slide bai giang toan a3

5/28/2018 slide bai giang Toan A3

1/19

ThS.on Vng Nguyn Slide bi ging Ton A3DH

Trang 1

TON CAO CP A 3 I HCTi liu tham kho:

1. Gio trnh Ton cao cp A3 Nguyn Ph Vinh HCN TP. HCM.2. Ngn hng cu hi Ton cao cp Nguyn Ph VinhHCN TP.HCM.3. Gii tch hm nhiu bin (Ton 3) Cng Khanh (chbin) NXBHQG TP. HCM.4. Gii tch hm nhiu bin (Ton 4) Cng Khanh (chbin) NXBHQG TP. HCM.5. Php tnh Vi tch phn (tp 2) Phan Quc Khnh NXB Gio dc.6. Php tnh Gii tch hm nhiu bin Nguynnh Tr (chbin) NXB Gio dc.7. Tch phn hm nhiu bin Phan Vn Hp, Lnh Thnh NXB KH v Kthut.8. Bi tp Gii tch (tp 2) Nguyn Thy Thanh NXB Gio dc.

Chng 1. HM SNHIU BIN S

1. KHI NIM CBN1.1. nh ngha Cho 2D . Tng ng :f D ,

( , ) ( , )x y z f x y=

duy nht, c gi l hm s2 bin x v y. Tp D c gi l MXca hm sv

{ }( ) ( , ), ( , )f D z z f x y x y D= = l min gi tr.

Nu M(x, y) th D l tp hp im M trong 2 sao chof(M) c ngha, thng l tp lin thng. (Tp lin thng Dl tn ti ng cong ni 2 im bt ktrong D nm honton trong D).

Hnh a Hnh b

Nu M(x, y) th D l tp hp im M trong2

sao chof(M) c ngha, thng l min lin thng (nu M, N thucmin D m tn ti 1 ng ni M vi N nm hon tontrong D th D l lin thng-Hnh a)).

Trtrng hp 2D = , D thng c gii hn bi 1ng cong kn D (bin) hoc khng. Min lin thng Dl n linnu D c gii hn bi 1 ng cong kn (Hnha); a linnu c gii hn bi nhiu ng cong kn rinhau tng i mt (Hnh b).

D l min ngnu M D M D , minm

nu M D M D

.Ch Khi cho hm sf(x, y) m khng ni g thm th ta hiuMXD l tp tt c(x, y) sao cho f(x, y) c ngha. Hm sn bin f(x1, x2,, xn) c nh ngha tng t.

VD 1.Hm sz = f(x, y) = x3y + 2xy2 1 xc nh trn 2 .

VD 2.Hm s 2 2( , ) 4z f x y x y= = c MXl hnhtrn ng tm O(0; 0), bn knh R = 2.

VD 3.Hm s 2 2( , ) ln(4 )z f x y x y= = c MXlhnh trn mtm O(0; 0), bn knh R = 2.

VD 4.Hm s ( , ) ln(2 3)z f x y x y= = + c MXl namp mbin d: 2x + y 3 khng cha O(0; 0).

1.2. Gii hn ca hm shai bin Hm slin tc Dy im Mn(xn; yn) dn n im M0(x0; y0) trong

2 ,

k hiu 0nM M hay 0 0( ; ) ( ; )n nx y x y , khi n +

nu ( ) 2 20 0 0lim , lim ( ) ( ) 0n n nn n

d M M x x y y

= + = .

Cho hm sf(x, y) xc nh trong min D (c thkhngcha M0), ta ni L l gii hn ca f(x, y) khi im M(x, y)dn n M0nu mi dy im Mn(Mnkhc M0) thuc Ddn n M0th lim ( , )n n

nf x y L

= .

K hiu:0 0 0( , ) ( , )

lim ( , ) lim ( )x y x y M M

f x y f M L

= = .

Nhn xt Nu khi 0nM M trn 2 ng khc nhau m dy

{f(xn, yn)} c hai gii hn khc nhau th0

lim ( )M M

f M

.

VD 5.Cho

2

2

2 3 1( , ) 3

x y xf x y xy

= + , tnh ( , ) (1, 1)lim ( , )x y f x y .

VD 6.Cho2 2

( , )xy

f x yx y

=+

, tnh( , ) (0,0)

lim ( , )x y

f x y

.

VD 7.Cho hm s2 2

3( , )

xyf x y

x y=

+.

Chng t( , ) (0,0)

lim ( , )x y

f x y

khng tn ti.

Hm sf(x, y) xc nh trong D cha M0, ta ni f(x, y)lin tc ti M0nu tn ti

0 0( , ) ( , )lim ( , )

x y x yf x y

v

0 00 0

( , ) ( , )lim ( , ) ( , )

x y x yf x y f x y

= .

Hm sf(x, y) lin tc trong D nu lin tc ti mi imM thuc D. Hm sf(x, y) lin tc trong min ng gii niD th t gi trln nht v nhnht trong D.

VD 8.Xt tnh lin tc ca hm s:

2 2, ( , ) (0,0)

( , )

0, ( , ) (0,0)

xyx y

x yf x y

x y

+= =

.


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Trang 2

2. O HM RING VI PHN

2.1. o hm ringa) o hm ring cp 1

Cho hm sf(x, y) xc nh trn D cha M0(x0, y0). Nuhm s1 bin f(x, y0) (y0l hng s) c o hm ti x = x0th ta gi o hm l o hm ring theo bin xca f(x,y) ti (x0, y0).

K hiu: 0 0( , )xf x y hay/

0 0( , )

xf x y hay 0 0( , )

fx y

x

.

Vy / 0 0 0 00 00

( , ) ( , )( , ) limx

x

f x x y f x yf x y

x

+ =

.

Tng tta c o hm ring theo y ti (x0, y0) l:

/ 0 0 0 00 0

0

( , ) ( , )( , ) lim

yy

f x y y f x yf x y

y

+ =

.

VD 1.Tnh cc o hm ring ca z = x4 3x3y2+ 2y33xy ti (1; 2).

VD 2.Tnh cc o hm ring ca f(x, y) = xy

(x > 0).VD 3.Tnh cc o hm ring ca cos

xz

y= ti ( ; 4) .

Vi hm n bin ta c nh ngha tng t.

VD 4.Tnh cc o hm ring ca2

( , , ) sinx yf x y z e z= .

b) o hm ring cp cao Cc hm sfx, fyc cc o hm ring (fx)x, (fy)y, (fx)y,(fy)xc gi l cc o hm ring cp hai ca f.

K hiu: ( ) 22

/ /

2x xxx x

f ff f f

x x x

= = = =

,

( ) 22

//

2y yy yy

f ff f f

y y y

= = = =

,

( )2

//

x xy xyy

f ff f f

y x y x

= = = =

,

( )2

//

y yx yxx

f ff f f

x y x y

= = = =

.

VD 5.Tnh cc o hm ring cp hai ca3 2 3 4yz x e x y y= + ti ( 1; 1) .

VD 6.Tnh cc o hm ring cp hai ca2

( , ) x yf x y xe = .

Cc o hm ring cp hai ca hm n bin v o hmring cp cao hn c nh ngha tng t.

nh l (Schwarz)

Nu hm sf(x, y) c cc o hm ring fxyv fyxlin tctrong min D th fxy= fyx.

2.2. Vi phna) Vi phn cp 1

Cho hm sf(x, y) xc nh trong2

D v0 0 0( , )M x y D , 0 0( , )M x x y y D+ + .

Nu sgia 0 0 0 0 0 0( , ) ( , ) ( , )f x y f x x y y f x y = + + c

thbiu din di dng:

0 0( , ) . .f x y A x B y x y = + + + ,

trong A, B l nhng skhng phthuc ,x y v

, 0 khi ( , ) (0,0)x y , ta ni f khviti M0.

Biu thc . .A x B y + c gi l vi phn cp 1 (ton

phn)ca f(x, y) ti M0(x0, y0) ng vi ,x y .

K hiu df(x0, y0).

Hm sf(x, y) khvi trn min D nu f(x, y) khvi timi (x, y) thuc D.

Nhn xt

Nu f(x, y) khvi ti M0th f(x, y) lin tc ti M0. T

0 0( , ) . .f x y A x B y x y = + + + , ta suy ra:

0 0 0 0( , ) ( , ) .f x x y f x y A x x+ = +

0 0 0 0

0

( , ) ( , )limx

f x x y f x yA

x

+ =

,

tng t 0 0 0 00

( , ) ( , )limy

f x y y f x yB

y

+ =

.

Vy / /0 0 0 0 0 0( , ) ( , ). ( , ).x ydf x y f x y x f x y y= + hay/ /

0 0 0 0 0 0( , ) ( , ) ( , )

x ydf x y f x y dx f x y dy= + .

Tng qut:/ /( , ) ( , ) ( , ) , ( , )

x ydf x y f x y dx f x y dy x y D= + .

VD 7.Tnh vi phn cp 1 ca 2 3 5x yz x e xy y= + ti (1; 1).

VD 8.Tnh vi phn cp 1 ca2 2( , ) sin( )x yf x y e xy= .

nh l

Nu hm sf(x, y) c cc o hm ring lin tc ti M0trong min D cha M0th f(x, y) khvi ti M0.

b) Vi phn cp cao

Vi phn cp 2:

( )

2 2

2

// 2 // // 2

( , ) ( , )

( , ) 2 ( , ) ( , )xyx y

d f x y d df x y

f x y dx f x y dxdy f x y dy

=

= + +.

