(p)) trong dinh nghia
cda f.X vk suf dung Ьё thiic [X, Y][F)] = ХГВД] - Y[X[p]] trong
bai to£n 1.2.
s6 hda dia phuong (Xem cdng thtfc trong: Dokn Quynh, "Hinh hoc
vi phkn’, NXBGD 1989, trang 59).
1.5.	Xdt tham s6 hda dia phuong r: U - M ma f(t) e r(U).
Oi4 вй X Ш vi. Vi aS the vift U, « S {)Ruj ,4 do M ccS tM viet
U, = UlW(p(t» = § mt»^ (r-Mt)) nen as the
chon f de trong X(t) [P] = § p(t) U|(Pit»IpJ c4c V,(pW)tpj - 0,
Tieng UjV»(t))[p] = 1; goi p dd la р& th£ thi pHt) = X(t)[pJ kha
vi fvl X khA vi)
- Di5u nguoc lai: hi£n nhikn.
D$t Uj = {(x, y) e R2 I x2 + y2 < 1} (i = 1, 2.,., 6)
Dat Uj = R2 (i = 0, 1, 2).
Cfiu trie da tap hai chifiu trdn RP2 xic djnh bdi hp don 4nh
{ Г, : Uj - RP2) (i = 0, 1, 2) sau dky:
70 : IJO - RP2. <x, y) -♦ (1 : x : y);
17 BTH
( e r,(U,)|
7. : I). - RP3, <x, y> vft : 1  y)i
т2 : Ъ2 - RP3, (x, У) —U : У : 1).
-Bly gia t« chdng minh X 14 vi phdi dia phuong. Xet bSt kl
digm A = <xo, x,. x2> e r.CU,) thl x: » 0 v4 r^U,)!! ISn can
(trtn S3) cua A, con г2(Ъ2) 14 14n can (trtn HP3) cdaJt(A>. Th cd:
. Ddi vdi ip C“"e шо”«
. Gii rt ditn M = n, a™ M --
S2. DA TAP RIMAN HAI CHIEU

Mfit mat, л la song anh til r,(U,) lOn r2( U2).
Mat khac, tut tr6n ta thfiy г"1 о л о tj la vi phOi, trong liic do'
1 va Г| cung la nhutng vi phOi.
DS vdi cac didm roi vho etc rt(U|) (i  2, 3, ..., 6) cung tuong tu.
VAy л 14 vi phdi dia phuong.
1.7. - Xdt j~. Motungt, vi
14 dong cau tuyen tinh trSn T M хйс dinh bdi
= B(p), Jp(B(p)) = -A(p). RO ring Jp true giao vg
X e Vec(V) va X = fA + gB; thg thl f, g Id cic him nhln trtn V
Tit dinh nghia cda J(X) Buy ra J(XXq) = Jq(X‘q)I = J (ItA-tgBi =
= J,(«q)A(q) +g(q)B(q>) - Hq)Jq<A<q)> + gtq) J (B(q>> =
“ f(q) B(q) - g(q) A(q) -(ГВ - gAl(q), V q e V, tdc
J (X) = -gA + IB trtn V.
Do dd J(X) e Vec(V). V»y Jp phu thudc nhin v4o p.
Mpi tu dOng cAu tuyftj tinh true giao «, сйа khdng gian v*ct ’
dit hai chi*. T?2 а,,* ’ _
H
M<?i tu dOng c4v
Oclit hai chiOu I?2 m4
^2 e2> ₽h4» cd pCeJ = £
jgj. ca sd true chuin mi i
2> Cd <J(X). y> . <j2a)i J(Y)> . < _x , J(y)> <x J(Y„
tinh bio t6n gdc <p cua khdng gian vecto Oclit phii li inh xa ddng
dang, nghia la <(«), р(Д)> = k<oc, 0>.
2.3. 1) Cac nh&n xa. Tu l&i giii bii 2. 1.1) Suy ra;
(2) Vdi bit ki 0 e ТрМ, {Д Jp$)} la he true giao thuin;
2.	5. SU dung tham sd hda (u, v) —r(u,v) = (cosucosv, sinu.cosv,
star) (0 « U S 2»,	« V t |> cia S2. ТУЯ, ((u „ es2|
0 4 U 4 2л, 04 v 4 | } inh xa лог cd inh li toin bd RP2, vi
thu hep tren {(u,v) e R2 | 0 < u< 2л, 0 < v < ^} thi лог li mdt
vi phdi 1ёп mdt bd phin cua RP2. Vi the dien tich cua RP2 li
2)	Chiing minh a) => b). Do (1), tU f bio ton hudng vi bio giac
suy ra {Tpf(<x), Tpf(Jp(«)} li he true giao thuin; do (2), {Tpf(o<),
J{(p) (Tpf(<x))} cung li he true, giao thuin. Tit do, sut dung them
nhin xet (3), di tdi Tpf(Jp(oc)) = Jf(p)(Tpf(o<)).
§3. DANG LI&N КЁТ VA D6 CONG GAOXO
CUA DA TAP RIMAN HAI CHlfcU
3)	Gid thi& co b). Do (2), {Tpf(<x), Jr(p)(Tpf(cx)} la he true giao
thuin. Do dd, kit hop vdi gii thiet b) suy ra {Tpf(o<), Tpf(Jp(o<)}
vfiy f bio t6n hudng.
T£r b) ebn suy ra dude f bio giac.
3.1.	1) Dd dii cua cung mdt cung p trdn (M, <,
(M, <~ » tuong ling la s = ) «AW.
s = /{Ik/'W,	dt . Do dd s' = llks.
2)	СЬо К Id mien compfc vdi bd trtn M. Nbd lai Id hibu.
I -* M x M, q »—»<p, q) vi ding ciu tuyin tinh
jj(M x M) = ImTpjp+ ImT-ip.
Due vao T(p (M x M) tich vd hudng <<, >> nhu sau. NSu
trong dang ciu ndi trin mi «k - Т^Д) + Tp Д) e TpM,
Дк e TpM; к = 1, 2) thi dit «<xp о^з.» = <02 ,	+
260
GrCO’P’u. <ri)’u
vi edng thUc tinh dien tich tren К ciia (M, <, >):
A = Jds = 2 J -J , (r.J’J .dudv
(ddi vdi (M, <~ >) cung tuong tiQ, ta thiy ring пёи gpi A li diin
tich trdn К ейа (M, < ~ >) thi A = kA.
 • Uj4P.
3.3. Su dung bai tain cufi chuong 1 va bbl loan 3.4. §3, Chuong 1П
1) TO f(t)
thi dang thiic = dw| trd
K01 л e2 hay Kike1 л ike2 = K01 A e2, nghia la
hoc vi pb»n-. NXBGD 1989, trang 261, W df P
ОАО HAM thuAn bi£n va chuvEn doi song song
THEN DA TAP RIMAN HAI CHlEU
E, (ЛИ)
в, VW = E2v=ct) = — u2°AU, tile 10 A
i phAn’, NXBGD 1989, trang 307-308) ap dung ddi
3)	Ihrtmg vecto t -* X(t) = A(cos(t + B) UjQHt)) + sinft + B)
tJ2 (p(t))) ( A, В la cac hang s6) la trudng vecto song song doc p.
trong do oj2} la dang Нёп k<t vdi (M, <,>) trong trudng muc tifeu
true chuan {Up U,) tren rtUi Goi {0‘, 0;) la trudng doi muc ti£u

