toan cao cap gioi han va lien tuc
TRANSCRIPT
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Trng i hc Bch khoa tp. H Ch MinhB mn Ton ng dng
-------------------------------------------------------------------------------------
Gii tch hm nhiu bin
Chng 1: Gii hn v lin tc
Ging vin Ts. ng Vn Vinh (2/2008)
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Mn hc cung cp cc kin thc c bn ca gii tch hm nhiu bin.Sinh vin sau khi kt thc mn hc nm vng cc kin thc nn tng:hm nhiu bin, gii hn kp v lin tc, o hm ring v vi phn, o
hm theo hng, khai trin Taylor, Maclaurint ca hm nhiu bin, ngdng ca o hm ring: phng trnh mt phng tip din, php vct,ng dng tm cc tr; cch tnh tch phn bi: bi 2, bi 3; tch phnng: loi 1, loi 2; tch phn mt: loi 1, loi 2 v cc ng dng hnhhc, c hc ca cc loi tch phn ny; tch phn suy rng ph thuctham s; trng vct.
Mc tiu ca mn hc Ton 3
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Gii hn v lin tc
o hm theo hng
ng dng ca o hm ring
Tch phn kp
Tch phn ng loi 1 v loi 2
Tch phn mt loi 1 v loi 2Trng vct
Tch phn bi ba
Tch phn ph thuc tham s
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Nhim v casinh vin.
i hc y (vng 20% trn tng s bui hc b cm thi!).
Lm tt c cc bi tp cho v nh.
c bi mi trc khi n lp.
nhgi, kimtra.
Thi gia hc k: hnh thc trc nghim (20%)
Thi cui k: hnh thc t lun + in kt qu (80%)
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Ti liutham kho
1. Cng Khanh, Ng Thu Lng, NguynMinh Hng. Gii tch nhiu bin.
NXB i hc qucgia
2.Ng Thu Lng, NguynMinh Hng. Bi tpton cao cp3.
4. James Stewart. Calculus, second edition, 2000.
5.www.tanbachkhoa.edu.vn
3. Cng Khanh. Giitch nhiu bin. NXB i hc qucgia
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Ni dung---------------------------------------------------------------------------------------------------------------------------
0.1Hm hai bin
0.2Cc khi nimtp trong Rn
0.4Gii hn
0.5lin tc
0.3Cc mt bchai
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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------
Nhit T ti mt imtrn b mttri t ti mt thi imtcho trc ph thucvo kinh x v v y ca imny. Chngta c thcoi T l mthm theo haibinx v y, k hiu
T = T(x,y)
V d
Thtch V ca mtbnh hnh tr ph thucvo bn knh yr v
chiucao h. Thc tta bit . Khi V l mthm haibintheo r v h:
2
V r h
2( , ) .V r h r h
V d
I H h i bi
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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------
D c gil minxc nh caf.
Cho . Hm haibinl mtnh x2D R
nh nghahm haibin
:f D R( , ) ( , )x y f x y
K hiu: ( , ).f f x y
{ | ( , ) : ( , )} E a R x y D a f x yMingi tr caf:
Nuf chobi biu thc i s: Minxc nhl tp hp tt ccc
gi tr cax v y, sao chobiu thcc ngha.
Mingi trl tp hp tt ccc s thcm hm c th nhn c.
I H h i bi
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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------
Minxc nh:
Hm haibinV d. 1
( , ) x y
f x yx y
{ }2( , ) | 1 0,D x y R x y x y
3 2 1(3,2) 6
3 2f
Hm haibinV d. 2 2
( , )f x y x y
Minxc nh: 2D R
Mingi tr: [0, )fE R
2 2 2 2( , ) ( ) ( ) 2( )f x y x y x y x y x y
2 2 2( , ) 2f x x x x x
I H h i bi
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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------
Minxc nh:
Hm haibinV d. ( , )1
xf x y
y
{ }2
( , ) | 1D x y R y
Hm haibinV d. 1
( , )1
f x yy
Minxc nh: { }2
( , ) | 1D x y R y
Mingi tr: {0}\fE R
Mingi tr: fE R
Hm haibinV d. neu
, neu
2 2
1
, ( , ) (0,0)( , )
0 ( , ) (0,0)
x ye x yf x y
x y
Minxc nh: 2D R
Mingi tr: [0,1)fE
II Tp trong R2
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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------
{ ) }2
0 0( , ) ( , ) | ( ,B M r M x y R d M M r
Hnh trn mtm , bn knh0 0 0( , )M x y 0r
{ }2 2 20 0( , ) | ( ) ( )x y R x x y y r
C mtln cn caM0 nm trntrong A, nghal ch cha nhng im caA. Khi M0 c gil imtrong ca tp A.