Vi phn cp n:

( )1 ( )

0

( , ) ( , ) ( , )k n k

nn n k n k n k

n x yk

d f x y d df x y C f x y dx dy

=

= = .


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Trang 3

VD 9.Tnh vi phn cp 2 ca 2 3 2 3 5( , ) 3f x y x y xy x y= + ti (2; 1).VD 10.Tnh vi phn cp 2 ca 2( , ) ln( )f x y xy= .

c) ng dng vi phn cp 1 vo tnh gn ng gi trhms

0 0

/ /

0 0 0 0 0 0

( , )

( , ) ( , ). ( , ).x y

f x x y y

f x y f x y x f x y y

+ +

+ + .

VD 11.Tnh gn ng1,02

0,97arctg .

2.3. o hm ca hm shp Cho hm sf(u, v), trong u = u(x) v v = v(x) l nhnghm sca x. Nu f(u, v) khvi ca u, v v u(x), v(x) kh

vi ca x th / /. .u vdf du dv

f fdx dx dx

= + . Vi , ,df du dv

dx dx dxl cc

o hm ton phn theo x. Nu hm sf(x, y) khvi ca x, y v y = y(x) l hm s

khvi ca x th / /.x ydf dy

f fdx dx= +

.

VD 12.Cho 2 22 , , sinxz u uv v u e v x= + = = . Tnhdz

dx.

VD 13.Cho 2 2 2( , ) ln( ), sinf x y x y y x= + = . Tnhdf

dx.

2.4. o hm ca hm sn Cho hai bin x, y tha phng trnh F(x, y) = 0 (*).Nu y = y(x) l hm sxc nh trong 1 khong no saocho khi thy(x) vo (*) ta c ng nht thc th y = y(x)l hm snxc nh bi (*).

VD 14.Xc nh hm sn y(x) trong phng trnh x2+ y2 4 = 0. o hm hai v(*) theo x, ta c:

// / /

/

( , )( , ) ( , ). 0 , ( , ) 0

( , )

xx y y

y

F x yF x y F x y y y F x y

F x y + = = .

VD 15.Cho 0x yxy e e + = . Tnh y .

VD 16.Cho 3 2 4( 1) 0y x y x+ + + = . Tnh y .

VD 17.Cho 2 2lny

x y arctgx

+ = . Tnh y .

Cho hm sn hai bin z = f(x, y) xc nh biF(x, y, z)) = 0, vi /( , , ) 0zF x y z ta c:

/ / /

//

/

/ / /

/

/

/

( , , ) ( , , ). ( , ) 0

( , , ) ( , ) ,

( , , )

( , , ) ( , , ). ( , ) 0

( , , ) ( , ) .

( , , )

x z x

xx

z

y z y

y

y

z

F x y z F x y z z x y

F x y zz x y

F x y z

F x y z F x y z z x y

F x y zz x y

F x y z

+ =

=

+ =

=

VD 18.Cho cos( )xyz x y z= + + . Tnh / /,x yz z .

3. CC TRCA HM HAI BIN S3.1. nh ngha Hm sz = f(x, y) t cc tr(a phng) ti imM0(x0; y0) nu vi mi im M(x, y) kh gn nhng khcM0th hiu f(M) f(M0) c du khng i. Nu f(M) f(M0) > 0 th f(M0) l cc tiuv M0l imcc tiu; f(M) f(M0) < 0 th f(M0) l cc iv M0l imcc i. Cc i v cc tiu gi chung l cc tr.VD 1.Hm sf(x, y) = x2+ y2 xy t cc tiu ti O(0; 0).

3.2. nh la) iu kin cn Nu hm sz = f(x, y) t cc trti M0(x0, y0) v ti hm sc o hm ring th:

/ /

0 0 0 0( , ) ( , ) 0

x yf x y f x y= =

.

Ch .im M0tha/ /

0 0 0 0( , ) ( , ) 0

x yf x y f x y= = c gi

l im dng, c thkhng l im cc trca z.b) iu kin .Gisf(x, y) c im dng l M0v co hm ring cp hai ti ln cn im M0.t 2 2

// // //

0 0 0 0 0 0( , ), ( , ), ( , )xyx yA f x y B f x y C f x y= = = .

Khi :+ Nu AC B2> 0 v A > 0 th hm st cc tiu tiim M0;

AC B2> 0 v A < 0 th hm st cc i ti im M0.+ Nu AC B2< 0 th hm skhng c cc tr(im M0c gi l im yn nga).+ Nu AC B2= 0 th cha thkt lun hm sc cc trhay khng (dng nh ngha xt).

3.3. Cc trtdoCho hm sz = f(x, y). tm cc trca f(x, y) trn MXD, ta thc hin cc bc sau:

Bc 1.Tm im dng M0(x0; y0) bng cch gii h:/

0 0

/

0 0

( , ) 0

( , ) 0

x

y

f x y

f x y

=

=.

Bc 2.Tnh 2// / /

0 0 0 0( , ), ( , )xyxA f x y B f x y= = ,

2

// 2

0 0( , )yC f x y AC B= = .

Bc 3.+ Nu > 0 v A > 0 th kt lun hm st cc tiu tiM0v cc tiu l f(M0);+ Nu > 0 v A < 0 th kt lun hm st cc i tiM0v cc i l f(M0).

+ Nu < 0 th kt lun hm skhng t cc tr.+ Nu = 0 th khng thkt lun (trong chng trnh hnchloi ny).

VD 2.Tm im dng ca hm sz = xy(1 x y).

VD 3.Tm cc trca hm sz = x2+ y2+ 4x 2y + 8.

VD 4.Tm cc trca hm sz = x3+ y3 3xy 2.

VD 5.Tm cc trca hm sz = 3x2y + y3 3x2 3y2+ 2.


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Trang 4

3.4. Cc trc iu kin Cho hm sz = f(x, y) xc nh trn ln cn ca imM0(x0; y0) thuc ng cong ( , ) 0x y = . Nu ti im M0

hm sf(x, y) t cc trth ta ni im M0l im cc trca f(x, y) vi iu kin ( , ) 0x y = .

tm cc trc iu kin ca hm sf(x, y) ta dngphng php khhoc nhn tLagrange.

Phng php khTphng trnh ( , ) 0x y = , ta rt x hoc y thvo f(x, y)

v tm cc trhm 1 bin.VD 6.Tm cc trca hm sf(x, y) = x2+ y2 xy + x + yvi iu kin x + y + 3 = 0.

VD 7.Tm cc trca hm sf(x, y) = xy vi iu kin:2x + 3y 5 = 0.

Phng php nhn tLagrange

Bc 1. Lp hm Lagrange:( , , ) ( , ) ( , )L x y f x y x y = + , l nhn tLagrange.

Bc 2.Gii h:'

'

'

0

0

0

x

y

L

L

L

=

=

=

im dng M0(x0; y0) ng vi 0.

Bc 3

Tnh 2 0 0( , )d L x y

2 2

'' 2 '' '' 2

0 0 0 0 0 0( , ) 2 ( , ) ( , )xyx yL x y dx L x y dxdy L x y dy= + + .

iu kin rng buc:

/ /

0 0 0 0 0 0( , ) 0 ( , ) ( , ) 0

x yd x y x y dx x y dy = + = (1)

v

(dx)2+ (dy)2> 0 (2).

Bc 4Tiu kin (1) v (2), ta c:+ Nu 2 0 0( , ) 0d L x y > th hm st cc tiu ti M0.

+ Nu 2 0 0( , ) 0d L x y < th hm st cc i ti M0.

+ Nu 2 0 0( , ) 0d L x y = th im M0khng l im cc tr.

VD 9.Tm cc trca hm sz = 2x + y vi iu kin x2+ y2= 5.VD 10.

Tm cc trca hm sz = xy vi iu kin2 2

18 2

x y+ = .

Chng 2. TCH PHN BI

1. TCH PHN BI HAI (KP)

1.1. Bi ton mu (thtch khi trcong) Xt hm sz = f(x, y) lin tc, khng m v mt mt trc cc ng sinh song song Oz, y l min phng ng Dtrong Oxy.

tnh thtch khi tr, ta chia min D thnh n phn khngdm ln nhau, din tch mi phn l Si(i=1,2,,n). Nhvy khi trcong c chia thnh n khi trnh. Trongmi Sita ly im Mi(xi; yi) ty . Ta c thtch Vicakhi trnhl:

1

( ; ) ( , )n

i i i i i i i

i

V f x y S V f x y S =

.Gi { }max ( , ) ,i id d A B A B S = l ng knhca iS .

Ta c:max 0

1

lim ( , )i

n

i i id

i

V f x y S

=

= .

1.2. nh ngha

Cho hm sz = f(x, y) xc nh trn min ng gii ni,o c D trong Oxy. Chia min D mt cch ty thnh nphn khng dm ln nhau, din tch mi phn l Si(i=1,2,,n). Trong mi Sita ly im Mi(xi; yi) ty . Khi

1

( , )n

n i i i

i

I f x y S=

= c gi l tng tch phnca hm

f(x, y) trn D (ng vi phn hoch Siv cc im Mi).

Numax 0

1

lim ( , )i

n

i i id

i

I f x y S

=

= tn ti hu hn, khng ph

thuc vo phn hoch Siv cch chn im Mith sIc gi l tch phn bi haica f(x, y) trn D.

K hiu ( , )D

I f x y dS= .

nh l.Hm f(x, y) lin tc trong min bchn, ng D thkhtch trong D. Nu tn ti tch phn, ta ni f(x, y) khtch; f(x, y) l hmdi du tch phn; x, y l cc bin tch phn.

