Красовский Н.Н. Теория управления движением

.------

~

......

....

...

.:.'

..;..,...:..;~,~

":;,'~·_I..

...

_

,.,,'

~

-,

......

':

~

..

".

OJ'

......

.........

,

-."

·t·.·.·.:

';

.

..c

......

..

~

••

(';.

;'1£1PW

~-Q~~~~-~DQ

k~'"""'·,~Q~,

'

...

...!...

·_

..

L.L~1

"

.......

,"

'

(j';

400

JI11HEl!fHhIE

OIIEPAI:(1111 B

BERTOPHhIX

IIPOCTPAHCTBAX

[rJI.

12

BeKTopaM

U

HOpMy

p*

[U],

paBHyIO

HOpMe

P*

[fIluJ.

TaKHM

nyTeu

npOCTpaHCTBO

.AI'

~2)

TaKme

npeBpam.aeTCH

B

HeKOTOpOe

HOpMHpOBaH-

HOe

npOCTpaHCTBO,

KOTopoe

6y.n;eM

060BHaqaTh

CHMBOJIOM

9iJ~.

IlTaK,

MId: nOJIYQ11JI11, QTO

Kam.n;oe

npOCTpaHCTBO

9iJ

n

C

llOpMOH

P[h]

nopom~

.n;aeT

npOCTpaHCTBO

5iJ~

C

HOpMOH

p*

[U].

IlpocTpaHcTBo

5iJ~

HaBIilBael'-

ell

COnp.ft:HCelUrbl.M K

npOCTpaHcTBy

5iJ

n

.

HopMa

p*

[u]

.n;JIH

BeKTopOB

U

JIB

npOCTpaHCTBa

9iJ~

HaBLIBaeTCH

HOpMOH,

conpHmeHHOH

KHopMe

p [h]. , .

PaCCMOTp11M

HeCKOJIhKO

npOCTIilX

np11MepOB

npOCTpaHCTB

9iJ

n

11

9iJ~.

IIpHMep

51.1.

IIYCTI> B rrpocTpaHcTBe

.AI'~)

BI>I6paHa

eBKJIJilp;OBa

HopMa

p [h] =

II

h

\I

(49.2),

orrpep;eJIJIIO~aJI

eBKJIJilp;OBO

HOpMJilpOBaHHOe

rrpocTpaHcTBo

$n'

BhI'IJilCJIJilM

COrrpJIJReHHYIO

IiOpMy

p*

[u].

COrJIaCHO

(50.4)

Jil

(51.3)

HMeeM:

n n

p.

[u]=max(l~

uAl

rrPH~h;<1).

(51.4)

t=1

,

JIerKo

aaMeTJilTI>,

'ITO B

(51.4)

MaKcHMyM

p;OCTJilraeTCJI

Ha

rpa~e

ccPePI>I

" h

\I

<

1,

T.

e.

npJil

h~

+

h:

+

...

+

h~

= 1.

CJIep;OBaTeJII>HO,

Tpe6yeTCJI

HaRTH

n

max

I

~1

uih

i

I

npH

~

h~=

1.

(51.5)

'""

i=1

COrJIaCHO

HepaBeHcTBy

KomH-ByHJIKOBCKOro

(CM.

,HanpJilMep,

[3*],

CTp.

209;

~i

[14*],

CTp.

496)

HMeeM:

I~I

I_I

~,

n n

'f

n

'I

(51.6)

l

~

uih

i

I<

(~

hr)

,.

(~

un

"

t=1 t=1

t=1

IlpH'leM

npH

hi

,=

A,

Ui

(i

=

1,

... ,n)

p;OCTUraeTCJI

paBeHCTBO.

CJIep;OBaTeJII>HO,

HCKOMI>IR

MaKcuMyM

(51.4)

p;OCTUraeTCJI

npu

hi

=

A,Ui,

U

BTOT

MaKcHMyM

onpe-

~eJIJIeTCJI

yCJIOBHeM

n n n

rrpH

1.

2

"'Q

U~

= 1

max

(I

~

uih

i

I)

= I

A,

I~

ur

LJ

.J

t '

t=1

i=1

o rKy)l;a cJie)l;yeT OKOHlfaTeJILHO, 'ITO

n il

p.

[u]=

(~

U~)

'=\lull,

(51.7\

'r

t~1

TaKHM 06paaOM,

CrrpaBep;JlHBO

CJIep;YIO~ee

aaKJIIO'leHHe.

e1j"

Ec,l/,u npocmpaJ-tcmoo

$n

nopOJlc(}eHO

eOI>Jtu(}OOOu

J-tOp.Mo!1

p[h] =

II

h

II.

mo

cOnpR,J«eH,fWe

npOcmpaHcmoo

$~

mal>J/Ce

onpe(}eJtR,emcR,

eOI>Jtuoooo!1

J-tOP.MOU

p·[u]

=

II

u

II.

,~

n

\'~.'_':1'

§ 51]

HOPMA

JI11HEl!fHOl!f OIIEPAI:(1111

401

.lIpHMep

51.2.

IIpep;nOJIOlRHM, 'ITO B

npOCTpaHCTBe

.A!'~1)

HOpMa

p [h] orrpep;e-

JIeHa

cPOPMYJIOR

(49.6),

T.

e.

p

[h]

= I

hi

1+

Ih

2

1+

...

+ Ih

n

I.

BI>I'lHCJIHM

conpll-

lReHHYIO

ROpMy p*

[u].

IIo

onpep;eJIeRUIO

UMeeM:

n n

p*[u]

=

max

(I

~

uih

i

I rrpH

~

I

hd

-<

1)

.

(51.8)

t;=1

t=1

MaKCHMYM

(51.8)

p;OCTHraeTCH,

O'leBH,D;HO,

rrpu

hj=

sgn Uj,

hi

= 0

(i

4=

0, rp;a

i-TOT

ROMep,

)l;JIlI KOTOporo I Uj I

=-=

max

I

ui

I,

H

BTOT

MaKcHMyM paBeH I Uj

I.

OTCIOP;

l

CJIep;yeT,

'ITO

p*

[u]

~

max

(I

Ull,

I u

2

1,

...

, I

Un

I).

(

51.9)

IIpuMep

51.3.

BOaI>MeM

HOpMHpOBaHHoe

rrpOCTpaHCTBO

$n

C

HOPMOH

P

[h]

=

max

(I

hi

I,

I h

2

I ,

...

, I h

n

i).

(51.10)

Tor)l;a cOnplllReHHall HopMa

p*

[u]

onpep;eJIlIeTClI

paBeHCTBOM

n

p*[u]=~

luil.

(5U1)

i=1

B 'leM

qHTaTeJII>

MOlReT

y6ep;UTI>CJI

caMOCTOJITeJILHO.

B

06m;eM

CJIyqae

P;JIJI

ROpMI>I

P [h]

(49.7)

COrrpJIJReHHall HopMa

p*

[u]

orr-

pe~eJIlIeTCJI

paBeHCTBOM

n

p*[u]

=

(~

lUi

Iq)1/q

(51.12)

t=1

rp;e

q -

'lHCJIO,

yP;OBJIeTBOpllIO~ee

yCJIOBUIO

1 1

-+-=1

p>1.

(51.13)

q p •

,IJ;oKaaaTeJII>CTBO

(51.12)

onUpaeTCJI Ha HepaBeHcTBo reJII>p;epa (CM., HarrpH-

Mep,

[14*],

CTp.

496),

KOTopoe

o606~aeT

HepaBenCTBO

KomH

- ByHJIKOBCKOro,

HCrrOJII>aOJ;laHHOe

BI>Ime

rrpu

p = q = 2 (CTp. 400).

Ha

)l;OKaaaTeJII>CTBe

paBeHCTBa

(51.12)

ap;eCL

OCTaHaBJIHBaTI>CH

He

6yp;eM.

B

3aKJIIOQeHIm

3aMeT11M

eII:(e,

"ITO

HOpMy

JIImeHHOH

onepan;HH

MomHO

Haxo.n;11Tb

11B

YCJIOB11H

P*[fIl]

=

sup

Iq>

[h] I

npH

h

-L

0

(51.14)

p[h]

-r- ,

KaK

:3'1'0

CJle.n;yeT

113

orrpe.n;eJIeHHH

BeJl11QHHIiI

p*

B

COOTBeTCTBHH

C

He-

paBeHCTBOM

(51.1).

HaKOHeI:(,CJle.n;yeT

3anOMHHTb

BalliHoe

HepaBeHCTBO

IfIl

[hll

~

p*

[fill· p [hl,

CJle.n;YIOII:(ee

TaKllie

cpa3Y

113

onpe.n;eJIeHHH

HOpMIiI

p*

[fill.

"'..

)"

••

:;;

.•

'-,

;

...•

'

..

,..

'

.•

,

.j'

•••.

'h~

__

t"

;.-

••

',.;;";.,--"'.:..

".:;.,:~

••

;:.

..

-:6"

".,.,

.

'.')'

':'.

';.

··

..

-:··.···,;;·"i.;iI+IF':

• ,

.:

••

~

••

'."

;:·'

,::·;;,;.-~:~·.··:z·,·

.';:_:·;~··

(fJ

JIHHEi'lHbIE

OIIEPAI(HH

B

BEKTOPHbIX

IIPOCTPAHCTBAX

[rn.

12.

402

§

52.

reOMeTpn1JeCKan

HHTepnpeTa~nn

,Il.ap;HM

rrOJIesHyIO

reOMeTpHlJecKyIO

HHTeprrpeTa~HIO

cooTHomeHHH,

CBHsaHHIilX

C

HOpMOH

JIHHeHHOH

orrepa~lIH

ep

[h] =

h'u

B

rrpocTpaH-

CTBe

5iJ

n

.

,Il.JIH

orrpep;eJIeHHOCTH

paccMoTpHM

CJIylJaH,

Korp;a

HopMa

p [h]

sa-

p;aHa

cJlOpMyJIOH

p

[h]

=

max

(Ih

l

l,

Ihzl,

.

..

,

Ihnl}·

(52

..

1)

ITYCTb

n = 2.

Ha

pHC.

52.1.

JIHHHH

(1)

Hs06pamaeT

rroBepxHocTb

ep;HHH1JHOH ccJlepbI p [h] < 1, a

ceMeHcTBo

rrapaJIJIeJIbHbIX

rrpHMbIx

(2)

Hs06pamaeT

rroBepxHocTH

ep

[h] =

h'u

= a =

const.

