пособие 6 по матану (2 курс)

M2: uxx = −(018, uxy =

36, uyy

= −(0108,

§- ç¨â J1 < 0, J2 =

u¢ x,⥫ìy -®, −¢x«î¡®©>

®ªà¥áâ-x®á⨠â®çª¨y

u x, y

−y <

§- ç¥- ª ª ¡®«ì訥

M1(0, 0) äã-ªæ ï u(x, y) ¯à¨-¨¬ ¥â

äã-ªæ¨ï

u 0),â®çª

¬¥-ì訥 u 0),¨¬¥§¥-â ç¨â, ¢ â®çª¥ M1

‚ â®çªu¥(x, y) -

¨¬¥¥â «®ª «ì-®£® íªáâ६㬠.

−3618

36

¢ áâଥ¥ ªá¨¬ã¬M2 äã-ªæ¨ï

«®ª «ì-ë© ¬ ª-

ᨬã¬. ‡-=ç¥648-¨¥>äã0.-ªæ¨¨‡- ç¨â¢

¯“á«®áâ¢- -¯®¢ª© íªáâৠ¤¥ç¨:¬ã¬-äã©â¨-ªæ¨¨íª ¬-㬮£¨å ¯¥uà(6¥¬, 3)¥--=ëå27..

¯

−108

¯

ãà¢ëà-¥- á¢ï§¨ f

(

1

, ...xn

x

ë

®à¤¨-®áâ 㤮 «¥â¢®àïîâ

¯à¨ ãá«®¢¨¨, çâ® ª®-

¬

ª®â®

g

1(

1

, ...xn

) = 0

(

1

, ...xn

x

, ..., gm

x

) = 0,

Ǒàאַ©ë¥ - ¬§ë¥â®¢ ¤овбпа¥ибвага¥-¨п¥¢¨вм¬г¬-:

¨ï¬¨§¨âì ¨§. ãà¢è¨å-¥-¨© á¢ï§¨

1á¢ï§¨,

ç¥à¥§ ¢¨âì-ë¥, ¯®¤áâ

¨å ¢

â®çª¨,

m ¯¥à¥¬¥--ëå

-¨î ¡¥§ãá«®¢- £® íª

äã-fªæ¨¨á¢¥®áâá⨠⥬ ááï¬ë¬

¤ çã ª - å® ¤¥-

1 Œ‘®áâá¨áâ¥â®¤ ¥-¬ëäã¨â-¥ªæ¨î‹‹«¥© £à£à-- :

n − m ¯¥à¥¬¥--ëå.

2

ˆ§

F (x

... xn) = f + λ1g1 + ...λmgm.

∂F

∂F

‹ £à - :

∂x1

= 0, ...,

∂xn = 0, g1

= 0, ... gm = 0 - ¢¬©â¨¥áâ®-® ¨â¥«¨

3

1

, ..., λm

ªà¨â¨ç¥áªãî â®çªã Œ = (

1

, ..., xn

λ

x

¤¨- вмвлгаªа¨в¨з¢- -¥¨пбª®©

¯®¤á⨧¯®«ã稢袨âì ). ¥©áï¯á¨áâ¥à¥¬¥¬ë¥--

-ëå©â¨Ǒத¨ää- ©¤¥--ë¥à¥¥ª®®à-æ¨à®¢

4

•

dx

, ..., dxm

.

©â¨1

5. Ǒ®¤áâ d¢¨âì2F . ¢

d2F

- ©¤¥--ë¥ §-

-¨ï λ1, ..., λm, x1, ..., xn, dx1, ..., dxm.

ä®à¬d2F, â®(¯®á«¥áâì¥ ¯®¯à¨¤áâ-¨¬-®¢ª¨)¥â ⮫쪯®«® ¯®«®¨â¥«ì¨â- ¥®¯à«ì ¥ë¤¥¥«§¥---ç¥ï¨ï- ¯à¨

6

,

-ᨬ㬠.

x

®âà¨æ®âà¨æ

- ï

d2F, â®

-

¥å

ª¢ ¤à â¨ç¢á

¤áâ-¨¬-®¢ª¨)¥â

-ë¥ §-

-ï¨ï

â® ªà¨â¨ç¥áª ï â®çª

Œ â®çª

ãá«®¢-®£® ¬ª¢

.