Ch 1) Nu chia D bi cc ng thng song song vi cc trcta th S

i= x

i.y

ihay dS = dxdy.

Vy ( , ) ( , )D D

I f x y dS f x y dxdy= = .

2) ( , ) ( , )D D

f x y dxdy f u v dudv= .


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Trang 5

Nhn xt

1) ( )D

dxdy S D= (din tch min D).

2) f(x, y) > 0, lin tc (x, y) D th ( , )D

f x y dxdy l th

tch hnh trc cc ng sinh song song vi Oz, hai ygii hn bi cc mt z = 0 v z = f(x, y).

1.3. Tnh cht ca tch phn kp Tnh cht 1.Hm sf(x, y) lin tc trn D th f(x, y) khtch trn D.

Tnh cht 2.Tnh tuyn tnh:

[ ( , ) ( , )]D D D

f x y g x y dxdy fdxdy gdxdy = ;

( , ) ( , ) ,D D

kf x y dxdy k f x y dxdy k = .

Tnh cht 3

Nu chia D thnh D1v D2bi ng cong c din tchbng 0 th:

1 2

( , ) ( , ) ( , )D D D

f x y dxdy f x y dxdy f x y dxdy= + .

1.4. Phng php tnh tch phn kp

1.4.1. a vtch phn lpnh l (Fubini)

Gistch phn ( , )D

f x y dxdy tn ti, vi

1 2{( , ) : , ( ) ( )}D x y a x b y x y y x= v vi mi

[ , ]x a b cnh2

1

( )

( )

( , )

y x

y x

f x y dy tn ti th:

2 2

1 1

( ) ( )

( ) ( )

( , ) ( , ) ( , )

y x y xb b

D a y x a y x

f x y dxdy f x y dy dx dx f x y dy

= =

.

Tng t, 1 2{( , ) : ( ) ( ), }D x y x y x x y c y d= th:

2 2

1 1

( ) ( )

( ) ( )

( , ) ( , ) ( , )x y x yd d

D c x y c x y

f x y dxdy f x y dx dy dy f x y dx

= =

.

Ch

1) Khi {( , ) : , } [ , ] [ , ]D x y a x b c y d a b c d= =

(hnh chnht) th:

( , ) ( , ) ( , )b d d b

D a c c a

f x y dxdy dx f x y dy dy f x y dx= =

(hon vcn).

2)1 2

{( , ) : , ( ) ( )}D x y a x b y x y y x= v

f(x, y) = u(x).v(y) th:2

1

( )

( )

( , ) ( ) ( )

y xb

D a y x

f x y dxdy u x dx v y dy= .

Tng t,1 2

{( , ) : ( ) ( ), }D x y x y x x y c y d= th:

2

1

( )

( )

( , ) ( ) ( )

x yd

D c x y

f x y dxdy v y dy u x dx= .

3) Nu D l min phc tp th ta chia D ra thnh nhngmin n gin nhtrn.

VD 1.Xc nh cn tch phn lp khi tnh tch phn

( , )D

I f x y dxdy= trong cc trng hp sau:

1) D gii hn bi cc ng y = 0, y = x v x = a.2) D gii hn bi cc ng y = 0, y = x2v x + y = 2.

VD 2.

TnhD

I xydxdy= vi D gii hn bi y = x 4, y2= 2x.

i thtly tch phn

2

1

( )

( )

( , )

y xb

a y x

I dx f x y dy= 2

1

( )

( )

( , )

x yd

c x y

I dy f x y dx=


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Trang 6

VD 3.i thtly tch phn trong cc tch phn sau:

1)

21 2

0

( , )x

x

I dx f x y dy

= ;

2)

23

1 0

( , )

y

I dy f x y dx= ;

3)2 2

1 3 1

0 1

9 9

( , ) ( , )x

x x

I dx f x y dy dx f x y dy= + .

1.4.2. Phng php i bina) Cng thc i bin tng qutnh l. Gisx = x(u, v), y = y(u, v) l hai hm sc cco hm ring lin tc trn min ng gii ni Duvtrong mpOuv. Gi {( , ) : ( , ), ( , ), ( , ) }

xy uvD x y x x u v y y u v u v D= = = .

Nu hm f(x, y) khtch trn Dxyv nh thc Jacobi( , )

0

( , )

x yJ

u v

=

trong Duvth:

( , ) ( ( , ), ( , ))

xy uvD D

f x y dxdy f x u v y u v J dudv= .

Trong :/ /

/ // /

/ /

( , ) 1 1

( , )( , )

( , )

u v

x yu v

x y

x xx yJ

u vu v u uy y

x yv v

= = = =

.

VD 4.Cho min Duvl hnh tam gic O(0;0), A(2;0), B(0;2)trong mpOuv. Gi min Dxyl nh ca Duvqua php binhnh g: (x, y) = g(u, v) = (u+v, u2v).

Tnh tch phn ca hm1

( , )1 4 4

f x yx y

=+ +

trn min

bin hnh Dxy= g(Duv).VD 5.Cho min Duvl phn thnh trn n vtrongmpOuv. Gi min Dxyl nh ca Duvqua php bin hnhg: (x, y) = g(u, v) = (u

2v

2, 2uv). Tnh tch phn ca hm

2 2

1( , )f x y

x y=

+trn min bin hnh Dxy.

VD 6.Tnh din tch hnh phng gii hn bi bn Parapol:y = x2, y = 2x2, x = y2v x = 3y2.

b) i bin trong ta cc

i bin:cos

sin

x r

y r

=

=, vi 0, 0 2r

hoc .

Khi , min Dxytrthnh:

1 2 1 2{( , ) : , ( ) ( )}

rD r r r r =

v

/ /

/ /

cos sin( , )

sin cos( , )

r

r

x x rx yJ r

y y rr

= = = =

.

Vy ta c:

2 2

1 1

( )

( )

( , ) ( cos , sin )

( cos , sin )

xy rD D

r

r

f x y dxdy f r r rdrd

d f r r rdr

=

=

.

Ch

1) i bin trong ta cc thng dng khi bin D lng trn hoc elip.

2) tm 1 2( ), ( )r r ta thaycos

sin

x r

y r

=

=vo phng

trnh ca bin D.3) Nu cc O nm trong D v mi tia tO ct bin D khngqu 1 im th:

( )2

0 0

( cos , sin ) ( cos , sin )

r

r

D

f r r rdrd d f r r rdr

= .

4) Nu cc O nm trn bin D th:2

1

( )

0

( cos , sin ) ( cos , sin )

r

r

D

f r r rdrd d f r r rdr

= .

5) Nu bin D l elip th t:cos

{( , ) : 0 2 , 0 1}sin

r

x r aD r r

y r b

= =

=.

VD 7.Biu din tch phn ( , )D

f x y dxdy trong ta cc.Bit min D l min phng nm ngoi (C1): (x 1)

2+ y

2= 1

v nm trong (C2): (x 2)2+ y2= 4.

VD 8.Tnh din tch hnh ellip:2 2

2 21

x y

a b+ .

VD 9.Tnh tch phn2 2( )x y

D

I e dxdy +

= vi D l hnh trn2 2 2

x y R+ .

VD 10.Tnh din tch min D gii hn bi:

y = x, 2 2 2 23 3x y x y x+ = + v 0y .

Cng thc Walliss

2 2

0 0

( 1)!!,

!!sin cos

( 1)!!. ,

2 !!

n n

nn

nxdx xdx

nn

n

= =

le

chan

.


7/19


Trang 7

MT SMT BC HAI TRONG KHNG GIAN Oxyz Trong khng gian Oxyz, mt bc hai l tp hp tt cccim M(x; y; z) c ta tha phng trnh:Ax2+ 2Bxy + 2Cxz+ Dy2+ 2Eyz + Fz2+ 2Gx + 2Hy+ 2Kz

+ L = 0.

Trong A, B, C, D, E, F khng ng thi bng 0. Cc dng chnh tc ca mt bc hai:1) 2 2 2 2x y z R+ + = (mt cu);

2)2 2 2

2 2 21x y z

a b c+ + = (mt elipxoit);

3)2 2 2

2 2 21

x y z

a b c+ = (hyperboloit 1 tng);

4)2 2 2

2 2 21

x y z

a b c+ = (hyperboloit 2 tng);

5)2 2 2

2 2 20

x y z

a b c+ = (nn eliptic);

6)2 2

2 22

x yz

a b+ = (parabolit eliptic);

7)2 2

2 22

x yz

a b = (parabolit hyperbolic yn nga);

8)

2 2

2 21x y

a b+ = (mt treliptic);

9)2 2

2 21

x y

a b = (mt trhyperbolic);

10) 2 2y px= (mt trparabolic).


8/19


Trang 8

2. TCH PHN BI BA

2.1. Bi ton mu (khi lng vt th) Gista cn tnh khi lng ca vt thV khng ngcht, bit mt (khi lng ring) ti P(x, y, z) l

( ) ( , , )P x y z = = . Ta chia V ty thnh n phn khng

dm ln nhau, thtch mi phn l Vi(i=1,2,,n). Trongmi Vita ly im Pi(xi; yi; zi) v ng knh ca Vildi. Khi lng V xp x:

1 1

( ) ( , , )n n

i i i i i i

i i

m P V x y z V = =

= .

Nu tn timax 0

1

lim ( , , )i

n

i i i id

i

x y z V

=

th l khi lng m

ca vt thV.