:RaK

MbI

SHaeM,

HopMa

JIHHeHHOH

orrepa:U:HH

ep

eCTb

HaH60JIbmee

rro

a6COJIIOTHOH BeJIH1JHHe

SHalJeHHe

::noH

orrepa:U:HH

Ha

cq,epe

ep;nHHlJHOro

pap;nyca.

OlJeBHp;HO,

MaK~

CHMyM

BeJIHlJHHbI

\ep

[h]\ =

.h

2

=

Ih'

ul

p;ocTHraeTcH

B '1'0'1-

Kax

h(l) H

h(Z)

,

rp;e

rrOBepx-

HOCTb p [h] = 1

KacaeTCH

:.

JlHHHH

ep

[h] = a

l

H

ep

[h] =

.,

= a

z

=

-a

l

•

(B

p;pyrHx

CJIylJaHX

TaKHX

TOlJeK

MO-

~

h,

meT

6bITb

H

60JIbme,

lJeM

~

]0

[h

J=~

const

p;Be, HO 3'1'0

Hecym;ecTBeHHO.)

~

J

]0[11]=0"

B

TOlJKaX

h(

\) H

h(2)

orrepa-

~

--p..<

(I)

:U:

HH

ep

[h]

rrpHHHMaeT

CO-

it:

it

'l

(2)

OTBeTCTBeHHO

HaH60JIbIIlee

a

l

H

HaHMeHbmee

(-a

l

)

SHa-

II"

1",

frhJ=-a:,~

lJeHHH.

Bo

Bcex

me

p;pyrHx

TOlJKaX

ccJlepbI p [h] <1

a6-

COJIIOTHOe

SHalJeHHe

orrepa-

PRe. 52.1.

:U:HH

ep

[h]

MeHbme,

lJeM

Ia

l

I·

,Il.eHcTBHTeJIbHO,

rrYCTb

TO'IKa

h,

OTJIH'IHaH

0'1'

TO'IeK

h(l) H

h(Z),

JIemHT

Ha

nOM

cq,epe.

ITpHMaH

ep

[h] =

h'u

= a,

rrpoxop;Hm;aH

'Iepes

TO'IKy

h,

pacrroJIomena

K

Ha'IaJIy

KoopP;IlnaT

6JIHme

(B

caMOM

06bI'I-

HOM

CMbICJIe),

'IeM

rrpHMbIe

ep

[h1 = +

aI'

rrpoxop;m:U:He

'Ie

pes

TO'IKIl

h(l) H

h(Z).

IT03TOMY

H

OKaSbIBaeTCH,

'ITO

SHa'IeHIle

a

orrepa:U:HIl

ep

[h]

B

TO'IHe

h

MeHbme,

'IeM

SHa'IeHHe

a

l

3TOH

me

OIIepa:U:HIl

B TOlJHe h(l).

I

TaHHM

06paSOM,

HopMa

p*

[ep]

orrepa~HH

ep

[h] =

h'u

- 3'1'0 Hau-

I

6Mbutee

SHa'IeHHe

a,

rrpH

HOTOpOM

rrpHMaH

ep

[h] = a

em;e

rrepeceKaeT

/

cq,epy

p [h] < 1.

ITo

aHaJIorliH

H B

06m;eM

CJIy'Iae

OHaSbIBaeTCH,

'ITO

HopMa

p*

[ep]

- 3'1'0

HaH60JIbIIlee

SHa'IeHHe

a,

rrpH

HOTOPOM

rHrrep-

rrJIOCHOCTb

ep

[h] = a

em;e

rrepeceHaeT

ep;HHH'IHyro

cq,epy

P [h] < 1.

ITpH

BbI'IHCJIeHHH

HOpM]",I

p*

[ep]

JIHHeMHOH

orrepa:U:HH

ep

[h] =

=

h'u

MomeT

OKasaTbCH

IIOJIeSHbIM

TaHme

CJIep;YIOm;HH

crroc06.

Pac-

',

..

·

"l

G

:;/

§ 52)

rEOMETPHqECKAH

llHTEPilPETAI(nrt

403

CMOTpHM

rHrreprrJIOCKOCTb

ep

[h] = 1.

STa

IIOBepXHOCTb

He

IIpOXOP;H'L

lJepes

TO'IKy

h = 0,

TaK

HaH

ep

[0] =

O.

C

p;pyroM

CTOpOHbI,

IIpH

II

h

11-+

00

06HSaTeJlbHO

lim

p [h] =

00,

KaHOBa

6bI

HH

6bIJIa

HopMa

p [h). OTCIOp;a

BbITeHaeT,

'ITO

Ha

rrOBepXHOCTH

ep

[h] =

h'u

= 1

He-

rrpepbIBHaH

q,yHH:U:HH

'tJ

= P [h]

p;OCTHraeT

CBoero

HaUoMeHbute30

SHa-

'IeHHH'tJ

=

pO

B

HeHoTopoH

TO'IKe

h =

hO.

ITPH

3TOM

pO

>0

B

CHJIy

Toro,

'ITO

hO

+

O.

IlTaK,

rrYCTb

'tJ

0

= P

[hOT

=

min

(p [h]

rrpH

ep

[h] = 1)

(ep

[hOT

= 1). (52.2)

OKasbIBaeTCH,

'ITO

p'

[fP]

=_1

(52.3)

p

° .

,Il.eHcTBHTeJIbHO,

P;JIH

Bcex

h,

rrpHHap;JIemam;HX

rHrreprrJIOCKOCTH

ep

[h] = 1,

BbIrrOJIHHeTCH

CJIep;yIOm;ee

HepaBeHCTBO;

ep

[h] = 1 <

<p*

[ep]

P

[h],

T.

e.

p

[h]

;>

1 /

p*

[c:pl.

IT03TOMY

1

1']0

=

min

(p

[h]

rrpH

ep

[h]

=

1);>

p*[cp]

. (52.4)

C

p;pyroH

CTOpOHbI,

rro

OIIpep;eJIeHHIO

HOPMbI

p*

[ep]

COrJIaCHO

(51.3)

cym;ecTByeT

3JIeMeHT

h=h(l)

C

ep;HHH'IHOH

HOpMOH

p [h(l)] =

1,

P;JIH

KOToporo

ep

[hell] =

p*

[c:p]

=

p*

[ep].

p

[h<l»).

(52.5)

ITYCTD

ep

[hell] =

~.

O'IeBHp;HO,

~

>

0,

TaK

KaK

MbI,KOHe'IHO,

pac-

cMaTpHBaeM

oIIepa:U:HIO

c:p

[hJ,

OTJIH'IHyIO

0'1'

TOJHp;eCTBeHHOrO

HyJIH.

ITOCTpOIIM

3JIeMeHT

h(Z)

=

~

h(l).

ITo

CBoHcTBaM

Op;Hopop;nOCTH

(50.2),

(49.4)

orrepa:U:HH

ep

[h] H

HOpMbI

P [h]

rrOJIy'IHM

Terrepb

HS (52.5)

ep

[h(Z)]

=

Tep

[h(l)l =

TP*[ep]·p

[Ml)] = p*[ep].p [h(Z)], (52.6)

rrpH'IeM

ep

[h<Z)]

= 1.

(52.7)

Y'IHTbIBaH,

HaHone:u:,

'ITO

rro

BbI60py

hO

BbIrrOJIHHeTCH

yCJIOBHe

pO

= P

[hOT

< p [h(Z)],

rrOJIy'IHM

HS (52.6) H (52.7)

nepaBeHCTBO

p*

[ep].

pO

< 1,

KOTopoe

BMeCTe

C (52.4) H

p;OKaSbIBaeT

(52.3).

ITOHCHHM

reOMeTpH'IeCKHH

CMbICJI

COOTHOIIleHHH

(52.3).

ITYCTb

P;JIH

orrpep;eJIeHHOCTH

HopMa

p [h]

sap;aHa

paBenCTBOM

n

p [h] =

~

Ihi

I·

(52.8)

i=l

,Il.orrycTHM,

'ITOrrpH

n = 2

Ha

pHC.

52.2

JIHnHH

(1)

H306pamaeT

rro-

BepxHocTb

ep

[h] =

1,

a

JIHHHH

(2)

Hso6pamaIOT

ceMeHCTBO

rrOBepxno-

CTeH p [h] =

1;.

lis

pHC.

52.2 B COOTBeTCTBHH C (52.3)

SaKJIIOlJaeM,

'ITO

--,-,~-,--,-~,,-,,-'

,..c'

..

'.'

,

','

-_.

..

> '

••

; ,

::.

~

,:'

•

.:;

...

:

....

'.

.'

."

0,

t.'·

,"!

404

mIHE:IilHbIE

OITEPAII11:M B

BEI<TOPlIbtX

I1POCTPAHCTBAX

[rn,

12

HopMa

p*

[<p]

paBHa

BeJIM'IMHe

1/

pO,

rp;e

pO

-

HaMMeHbmee

3Ha'Ie-

HMe

£ =

pO,

rrpR

KOTOpOM

ITOBepXHocTb

p [h] - £

em;e

rrepeceKaeT

npHMyIO

<p

[hl = 1.

AHaJIOrR'IHyIO

reOMeTpR'IecKyIO

KapTRHy

MomHO

rrpep;CTaBJIHTb

ce6e

rrpR

n > 2 H B

p;pyrHX

CJIyqaHx

HOpMbI

p [hI.

B

rrpocTpaHcTBe

5i3

n

BeJIM'IRHa

2

p [h 1

MMeeT

CMbICJI

paCCmOR.HUR.

TO'IKH

1

°

{ht} 01'

Ha'Ilwa

Koopp;RHaT

h =

O.

II09TOMY

MomHO

CKa3aTb,

'ITO

liopMa onepaq,uu

<p

[hl =

h'u

eCl'b

.I

ij

8e.tf,U'If,UHa

p*

[<pl

= 1

fpo,

o6pamJ-taR.

paCCmOR.HUlO

p =

pO

8unepMocr;,ocmu

:1

<p

[hl = 1 om

Ha'lf,a.tLa

r;,oopouHam

11

f

h = 0

•

"

lIa

Bcero

CKaaaHHoro

MomHO

"

if

~~

cp;eJIaTb

CJIep;yIOm;RM

BbIBOp;.

Byp;eM

1;1

rOBopRTb

(He

COBceM

aKKypaTHo,

[:

,;

MomeT

6bITb)

,

'ITO

c<jJepa

p [hl

-<

8

r,

!::

R

rRrreprrJIOCKOCTb

<p

[hl =

~

r;,aca-

lOmCR.

p;pyr

p;pyra,

eCJI1I OHM

RMeIOT

U

:

,:

;l

o6m;Re

TO'IKH

h,

no

rrpR

9TOM

Ha

PMC.