Hnh trn mny cng gil mtr-ln cn caM0v mi tpcon caR2 chamtr-ln cnno caM0 gil mtln cn caM0.
Xt mt im v mt tp . C th xyra ba trng hp loitrnhau sau y:
2
0M R
2
A R
C mtln cn caM0 nm trnngoi A, nghal hon ton khng chaimno ca A. Khi M0l mt imtrong ca phnb ca A.
Bt kln cnno caM0 cngc c nhng im caA v nhng imkhng thuc A. Khi M0l mt imbin ca A.
II Tp trong R2
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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------
Ch . 1) imtrong ca A l mt im thuc A.
2) imbin ca A c th thuc hockhng thuc A.
Mt tp hp c gil mnu mi im thucn ul imtrong can.
Mt tp hp c gil ng nu mi imkhng thucn ul imtrongca phnb can.
Mt tp hpl ng nu phnb can l m.
Mt tp hpl m nun khng cha imbin no can.
II Tp trong R2
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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------
imM0 c gil im t caA, nu miln cn caM0 u chav sim ca A.
imM0l im t ca tpA, nu miln cn can c chat nht mtim ca A khc viM0.
Ch . 1) im tc th thuc A, c thkhng thuc A.
2) C nhng tp hpkhng l tp ng, cngkhng l tp m.
Mt tp hpl ng nun cha tt ccc imbin can.
II Tp trong R2
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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------
V d.
Xt tp hpcc imtrong mt phng. Cho tp hp A
2 2 2
, 1A x y R x y
4. Tp A l tp m.
1.Tt ccc imtrong ca A: 2 2 2, 1x y R x y
2. Tt ccc imbin ca A: 2 2 2, 1x y R x y
3. Tt ccc im t ca A: 2 2 2, 1x y R x y
II Tp trong R2
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II. Tp trong R---------------------------------------------------------------------------------------------------------------------------
3. Tm tt ccc im t ca A.
V d.
Xt tp hpcc imtrong mt phng. Cho A l tp hpcc im nmtronghnh trn n vm ta cc iml cc s hu t.
{ }2 2 2( , ) | 1A x y xQ y
4. Tp A nghay m.
1. Tm tt ccc imtrong ca A.
2. Tm tt ccc imbin ca A.
p s: 1) Khng c imtrong
2) Tp imbin v im t bngnhau
{ }2 22( , ) | 1RA x y x y Tp imbin
4) A khng ng,khng m.
II Tp trong R2
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II. Tp trong R---------------------------------------------------------------------------------------------------------------------------
3. Tm tt ccc im t ca A.
V d.
Xt tp hpcc imtrong mt phng. Cho tp hp A
2n+3,n+1
211A Rn
4. Tp A nghay m.
1. Tm tt ccc imtrong ca A.
2. Tm tt ccc imbin ca A.
p s: 1) Khng c imtrong
2) C mt imbin l (1,2).
4) A khng ng,khng m.
3) C mt im tl (1,2).
III Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
T chngtrnh ton A2, v mt bchai:
Phngtrnh tngqut ca mt bchai trong h ta Descartes 0xyz l
2 2 2 2 2 02Ax By Cz Dxy Ex Gxz Fyz Hy Kz L
1) a dngtonphng(mu ) v dngchnh tc bng bin i trcgiao
2) Tm phpbin i,
xcnh trc ta mi
.
3) Vhnh.
III Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Tp hp tt ccc im(x,y) ca minxc nhDf, sao cho f(x,y) = k c gil
ng mc,trong k l hng scho trc.
k= 0
k= 1
k= 2
k= 3
k= 4
Xt th cahm s: 2 2z x y
III Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Mtparaboloid elliptic2 2
2 2
x yz
a b
III Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Mtparaboloid elliptic2 2
2 2
x yz
a b
III Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Mtparaboloid elliptic 2 2( 1) ( 3) 4z x y
III. Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Mtparaboloid elliptic 2 2y x z
III. Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
Mtellipsoid2 2 2
2 2 2 1
x y z
a b c
III. Cc mt bc hai
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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
MtParaboloid hyperbolic2 2
2 2
x yz
a b
III. Cc mt bc hai
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. Cc t bc a---------------------------------------------------------------------------------------------------------------------------
MtParaboloid hyperbolic
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
MtParaboloid hyperbolic 2 2y z x
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
MtHyperboloid 1 tng2 2 2
2 2 2 1
x y z
a b c
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
MtHyperboloid hai tng2 2 2
2 2 2 1
x y z
a b c
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
Ta thy vi mik, ng mclun l ngtrn bn knhbng1.
k= 0
k= 1
k= 2
k= -2
k= -1
Xt th cahm s: 2 2 1x y
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
Mt tr: trongphngtrnh thiu hocx, hocy, hocz.
2 2
2 2 1
x y
a b
III. Cc mt bc hai
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---------------------------------------------------------------------------------------------------------------------------
Mt tr: 2 2 4x z
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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---------------------------------------------------------------------------------------------------------------------------
Mt tr 2y x
x
z
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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Mt tr 2z x
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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Mt tr 22z x
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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Mt tr 22z x
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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Mtnn hai pha2 2 2
2 2 2
x x x
a b c
III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------
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Mtnn hai pha
IV. Gii hn---------------------------------------------------------------------------------------------------------------------------
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nh ngha gii hnkp
Cho hm haibin , sao cho l im t caDf.( , )f f x y 2
0 0 0( , )M x y R 0M
Ta ni gii hn cafkhi (x,y) dn n imM0bng , nugi tr caf(x,y) tingn n ty thchbngcch ly nhng im(x,y) gn imM0, nhngkhngtrng viM0.
aa
0 0( , ) ( , )lim ( , )
x y x yf x y a
2 20 0 0 00, 0: ( , ) ,( , ) ( , ), ( ) ( )fx y D x y x y x x y y
Khi ( , ) .f x y a
K hiukhc ca gii hnkp: 00
lim ( , )x x
y x
f x y a
. n------------------------------------------------------------------------------------------------------------------------
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---
Tnh cht ca gii hn
( , ) ( , ) ( , ) ( , ) ( , ) ( , )1. lim [ ( , ) ( , )] lim limx y a b x y a b x y a bf x y g x y f g
( , ) ( , ) ( , ) ( , ) ( , ) ( , )2. lim [ ( , ) ( , )] lim lim
x y a b x y a b x y a bf x y g x y f g
( , ) ( , )
( , ) ( , ) ( , ) ( , )( , ) ( , )
lim ( , )( , )
lim , lim 0( , ) lim ( , )
3. neu
x y a bx y a b x y a b
x y a b
f x yf x y
gg x y g x y
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , )lim lim , lim .
4. neu vath
x y a b x y a b x y a b
f x y g x y h x yf h M g M
. n---------------------------------------------------------------------------------------------------------------------------
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---
V d
Tm gii hn nu tn ti hoc chngminh khng tn ti.
( , ) (0,0)1lim sin
x y
I x yx
1 10 | ( , ) | sin | | sin | | f x y x y x y x y
x x
0
( , ) (0,0)
1lim sin 0.
x y
x yx
. n---------------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn nu tn ti, hoc chng t gii hnkhng tn ti.
2
2 2( , ) (0,0)
3lim
x y
x yI
x y
2 2
2 2 2 2
30 | ( , ) | 3 | |, v 1.
x y xf x y y
x y x y
0
2
2 2( , ) (0,0)
3lim 0.
x y
x y
x y
. n---------------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.
2 22 2( , ) (0,0)
2lim
x y
x yIx y
Chndy 1
( , ) ,0 (0,0)
nn nx y
n
Khi 1
( , ) ,0 1.
n nf x y fn
Chndy thhai ' ' 1
( , ) 0, (0,0)
nn nx y
n
Khi ' ' 1( , ) 0, 2.
n nf x y fn
Vy tn tihai dy dn n(0,0) nhnggi tr cafti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.
. n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.
2 2( , ) (0,0)lim
x y
xyIx y
Chn y kx
Khi ( , ) , .1
kf x y f x kx
k
f(x,y) l mt i lng ph thucvok, mkthay inn khng tn ti gii hn.
Ch . Chny =kx, tcl tin n(0,0)bng nhng ng thng.
Phngphp ny khng thdng tm gii hn cady.
. n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.