2.2. nh ngha Cho hm sf(x, y, z) xc nh trong min o c V cakhng gian Oxyz. Chia min V mt cch ty thnh n phnkhng dm ln nhau, thtch mi phn l Vi(i=1,2,,n).Trong mi Vita ly Pi(xi; yi; zi) ty v lp tng tch phn

1

: ( , , )n

n i i i i

i

I f x y z V=

= .

Numax 0

1

lim ( , , )i

n

i i i id

i

I f x y z V

=

= tn ti hu hn, khng

phthuc vo cch chia V v cch chn im Pith sthcI c gi l tch phn bi baca f(x, y, z) trn V.

K hiu ( , , )V

I f x y z dV= .

nh l.Hm f(x, y, z) lin tc trong min bchn, ng Vth khtch trong V. Nu tn ti tch phn, ta ni f(x, y, z) khtch; f(x, y, z) lhm di du tch phn; x, y, z l cc bin tch phn.Nhn xt1) Nu chia V bi cc ng thng song song vi cc trcta th Vi= xi.yi.zihay dV = dxdydz.

Vy ( , , ) ( , , )V V

I f x y z dV f x y z dxdydz= = .

2) Nu ( , , ) 0f x y z trn V th ( , , )V

I f x y z dxdydz= l

khi lng vt thV, vi khi lng ring vt cht chim

thtch V l f(x, y, z).Nu f(x, y, z) = 1 th I l thtch V.

3) Tch phn bi ba c cc tnh cht nhtch phn kp.

2.3. Phng php tnh tch phn bi ba2.3.1. a vtch phn lpa) Gismin V c gii hn trn bi mt z = z2(x, y), giihn di bi z = z1(x, y), gii hn xung quanh bi mt trc ng sinh song song vi trc Oz. Gi D l hnh chiuca V trn mpOxy.Khi :

2

1

2

1

( , )

( , )

( , )

( , )

( , , ) ( , , )

( , , )

z x y

V D z x y

z x y

D z x y

f x y z dxdydz f x y z dz dxdy

dxdy f x y z dz

=

=

.

Nu 1 2{( , ) : , ( ) ( )}D x y a x b y x y y x= th:2 2

1 1

( ) ( , )

( ) ( , )

( , , ) ( , , )

y x z x yb

V a y x z x y

f x y z dxdydz dx dy f x y z dz= .

Nu 1 2{( , ) : ( ) ( ), }D x y x y x x y c y d= th:2 2

1 1

( ) ( , )

( ) ( , )

( , , ) ( , , )

x y z x yd

V c x y z x y

f x y z dxdydz dy dx f x y z dz= .

b) Gi D l hnh chiu ca V trn mpOxz.Gismin V c gii hn (theo chiu ngc vi tia Oy)bi hai mt y = y2(x, z) v mt y = y1(x, z), gii hn xungquanh bi mt trc ng sinh song song Oy.Khi :

2

1

2

1

( , )

( , )

( , )

( , )

( , , ) ( , , )

( , , )

y x z

V D y x z

y x z

D y x z

f x y z dxdydz f x y z dy dxdz

dxdz f x y z dy

=

=

.

Nu 1 2{( , ) : , z ( ) ( )}D x z a x b x z z x= th:2 2

1 1

( ) ( , )

( ) ( , )( , , ) ( , , )

z x y x zb

V a z x y x zf x y z dxdydz dx dz f x y z dy

=

. Nu 1 2{( , ) : ( ) ( ), e }D x z x z x x z z f= th:

2 2

1 1

( ) ( , )

( ) ( , )

( , , ) ( , , )

x z y x zf

V e x z y x z

f x y z dxdydz dz dx f x y z dy= .


9/19


Trang 9

c) Gi D l hnh chiu ca V trn mpOyz.Gismin V c gii hn (theo chiu ngc vi tia Ox)bi hai mt x = x2(y, z) v mt x = x1(y, z), gii hn xungquanh bi mt trc ng sinh song song Ox.Khi :

2

1

2

1

( , )

( , )

( , )

( , )

( , , ) ( , , )

( , , )

x y z

V D x y z

x y z

D x y z

f x y z dxdydz f x y z dx dydz

dydz f x y z dx

=

=

.

Nu 1 2{( , ) : , z ( ) ( )}D y z c y d y z z y= th:2 2

1 1

( ) ( , )

( ) ( , )

( , , ) ( , , )

z y x y zd

V c z y x y z

f x y z dxdydz dy dz f x y z dx= .

Nu1 2

{( , ) : ( ) ( ), e }D y z y z y y z z f=

th:2 2

1 1

( ) ( , )

( ) ( , )

( , , ) ( , , )

y z x y zf

V e y z x y z

f x y z dxdydz dz dy f x y z dx= .

c bit

Nu{( , , ) : , c , e }

[ , ] [ , ] [ , ]

D x y z a x b y d z f

a b c d e f

=

=

th:

( , , ) ( , , )

fb d

V a c e

f x y z dxdydz dx dy f x y z dz= .

VD 1.Tnh tch phn 8V

I xyzdxdydz= vi

V = [1, 2] [1, 3] [0, 2].

VD 2.Tnh tch phn lp2

1 1 2

1 0

(4 )

x

I dx dy z dz

= + v dng

min ly tch phn V.

VD 3.Tnh tch phnV

I ydxdydz= vi V gii hn bi

x + y + z 1 = 0 v 3 mt phng ta .

2.3.2. i bin tng qut

t

( , , )

( , , )

( , , )

x x u v w

y y u v w

z z u v w

=

= =

v

/ / /

/ / /

/ / /

( , , )

( , , )

u v w

u v w

u v w

x x xx y z

J y y yu v w z z z

= =

.

Giscc hm x, y, z c o hm ring lin tc trong minng, gii ni o c Vuvwtrong khng gian Ouvw v

0J th:

( , , )

( ( , , ), ( , , ), ( , , )). .

uvw

V

V

f x y z dxdydz

f x u v w y u v w z u v w J dudvdw=

.

VD 4.Tnh tch phn ( )VI x y z dxdydz= + +

vi

: 2V x y z x y z x y z + + + + + + .

VD 5.Tnh thtch ca khi elipxoit2 2 2

2

2 2 2:x y z

V Ra b c

+ + .

2.3.3. i bin trong ta tr

t

cos

sin

x r

y r

z z

=

= =

, vi

0, 0 2r hoc

.

Ta c:/ / /

/ / / 2

/ / /

( , , )sin

( , , )

r

r

r

x x xx y z

J y y y rr

z z z

= = =

.

Khi ta c:

( , , ) ( cos , sin , ). .

r zV V

f x y z dxdydz f r r z r drd dz

= .

VD 6.Tnh thtch khi V gii hn bi cc mt2 2 4x y z+ = , 2 2 2x y+ v z = 0.

VD 7.Tnh tch phn 2 2

V

I z x y dxdydz= + vi V l

min hnh trgii hn bi: 2 2 2x y y+ = , z = 0 v z = 1.

VD 8.Tnh tch phn 2 2 2( )V

I x y z dxdydz= + + vi V l

min hnh nn gii hn bi cc mt: 2 2 2x y z+ = v z = 1.

2.3.3. i bin trong ta cu

t

sin cos

sin sin

cos

x r

y r

z r

=

=

=

, vi

0, 0 2 , 0r hoc

.

Ta c:/ / /

/ / /

/ / /

cos sin 0( , , )

sin cos 0( , , )

0 0 1

r z

r z

r z

x x x rx y z

J y y y r rr z

z z z

= = = =

.

Khi ta c:

2

( , , )

( sin cos , sin sin , cos ). sin .

r

V

V

f x y z dxdydz

f r r r r drd d

=

.


10/19


Trang 10

3. NG DNG CA TCH PHN BI

VD 9.Tnh tch phn2 2 2

1

V

I dxdydzx y z

=+ +

vi V l

min gii hn bi cc mt cu:2 2 2 1x y z+ + = v 2 2 2 4x y z+ + = .

VD 10.Tnh tch phn2 2

( )V

I x y dxdydz= +

vi V lmin gii hn bi: 2 2 2 4x y z+ + v 0z .

3.1. Din tch, thtch(xem nhn xttch phn bi hai, ba).3.2. Gi trtrung bnh ca hm strn min ng Gi trtrung bnh ca hm sf(x, y) trn min ng D l:

1( , )

( )D

f f x y dxdyS D

= .

VD 1.Tnh gi trtrung bnh ca f(x, y) = xcosxy trong

hnh chnht 0 x , 0 1y . Gi trtrung bnh ca hm sf(x, y, z) trn min ng

l:1

( , , )( )

f f x y z dxdydzV

=

.

VD 2.Tnh gi trtrung bnh ca f(x, y, z) = xyz trong hnhlp phng [0, 2] [0, 2] [0, 2].

3.3. Khi lng Cho mt bn phng chim min D ng trong Oxy c khilng ring (mt khi lng) ti im M(x, y) thuc D lhm ( , )x y lin tc trn D. Khi lng ca bn phng l:

( , )D

m x y dxdy= .

Cho mt vt thchim min V ng trong Oxyz c khilng ring ti im M(x, y, z) thuc V l hm ( , , )x y z

lin tc trn V. Khi lng ca vt thl:

( , , )V

m x y z dxdydz= .

VD 3.Tnh khi lng bn phng chim min D gii hnbi 2 2 4x y+ , 0x v 0y . Bit khi lng ring l

hm ( , )x y xy = .