52.2.

rrJIOCKOCTH

<p

[hl =

~

HeT

TO'IeK,

JIe-

mam;HX

B

06JIaCTR

p [hl <a.

IIYCTb

I

I,

B

rrpOCTpaHCTBe

5i3

n

aap;aHa

orrepaIIHH

<p

[hl.

HopMa

9TOM

orrepa-

I'

I~

IIlm;

MomeT

6bITb

orrpep;eJIeHa

CJIep;yIom;HMR

p;ByMH

crroco6aMH:

['"

!'

1)

p*

[<pl=

lal,

eCJIR

rrJIOCKOCTb

<p

[hl=

a

eCTb

KaCaTeJIbHaH

K

!~

!I'll

ep;lIHR'IHOM

c<jJepe p [hl

-<

1;

2)

p*

{<pl

= tlpo,

rp;e

pO

-

pap;Myc

c<jJepbI p [hl

-<

pO,

KOTopaH

r;,acaemcR.

rHrreprrJIOCKOCTR

<p

[hl = 1.

~ft!

,

i·

'

Ii

§ 53.

OTAeJIeHUe

BbIllYKJIblX

MHomeCTB

I

\

3ap;a'Ia

06

OTp;eJIeHUH

BbIrrYKJIbIX

MHomeCTB

HrpaeT

BamHyIO

pom.

BO

MHorRX

Borrpocax

MaTeMaTlIKR,

CBHaaHHbIX

C

paCCMOTpeHHeM

JIH-

HeMHbIX

orrepaIIHM.

B

'IaCTHOCTB,

<jJYHp;aMeHTaJIbHaH

meopeMa

Xa-

Ha

-

Baflaxa

[6* l,

HBJIHIOm;aHCH

6aaoM

P;JIH

pemeHlIH

TaK

HaabIBae-

MOM

np06JLeMbl

MOMeNmo8,

Ha

KOTOpOM

OCHOBaHa

60JIbmaH

'IaCTb

BbI-

BOP;OB

HameH

MOHorpa<jJHH,

yp;06Ho

HHTeprrpeTHpyeTcH

reOMeTpH'IeCKH

B

BHp;e

TeopeMbI

06

OTp;eJIeHRH

MHomeCTB.

B

p;aHHOM

rraparpa<jJe

MbI

06cYP;HM

rrp06JIeMy

P;JIH

CJIy'IaH

KOHe'IHOMepHoro

rrpOCTpaHCTBa

~n'

IIpemp;e

Bcero,

p;ap;HM

onpep;eJIeHUe

8bmY'fi,JL080

MHomeCTBa,

KOTO-

poe

HBJIHeTCH

MaTeMaTH'IeCKOH

<jJopMaJIHaaIIHeH

RHTyHTHBHoro

npeA-

CTaBJIeHHH

0

BhIrryKJIOCTH.

_ ,

IIYCTb

h(l)

H

h(2)_

p;Be

TO'IKH

B

rrpOCTpaHCTBe

.#'

n'

OTpe3KOM,

coe-

p;HHHIOm;HM

TO'IKH

h(l)

H

h(2)

,

Ha3hIBaeTCH

cOBoKyrrHocTb

Bcex

TO'IeK

BHAa

h =

'Ah(l)

+

(1

-

'A)

h(2),

rp;e

0

-<

'A

-<

1.

",

p

,':- !

,,",',.1

§

50]

OTgEJIEHHE

,BhlITYHJIhIX

MlIOiH~C'rB

405

MHomeCTBO

.At B

JIHHeHHOM

npOCTpaH<;:TBe

~

n

HaabIBaeTCH

BhIrryK-

JIhIM,

eCJIH

P;JIH'

BCHKOH

rraphI

TO'IeI<

H3

.At

COep;lIHHIOm;HH

HX

OTpe30K

co,n;epmHTCH

B .At. .

B

,n;ByMepHoM

(~2)

H

TpeXMepHOM

(~3)

rrpOCTpaHCTBax

rrOHHTHe

OTpe3Ka

COBrra,n;aeT C

06hI'IHbIM

reOMeTpH'IeCKHM

rrOHHTHeM

OTpe3Ka

[h(1)

,

h(2)

1

rrpHMoH,

rrpoxOp;Hm;eH

'Iepea

TO'IKH

h(1),

h(2).

BbIrrYKJIbIMH

MHomeCTBaMR

B

rrpOCTpaHCTBe

~

3

HBJIHIOTCH,

CTH

TO'IeK,

06paaYIOm;HX

OTpeaOK,

!!l

rrpHMyIO,

rrJIOCKOCTb,

TpeyrOJIbHHK,

Ky6,

map

H

T.

p;.

HarrpOTliB,

cOBoKyrrHOCTb

TO'IeK,

aaKJIIO'IeHHbIX

Memp;y

p;ByMH

KOHIIeHTpnqeCKHMH

c<jJepaMH,

BbIrrYKJIbIM

MHomeCTBOM

He

HBJIHeTCH.

OTMeTHM,

'ITO

B

rrpOCTpaHCTBe

~

n

rrpH

JII060M

BbI60pe

MeTpHKH

p {hl

c<jJepa

p {h]

-<

8

HBJIHeTCH

BbIrrYKJIhIM

MHomeCTBOM.

CrrpaBep;-

JIHBOCTb

9Toro

yTBepmp;eHHH

BbITe-

KaeT

Herrocpep;CTBeHHO

H3

orrpep;eJIe-

HHH

BbIrryKJIOrO

-MHomeCTBa

H

H3

CO-

OTHomeHHH

(49.4) P;JIlI

HOpMbI.

PaccMoTpHM

Terrepb

B

rrpOCTpaH-

CTBe

~

n ,n;Ba

KaKBx-JIH60

MHomeCT-

Ba

TO'IeK.At

1

R

.At

2

.

rOBOpHT,

'ITO

JIH-

HarrpHMep,

cOBoKyrrHO-

h

z

J

PJilC,

5.3.1.

HeHHaH

orrepaIIHH

<p

[hl

paaOe.tf,R.em

9TH

MHomeCTBa,

eCJIH

cym;ecT-

ByeT

TaKoe

'IHCJIO

a,

'ITO

BhIrrOJIHHIOTCH

HepaBeHCTBa:

<p

[hl

-<

a

npH

Bcex

h

H3

.At

1

,

(53.1)

<p

[hl > a

rrpB

Bcex

h

H3

.At

2

.

(53.2)

Y

CJIOBlIH

(53.1), (53.2)

MomHO

HarJIHp;HO

HCTOJIKOBaTb

CJIep;yIO-

m;HM

06pa30M.

IIYCTb,

HarrpHMep,

n =

3.

Torp;a

ypaBHeHHe

<p

[hl =

=

uIh

l

+ u

2

h

2

+ u

3

h

3

= a

orrpe,n;eJIHeT

B

rrpOCTpaHCTBe

~

3

rrJIOCKOCTb.

HepaBeHcTBa

(53.1), (53.2)

03Ha'IaIOT,

'ITO

MHomeCTBa

.At

l

H

.At

2

pac-

~OJIOmeHbI

rro

pa3HhIe

CTOpOHbI

01'

9TOH

rrJIOCKOCTH,

XOTH,

MomeT

6hITb,

Ii:

HMeIOT

C

HeH06m;He

TO'IKH

(pHC.

53.1).

IIYCTb

B

rrpOCTpaHCTBe

JV'n

BBep;eHa

MeTpHKa,

11

MhI

HMeeM,

CJIep;OBaTeJIbHO,

JIHHeHHOe

HOpMHpO-

BaHHoe

rrpOCTpaHCTBO

5i3

n

.

RaK

06hI'IHo,

omr;,pblmou

c<jJepoH

B

rrpo-

CTpaHCTBe

5i3

n

C

IIeHTpOM

B

TO'IKe

h(l)

H C

pap;HycoM

8

6y,n;eM

HaahIBaTb

MHomeCTBO

TO'IeK

h

Ha

5i3

n

,

y,n;OBJIeTBOpHIOm;HX

yCJIOBHIO

p [h -

h(l)l<

<8

(8

> 0).

TaKlIe

c<jJephI

6yp;eM

Ha3hIBaTb

8-0KpeCTHOCTbIO

TO'I-

KH

h(I).

Torp;a

TO'IKa

h(l)

HBJIHeTCH

no

orrpe,n;eJIeHHIO

BHyTpeHHeii:

TO'IKOH

KaKOrO-JIH60

MHomeCTBa.At,

eCJIH

OHa

co,n;epmHTCH

B

J(.

BMeCTe

C

HeKOTopOH

CBoeH

8-0KpeCTHOCTbIO.

.,

""'",'

',,",

" •

......:....~-~:.;~

•

.',.,';"

.••...•

" ;

.••

~

::.

::;:,;,';

:-.,

,.0,

••

'.".

.

.••••.

.'.'.:

;;:Yi.i:.;,.

II

.j

,j

h

I

'i

,

~

:~

L:

t,

~

(

C?J

..

<:_;>'

406

JUIHE:Y.I:HbIE

OIIEPAn;HH

B

DEH'l'OPBbIX

IlPOCTPABCTBAX

[rJI.

12

CrrpaBep;JUIBa

CJIep;yIOm;aH

BaiImaH

TeopeMa

([6*],

CTp.

416;

[7*],

CTp.

42).

Teo

p e M a 53.1.

II

ycmb

.At

l

U .At

z

-

8bmyr.J£bLe

MHOJICeCm8a

8

!liJ

n

ma1>ue,

'l(,mo

8

.At

l

lMtelOmCR.

8HympeHHue

mo'l(,/'l,U,

a .At

z

He

co8epJICum

flU

08HOU

8Hympemwu

mo'l(,/'l,U

MHOJICeCm8a

.At

l

.

To~8a

CYUfecm8yem .ltU-

fleUHaR.

onepalfuR.

cp

[hl,

paa8e.ttR.lOUfaR.

amu

MHOJICeCm8a.

CrrpaBep;JIlIBOCTb

TeopeMbI

53.1Mb!

rrpHMeM

oe3

p;ORa3aTeJIbCTBa.