32 6( , ) (0,0)
lim
x y
xyIx y
Chndy 1
( , ) ,0 (0,0)
nn nx y
n
Khi 1
( , ) ,0 1.
n nf x y fn
Chndy thhai ' '3
1 1( , ) , (0,0)
n
n nx ynn
Khi ' ' 31 1 1( , ) , .2n nf x y f
nn
Vy tn tihai dy dn n(0,0) nhnggi tr ca fti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.
. n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.
2 22 2 2( , ) (0,0)
lim( )x y
x yIx y x y
Chndy 1
( , ) ,0 (0,0)
nn nx y
n
Khi 1
( , ) ,0 0.n nf x y fn
Chndy thhai ' '2
1 1 1( , ) , (0,0)
nn nx y
n n n
Khi ' ' 21 1 1 1( , ) , .2n nf x y f
n n n
Vy tn tihai dy dn n(0,0) nhnggi tr ca fti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.
. n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.
3( , ) (0,0)lim
1 1x yxyI
xy
t t xy Khi 0t
30lim
1 1t
tI
t
3
. n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.2
2( , ) (0,0)lim
9 3x yx yI
x y
t 2t x y Khi 0t
0lim
9 3t
tI
t
6
.
n------------------------------------------------------------------------------------------------------------------
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V d
Tm gii hn (nuc) hoc chng tkhng tn ti.4
2 2 2( , ) (0,0)lim
( )x yxyI
x y
S dng h ta cc, t cos ; sinx r t y r t
4 4
40
cos sinlimr
r t r t I
r
4
0lim( cos sin )r
I r t t
Khi th0; 0x y 0r
0I
.
n c------------------------------------------------------------------------------------------------------------------
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nh ngha
Hm sf(x,y) c gil lin tc ti , nu0 0( , )x y
0 0
0 0( , ) ( , )
lim ( , ) ( , )x y x y
f x y f x y
Hm c gil lin tc nun lin tc ti mi imm n xc nh
Tng, hiu,tch cahai hm lin tcl lin tc.
Thng cahai hm lin tcl lin tc nuhm mukhc 0.
Hp cahai hm lin tcl lin tc (ti nhng imthch hp).
.
n c------------------------------------------------------------------------------------------------------------------
h h
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nh nghaCc hm sau y c gilhm s cp c bn:
1) Hm hng; 2) hm m; 3) hm ly tha; 4) hm lnggic;5) hm lnggic ngc; 6) hm logarit.
nh ngha
Hm thu c tcc hm s cp c bn bng hu hncc phpton: cng, tr,nhn, chia, khai cn c gilhm s cp.
nhl
Hm s cplin tc ti nhng imm n xc nh.
. n c---------------------------------------------------------------------------------------------------------------------------
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V d
Tm cc imgin on cahm sau
( , ) xy
f x yx y
Suy ra nhng imgin on cahm sl ng thngx + y = 0.
yl hm s cp c bnnn lin tc ti nhng imm n xc nh
.
n c---------------------------------------------------------------------------------------------------------------------------
V d
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V d
Khost tnh lin tc cahm sau:
3 3
2 2sin( ) , ( , ) (0,0)
( , )
0, ( , ) (0,0)
x y x yf x y x y
x y
3 3 0
3 3
sin( ) sin1
tx y t
tx y
3 3 3 3 3 3
2 2 3 3 2 2
sin( ) sin( )x y x y x y
x y x y x y
3 3
2 20 | | | |
x yx y
x y
( , ) (0,0)lim ( , ) 1.0 0 (0,0)
x yf x y f
Vyhm cho lin tc ti mi imtrong mt phng.
Suy raf lin tc ti(0,0).
.
n c---------------------------------------------------------------------------------------------------------------------------
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V d
Tm tt ccc gi tr caa hm slin tc ti im(0,0)
2 2
2 2, ( , ) (0,0)
( , )
, ( , ) (0,0)
x yx y
f x y x y
a x y
Ta c khng tn ti.( , ) (0,0)
lim ( , )x y
f x y
Vyhm khng lin tc ti(0,0). Khng tn tia.