3.4. Momen tnhnh ngha Momen tnh ca mt cht im c khi lng m t tiim M(x, y) trong Oxy i vi trc Ox, Oy theo thtl:

My=0= my, Mx=0= mx. Momen tnh ca mt cht im c khi lng m t tiim M(x, y, z) trong Oxyz i vi cc mt phng ta Oxy, Oyz, Oxz theo thtl:

Mz=0= mz, Mx=0= mx, My=0= my.

Cng thc tnh Momen tnh ca bn phng chim din tch D trong Oxyc khi lng ring ti im M(x, y) l hm ( , )x y lin

tc trn D l:

0 0( , ) , ( , )y xD D

M y x y dxdy M x x y dxdy = =

= = .

Momen tnh ca vt thchim min V trong Oxyz c khilng ring ti im M(x, y, z) l hm ( , , )x y z lin tc

trn V l:

0

0

0

( , , ) ,

M ( , , ) ,

M ( , , ) .

z

V

x

V

y

V

M z x y z dxdydz

x x y z dxdydz

y x y z dxdydz

=

=

=

=

=

=

3.5. Trng tm Cho bn phng chim din tch D trong Oxy c khi lngring ti im M(x, y) l hm ( , )x y lin tc trn D. Khi

, ta trng tm G ca bn phng l:

( , )1

( , ) ,( , )

( , )1

y ( , ) .( , )

DG

D

D

DG

D

D

x x y dxdy

x x x y dxdymx y dxdy

y x y dxdy

y x y dxdymx y dxdy

= =

= =

Khi bn phng ng cht th ( , )x y l hng snn:

1 1, y

( ) ( )G G

D D

x xdxdy ydxdyS D S D

= = .

Cho vt thchim thtch V trong Oxyz c khi lngring ti im M(x, y, z) l hm ( , , )x y z lin tc trn V.

Khi , ta trng tm G ca vt thl:1

( , , ) ,

1y ( , , ) ,

1( , , ) .

G

V

G

V

G

V

x x x y z dxdydzm

y x y z dxdydzm

z z x y z dxdydzm

=

=

=

Khi vt thng cht th ( , , )x y z l hng snn:

1,

1y ,

1z .

G

V

G

V

G

V

x xdxdydzV

ydxdydzV

zdxdydzV

=

=

=

.

VD 4.Tm ta trng tm hnh phng D gii hn bi

0, 0, 1x y x y + . Bit ( , ) 2x y x y = + .VD 5.Tm ta trng tm ca vt thng cht chim thtch V gii hn bi mt nn 2 2 2z x y= + , 0z v mt cu

2 2 2 1x y z+ + = .


11/19


Trang 11

3.4. Momen qun tnhnh ngha Momen qun tnh ca mt cht im c khi lng m tti im M(x, y) i vi trc Ox, Oy v gc ta O theothtl:

Ix= my2, Iy= mx

2v IO= Ix+ Iy= m(x2+ y2).

Momen qun tnh ca mt cht im c khi lng m tti im M(x, y, z) i vi trc Ox, Oy, Oz v gc ta Otheo thtl:

Ix= m(y2+ z

2), Iy= m(x

2+ z

2), Iz= m(x

2+ y

2)

v IO= Ix+ Iy+ Iz= m(x2+ y

2+ z

2).

Momen qun tnh ca mt cht im c khi lng m tti im M(x, y, z) i vi cc mt phng ta Oxy, Oyz,Oxz thtl:

Iz=0= mz2, Ix=0= mx

2, Iy=0= my2.

Cng thc tnh

Cho bn phng chim din tch D trong mpOxy c khilng ring ti im M(x, y) l hm ( , )x y lin tc trn D.

Khi :

( )

2

2

2 2

( , ) ,

( , ) ,

( , ) .

x

D

y

D

O

D

I y x y dxdy

I x x y dxdy

I x y x y dxdy

=

=

= +

Cho vt thchim min V trong Oxyz c khi lng ringti im M(x, y, z) l hm ( , , )x y z lin tc trn V. Khi

:( )

( )

( )

( )

2 2

2 2

2 2

2 2 2

( , , ) ,

( , , ) ,

( , , ) ,

( , , )

x

V

y

V

z

V

O

V

I y z x y z dxdydz

I x z x y z dxdydz

I x y x y z dxdydz

I x y z x y z dxdydz

= +

= +

= +

= + +

v

2

0

2

0

2

0

( , , ) ,

( , , ) ,

( , , ) .

z

V

x

V

y

V

I z x y z dxdydz

I x x y z dxdydz

I y x y z dxdydz

=

=

=

=

=

=

VD 6.Tnh Ix, Iyca hnh D gii hn bi y2= 1 x, x = 0,

y = 0. Bit khi lng ring l ( , )x y y = .

VD 7.Tnh IOca hnh trn2 2 2 0x y Rx+ .

Bit 2 2( , )x y x y = + .

Chng 3. TCH PHN NG TCH PHN MT1. TCH PHN NG LOI I

1.1. nh ngha

Gisng cong L trong mt phng Oxy c phngtrnh tham s ( )x x t= , ( )y y t= vi a t b v f(x, y) l

hm sxc nh trn L. Chia L thnh n cung khng dm lnnhau bi cc im chia ng vi 0 1 ... na t t t b= < < < = .

Gi di cung thi li

s . Trn cung thi ly im

( , )i i iM x y . Tng 1 ( , )

n

n i i i

iI f x y s

== c gi l tng tch

phn ng (loi 1)ca hm f(x, y) trn ng cong L.

Gii hn0

1

lim ( , )i

n

i i imax s

i

f x y s

=

tn ti c gi l tch

phn ng loi 1ca f(x, y) trn ng cong L.

K hiu l ( , )L

f x y ds .

Nhn xt1) Tch phn ng loi 1 c tt ccc tnh cht ca tchphn xc nh.2) Tch phn ng loi 1 khng phthuc vo chiu ca

L:

( , ) ( , )

AB BA

f x y ds f x y ds= .

1.2. Phng php tnha) ng cong L c phng trnh tham s

Nu L c phng trnh ( )x x t= , ( )y y t= vi a t b

th:

( ) ( )2 2

/ /( , ) ( ( ), ( ))

b

t t

L a

f x y ds f x t y t x y dt= + .

Nu L trong khng gian c phng trnh ( )x x t= ,( )y y t= , ( )z z t= vi a t b th:

( ) ( ) ( )2 2 2

/ / /( , , ) ( ( ), ( ), ( ))b

t t t

L a

f x y z ds f x t y t z t x y z dt= + + .

b) ng cong L trong Oxy c phng trnh tng qut

Nu L c phng trnh ( )y y x= vi a x b th:

( )2

/( , ) ( , ( )) 1b

x

L a

f x y ds f x y x y dx= + .

Nu L c phng trnh ( )x x y= vi a y b th:

( )2

/( , ) ( ( ), ) 1

b

y

L a

f x y ds f x y y x dy= + .


12/19


Trang 12

c bit Nu L c phng trnh y = (hng s) vi a x b th:

( , ) ( , )b

L a

f x y ds f x dx= .

Nu L c phng trnh x = (hng s) vi a y b th:

( , ) ( , )

b

L a

f x y ds f y dy= .

c) ng cong L trong ta cc Nu L c cho trong ta cc ( )r r= vi

th ta xem l tham s. Khi , phng trnh ca L l

( )cosx r = , ( )siny r = , v:

( )2

2 /( , ) ( ( )cos , ( ) sin )L

f x y ds f r r r r d

= + .

VD 1.TnhL

zds vi L l ng xon c trtrn xoay c

phng trnh cosx a t= , siny a t= , z bt= , 0 2t .

VD 2.Tnh ( )L

x y ds+ vi L l tam gic c cc nh

O(0; 0), A(1; 0), B(0; 1).

VD 3.TnhL

xyds vi L l phn giao tuyn gia mt2 22 2z x y= v 2z x= nm trong gc phn tm th

nht tim A(0; 1; 0) n B(1; 0; 1).

1.3. ng dng

1) di cung L lL

ds , vi f(x, y) = 1 hoc f(x, y, z) = 1.

2) Nu dy vt dn c hnh dng L v hm mt khilng ( , )x y phthuc vo im M(x, y) trn L th khi

lng ca dy vt dn l ( , )L

m x y ds= .

Trng tm G ca L l:1

( , )G

L

x x x y dsm

= ,1

( , )G

L

y y x y dsm

= .

3) Nu dy vt dn c hnh dng L v hm mt khilng ( , , )x y z phthuc vo im M(x, y, z) trn L th

khi lng ca dy vt dn l ( , , )L

m x y z ds= .

Trng tm G ca L l:1

( , , )G

L

x x x y z dsm

= ,1

( , , )G

L

y y x y z dsm

= ,

1( , , )

G

L

z z x y z dsm

= .

VD 4.Tnh di cung trn 2 2 2 0x y x+ = nm trong

gc thnht tA(2; 0) n1 3

;2 2

B

.

VD 5.Cho mt dy thp dng na ng trn trong mpOyzvi phng trnh 2 2 1y z+ = , 0z . Bit mt khi lng

( , , ) 2x y z z = .

Tm khi lng v trng tm ca dy thp.

2. TCH PHN NG LOI II2.1. Bi ton mu

Tnh cng sinh ra do lc ( )F F M=

tc dng ln cht

im M(x, y) di chuyn dc theo ng cong L.Nu L l on thng AB th cng sinh ra l

( ). cos ,W F AB F AB F AB= =

.

Chia L thnh n cung nhbi cc im chia 0 1, ,..., nA A A .