OTMeTlIM

TOJIbRO

em;e,

'ITO

eCJIII

P;JIH

MHOiI\eCTBa

..;ttl

TOqRa

h = Q

HBJIHeTCH

BHyTpeHHeH,

TO

qlICJIO

a B

yCJIOBHHX

(53.1), (53.2)

OTJIlIqHO

01'

HyJIH

H,

oOJIee

Toro,

P;JIH

JIIOOOrO

a >0

cym;eCTByeT

orrepan;HH

cp,

yp;OBJIeTBOpHIOm;aH

8TlIM

YCJIOBlIHM.

11

liz

2

h,

PHC.

53.2.

PHC.

53.3.

HarJIHp;HbIH

CMbICJI

C<pOpMyJIlIpOBaHHoH

TeopeMbI

(HarrpHMep,

rrplI

n =

2)

TaROB.

ECJIlI

MHomeCTBa

.At

l

II

.At

z

HBJIHIOTCH

BbIrrYRJIbIMH,

He

rrepeceRaIOTCH

CBOlIMH

BHyTpeHHoCTHMH,

HO,

MomeT

ObITb,

JIHrnb

Ra-

caIOTCH

.CBOHMlI

rpaHlIn;aMlI,

TO

Bcerp;a

MomHO

rrpOBeCTlI

rrpHMYIO

cp

[h]

= ulh

l

+ uzh

z

=

a,

ROTopaH

pa3p;eJiHeT

:3TlI

MHomeCTBa

(CM.

plIC.

53.2, 53.3).

II3

plIC.

53.3, rp;(:)

rrplIBep;eH

rrplIMep

HeOTp;eJIlIMbIX

MHomeCTB,

BlIP;-

HO,

RaROH

CMbICJI

lIMeeT

B

TeopeMe

53.1

rrpep;rrOJIOmeHHe

0

BbIrrYRJIOCTlI

.At

l

II

.At

z

.

B

p;aJIbHeHrneM

HaM

BCTpeTlITCH

CJIyqaH,

Rorp;a

MHomeCTBO

.Atz,

HBJIHHCb

BbIrrYRJIbIM,

OOJIap;aeT

R

TOMy

me

CJIep;YIOm;lIM

CBOHCTBOM:

eCJIlI

TOqRlI

he

l

)

II

hM

cop;epmaTCH

B .At

z

' TO

II

Bcn

npR.MaR.

h =

'AMI)

+

(1

-

'A)

h(Z)

(-

00

<

'A

<

00)

cop;epiImTcH

B.At

z

.

TaRoe

MHomeCT-

BO

Ha3bIBaeTCH

n.ltOC1>UM.

B

3TOM

CJIyqae

JIlIHeHHaH

orrepan;lIH

cp

[hl,

o

ROTopoii:

IIp;eT

peqb

B

TeopeMe

53.1

,yp;OBJIeTBOpHeT

Ha

MHOmeCTBe

..4t

z

CTporoMy

paBeHcTBy:

cp

[h] = a

rrplI

Bcex

h

lI3

.At

z

, (53.3)

eCJIH

TOJIbRO

3TO

paBeHCTBO

BbIrrOJIHHeTCH

XOTH

ObI B OP;HOH

TOqR6

lI3

.At

z

.

t

D,

\.~

...

~

§

53]

OTP;EJIEHHE

BbIIIYRJIbIX

MHOJRECTB

407

B

caMOM

p;eJIe,

rryCTb

B

TOqRe

h(l)

lI3

..4t

z

lIMeeM

cp

[h(l)] = a

II

orre-

pan;lIH

cp

OTp;eJIHeT

Jttz.

IIpep;rroJlomHM,

BorrpeRH

(53.3),

'ITO

B

HeRO-

TOpOii:

TOqRe

h(Z)

H3..4t

z

cp

[h<Z)]

=

~

>a.

(53.4)

TaR

RaR

MHomeCTBO

.At

z

rrJIOCROe,

TO B

HeM

COp;epJRHTCH H

JIIOOaH

.

TO'IRa

h =

'A

h(l)

+

(1

-

'A)

h(Z)

(-

00

<

'A

< 00).

IIMeeM:

cp

[h] =

'Acp

[h(l)] +

(1

-'A)

cp

[h(Z)]

=

a'A

+

(1

-

'A)

~.

(53.5)

HplI

'A

> 1

H3

(53.5), (53.4)

rrOJIy'IlIM

HepaBeHCTBO

cp[h(l)]

<

ex,

'ITO

rrpoTlIBOpe'IHT

(53.2).

IIoJIY-

lJeHHOe

npoTlIBOpeqlIe

p;ORaabI-

BaeT

Harne

yTBepmp;eHlIe.

IIplIMepaMlI

rrJIOCRlIX

MHO-

meCTB

B

npOCTpaHCTBe

.AI'

3

HB-

JIHIOTCH

IIpHMaH

II

IIJIOCROCTb.

HCHo,

'ITO

rrJIOCROCTb

cp

[h] =

=ulh

l

+ u

Z

h

2

+ u

3

h

3

=

a,

OT-

p;eJIHIOm;aH

3TlI

nJIOCRlIe

MHOJRe-

CTBa

01'

HeROToporo

BbIrryRJIOrO

MHomeCTBa

.At

l

,

rrapaJIJIeJIbHa

lIM

HJIH

HX

cop;epmlIT

(pHC.

53.4).

HaM

nOHap;ooHTcH

TeopeMa

00

oTp;(:)JIeHHH

BbIIIYRJIbIX

MHomeCTB

B

oeCROHeqHOMepHbIx

<PYHRD;HOHaJIbHbIX

rrpocTpaHcTBax.

::ha

TeopeMa

oyp;eT

OOCyrRp;eHa

HHme,

B § 61.

Oocymp;eHHe

aap;alJH

00

OTp;eJIeHHH

MHomeCTB

B

ROHelJHOMepHbIx

npocTpaHcTBax,

H3JIOJReHHOe

B

p;aH-

HOM

rraparpa<p

e

,

HMeeT

BBOP;HbIH

xapaRTep.

3aMeTHM

ern;e,

'lJ:TO

B

1<0-

HeqHOMepHbIX

rrpOCTpaHCTBax

jfI'

n

rrpep;rrOJIOmeHlIe

0

cyu~ecTBOBaHHlI

y

MHomeCTBa

..;ttl

BHyTpeHHeii:

TOlJRH

He

HBJIHeTCH

ocooeHHO

cym;ecTBeH-

HbIM

P;JIH

B03MomHOCTH

lIX

pa3p;eJleHHH

rHUeprrJIOCROCTbIO

<p[h)

=

Ii

(CM.

[93],

CTp.

781,.

Op;HaRo

3TO

npep;rrOJIOmeHHe

BamlfO

B

oeCROHeqHO-

MepHOM

CJIylJae,

ROTOpbIH

paccMaTpHBaeTcH

B § 16.

Ii.'!

'~

0

\:

I

h

z

h,

Puc. 53.4.

-'

··_~_·_c:l-·.~·.·

-"-

..

_':~"'

••

'..

.

...•

'j"N'

"~l~

,;.,,2,:.

';:'

.•.

;

••••.•.

';

..

:.•.•

:1..>.::::

::.

'

..

~':~':'.:~.':

":.'~':~'.:::.:""'-.:;:

<

.....

:

..

:..

:.;.::

..

\_,:l.:_.~

..•.

:.:.:.

:'.:.

:.':.~·~'!r11'.i:

""

,

,"J

C

r.l£

a e a 13

HOPMHPOBAHHbIE llPOCTPAHCTBA«J)'YHIUJ;HII

§ 54. llpeABapRTeJlbHbIe saMMaHHH

B 9TOD: H

cJIep;YIO~eD:

rJIaBaX

npHBOAHTCH

HeKOTopLIe

CBep;eHHH

H3

TeopHH

tPYHKIJ;HH,

COCTaBJIHIO~He

tPYHp;aMeHT

TOH

TeopHH

ynpaBJIeHHH

H

Ha6JIIOp;eHHH,

KOTopaH

pa3BHBaeTcH

BO

II

H B

III

lIaCTHX

KHHrH.

BOJIbillHHCTBO

tPaKToB,

KaK

OTMelleHO

BO

BBep;eHlIH,

MbI

npHBop;HM

6e3

p;oKa3aTeJIbCTBa,

aneJIJIHpyH

K

HX

HarJIHp;HoMy

CMbICJIY·

IIMeHHo

AJIH

9TOH

IJ;eJIH H

6bIJIH

paccMoTpeHbI

B §

49-54

aHaJIOrHlIHble

nOHH-

THH

P;JIH

60JIee

npHBbIlIHbIX

KOHellHOMepHbIX

npocTpaHcTB.

Byp;eM

paccMaTpHBaTb

Be~eCTBeHHbIe

lIHCJIOBbIe

tPYHKIJ;HH h

('r),

i",

onpep;eJIeHHbIe

Ha

OTpe3Ke

[toc,

till.

MomHO

TpaKToBaTb

KaK

BeKTOp

h =

Ramp;ylO

TaKyro

tPYHKIJ;HIO h

('r)

{h't},

HMelO~HH

6ecKoHellHoe

KO-

JIHlIeCTBO

KOMnOHeHT

hi

= h

('r),

cooTBeTcTBYIO~HX

BceB03Mom-

HbIM

3HalleHHHM

't'

H3

oTpe3Ka

[t

oc

, till.

IIcXOAH

H3

TaKoH

TpaKToBKH

~l

:1

tPYHKIJ;HH h

('t')

= {h't},

Ha

HHX

MomHO

pacnpocTpaHHTb

Te

nOHHTHH

H

pe3yJIbTaTbI,

KOTopble

6bIJIH

H3JIOmeHbI

BbIille

P;JIH

KOHellHoMepHbIx

!~

~"

BeKTOpOB

h =

{hI}'

1I3YlIeHHe

9THX

BonpocoB

BXOP;HT B

3ap;ally

Kypca

tPYHKIJ;HOHaJIbHOrO

aHaJIH3a,

KOTOpblH

H3JIaraeTCH,

HanpHMep,

B

KHH-

rax

[6*, 7*, '9*, 14*l.

,JJ;JIH

OBJIap;eHHH

tPYHKJ:~HOHaJIbHbIM

aHaJIII30M

Tpe6yeTcH

cy~ecTBeHHoe

pacillHpeHHe

3HaHHH

no

TeopHH

tPYHKIJ;HH

p;eHCTBHTeJIbHOrO

nepeMeHHoro

no

cpaBHeHHIO

co

CTaH,n;apTHbIM

MaTe-

MaTHlIeCKHM

KypCOM.