J <L4 = J K01 л 02 = Jtyo
r(C) r(C)	r(C)
(2)

r(C, r.;l
ТЙ (1), (2), (3) suy ra = Jk^0
4.6. t -* />(t) thufic mi€n ma tai day xac dinh trudng muc Нёи
true chu£n {Up U2} ; p = ^(to) ; ai^1 la dang Нёп ket vffi (M,
<,>) trong (U,. U;>; i-Xm = fl'1'’и, (Л1' - ----
Trudng nay song song doc P
(D2p' chi dao ham ri6ng theo bien thU hai).
Gii thia : X(t, t) = X(t) «=TP‘ct,»; =	H(«, 0 = r(‘>- v“
cac gii thiet do', di£u cin chdng minh.
(to) = lim----------—-------co nghia la

0) +	= lim—rz
‘Ч
(2)
Hay chiing minh (1)
, ph» tbuK t »h«4J luhn g6m gito	W
/(t)
55. CUNG TRAC D|A TR^ DA TAP RIMAN HAI CHl£'J
||x||hAng(doc/>).
BC tai C 14 у (vi cung BC la cung tidn trAc*Jia) Vfty I
сйа chuyfin dbi song song U2(C) doc chu vi tam gi6c CAB l d£ gdc
gifta t(U2(C)) va у (trong TCS2) bing gdc giixa U2(B) va $ (trong

TMtbi Wl’’,, ” й)‘й ь
W,,	1 X.
Do dd	(4) - f W-
И д 1» eung trie dia ndn vd phai bAng khdng Mi ’«
v~( = e I,, do dd vd trii bang khdng ddi vdi V t,e J,, Me
la Pj la cung trac dia.
Mat khAc, /»Jj = P-
Di6u do trAi vdi giA thift ring P la cung trie dja t& dai.
Vay p t6i dai.
5.5. GiA si Wj1 la dang lifin ket cua M trong {Up U2} va , G }
la twang ddi moo tidu cua (Up Up. Cd the' via »p = f'e1 + Л2
Khi dd : d»2' - di> Л S' + f dd' + df2 A e2 + f2dfi2
= dr1 Л s' + r1 (-«J1 л e2) + dPa e1 + P. (-»/ Л «>).
Ij
MVC Ltic
niXNl - rtM TAT Lf тнитйт vA de bAi tap
Chuang I
PH& TiNH С1Л1 TfCH TRONG KHONG GIAN OCL1T ЕП
VA HINH HOC VI PHAN ClIA En In = 2,3)

d«J(Up U2) = -df1 (Uj) + r‘. (-0>/<йр) + df2(U,) +
+ Г2 («,:(U2)) = -Vj If1] - f'- «2*ГОр + U, if2) - ft»,2(U2>
Nhung «РГОР = f, «2Wp = f2 ; mat khac, vi kp kj 14 ode
(k« * S “ S“y “
к,* - «2‘ГОР - -P, k2 = - «p(Up « -i2.
Do dd d«p (Up U2) = U2 [kp - k2 - UJkj] - k2
. U2Ik.) -	- k2 - 1$1)
Th lai ed: d«2‘ = KS'dd2,
d»p(Up u2> = Ks' л e2 (u„ up = k.
Tit (1) va (2) soy ra dang thde eSn chdng minh:
к = u2 (kp - upy - k2 - kj.
Chuang II
DUONG TRONG En (n = 2,3)
Cung hinh hoc vi da tap mOt chita
Chuang III
MAT TRONG E3
Chuang TV
DA TAP RIMAN HAI CHlfeU


PHAN U - GlAl НОДС TRA Ldl
РНЁТ TfNH GIA! TfCH TRONG KHONG GIAN O'CL IT e"
Chuang II
BUfoiG TRONG En (n = 2,3)
Chuang III
MAT TRONG E3
Chuang IV
DA ТДР RIMAN HAI CHlfeU