Trn mi cung 1i iA A ly im Mi(xi, yi) ty . Chiu

( )i

F M

v 1i iA A

ln trc Ox, Oy ta c:

( ) ( , ). ( , ).i i i i iF M P i Q j = +

v 1 . .i i i iA A x i y j = +

.

Khi , cng W sinh ra:

[ ]

1

1 1

1

( )

( , ) ( , ) .

n n

i i i i

i i

n

i i i i i i

i

W W F M A A

P x Q y

= =

=

=

= +

Vy

[ ]1 0 1

lim ( , ) ( , )i i

n

i i i i i imax A A

i

W P x Q y =

= + .

2.2. nh ngha Cho hai hm P(x, y), Q(x, y) xc nh trn ng cong L.Chia L thnh n cung nhbi cc im chia 0 1, ,..., nA A A .

Trn mi cung 1i iA A ly im Mi(xi, yi) ty . Gi

( )1 ,i i i iA A x y =

. Tng

[ ]1

( , ) ( , )n

n i i i i i i

i

I P x Q y =

= + c gi l tng tch

phn ng (loi 2)ca hm P(x, y) v Q(x, y) trn ngcong L.

Gii hn1 0

limi i

nmax A A

I

tn ti c gi l tch phn ng

loi 2ca P(x, y) v Q(x, y) trn ng cong L.

K hiu l ( , ) ( , )L

P x y dx Q x y dy+ .

Nhn xt

1) Tch phn ng loi 2 c tt ccc tnh cht nhtchphn xc nh.

2) Tch phn ng loi 2 phthuc vo chiu ca L v khithay i chiu th ( )1 ,i i i iA A x y =

i du, do khi vit

tch phn ta cn ghi r im u v cui:

( , ) ( , ) ( , ) ( , )

AB BA

P x y dx Q x y dy P x y dx Q x y dy+ = + .


13/19


Trang 13

3) Tnh ngha tng tch phn, ta c thvit:

( , ) ( , ) ( , ) ( , )

AB AB AB

P x y dx Q x y dy P x y dx Q x y dy+ = + .

Nu L l ng cong phng, kn ly theo chiu dng(ngc chiu kim ng h) th ta dng k hiu:

( , ) ( , )

L

P x y dx Q x y dy+ .

nh ngha tng t:

( , , ) ( , , ) ( , , )L

P x y z dx Q x y z dy R x y z dz+ + .

2.3. Phng php tnha) ng cong L c phng trnh tham s Nu L c phng trnh ( )x x t= , ( )y y t= th:

/ /

( , ) ( , )

( ( ), ( )) ( ( ), ( )) .B

A

AB

t

t t

t

P x y dx Q x y dy

P x t y t x Q x t y t y dt

+

= +

Nu L trong khng gian c pt ( )x x t= , ( )y y t= , ( )z z t= :

/ /

/

( , , ) ( , , ) ( , , )

( ( ), ( ), ( )) ( ( ), ( ), ( )) .

( ( ), ( ), ( ))

B

A

AB

t

t t

t t

P x y z dx Q x y z dy R x y z dz

P x t y t z t x Q x t y t z t ydt

R x t y t z t z

+ +

+=

+

b) ng cong L trong Oxy c phng trnh tng qut

Nu L c phng trnh ( )y y x= th:

/

( , ( )) ( , ( )).

B

A

x

xxAB

Pdx Qdy P x y x Q x y x y dx + = + .

Nu L c phng trnh ( )x x y= th:

/( ( ), ). ( ( ), )B

A

y

y

yAB

Pdx Qdy P x y y x Q x y y dy + = + .

c bit Nu L c phng trnh y = (hng s) th:

( , ) ( , ) ( , )B

A

x

xAB

P x y dx Q x y dy P x dx+ = .

Nu L c phng trnh x = (hng s) th:

( , ) ( , ) ( , )B

A

y

yAB

P x y dx Q x y dy Q y dy+ = .

VD 1.TnhL

xdy ydx vi L l elip2 2

2 21

x y

a b+ = ly theo

chiu dng.

VD 2.Tnh ( ) ( )L

x y dx x y dy + + vi L l ng ni

im O(0; 0) vi A(1; 1) trong cc trng hp:a) ng thng y = x;b) ng y = x2;

c) ng y x= .

VD 3.TnhL

dx ydy dz + vi L l ng xon c trtrn

xoay c phng trnh cosx t= , siny t= , 2z t= tim

A(1; 0; 0) n (0; 1; )B .

2.4. Cng thc Green (lin hvi tch phn kp) Cho min D l min lin thng, bchn, c bin L Jordan

kn trn tng khc. Chiu dng ca L l chiu m khi dichuyn ta thy min D nm vpha tay tri.

Nu cc hm sP(x, y) v Q(x, y) c cc o hm ringcp 1 lin tc trn D th:

( )/ / ( , ) ( , )x yD L

Q P dxdy P x y dx Q x y dy = + .

Hqu1

( )2

D

S D xdy ydx

= .

VD 4.Tnh din tch ca elip2 2

2 21

x y

a b+ = .

VD 5.Tnh 2 2( ) ( 2 )y

L

xarctgx y dx x xy y e dx

+ + + + , vi L

l 2 2 2 0x y y+ = .

VD 6.Tnh 2 2L

xdy ydx

x y

+ trong cc trng hp:a) L l ng cong kn khng bao gc O;b) L l ng cong kn bao gc O.

2.5. iu kin tch phn ng khng phthuc ngly tch phnnh l Giscc hm sP(x, y), Q(x, y) v cc o hm ringcp 1 ca chng lin tc trong min n lin D. Khi , bnmnh sau tng ng:1)

/ /, ( , )y x

P Q x y D= .

2) ( , ) ( , ) 0L

P x y dx Q x y dy+ = dc theo mi ng kn L

nm trong D.

3)

( , ) ( , )

AB

P x y dx Q x y dy+ , trong AB nm trong D, ch

phthuc vo hai mt A, B m khng phthuc vo ngni A vi B.4) Biu thc P(x, y)dx + Q(x, y)dy l vi phn ton phn cahm u(x, y) no trong min D.


14/19


Trang 14

3. TCH PHN MT LOI I

Hqu Nu P(x, y)dx + Q(x, y)dy l vi phn ton phn ca hmu(x, y) no trong min D, ngha l / /, ( , )

y xP Q x y D=

th:

( , ) ( , ) ( ) ( )

AB

P x y dx Q x y dy u B u A+ = .

VD 7.Tnh2 2 2 2

L

x y x ydx dyx y x y

+++ +

vi L l ng trn

tng khc ni A(1; 1) v B(2; 2) nm trong min D khngcha gc ta O.

3.1. nh ngha Cho hm sf(x, y, z) xc nh trn mt S. Chia S mt cchty thnh n phn khng dm ln nhau, din tch mi phnl Si(i=1,2,,n). Trong mi Sita ly im ( , , )i i i iM

ty v lp tng tch phn1

( , , )n

n i i i i

i

I f S =

= .

Nu max ( ) 0 1lim ( , , )i

n

i i i id Si

I f S == tn ti hu hn, khngphthuc vo cch chia S v cch chn im Mith sIc gi l tch phn mt loi 1ca f(x, y, z) trn S.

K hiu ( , , )S

I f x y z dS= .

3.2. Phng php tnha) Chiu S ln Oxy Nu S c phng trnh z = z(x, y) v S c hnh chiu trnOxy l D th:

( ) ( )

2 2/ /

( , , ) ( , , ( , )) 1 x yS D

f x y z dS f x y z x y z z dxdy= + +

.

b) Chiu S ln Oxz Nu S c phng trnh y = y(x, z) v S c hnh chiu trnOxz l D th:

( ) ( )2 2

/ /( , , ) ( , ( , ), ) 1

x z

S D

f x y z dS f x y x y z y y dxdz= + + .

c) Chiu S ln Oyz Nu S c phng trnh x = x(y, z) v S c hnh chiu trnOyz l D th:

( ) ( )2 2

/ /( , , ) ( ( , ), , ) 1

y z

S D

f x y z dS f x y z y z x x dydz= + + .

VD 1.TnhS

I zdS= , trong S l phn mt nn2 2 2

z x y= + vi 0 1z .

VD 2.Tnh 2 2 2( )S

I z x y dS= + , trong S l phn mt

cu 2 2 2 4x y z+ + = vi 0x , 0y .

4. TCH PHN MT LOI II

3.3. ng dng ca tch phn mt loi 1

1) Din tch mt S lS

dS .

2) Nu mt S c hm mt khi lng l ( , , )x y z th

khi lng ca mt S l:

( , , )S

m x y z dS = .

Khi , ta trng tm G ca mt S l:1 1

( , , ) , y ( , , ) ,

1( , , ) .

G G

S S

G

S

x x x y z dS y x y z dSm m

z z x y z dSm

= =

=

4.1. nh ngha

4.1.1. Mt nh hng Mt trn S c gi l mt nh hngnu php vector

n v n

xc nh ti mi im M thuc S (c thtrbinS) bin i lin tc khi M chy trn S. Mt nh hng c

hai pha, pha m nu ng trn th n

hng tchn lnu l pha dng, ngc li l pha m.

Hng ca bin S l hng ngc chiu kim ng hkhi

nhn tngn ca n

.

Khi mt S khng kn, ta gipha trnl pha m n

lp vitia Oz gc nhn, ngc l lpha di.Khi mt S kn ta gipha trong vpha ngoi. Mt trn tng khc S l nh hng cnu hai phntrn bt kca S ni vi nhau bi ng bin C c nhhng ngc nhau.