MbI

orpaHHlIHBaeMcH

3,n;eCb ,n;OBOJIbHO

nOBepx-

,

~

HOCTHblM

H3JIOmeHHeM

OCHOBHblX

nOHHTHH

H

tPaKToB,

,n;OCTaTOllHblM

,n;JIH

HaillHX

IJ;eJIeH.

CJIe,n;yeT

HMeTb

B

BH,n;y,

liTO,

HeCMoTpH

Ha

60JIbillOe

KOJIHlIeCTBO

MaTeMaTHlIeCKHX

TOHKOCTeH,

B03HHKalO~HX

npH

pa60Te

C

«6eCKOHellHOMepHbIMH

BeKTOpaMH»

{h't}(t

oc

-<

't'

-<

tll),3,n;eCb

coxpaHH-

lOT

CBOH CMblCJI

MHorHe

OCHOBHbIe

3aKOHOMepHOCTH,

xapaKTepHble

,n;JIH

KOHellHOMepHblX

BeKTopOB

{hil.

qHTaTeJIb,

MaJIO

3HaKoMbIH

C

tPYHKIJ;H-

.,

Ii:

OHaJIbHbIM

aHaJIH30M,

HO

XOpOillO

pa306paBillHHcH

B

MaTepHaJIe

npe,n;bI-

,n;y~HX

naparpatPoB

9TOH

lIaCTH

KHHrH,

MOiKeT

npH

H3yqeHHH

3a,n;all

06

ynpaBJIeHHH

H 0

Ha6JIIO,n;eHHH

yBepeHHo

pa60TaTb

C

06~HMH

nOHH-

THHMH,

paccMaTpHBaeMblMH

B

§§

55-61.

ITPH

9TOM

,n;OCTaTOllHO

orpa-

HHlIHTbCH

TeMH

tPYHKIJ;HOHaJIbHblMH

Mo,n;eJIHMH,

KOTopble

6y,n;yT

pa306-

paHbI

B

9THX

naparpatPax.

ITPH

meJIaHHH

pacillHpHTb

Kpyr

9THX.

Mo,n;eJIeH CJIe,n;yeT

YllHTblBaTb

HeKOTopbIe

,n;onOJIHHTeJIbHble

06CTOHTeJIbCT-

Ba,

CBH3aHHbIe

C

6ecKoHellHoMepHocTblO

BeKTopOB

{h't}.

HaH60JIee

cy~ecTBeHHbIe

H3

9THX

06CTOHTeJIbCTB

6y,n;yT

OTMellaTbCH

no

Xo,n;y

~l

~~J

§ 51,]

nPE)J;BAPHTEJIbHbIE

3AMEqAHHH

409

H3JIOiKeHHH

MaTepHaJIa.

3aMeTHM

e~e,

liTO

MaTepHaJI

9TOH

H

CJIep;ylO-

~eH

rJIaB

BameH

,n;JIH

060CHOBaHHH

H

nOHHMaHHH

BHyTpeHHero

Mexa-

HH3Ma

Tex

TeopeM,

KOTopbIe

,n;aHbI B

KHHre

,n;JIH

3a,n;all

06

ynpaBJIeHHH

H

Ha6JIIO,n;eHHH.

Op;HaKo

OCHOBHbIe

BblBO,n;bI 0

pemeHHH

9THX

3ap;alJ

He

co,n;epmaT

KaKHX-JIH60

nOHHTHH,

BbIXO,n;H~HX

cy~ecTBeHHO

3a

paMKH

pHp;OBoro

Kypca

MaTeMaTHlJeCKOrO

aHaJIH3a.

IITaK,

paCCMOTpHM

HeKOTopoe

MHOiKecTBo.Al'

tPYHKu;HH

h

('t'),

3ap;aH-

HbIX

Ha

OTpe3Ke

[toc,

till. ,JJ;JIH

9THX

tPYHKIJ;HH

eCTeCTBeHHbIM

06pa30M

onpep;eJIeHbI

OnepaIJ;lIlI

CJIOmeHHH

H

YMHOiKeHlIH

Ha

1I1ICJIO:

tPYHKIJ;lIH

f

('r)

ecTO

cyMMa

tPYHKIJ;HH h

('t')

H g

('t'),

eCJIH

B KaiK,n;OH

TOllKe

't'

1I3

[toc,

tJ3]

3HalleHHe

tPYHKu;lIlI

f

('t')

paBHO

cyMMe

3HalleHHH

h

('t')

1I g

('t');

tPYHKIJ;HH f

('t')

eCTb

np01I3Bep;eHlIe

'A

h

('t')

(lIJIlI

h ('t')'A) 1I1ICJIa

'A

Ha

tPYHKIJ;lIlO h

('t'),

eCJIlI

B

KaiKp;OHTOllKe

't'

H3

[toc,

til]

3HalleHHe

t

('r)

paBHO

np01I3Bep;eHHIO

'A

Ha

3HalleHlIe

h

('t').

fOBOpHT,

liTO

MHomeCTBO

.AI'

tPYHKu;lIH

h ('t')(t

oc

-<

't'

-<

til)

!IBJIHeTCH

JIHHeHHbIM

tPYHKIJ;lIOHaJIb-

HbIM

npocTpaHcTBoM,

eCJIlI,

KaKOBbI

6bI

HH

6bIJIH

P;BetPYHKIJ;lIH

h(l)('r)

1I

h(2)

('r),

cop;epiKa~HecH

B .AI', 1I

KaKOBbI

6bI

HH

6bIJIlI

Be~eCTBeHHbIe

1I1ICJIa'A

1I

v,

tPYHKIJ;lIH

h

('t')

=

'Ah(l)

('t')

+

Vh(2)

('t')

TaKme

cop;epiKlITcH

B

.AI'.

RaK

MbI

BH,n;lIM,

onpep;eJIeHHe

JIlIHeHHOro

npocTpaHcTBa

.AI'

tPYHKIJ;HH h

('t')

=

h't

(t

oc

-<

't'

-<

til)

no

tPopMe

cOBeprneHHo

COBnap;aeT

C

orrpe,n;eJIeHHeM

JIlIHeHHOro

npocTpaHcTBa

.AI'

n

BeKTopoB

h =

{hI}

(i

=

1,2,

... , n). B

p;aJIbHeHilleM

JIHHeHHbIe

tPYHKIJ;lIOHaJIbHbIe

npo-

CTpaHCTBa

6yp;eM

0603HallaTb

ClIMBOJIOM

.AI'

(t

....

t

iJ

],

eCJIlI

,6yp;eT

BamHO

nop;lIepKHyTb,

Ha

KaKOM

OTpe3Ke

OClI

't'

orrpep;eJIeHbI

tPYHKIJ;lIH

h

('t'),

HBJIHIO~HeCH

9JIeMeHTaMH

9Toro

npocTpaHcTBa.

I1Hor,n;a

,n;JIH

0603:'

HalleHlIH

JI:fIHeHHOro

npocTpaHcTBa

.Al'u

oc

•

t

iJ]6y,n;eM

e~e

yrroTpe6JIHTb

CHMBOJI

.AI'

{h

('t')},

eCJIlI

BaiKHO

6y,n;eT

OTMeTlITb

6yKBY

h,

BbI6paHHylO

P;JIH

0603HalleHlIH

9JIeMeHTOB

:3Toro

npocTpaHcTBa.

HaKOHeIJ;,

B

Tex

CJIyqaHx,

Korp;a

BamHO

YKa3aTb

H

0603HalleHlIe

9JIeMeHTOB

p;aHHoro

npocTpaHcTBa

1I

OTMeTlITb

OTp630K,

Ha

KOTOpOM

3ap;aHbI

tPYHKIJ;lIH,

6y,n;eM

rrHcaTb

.AI'{h

('t');

t

oc

-<

't' < til}. B

Tex

CJIyqaHx,

KorAa

He

p;OJIm-

HO

B03HHKaTb

Hep;opa3yMeHlIH,

6yp;eM

0603HallaTb

tPYHKIJ;HOHaJIbHOe

rrpOCTpaHCTBO

rrpOCTO

6yKBOH.AI'.

ITo

aHaJIOrlIlI

C

KOHellHOMepHbIM

CJIy-

lIaeM

6y,n;eM

lIHorp;a

Ha3b1BaTb

9JIeMeHTbI

h

('r)

rrpOCTpaHCTBa

.AI'[toc.tiJl

TOllKaMH

H3

.AI'.

'

IIplIMepaMH

JIlIHeHHbIX

npOCTpaHCTB

.AI'[toc.tiJl

HBJIHIOTCH,

HarrpHMep,

CJIep;YIO~He

MHOiKeCTBa

tPyHKIJ;HD::

(1)

MHomecTBo

Bcex

B03MomHblX

Be~eCTBeHHbIX

tPYHKIJ;HH h

('t')

(t

oc

< 't' < til).

(2)

MHomecTBo

tPYHKIJ;lIH

h

('t')

rp;e

g(i)

(.)

-

KaKHe-JIH60

m

BlIp;a

m

=

~

'Aig(i)

('t'),

(54.1)

i=l

orrpeAeJIeHHblX

tPyHKIJ;Hii,

HarrpHMep

~~,"

......

.

..

,....

~,."

'"

'.;

"

'".

··~···,:·,:.i.'·

:.:

"

."

":'

..

',:;,

'::':'

;,.:::

..

:;..;.,

".'

:.,.:".:.

','

··

..

·::·.·I·

..

:::'::·.:;··.:·::

..

~:::;~:'4f:?:'

"·

.:

:'

..

C

410

l-IOPMlIPOBAHHhIiil

npOCTPAHCT1iA

<J>YtIH.QH:t'1

[rJl.

i 3

MoweT

6LITI.

g(i)

('t) =

-r

i

,

a

'Ai

-

BceBoaMowHLIe

Bem;ecTBeHHI.le

'lHC-

JIa.

)l;eHCTBHTeJII.HO,

O'leBHp;HO,

cYMMa

JII06LIX

P;BYX «PYHKIJ;HH

BHp;a

(54.1) H

rrpoHaBep;eHHe

JII060H

«PYHKIJ;HH h

(-r)

BHp;a

(54.1)

Ha

JII060e

'lHCJIO

'A

CHOBa

HBJIHeTCH

«pyHKIJ;HeH

TaKoro

we

BHp;a.

(3)

MHoweCTBO

Bcex

«PYHKIJ;HH h

(-r),

HerrpepLIBHLIX

Ha

OTpeaKe

[toc,tll).