VI. Bi tp----------------------------------------------------------------------------------------------------------------
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1. Tm minxc nhv mingi tr cacc hm
2 21) ( , ) 4f x y x y
2
2) ( , )x y
f x y e
2 2
1
3) ( , ) x yf x y e
24) ( , ) ln( 4 8)f x y y x
5) ( , ) arcsiny
f x yx
2
16) ( , )f x y
x y
VI. Bi tp----------------------------------------------------------------------------------------------------------------
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2. Vcc mt bchai sau:
1) 3z
2 22) 2z x y
2 23) 1z x y
2
4) z x
25) 1z y
2 2
6) 1
9 4
x y
27) 1y x
2 28) 1z x y
2 29) 1z x y
10) 1x y z
11) 2z x
2 212) 2 2x y x y
2 213) 2x y z
2 2 214) 82x y z
VI. Bi tp----------------------------------------------------------------------------------------------------------------
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3. Vcc khisau:
2 21) ; 4x y z z
2 22) ; 1.x y z x z
2 23) 1; 1; 4.x y z z
2 2 2 2 2 2
4) 1; ; 4x y x y z x y z
25) 1 ; 0; 0; 2.y x y z z
2 22 2 2
6) 1; 4; 0.
9 4
x y
x y z z
2 27) 4 ; 2; 0; 4.z y z y x x
VI. Bi tp----------------------------------------------------------------------------------------------------------------3 khi ( i h )
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3. Vcc khisau (tiptheo)
2 28) ; 2; 2x y x x z x z
2 2 2 29) 2 ; ; 1; ;2 .y x y x x z x y z x y
210) 1 ; 3 ; 5; 0y x z x y z
2 2 2 211) ; 1 1z x y z x y
2 2 212) ; ; 1; 0.z x y y x y z
13) ; 2 ; 0; 6.y x y x z x z
2 2 2 214) 1; 2; 4; 1.x x y z y z
VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k
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4. Tm cc gii hnkp2
2 4( , ) (0,0)
1) limx y
x y
y x
( , ) (0,0)
12) lim os
x yy c
y x
3
3( , ) (0,0)
3) limx y
x y
x y
( , ) (0,0)
1 14) lim sin sin
x yx y
y x
2 2
2 2( , ) (0,0)5) limx y
x y xy
x xy y
2 2 2
2 2( , ) (0,0)
( )6) lim
1 os( )x y
xy x y
c x y
VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k (ti th )
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4. Tm cc gii hnkp (tiptheo)
2 2
1/
( , ) (0,0)
7) lim 1 x y
x y
xy
2 2
1/( )2 2
( , ) (0,0)8) lim os
x y
x yc x y
( , ) ( , )
9) lim sinx y
xy
xy
2 2 2 2
4 4 2 2 2 26( , ) ( , )
610) lim
2(1 )x y
x y x y
x y x y x y
2 2
( , ) (0,0)
111) lim ( )sin
x y
x y
xy
( , ) (0,0)12) lim
x y
x
x y
VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k (ti th )
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4. Tm cc gii hnkp (tiptheo)2 2
2
( , ) (2,1)
413) lim
2 2 4x y
x y
x x xy y
( , ) ( , )14) lim (1 )
x
x y k
y
x
2 2( , ) ( , )
15) limx y
x y
x y
( , ) (0,2)
sin( )16) lim
x y
xy
x
2 2( , ) (0,0)
17) limx y
x y
x xy y
2 2 2
( , ) (0,0)18) lim ln( )
x yx x y
VI. Bi tp----------------------------------------------------------------------------------------------------------------5 Tm cc im gin on
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5. Tm cc imgin on
2 21) lnz x y
2 2
12)
1z
x y
2
13)
( )
z
x y
1
4) coszxy
3
2 25)
xz
x y
2
2 26)
xz
x y
VI. Bi tp----------------------------------------------------------------------------------------------------------------5 Tm cc im gin on (tip theo)
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5. Tm cc imgin on (tiptheo)
2 2
2 2
2 2
, 07)
0, 0
x
x yx yz
x y
3 3
, 08)
3, 0
x y
x yz x y
x y
2
sin( ), 0
9)
, 0
xyzz
zu
x z
VI. Bi tp----------------------------------------------------------------------------------------------------------------6 Tm tt c cc gi tr m hm s lin tc ti (0 0)
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6. Tm tt ccc gi trm hm slin tc ti(0,0).
3 22 2
2 2
2 2
, 01)
, 0
x xy
x yx yz
m x y
22 2
4 2
2 2
, 02)
, 0
x yx y
x yz
m x y
2 2
2 2
2 2
2
, 03)
, 0
xy
x yx yz
m x y