4.1.2. nh ngha tch phn mt loi 2

Cho hm sf(x, y, z) xc nh trn mt nh hng, trntng khc S. Chia S mt cch ty thnh n phn khngdm ln nhau, din tch mi phn l Si(i=1,2,,n). Trongmi Sita ly im ( , , )i i i iM ty .

Gi Dil hnh chiu ca Siln Oxy km theo du dngnu Sic nh hng trn, ngc li l du m.

Lp tng tch phn ( )1

( , , ).n

n i i i i

i

I f S D =

= .

Nu ( )max ( ) 0

1

lim ( , , ).i

n

i i i id S

i

I f S D

=

= tn ti hu hn,

khng phthuc vo cch chia S v cch chn im MithsI c gi l tch phn mt loi 2ca f(x, y, z) trn mtnh hng S.

K hiu ( , , )S

f x y z dxdy .


15/19


Trang 15

Tng t, khi chiu S ln Ozx v Oyz ta c

( , , )S

f x y z dzdx v ( , , )S

f x y z dydz .

Kt hp c3 dng trn ta c tch phn mt loi 2cacc hm P, Q, R trn S:

( , , ) ( , , ) ( , , )S

P x y z dydz Q x y z dzdx R x y z dxdy+ + .

Nhn xt Nu i hng ca mt S th tch phn i du. Nu S kn th tch phn cn c k hiu l:

S

Pdydz Qdzdx Rdxdy+ + .

4.2. Lin hvi tch phn mt loi 1 Cho mt nh hng trn tng khc S c php vector n

v n

. Gi , , ln lt l gc hp bi n

vi cc tia

Ox, Oy, Oz. Khi :

( cos cos cos ) .

S

S

Pdydz Qdzdx Rdxdy

P Q R dS

+ +

= + +

Trong :

( ) ( )

( ) ( ) ( ) ( )

2 2/ /

2 2 2 2/ / / /

1cos ,

1

1 1cos , cos .

1 1

y z

x z x y

x x

y y z z

=

+ +

= =

+ + + +

4.3. Phng php tnha) Nu S c hnh chiu n trln Oxy l min Dxyv cphng trnh z(x, y) th:

( , , ) ( , , ( , ))

xyS D

R x y z dxdy R x y z x y dxdy= .

(du + hay ty thuc vo mt pha trn hay di).b) Nu S c hnh chiu n trln Oxz l min Dxzv cphng trnh y(x, z) th:

( , , ) ( , ( , ), )

xzS D

Q x y z dzdx Q x y x z z dzdx= .

c) Nu S c hnh chiu n trln Oyz l min Dyzv cphng trnh x(y, z) th:

( , , ) ( ( , ), , )

yzS D

P x y z dydz P x y z y z dydz= .

VD 1.TnhS

zdxdy , vi S l pha ngoi ca mt cu2 2 2 2

x y z R+ + = .

VD 2.Cho ( ) ( ) ( )S

I y z dydz z x dzdx x y dxdy= + + ,

vi S l pha ngoi ca mt nn 2 2 2x y z+ = , 0 4z .

Chuyn tch phn vloi 1 ri tnh I.

4.4. Cng thc Stokes Cho S l mt nh hng trn tng khc c bin S trntng khc v khng tct. GisP, Q, R l cc hm c ohm ring lin tc trong min mcha S. Khi :

( ) ( ) ( )/ / / / / / .S

y z z x x y

S

Pdx Qdy Rdz

R Q dydz P R dzdx Q P dxdy

+ +

= + +

(Hng ca S l hng dng ph hp vi hng ca S).

VD 3.TnhC

ydx zdy xdz+ + , vi C l ng trn giao

ca mt cu 2 2 2 2x y z R+ + = v mt phng 0x y z+ + =

v hng tch phn trn C l hng dng khi nhn tngn

tia Oz.

4.5. Cng thc Gauss Ostrogradski

Cho V l mt khi gii ni vi bin S trn tng khc. GisP, Q, R l cc hm c o hm ring lin tc trong minmcha V. Khi :

( )/ / /x y zS V

Pdydz Qdzdx Rdxdy P Q R dxdydz+ + = + + .

(Tch phnS

ly theo pha ngoi ca S).

VD 4.Tnh 3 3 3

S

x dydz y dzdx z dxdy+ + , vi S l pha

ngoi ca mt cu 2 2 2 2x y z R+ + = .

Chng 5. PHNG TRNH VI PHN HPHNG TRNH VI PHN CP I

1. KHI NIM CBN VPHNG TRNH VI PHN Mt phng trnh cha o hm hoc vi phn ca 1 hocvi hm cn tm c gi lphng trnh vi phn.

VD 1. y 2y = 1; (x + y)dy 2ydx = 0;

2y 3y + y = 0; 2 0dy dz

dx dx+ = .

Cp cao nht ca o hm cha trong phng trnh viphn (ptvp) c gi l cp ca ptvp .

VD 2. y = 3x v 2dy

xdx

= l ptvp cp 1;

y + 4y 3y = 0 l ptvp cp 2.

Dng tng qut ca ptvp cp n l F(x, y, y,, y(n)) = 0(*),nu t(*) ta gii c theo y(n)th ptvp c dng:

y(n)

= f(x, y, y,, y(n1)

). Nghim ca ptvp F(x, y, y,, y(n)) = 0 trn khong K l 1hm sy = (x) xc nh trn K sao cho khi thay y = (x)vo ptvp ta c ng nht thc trn K.Phng trnh vi phn c v snghim sai khc hng sC. Gii phng trnh vi phn l tm tt ccc nghim ca n. thca nghim y = (x) c gi l ng cong tchphn.

VD 3.Cc hm sy = ex, y = ex, y = C1ex+ C2e

xu l

nghim ca y y = 0.


16/19


Trang 16

2. PHNG TRNH VI PHN CP 1

2.1. Khi nim cbn vphng trnh vi phn cp 1 Phng trnh vi phn cp 1 l phng trnh c dng tngqut F(x, y, y) = 0 (*), nu t(*) ta gii c theo y th

y = f(x, y).

Gii ptvp cp 1 vi iu kin u y(x0) = y0l i tmnghim tha iu kin u, hay tm 1 ng cong tch phnca ptvp i qua im M0(x0; y0).

VD 1.Gii ptvp 0y x = , bit ng cong tch phn iqua im M(2; 1).

Nghim ca ptvp cha hng sC l nghim tng qut,nghim cha hng sC0cthl nghim ringv nghimkhng nhn c tnghim tng qut l nghim kd.

VD 2.

Tm nghim kdca ptvp 21y y = .

VD 3.Tm ptvp ca hng cong y = Cx2.

2.2. Mt sphng trnh vi phn cp 1 cbn2.2.1. Phng trnh vi phn cp 1 c bin phn ly Ptvp c bin phn ly c dng:

( ) ( ) (1)f x dx g y dy+ .

Phng php gii

Ly tch phn hai v(1) ta c nghim tng qut:( ) ( )f x dx g y dy C+ = .

VD 4.Gii ptvp2 2

01 1

xdx ydy

x y+ =

+ +.

Ch

Ptvp1 1 2 2( ) ( ) ( ) ( ) 0f x g y dx f x g y dy+ = (1) c a v

dng (1) nhsau:+ Nu g1(y0) = 0 th y = y0 l nghim ca (1).+ Nu f2(x0) = 0 th x = x0l nghim ca (1).+ Nu 1 2( ) 0, ( ) 0g y f x th:

1 2

2 1

( ) ( )(1') 0

( ) ( )

f x g ydx dy

f x g y + = (dng (1)).

VD 5.Gii ptvp ( 2)y xy y = + .

VD 6.Gii ptvp 2 3( 1) ( 1)( 1) 0x y dx x y dy+ + = .

VD 7.Gii ptvp xy + y = y2tha iu kin u1

(1)2

y = .

2.2.2. Phng trnh vi phn ng cp cp 1

Hm hai bin f(x, y) c gi l ng cp bc n nu vi

mi k > 0 th f(kx, ky) = kn

f(x, y). Chng hn, cc hm

( , )2 3

x yf x y

x y

=

+,

2

( , )2 3

x xyf x y

x y

=

+, f(x, y) = x2+ xy l

ng cp bc 0, 1, 2 tng ng.

Cho hm f(x, y) ng cp bc 0 hay ( , )y

f x yx

=

.

Khi , ptvp ng cp c dng:( , ) (2)y f x y = .

Phng php gii

ty

u y u xux

= = + .

(2) ( )( )

du dxu xu u

u u x

+ = =

( )( ) 0u u x

(ptvp c bin phn ly).

VD 9.Gii ptvp2 2

x xy yy

xy

+ = .

VD 10.Gii ptvpx y

yx y

+ =

vi iu kin u y(1) = 0.

2.2.3. Phng trnh vi phn ton phn Cho ptvp c dng: ( , ) ( , ) 0 (3)P x y dx Q x y dy+ = vi iu

kin / /x y

Q P= trong min phng D. Nu tn ti hm u(x, y)

sao cho du(x, y) = P(x, y)dx + Q(x, y)dy th (3) c gi lptvp ton phn. Nghim tng qut ca (3) l u(x, y) = C.

Phng php giiBc 1.T(3) ta c /xu P= (3a) v

/

yu Q= (3b).

Bc 2.Ly tch phn (3a) theo x:

( , ) ( , ) ( , ) ( )u x y P x y dx x y C y= = + (3c),

vi C(y) l hm theo bin y.

Bc 3.o hm (3c) theo y:/ / ( )y y

u C y = + (3d).