ɇǬ0

=

,

x

6

Ǒਬáâì¢ â®çª¥à¥23Œ.¯а¨(¯®б«Œг¥-в¨¬¤®¬¢¥-¯®¤бв®вм¨бª«оз£® нªбва-®¢ª¨)¯®«®¥¥¬г¬-¨п-¨в¥®¯а-«ма¥в.¥¥¬-¤л¥¥--«¥,¥ле--ª-п®ва©в¨¤ажнªбвавв¨¥«м¥-¬ã¬ë¥ ä®à¬§äã- ç-¥ªæ¨¨-,¨ï,â®

¥â®

d F

u = x + y + z2

á¨áâ¯à¨ ¥¬ãá«®¢¨ïå á¢ï§¨:

z − x

= 1,

•¥è¥-¨¥: •¥è

®â-®á¨â¥«ì-½

y − xz

.

®

= 1

y ¨ zâ®à®©- å ¤¨¬ z = x + 1, y = x2 +

ï í⨠¢ëà ¥-¨ï áâࢥ¥¬ã¬-á⢮ ¤«ï

u

, ¯à¨å®¤¨¬ ª äã-ªæ¨¨

x +-®©1. Ǒ®¯à¤á⥬¥--¢«ï©:

® ¡¥§ãá« ¢ ®¬ íªáâàu¥(¬ã¬x) =.2’¥ x.2ª+. 4x + 2, ¤«ï ª

à áᬠâਢ ¥¬ § ¤ çã

′

=.

1, â®

(

) ¨¬¥¥â

¥¤¨-á⢥--ãî â®çªã ¢®§¬® -®£®u íª= 4(x +1)

0Ǒ®áª®«ìªã¯à¨x = −

′′

u

x

u

(−1) = 4 > 0,

u

x

+ y −

2

u

x

umin = 0 ¯à¨ x = 0

¯à¨ y = 1.

â®ç− (¥yª −íªáâà

3

2

1)2¥;¬ã¬

-¥â.

4

u

(x-−¥áâà®y +£1)¨©2¬¨; -¨¬ã¬

u = 0 ¢ â®çª å ¯àאַ© x − y + 1 = 0.

u x − xy + y2 − 2x + y;

5

umin = −108¯à¨ x ᨬã¬=1¯à¨ y = 0.

u x2y3(6 − x − y)

0

umax

¯à¨ x = 2 ¯à¨ y = 3;-¥áâண¨© ¬¨-¨¬ã¬ z = 0 ¯à¨ x = 0,

6)

< y < 6; -¥áâண¨© ¬ ª

z = 0 ¯à¨ x = 0, −∞ < y < 0

6 < y < +∞.

u

x3

y3 − 3xy;

¯à¨

umin = −

¯à¨ x1 = −1,y1 = −1 ¨ ¯à¨ x2 = 1,y2 = 1; íªáâ६㬠-¥â

7)

x = 0, y = 0.

u x4

y4 − x − 2xy − y2;

8

umin

= −1 ¯à¨ x = 1 ¨ ¯à¨ y = 1.

u

xy +

50

+

(x > 0, y > 0);

1

1

x

20

y

(9

umin = 30

x

5

y

2.

1 −

p

u

x2 + y ;

0

umax = 1 ¯à¨ x = 0 ¨ ¯à¨ y =0);.

q

1

u

− a2

3−ab

b2

a x

y

3ab

x = y

xy

x

y

(

> 0, b >

umin

1

1

1

√3 ;umax =

1

1

1

1

=

−

√

3

¯à¨ a

=

−

b =

±

√

3

¯à¨ a

b

=

±

√

3

.

u

x2−y

e ᥤ«®(5 − 2x + y);

2

u = e3 ¯à¨ x = 1, y = 256− .

u xyz(1 − x − y − z)

3

umax = u(1/4, 1/4, 1/4) = 1/

x2 + y2

2= −1/8

u 2

− 4x + 2z2

y

z

(14)

umin

u(1/4, 1/24, 12/4)

Žâ¢¥â:

−(x +y +z )

u = (x + y + 2z)e

3

3

• ©â¨ ãá«®¢-ë¥

√

√

√

q e

√

−

√

√

−q e

u

max

u

2

3 ,

2

3 ,

3

) =

; u

= u 2

3 ,

2

3 ,

3 ) =

íªáâ६ã¬ë äã-ªæ¨¨min

-®(15)£® ãà

- -¨ï á¢ï§¨:

u = f (x; y) ®â-®á¨â¥«ì-® § ¤ --

u x ¬¨-yᨬã¬, x

y −

7)

u(18/13; 12/13) = 36/13.