(4)

MHowecTBo

~

Bcex

HerrpepLIBHLIX

«PYHKIJ;HH h

(-r),

HMeIOm;HX

B

HHTepBaJIe

(t

..

,

til)

HerrpepLIBHYIO

rrpoHaBop;HYlO

h

(-r).

HarrpoTHB,

.

HarrpHMep,

MHoweCTBO

.At «PYHKIJ;HH h

(-r),

KYCO'lHO-He:

rrpepLIBHLIx

Ha

[t

oc

, til],

Kawp;aH

Ha

KOTOpLIX

HMeeT

Ha

:3TOM

OTpeaKe

JIHmI.

OP;HY

TO'lKY

paapLIBa,

JIHHeHHOro

rrpocTpaHcTBa

.;Y'[toc.tll]

He

06paayeT,

TaK

KaK

CYMMa

TaKHX

P;BYX «PYHKIJ;HH

MoweT

HMeTI.

p;Be

TO'l-

KH

paapLIBa

H,

CJIep;OBaTeJII.HO,

He

6yp;eT

cop;epwaTI.CH

B .At.

BLI60p

Toro

HJIH

HHoro

rrpocTpaHcTBa

.;Y'[tCl.,tll]

orrpep;eJIHeTCH

Kaw-

P;LIH

paa

yCJIOBHHMH

TOH

aap;a'lH,

rp;e

HCrrOJII.aYIOTcH

:3JIeMeHTLI h

(-r)

:3TOrO

rrpocTpaHcTBa.

HapHp;y

C

'lHCJIOBLIMH

«PYHKIJ;HHMH h

(-r),

aap;aHHLIMH

Ha

oTpeaKe

[t

oc

, till H

rrpHHHMaIOm;HMH

CKaJIHpHLIe

aHa'leHHH

h

(-r),

MLI

6yp;eM

paccMaTpHBaTI.

TaKwe

BeKTop-«PYHI\IJ;HH

h

(-r),

aap;aHHLIe

Ha

oTpeaKe

[toc,

till H

rrpHHHMaIOm;He

BeKTopHI.le

aHa'leHHH

h

(-r)

= {hi

(-r)}

I"

I

(i

= 1, 2,

...

, r).

)l;JIH

TaKHX

BeKTop-«PYHKIJ;HH

h

(-r)

=

{'hi

(-rn

TaK-

I

we

eCTeCTBeHHLIM

06paaoM

orrpep;eJIHIOTCH

orrepaIJ;HH

CJIOWeHHH

H

yM-

I

,

f~'

I

.

HoweHHH

Ha

'lHCJIO:

f

(-r)

=

{ii

(-r)}

= h

(-r)

+ g

(-r)

=

{hi

(-r)

+

: i

+ gi (-rn, f

(-r)

=

{ii

(-r)}

=

'Ah

(-r)

=

{'Ah

i

(-r)}.

AHaJIOrH'lHLIM

06pa-

aOM

MHoweCTBO

.;Y'

BeI\TOp-«PYHKIJ;HH

h

(-r)

={h

i

(-rn

(toc

<

-r

<

til)

HB-

JIHeTCH

JIHHeHHLIM

«PYHKIJ;HOHaJII.HbIM

rrpocTpaHcTBoM,

eCJIH,

KaKOBLI

6LI

HH

6LIJIH

p;Be

BeI\TOp-«PYHKIJ;HH

h(1)(-r)

= {h(f)

(-r)}

H

h(2)

(-r)

=

= {h(f)

(-r)},

cop;epwam;HecH

B,ff,

H

KaI\OBLI

6LI

HH

6LIJIH

Bem;ecTBeH-

HLIe

'lHCJIa

'A

~

",

BeI\TOp-«PYHKIJ;HH

h

(-r)

=

').,h(l)

(-r)

+

"h(2)

(-r)

TaI\-

we

cop;epwHTcH

B

.;Y'.

IIpHMepoM

TaI\OrO

rrpocTpaHcTBa

.;Y'

MoweT

CJIYWHTI.

MHoweCTBO

r-MepHLIx

BeKTop-«PYHKIJ;HH

h

(-r)

= {hi (-rn

(i

= 1,

...

,

r;

toc

<

-r

< til),

Kawp;aH

COCTaBJIHIOm;aH

hi

(-r)

KOTOpLIX

HBJIHeTCH

«PYHK-

n;HeH,

HerrpepLIBHoH

Ha

[toc,

til).

BOJII.maH

'laCTI.

Tex

CBOHCTB «PYHKIJ;HOHaJII.HLIX

rrpocTpaHcTB,

KO-

TopLIe

paccMaTpHBaIOTcH

HHwe,

6yp;eT

OTHOCHTI.CH B

paBHOH

Mepe

KaK

I\

JIHHeHHLIM

rrpocTpaHcTBaM

.;Y',

COCTOHm;HM

mi

CKaJIHpHLIX

«PYHK-

n;HH

h

(-r)

(t

..

<

-r

< tlJ),

TaK

H K

JIHHeHHLIM

rrpOCTpaHCTBaM.;Y',

COCTOHm;HM

Ha

BeKTop-«PYHKIJ;lIH

h

(-r)

= {hi

(-r)}

(i =

1,2,

...

,

r;

toc

<

-r

< tlJ)'

IIO:3TOMY

MhI

'laCTO

TaM,

rp;e

:3'1'0

He

BawHO,

He

6yp;eM

crreIJ;HaJII.HO

orOBapHBaTI.,

0

KaKOM

HMeHHO

JIHHeHHOM

«PYHKIJ;HOHaJII.-

HOM

rrpOCTpaHcTBe,Al'

{h

(-r);

toc

<

-r

<

til}

(CI\aJIHpHOM

H

BeKTopHOM)

Hp;eT

pe'II.,

yrroTpe6JIHH

B

060HX

CJIY'laHX

Op;HH H

Te

we

060aHa'leHHH

.

JV'[toc.

til]'

.;Y'

{h

(-r);

toc

<:

-r

<

til}

P;JIH

rrpOCTpaHCTlil

1I

PJ;JIH

laX

:3JIeMeHTOB

C

·''

':.':!

'~...:

>'

§ 55]

lIHTErPAJI

JIEBErA

411

h

(-r).

TaM,

rAe

paaJIlI'lHH

Mewp;y

BeKTOpHI.IMl'I

H

CKaJIHpHblMli

~Ha'le

HHHMH

«PYHKIJ;HH h

(-r)

6YP;YT

cym;ecTBeHHLI,

()TO

06CTOHTeJII.CTBO

6yp;eT

crreIJ;lIaJII.HO

oroBapHBaTI.cH

1I

rrOA'lepKlIBaTI.cH.

3aMeTHM

B

aaKJIIO'leHHe

:3TOrO

rraparpa«pa,

'ITO

paCCMOTpeHHLIe

I

3AecI.

«PYHKIJ;HOHaJII.HI.le

rrpOCTpaHCTBa

.At'[t

<t.

til]

YP;OBJIeTBOpHIOT,

O'le-

BHP;HO,

BceM

yCJIOBHHM

I H

II,

yKaaaHHLIM

BLIme,

B § 49,

rrpH

lIaY'le-

HHH

KOHe'lHOMepHLIX

JIHHeHHLIX

BeKTopHLIX

rrpOCTpaHCTB

H

xapaKTe-

pHaYIOIII;lIM

06m;ee

a6cTpaKTHoe

rrOHHTHe

JIHHeHHOro

rrpocTpaHcTBa.

3'1'0

BaWHoe

06CTOHTeJII.CTBO

H

orrpaBp;LIBaeT

3p;ecI.

yrroTpe6JIeHHe

Tep-

MHHa

JIHHeHHOe

rrpOCTpaHCTBO

.;Y'[t

...

tll]«PYHKIJ;lIH

h

(-r),

§ 55. HHTerpan: JIe6era

}

IIpewp;e

'leM

rrpHcTyrrHTI.

K

uaY'leHHIO

«PYHKIJ;1I0HaJII.HLIX

rrpo-

CTpaHCTB

.;Y'[t

...

tflJ'

MI.I

B

HaCTOHm;eM

rraparpa«pe

paCCMOTpHM

COBceM

KpaTKO

H B

orrHCaTeJII.HOH

«popMe

rrOHHTHe

HHTerpaJIa

JIe6era,

6ea

yrrOMHHaHHH

0

KOTOpOM

6LIJIO

6LI

TPYP;HO

060HTHCI..

IIpH

()TOM

6yp;eM

CJIep;OBaTI.

H3BeCTHOMY

KypCy

[20*).

IIoJIHoe

HaJIOWeHHe

Borrpoca

'lH-

TaTeJII.

MoweT

HaHTH

B

yrroMHHYTOH

MOHorpa«pHH

[20*1,

a

TaK1Ke

B

KHH-

re

[9*1,

'l.

2.

MCrrOJII.aOBaHHe

HHTerpaJIa

JIe6era

OKaaI.IBaeTCH

He06-

XOP;HMLIM

BBHAY

Toro,

'lTO

MHorHe

pa3pLIBHLIe

«PYHI\IJ;HH

p;awe

co

cpaB-

HHTeJII.HO

HeCJIOWHOH

CTpyKTYPOH

HBJIHIOTCH

HeHHTerpHpyeMhIMH

B

06hI'lHOM,

aHaKOMOM

Ha

Kypca

MaTeMaTH'leCKOrO

aHaJIHaa

CMLICJIe

HJIH,

I\aK

rrpHHHTO

rOBopHTI.,

B CMLICJIe

PHMaHa.

OAHaKO

B rrpOIJ;eCce

Teope-

TH'leCKHX

paCCMOTpeHHH

OrrepaIJ;HH

HHTerpHpOBaHHH

rrpHXOAHTCH

rrpo-

HaBop;HTI.

H

Hap;

p;OCTaTO'lHO

CJIOWHLIMH

paapLIBHLIMH

«pyHKIJ;HHMH.

IIpH

:3TOM

OKaaI.IBaeTCH,

'lTO

P;JIH

mHpOKoro

KJIaCCa

paapLIBHLIX

«PYHK-

IJ;HH

rrpaBOMepHOCTI.

TaKHX

OrrepaIJ;HH

MOWHO

BrrOJIHe

CTporo

060CHO-

BaTI..

Ho

P;JIH

:3TOrO

rrpHXOAIITCH

BBOP;HTI.

HOBoe

orrpeAeJIeHHe

HHTe-

rpaJIa,

60JIee

06m;ee,

'leM

orrpeAeJIeHHe

ero

rro

PHMaHY.