Bc 4.So snh (3b) v (3d) ta tm c C(y), thay vo(3c) ta c u(x, y).

VD 11.Cho phng trnh vi phn:2 2(3 2 2 ) ( 6 3) 0y xy x dx x xy dy+ + + + + = (*).

a) Chng t(*) l ptvp ton phn.b) Gii ptvi (*).

VD 12.Gii ptvp ( 1) ( ) 0yx y dx e x dy+ + + = .


17/19


Trang 17

2.2.4. Phng trnh vi phn tuyn tnh cp 1 Phng trnh vi phn tuyn tnh cp 1 c dng:

( ) ( ) (4)y p x y q x + = .

Khi f(x) = 0 th (4) c gi l ptvp tuyn tnh cp 1 thunnht.

Phng php gii (phng php bin thin hng sLagrange)

Bc 1.Tm biu thc( )

( )p x dx

A x e

= .

Bc 2.Tm biu thc( )

( ) ( ).p x dx

B x q x e dx= .

Bc 3.Nghim tng qut l [ ]( ) ( )y A x B x C= + .

Ch

Khi tnh cc tch phn trn, ta chn hng sl 0. Phng php bin thin hng sl i tm nghim tngqut ca (4) di dng:

( )

( )p x dx

y C x e= .

VD 13.Gii pt 2 0y x y = tha iu kin x = 3, y = e9.

VD 14.Gii pt sincos xy y x e + = .

VD 15.Gii pt 2( )y x y y + = .

2.2.5. Phng trnh vi phn Bernoulli Phng trnh vi phn Bernoulli c dng:

( ) ( ) (5)y p x y q x y + = .

Khi = 0 hoc = 1 th (5) l tuyn tnh cp 1. Khi p(x) = q(x) = 1 th (5) l phng trnh c bin phn ly.

Phng php gii (vi khc 0 v 1)+ Vi 0y , bin i:

1(5) ( ) ( ) ( ) ( )y y

p x q x y y p x y q xy y

+ = + = .

+ t 1 (1 )z y z y y = = th

(5) (1 ) ( ) (1 ) ( )z p x z q x + = (pt tuyn tnh cp 1).

Ch Phng trnh Bernoulli lun c nghim kdl y = 0.

VD 16.

Gii ptvp 2y

y xyx + =

vi iu kin x = 1, y = 1.

VD 17.Gii ptvp 3 22y xy x y = .

VD 18.Gii ptvp 3 sin 2dy dy

x y y xdx dx

+ = .

3. PHNG TRNH VI PHN CP 2

3.1. Cc dng phng trnh cbn3.1.1. Phng trnh khuyt y v y Dng phng trnh:

( ) (1)y f x = .

Phng php gii Ly tch phn hai v(1) hai ln.

VD 1.Gii ptvp 2y x = .

VD 2.Gii ptvp 2xy e = vi7 3

(0) , (0)4 2

y y= = .

3.1.2. Phng trnh khuyt y Dng phng trnh:( , ) (2)y f x y = .

Phng php gii t z = y a (2) vphng trnh tuyn tnh cp 1.

VD 3.Gii ptvpy

y xx

= .

VD 4.Gii ptvp ( 1) 01

yy x x

x

=

vi

y(2) = 1 v y(2) = 1.

3.1.3. Phng trnh khuyt x Dng phng trnh:

( , ) (3)y f y y = .

Phng php gii

t .dz dz dy dz

z y y z zdx dy dx dy

= = = = = a vpt

bin sphn ly.

VD 5.Gii ptvp ( )2

2 1yy y = + .

VD 6.Gii ptvp 2 (1 2 ) 0y y y + =

vi1(0) 0, (0)

2y y= = .

3.2. Phng trnh vi phn cp 2 tuyn tnh vi hshng3.2.1. Phng trnh thun nht Dng phng trnh:

1 20 (4)y a y a y + + = (a1, a2l cc hng s).

Phng php gii Xt phng trnh c trng ca (4): 2

1 20k a k a+ + = (5).

1) Trng hp 1:(5) c hai nghim thc phn bit k1, k2.Khi , (4) c hai nghim ring 1 21 2,

k x k xy e y e= = v

nghim tng qut l 1 21 2k x k xy C e C e= + .


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Trang 18

2) Trng hp 2:(5) c nghim kp thc k.Khi , (4) c hai nghim ring 1 2,

kx kxy e y xe= = v

nghim tng qut l 1 2kx kx

y C e C xe= + .

3) Trng hp 3:(5) c hai nghim phc lin hpk i = . Khi , (4) c hai nghim ring

1 2cos , sinx xy e x y e x = = v nghim tng qut:

( )1 2cos sinx

y e C x C x

= + .VD 7.Gii ptvp 2 3 0y y y + = .

VD 8.Gii ptvp 6 9 0y y y + = .

VD 9.Gii ptvp 2 7 0y y y + + = .

3.2.2. Phng trnh khng thun nht Dng phng trnh:

1 2( ) (6)y a y a y f x + + = (a1, a2l cc hng s).

Phng php gii Nu (4) c hai nghim ring y1(x), y2(x) th (6) c nghimtng qut l 1 1 2 2( ) ( ) ( ) ( )y C x y x C x y x= + .

tm C1(x) v C2(x), ta gii hWronsky:

1 1 2 2

1 1 2 2

( ) ( ) ( ) ( ) 0( ) ( ) ( ) ( ) ( )

C x y x C x y x

C x y x C x y x f x + = + =

.

VD 10.Gii ptvp1

cosy y

x + = .

nh l Nghim tng qut ca (6) bng tng nghim tng qut ca(4) vi 1 nghim ring ca (6).

VD 11.Cho phng trnh vi phn:22 2 (2 ) xy y y x e + = + (*).

a) Chng t(*) c 1 nghim ring l 2 xy x e= .

b) Tm nghim tng qut ca (*).

VD 12.Tm nghim tng qut ca ptvp:2sin 2 4cos2y y x x + = +

bit 1 nghim ring l cos2y x= .

nh l (nguyn l chng nghim)

Cho ptvp1 2

( ) ( ) ( ) ( ) (9)y p x y q x y f x f x + + = + .

Gisy1(x) v y2(x) ln lt l nghim ring ca

1( ) ( ) ( )y p x y q x y f x + + = , 2( ) ( ) ( )y p x y q x y f x + + =

th y(x) = y1(x) + y2(x) l nghim ring ca (9).

VD 14.Tm nghim tng qut ca ptvp 22cosy y x = .Bit:

1y y = c nghim ring 1y x= , cos2y y x = c

nghim ring 22 1

cos2 sin 210 10

y x x= .

4. HPHNG TRNH VI PHN CP 1

4.1. Khi nim cbn

Hphng trnh vi phn chun tc cp 1c dng:/

1 1 1 2

/

2 2 1 2

/

1 2

( , , ,..., )

( , , ,..., )

.................................

( , , ,..., )

n

n

n n n

y f x y y y

y f x y y y

y f x y y y

=

= =

,

trong x l bin sc lp v y1(x), y2(x),, yn(x) l cchm scn tm.

Bn hm s 1 2( , , ,..., ), 1,i i ny x C C C i n= = tha hptvp

l nghim.

Mi ptvp cp n dng ( ) ( 1)( , , , ..., )n ny f x y y y = u c th

a vdng hptvp chun tc cp 1 bng cch t( 1)

1 2, ,..., n

ny y y y y y

= = = .

Khi , ta c h:

/

1 2

/

2 3

/

1

/

1 2

..........

( , , ,..., )

n n

n n

y y

y y

y y

y f x y y y

=

=

= =

.

4.2. Phng php giia) Phng php kha vphng trnh vi phn cpcao

VD 1.Gii hphng trnh:5

4 5

y y z

z y z

= +

= +.

VD 2.Gii hphng trnh:y z

z y

=

=.

b) Phng php ma trn

/ /1 11 1 12 2 1 11

/ /22 21 1 22 2 2 2

//

1 1 2 2

...

...

.....................................................

...

n n

n n

nnn n n nn n

y a y a y a y yy

yy a y a y a y yA

yyy a y a y a y

= + + +

= + + + = = + + +

,

vi ( )ijA a= .Gisphng trnh c trng det( ) 0A I = c n nghim

phn bit , 1,i

i n = .

Vi mii

c vector ring1 2

( , ,..., )i i ni

p p p .


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Trang 19

Khi , hptvp c hnghim cbn l:1 1 1

2 2 2

11 11 21 21 1 1

12 12 22 22 2 2

1 1 2 2

, ,...,

, ,...,

................................................................

, ,...,n n n

x x x

n n

x x x

n n

x x x

n n n n nn nn

y p e y p e y p e

y p e y p e y p e

y p e y p e y p e

= = =

= = = = = =

v nghim tng qut l

1 1 11 2 12 1

2 1 21 2 22 2

1 1 2 2

...

...

.................................................

...

n n

n n

n n n n nn

y C y C y C y

y C y C y C y

y C y C y C y

= + + +

= + + +

= + + +

.

VD 3.Gii hphng trnh:2

4 3

y y z

z y z

= +

= +.

c bit Hptvp c dng

11

22

/11 1 11 11

/22 2 22 22

/

0 ... 0

0 ... 0

... ... ... ... ... ...... ...

0 0 ... nn

x

x

xnn n nn nn

y yy C e

y yy C e

y yy C e

= =

.

VD 4.Gii hphng trnh: 3

2

y y

z z

u u

=

= =

.

..Ht

slide bai giang toan a3

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