8

u(2; 1) = 3.

(19)

u(1; 0) = 0, ¬ ªá¨¬ã¬ u(1/3; 1/3) = 1/27

Žâ¢=¥â:os2

x-+

os2

4 = 0;

u

y, x − y − π/

1/√

2, ¬ ªá¨¬ã¬ë

Žá-ª

u(5π/8 + πk; 3π/8 + πk) = 1

−

u(π/8 +

Ǒà¨-⮢¨«¨¯®âà-®©¯à¨¢¥‡áᬮâàä¡€-®áâìà¬ã«®©,Œ¥á⨅‘•¥-¢ë¯®«—¥¨¨€£€®¨á¯®«ì§ãǑ¢ëà‘’•›Œˆª-…¨âì¤ã,…2¥.-¡®«¥¨©¬…••›•¬®©¥-¥á¥ã¥Ǒ•Žˆ‡¬ç¯à¨¯ã¤®¡¥áâà3¥-í⮬,.¬-ë‚¥--¨ëå,Ž„•›Œˆ›•¯à®¨§¢®¤ï¢«ï¤«ïç⮡뤀¥âáì…ä®à¬ã«ã¯à®áâ¨âì묨•ˆŸ•¥©è¥£ç®á⮨§¤«ï碮§à¥¯à®-¨ï¥.

¨ ¥ ®

®¬ã‚ - äã ã - -

πk; −π/8 + πk) = 1 + 1/√

¨§¢®¤-

á« -®© äã-ªæ ¨:

¯à¨çá⥠-ë¥ ¯à®¨§¢®¤-ë¥ ∂z ¨ ∂z

á«® -®©

u0, v0)

’¥®à¥¬

1 (¯à®¨§¢®¤¨ä- ï á«® -®© äã-ªæ¨¨). …᫨

-ªæ¨¨ x

x u, v ,

y

= y u, v)

ä¥à¥-æ¨àã¥¬ë ¢ â®çª¥ (u0, v0),

-ªæ¨ï z

=

(

¢ â®çª) ¤¨дд¥ ¥(а¥-ж¨агбгй¥¬¥бв¢говв®зª¥ (

0

, y

0), £¤¥

x

0 =

(

0

, v

0),

y

0 =

y

(

0

, v

0),

fâ®x, y

x

x

u

u

(1)

äã-ªæ¨¨

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

¬

∂u

∂v

∂z = ∂z ∂x + ∂z ∂y

∂z = ∂z ∂x + ∂z ∂y

A = F (x, y, z, ∂z , ∂z ) âॡã-

--®,¤«ï¨¬¥«î¡®îâ ¬¥£áâ®® ç¨á«"®¡à¯¥àâ¥-¬ë¥--¥"ëåà.¢¥-á⢠:

∂y ∂v

(2)

…€áâ- ¥«®á⢣¨ç¥ ∂u

∂x ∂u

∂y ∂u

∂v

∂x ∂v

∂u , ∂u , ∂v , ∂v

. Ǒà¨

∂z = ∂z

∂u

+ ∂z ∂v

∂z = ∂z ∂u + ∂z ∂v

’ •¯ 1áá. ‚®âਬ¤¨ää∂x¤¢¥à¥-⨯æ¨∂u ∂x«ìç-áâ®®¬∂v¢ëà¢áâà∂x ¯¥ç¥à-îé¨åá﨨¥¬∂y¥¢¨¤∂u§ ∂y¤

ç. ∂v ∂y

¥âáï § ¬¥-¨âì ⮫쪮 -¥§ ¢¨á¨¬ë¥

--ë¥

∂x

∂y

€«1.£®à¨â¬.• ©â¨ à¥è¥-¨ï:

x ¨ y - -®¢ë¥ ¯¥à¥¬¥--ë¥

u

v

) ¥á«

¤∂x -ë∂y ¢ëà∂x

∂y¥-¨ï "-í⮬ëå" ¯¥à¥¬¥--ëå ç¥à¥§ "áâ àë¥":