TaKoe

orrpep;e-

JIeHHe

6LIJIO

p;aHO

A.

JIe6eroM

B

Ha'laJIe

XX

BeKa.

CeH'laC

H3BeCTHLI

paaJIH'lHLIe

rrop;xop;LI

K

rrOCTpoeHHIO

HHTerpaJIa

JIe6era.

3p;ecI.,

I\aK

ywe

O')'Me'laJIOCI.,

MLI

H36HpaeM

rrYTI.,

orrHcaHHLIH

B

KHHre

[20*1.

3THM

rryTeM

HHTerpaJI

orrpep;eJIHeTCH

CJIeP;YIOm;HM

06paaoM.

CHa'laJIa

paCCMOTpHM

cTyrreH'laTLIe

«PYHKIJ;HH,

orrpep;eJIeHHLIe

B

KO-

He'lHOM

HJIH

6eCKOHe'lHOM

HHTepBaJIe

(a,

b),

'1',

e.

TaKHe

«PYHKIJ;HH, 1{Q-

TopLIe

rrpHHHMalOT

rrOCTOHHHLIe

aHa'leHHH

Ck

Ha

HHTepBaJIaX

i

k

KOHe'l-

HOH

P;JIHHLI I i

k

I,

BaHTLIX

B

KOHe'lHOM

'lHCJIe,

H

paBHLIe

HYJIIO

BHe

:3THX

HHTepBaJIOB

(pHC.

55.1).

B

KOHIJ;ax

HHTepBaJIOB

i

k

3Ha'leHHH

qlYHKIJ;HH

MoryT

6hITI.

KaKHMH

yrop;Ho

-

:3TO

Hecym;ecTBeHHO.

)l;JIH

Kawp;oH

cTyneH'laTOH

«PYHKIJ;HH

HHTerpaJI

OUpep;eJIHeTCH

06LI'lHhIM

o6pa30M

KaK

CYMMa

I =

~Ck

Ii

k

I

(55.i)

J,;

-'-----""--'-'--'-'-'

-'.-.

~"~'

. .

..

.

""

..

' .. . ...

.'

".".-

'"

'.':':."

C

ern. 13

lIOPMHPOBAHHbIE

npOCTPAHGTBA

<I>YHRIVI'fl

412

H

HMeeT,

CJIe,n;OBaTeJIbHO,

KaK

H

nOJIOmeHO,

CMbICJI

nJIOm;a,n;H

(c

y'IeTOM

3HaKOB

op,n;HHaT

Ck)'

C

nOMOm;bIO

npe,n;eJIbHOrO

nepexo,n;a

nOHHTHe

HH-

TerpaJIa

pacnpOCTpaHHeTCH

Ha

cPYHRIJ;HH

OOJIee

oOm;eH.

npHpo,n;bI.

IIpH

8TOM,

o,n;HaKO,

nOTpeoye.TCH

O,n;HO

HOBoe

nOHHTHe,

KOTopoe

MbI

CeH'IaC

onpe,n;eJIHM.

PaCCMOTpHM

HeKOTopoe

MHomeCTBO

@

TO'IeK

Ha

'IHCJIOBOH

npHMoiL

fOBOpHT,

'ITO

8TO

MHomeCTBO

HMeeT

uy.aeeylO

Mepy,

eCJIH

Bce

TO'IKH

H3

@

MomHO

nOKpbITb

CHCTeMOH

HHTepBaJIOB,

CyM-

MapHaH

,n;JIHHa

KOTOpbIX

He

npeBocxo,n;HTe,

KaKHM

ObI

MaJIbIM

HH

ObIJIO

Hanepe,n;

3a,n;aHHoe

'IHCJIO

e

>0.

r;

__

.

IIpHMepaMH

MHomeCTB

HYJIe-

BOH

MepbI

HBJIHroTCH:

(1)

JIro-

C,~

I _

II

I

e"

ooe

KOHe'IHOe

MHomeCTBO

TO-

'IeK

T

1

, T

2

,

...

,

Tk;

(2)

JIroooe

,1

(U

\II

i I I ...

C'IeTHOe

MHomeCTB

0

TO'IeK

T

1

,

ill-

. b

'C'

.

'n

. T

2

,

...

,

Tk""

B

caMOM

,n;eJIe,

B

OOOHX

CJIy'IaHX

Heooxo,n;HMOH

cHcTeMoH

HHTepBaJIOB

MoryT,

PRe. 55.1.

O'IeBH,n;HO,

CJIymHTb

HHTepBa-

I

JIbI

BH,n;a

(Tk

- e/2

k

,

Tk

+e/2

k

).

l~.

fOBOpHT,

'ITO

HeKOTopoe

yCJIOBHe

BbInOJIHHeTCH

n01J,mu

eclO{)y

Ha

He-

KOTOpOM

HHTepBaJIe,

eCJIH

8TO

yCJIOBHe

BbInOJIHHeTCH

,n;JIH

Bcex

TO'IeK

H3

,n;aHHoro

HHTepBaJIa,

3a

HCKJIro'IeHHeM,

MomeT

ObITb,

KaKorO-TO

MHomeCTBa

@

HyJIeBOH

MepbI.

Tenepb

HHTerpaJI

JIeoera

CTpOHTCH

CJIe,n;yrom;HM

00pa30M.

Pac-

CMOTpHM

KaKyro-JIHOO

uey6blealOUfYlO

nOCJIe,n;OBaTeJIbHOCTb

{'¢k

(T)}

(k =

1,2,

...

),

COCTOHm;yro

H3

cmyneu1J,arn,ux

tjiyw>/fuu

'¢k

(T)

H

cxo-

,n;Hm;yrocH

nO'ITH

BCro,n;y

K

HeKoTopoH

cPYHK~HH

'¢

(T),

T.

e.

'¢k

(T)

-<

'¢k+l

(T)

H

lim

'¢k

(T)

= '¢

(T)

npH

Bcex

T,

KpOMe,

MomeT

ObITb,

MHO-

.

k->-oo'

meCTBa

TO'IeK

T

HyJIeBOH

MepbI.

IIpe,n;noJIomHM

em;e,

'ITO

,n;JIH

,n;aHHOH

nOCJIe,n;OBaTeJIbHOCTH

{'¢k

(T)}

nOCJIe,n;OBaTeJIbHOCTb

HX

HHTerpaJIOB

I

['¢k

(T)l,

3a,n;aHHbIX

paBeHCTBaMH

(55.1),

orpaHfleHa.

Bce

cPYHK~HH

'¢

(T),

KOTopbIe

MomHO

nOJIyqHTb

KaK

onHcaHHbIe

TOJIbKO

'ITO

npe,n;e-

JIbI

no,n;OOHbIX

nOCJIe,n;OBaTeJIbHOCTeH

{'¢k

(T)},

COCTaBHT

HeKOTopbIH

KJIaCC

~(1).

II

0

onpe{)e.aeuulO

,n;JIH

cPYHK~HH

'¢

(T)

H3

8(1)

nOJIararoTl

i

b b

11

~

'¢'('t) dT:'

lim

~

"'k

(T:)

dT:.

(55.2)

a k->oo

a.

'I

,

AJIH

0003Ha'IeHHH

HRTerpaJIa

JIeoera

MhI

HCnOJIb3yeM

OOm;HH

3HaK

onpe,n;eJIeHHOrO

HHTerpaJIa.)

AaJIee,

paCCMaTpHBaeTCH

em;e6oJIee

mH-

POKHH

-n.aacc

tjiyur>/fUU

~(2),

cocmOJW,fuu

ua

paauocmeu

ecee03MOJICHblX

.i

'1

I

livit1£lfUU

''P

(T),

npuHq,Qlte;HfaUfUX

(/(1),

AJIH

<PYIfK~HH

t

(T)

113

?(3/

;!r

."

'

..

.'

...

-:

..'::' "

:....

..,"

":,'

;

'.

',

..'

';"

:

·.·.··

,'.',

',,',

c·

.'/

§ 65]

HHTErPAJI

JIEBEfA

413

HHTerpaJI

onpe{)e.aJLemCJL

paBeHCTBOM

b b b

~

f

(T)

d't =

~

(55.3)

,¢(1)

(T:)

d't -

~

,¢(Z)

('t) d't,

d a a

r,n;e

",(1)

(T)

H

,¢(2)(T)

-

8TO

KaK

pa3

Te

cPYHK~HH

H3

/1(1),

KOTOpble

onpe-

,n;eJIHroT f

(T)

KaK

pa3HOCTb

f

(T)

~

'¢(1)

(T)

-

'¢(2)

(T).

8THM

H

3aBep-

maeTCH

nOCTpOeHHe

HHTerpaJIa

JIeoera.

IIMeHHo,

Kam,n;yro

cPYHK~Hro

f(T)

H3

KJIaCCa

~(2)

Ha3bInaroT

cPyIiK~HeH

uumeepupyeMou

(no

JIeoery),

a

BeJIflHHY

(55.3),

onpe,n;eJIeHHYro

,n;JIH

Hee,

HCXO,n;H

H3

paBeHCTB

(55.1) H (55.2),

Ha3bIBaroT

ee

uumeepa.lLOM

JIe6eea.

qHTaTeJIH

He

,n;OJImHa

BBO,n;HTb B

3aOJIym,n;eHHe

Ta

Kamym;aHcH

npOCTOTa,

C

KOTOpOH

nOCTpoeH

BbIme

HHTerpaJI

JIeoera

(55.3).

)J;eJIo

B

TOM,

'ITO

MbI

COBepmeHHO

OCTaBHJIH

B

CTopOHe

CTporoe

MaTe¥aTH'IeC-

Koe

OOOCHOBaHHe

,n;aHHoro

onpe,n;eJIeHHH.

RpOMe

Toro,

O'IeBH,n;HO,

'ITO

,n;aHHoe

BbIme

onpe,n;eJIeHHe

HHTerpHpyeMocTH

cPYHK~HH

f

(T)

H

OIIH-

caHHbIH

TOJIbKO

'ITO

cnocoo

BbI'IHCJIeHHH

HHTerpaJIa

OT

nOH

cPYHK~HH

y,n;OOHbI

JIHmb

C

n03H~HH

TeopHH,

HO

npaKTH'IeCKaH

paOOTa

C

HHTe-

rpaJIOM

JIeoera

TpeoyeT

Cepb63HOrO

,n;onOJIHHTeJIbHOrO

H3yqeHHH.

O,n;HaKO

Ha

8TH

X

Bonpocax

MbI

OCTaHaBJIHBaTbCH

He

oy,n;eM.