u =

uáâ¢(x,¥--y),® v¨§=¤ v--(x,ëåy),à ⮢¥-âàá⥢¡ã. ¥¬ë¥ ¢¥«¨ç¨-ë - 室ïâáï -¥¯®á।-

(1), ¯®«ãç âì ¢ëà -¨ï ¤«ï

∂x

∂y

∂x

∂y ¢ ä®à¬ã«ë

xé¨(¯®u,¥ v), y = y(u, v), â®, ¯à®¤¨ää¥à¥-æ¨à®¢®¢ë¥ ¤ --ë¥ ®â®à®©¢¥-áâ¢

x

¯® x

y ª ¤ ¥, à áᬠâਢ ï ¯à¨ í⮬¯à®¨§¢®u v ª ª ¢¥«¨ç¨-ë, § ¢¨áï-

2

âà¤á⥡ãx¢¨â쥬®«ïïy¥,.¯®«ãç«ã稬¥--á¨áâë¥ ¢ë६ã ç¥âëà-¨ï¥å¤«ïãà

- -¨©, ¨§ ª

- ©¤¥¬

∂u , ∂u

∂v ,

∂v

∂z

∂z

Ǒ®¥. ¢‚¨¤¨ääåï-ã®á«®¢¨¥à¥¢ëà-樥,«ì¥--®¬¨¥. ¢ëà áâ¥-ç-¥à먨॥§¢¯-¥¤à¥¬¥¤--¯¥ë客ë६.祥¥--à¥ë§ ¥-®¢ë¨ ¥ .¯¥à¥

’¨¯¬¥--43. ëˆá¯®«ì§ã2

¨áª«îç¨âì∂x ∂y

u

v

∂z ) âà¥-

¡ã¥âáï

¬¥-¨âì -¥§

á¨¬ë¥ ¯¥à¥¬¥--ë¥

A = F (x, y, z, ∂z ,

∂x

∂y

x ¨ y -

1.

¯¥à¥¬¥--ë¥ u ¨

Ǒਬ¥-à¢ë©ªæ¨î¥-à¨âì¥è(£¥ä®à¬ã«ë-â¨ï:)ª®©- äã¥(2)-, ªæ¨îª ä㨢ëè-(§ªæ¨¨¤¥ )ç. å ⨯

v€,«21£®à¨â¬äã

z x, y

w u, v

w:

∂w = ∂w ∂u

+ ∂w ∂v

∂w

= ∂w

∂u

+ ∂w

∂v

á¨á⥯®¤áâ¬ë ¢¨âì á®«ã祫ãç¥--ë¥

¢ëà ¥- ï ¤«ï ,

∂x

∂u ∂x

∂v ∂x

∂y

∂u ∂y

∂v ∂y

∂u , ∂u , ∂v , ∂v

3 Ǒத¨ää¥à¥-æ¨à®¢¤¥©á⢨ﯮ

∂x

∂y ∂x ∂y .

-®¢®© äã-ªæ¨¨

x ¨ ¯® y ¤ ¯à®¨§¢®--®¥ ãá«®¢¨¨ ¢ëà -¨¥

«ï

¢ëà -¨¥ ¤«ï w (-

§ ¡ëâì, çâ® z § ¢¨á¨â ®â x

y!), ¯®«ãç¨âì ¥é¥ ®¤®©-

4

∂w

∂w

¯ã-ªâ å 2

3,

¨§ ¯®«ãç¥--

пвмга ¢-ëà-¨©.--¨ï,©â¨¯®«ãç¥--ë¥

Ǒà¨à¤¢ãå-

∂x

∂y

¨ ∂z .

∂z

¨.¥вм¢ла¡г¤гв¢¨вмǑа¤в¤п-¥-¥гб«®¢¨®®¡а®¬¥¯®«гз¢ла-¢ла¨§®¢¥¥¯а®¨§¢®,¥¨бª«оз¨вмвм-л-л-¨га¢ла¨¨¤«п.¢ла¥¥ ¢бва-лег-¨-¥.¥з¨п-¨повбпал穤¥¥а¥¥¯--§¥-ле¥¬¨§¢®¢л¤¯¥--¥ëåà¢ë寤ë¥-¥àë.祥¥¬à¯à®¨§¢®¢â¥¥--§ -à®ë®¢ë¥£®¤«ï¯®à拉-ëå,¯¢á¥à¥¥¨å--

Ǒਬ¢å®¤ïé¨å⪬.¥¤,--…65.,¯®¢â®à.᫨¯®ªëˆá¯®«ì§ãǑ®¤á⥥à 1-¨á

∂x

∂y

x