HaM

ObIJIO

BamHO

,n;aTb

3,n;ecb

JIHmb

caMoe

Oom;ee

npe,n;cTaBJIeHHe

0

MaTeMaTH'IeC-

KOH

KOHCTPYK~HH

HHTerpaJIa

JIeoera.

AeJIo

B

TOM

em;e,

'ITO

HHTerpaJIbI

JIe6era

H

PHMaHa

qaCTO

COBna,n;aroT.

TaK,

,n;OKa3bIBaeTCH,

'ITO

eCJIH

,n;JIH

HeKOTopOH

OrpaHH'IeHHOH

Ha

[a, b1

cPYHK~HH

h

(T)

cym;ecTByeT

HHTerpaJI

PJiIMaHa,

TO

HHTerpaJI

JIeoera

8TOH

cPYHK~HJiI

TaKme

cym;ecT-

ByeT

JiI

paBeH

HHTerpaJIy

PHMaHa.

B

'IaCTHOCTH,

aTO

paBeHCTBO

cnpa-

Be,u;JIJiIBO,

HanpHMep,

,n;JIH

KyCO'IHO-HenpepbIBHbIX

cPYHK~HH

h

(T),

C

KOTOpbIMH

HaM

H

npH,n;eTCH

JiIMeTb

,n;eJIO B

,n;aHHOH

KHHre.

CJIe,n;OBaTeJIb-

HO,

npar>mU1{ecr>oe

BbI'IJiICJIeHHe

Tex

HHTerpaJIOB,

C

KOTOPbIMJiI

MbI

cPaK-

TfleCKH

BCTpeTHMCH,

Ha,n;JIemJiIT

BbInOJIHHTb

no

H3BeCTHbIM

'IHTaTeJIro

npaBHJIaM,

BHe

3aBHCHMOCTJiI

OT

Toro,

KaKOH

no

HHTerpaJI

-

JIeoera

JiIJIJiI

PJiIMaHa.

3aMeTJiIM

em;e,

'ITO

JiIHTerpaJI

JIeoera

Booom;e

06JIa,n;aeT

CBOHCTBaMJiI,

aHaJIOrJiI'IHbIMH

JiI3BeCTHbIM

CBoHcTBaM

HHTerpaJIa

PJiIMa-

Ha,

XOTH

JiIHTerpaJI

JIeoera

H

HBJIHeTCH

nOHHTJiIeM

OOJIee

OOm;JiIM

JiI

cym;ecTByeT

,n;JIH

OOJIee

mHpOKoro

KJIaCCa

cPYHK~JiIH,

'IeM

JiIHTerpaJI

PJiIMaHa.

Bo

BCHKOM

cJIyqae,

'IJiITaTeJIb,COBepmeHHO

He

3HaKOMbIH

C JiIH-

TerpaJIOM

JIeoera,

npJiIHHB

K CBe,n;eHJiIro

MaTepJiIaJI

,lToro

naparpa4>a,

MomeT

npJiI

'ITeHJiIJiI

OCHOBHoro

MaTepJiIaJIa

KHJiIrJiI

JiIrHOpJiIpOBaTb

pa3-

JIJiI'IJiIe

Mem,n;y

JiIHTerpaJIOM

JIeoera

JiI

PJiIMaHa,

TaK

KaK

aTO

pa3JIJiI'IJiIe

BamHO

JIJiImb

,n;JIH

CTpororo

OOOCHOBaHliIH

Tex

HymHhIx

HaM

CBOHCTB

cPYHK~JiIOHaJIbHbIX

npOCTpaHCTB,

nOHHTb

CMbICJI

KOTOpbIX

MomHO

JiI

oe3

npOHJiIKHOBeHJiIH

B

,n;eTaJIH

,noro

CTpororo

OOOCHOBaHHH

•

OCTaHOBJiIMCH

B

3aKJIro'IeHJiIe

em;e

Ha

O,n;HOM

nOHHTJiIH,

CBH3aHHOM

C

JiIHTerpaJIOM'

JIeoera.

PaCCMOTpJiIM

HeKOTopoe

MHomeCTBO

ff>

TO'IeK

JJa

JiIHT8,pBaJI8

(a,

b),

JiI

nycTb

WyHK~JiIH

f ('t),

~oTopaJ!

npJiIHll14aeT

HQ

<.

~·~.:~~:

..

:;·:.~:i.~:~W4~

'"

-K3r=r"7?i2'-oJ;oT'~""

"

••

"

.;,.;

, ,

'.

";",'';''''';''".

".";.:.,.

,":.

.

•...

".'

,...

..

.•.

.',.,

.

.'

:.

..

':

':,:

..

:.•

CD

~:..:

~J

41ft

HOPMHPOBAHHhIE

IIPOCTP

AHCTBA

<I>YHIUJ,Hff [rJIo 13

:!P

3Ha'IeHHe

1 H

3Ha'IeHlIe

°

BHe

fJU,

eCTh

Ul-tmezpupye.ltMI.

rjJYH'K,/fUfl,

Tor,n;a,

HCXO,n;H

H3

onHcaHHoro

BliIllie

cnoc06a

nOCTpoeHHH

HHTerpaJIa

(55,3),

eCTeCTBeHHO

C'IHTaTh,

'ITO

HHTerpaJI

(55.3)

HMeeT

CMhICJI

IIJIOlll,a,n;H

TOH

<pHryphI,

KOTopaH

HMeeT

BhICOTY

h = 1 H

onHpaeTCH

Ha

MHomeCTBO

fJU

KaK

Ha

OCHOBaHMe.

Ho,

C

,n;pyroH

CTOpOHhI,

eCTeCTBeHHO

npMHHTh,

'ITO

~Ta

rrJIOlll,a,n;h

eCTh

npOII3Be,n;eHHe

BliICOThI

h = 1

Ha

«CyM-

MapHylO»

,n;JIHHy

f..I.

(fJU)

MHO-

t

meCTBa

fJU,

T.

e.

b

~

f

(T)

dT

=

f..I.

(fJU).

(55.4)

a'

B

COOTBeTCTBIIH

co

CKa3aH-

HhIM

BeJIH'IHHy

f..I.

(fJU),

on-

pe,n;eJIeHHylO

paBeHCTBOM

'&'

g>

MHomeCTBa

fJU,

a

caMO

MHO-

P:ne.

55.2.

meCTBO

fJU

Ha3hIBalOT

H3-

MepHMhIM.

BamHo

OTMe-

tl

0

-----

/---~

b

(55.4),

Ha3hlBalOT

MepoH

I,..",

THTh

TpH

06CTOHTeJIhCTBa:

(1)

f..I.

(fJU)

= °

Tor,n;a

II

TOJIhKO

Tor,n;a,

Kor,n;a

fJU

HBJIHeTCH

MHomeCTBOM

HYJIeBOH

MephI

B

onpe,n;eJIeHHOM

paHhllie

CMhICJIe;

(2)

H3MepHMble

MHO-

meCTBa

COCTaBJIHIOT

BeChMa

06lliMPHhIH

KJIaCC,

HO

He

HC'IepnhIBalOT

Bcex

MHomeCTB,

IWTophIe

MomHO

nOCTpOHTh

H3

TO'IeK

Ha

'IHCJIOBOH

rrpHMoH;

(3)

MomHO

6hIJIO

6bI,

Ha060poT,

onpe,n;eJIHTh

CHa'IaJIa

nOHHTHe

H3MepHMoro

MHomeCTBa

Hero

Mepy,

a

nOTOM

orrpe,n;eJIHTh

HHTerpaJI.

TaKOH

nYTh

onpe,n;eJIeHHH

IIHTerpaJIa

JIe6era.HBJIHeTCH

06lll,errpHHHThIM.

§ 56.

CT~H)wpTHbre

HOPMbI

B

«pyrmD;HOHaJIbHbIX

npOCTpaHC'l'BaX

B

aTOM

rraparpa<pe

paCCMaTpHBalOTCH

BeJIH'IHHhI,

aHaJIOrM'IHhIe

TeM,

KOTophIe

paHee

6hIJIH

BBe,n;eHhI ,n;JIH

KOHe'IHOMepHoro

CJIy'IaH

B

§ 49.

nYCTh

MhI

HMeeM

HeKOTopoe

JIHHeHHOe

npoCTpaHCTBo.Ai'

{h

(t);

t

a

<;;

1:'

<;;

t~}.

CKameM,

'ITO

P;JIH

aJIeMeHTOB

h

aToro

rrpOCTpaHCTBa

orrpep;eJIeHa

HopMa

p

[hl,eCJIH

Kam,n;oH

<PYHKD;HH h

(1:')

H3.Ai'

conOCTaB-

JIeHO

HeoTpHD;aTeJIhHOe

'IIICJIO

p [h],

yp;OBJIeTBopHIolll,ee

CJIe,n;ylOlll,IlM

yCJIOBHHM:

1)

p [h] = 0,

eCJIIl

h

(1:')

=°

HJIH

eMIl

h

(1:'1

HecYUfecm-

eel/HO

OTJIM'IHa

01'

Tomp;eCTBeHHoro

HyJIH;

2) p [h] >

0,

eCJIH

<PYHKD;HH h

(1:')

cYUfeCmeeHHO

OTJIM'lHa

01'

TOi-Kp;eCTBeHHoro

HyJIH;

l

(56.1)

3) p

[h(l)

+

h(2)]

<;;

p

[hell]

+p [h(2)];

4) p ["-h] =

)I.,

Ip [hl,

0;

~>

§ 56]

HOPMhI

B

<'t>YHRUHOBAJIhHh!X

ITPOCTPAHCTBAX

415

KaKOBhI

6liI

HH

6hIJIH

h,

h(l),

h(2)

H3

.Ai'

H

'IIICJIO

')"

(CM.

,n;JIH

cpaBHeHHH

onpep;eJIeHHe

HOpMhI

n-MepHoro

BeKTopa

h =

{h;}

B § 49,

CTp.

396).

JIHHeHHoe

<PYHKD;HOHaJIbHOe

npocTpaHcTBo

';y'

Ha3hIBalOT

HOpMUpO-

eaHHbLM

npocmpaHcmeOllt,

eCJIH

.,n;JIH

ero

aJIeMeHTOB

h

(1:')

(tIL

<;;

1:'

<;;

t~)

BBe,n;eHa

HopMa

co

CBOHCTBaMH

(56.1).

CBoHcTBa

3),

4)

aHaJIOrH'IHliI

no

cBoeMy

CMhICJIY

cooTBeTcTBylOlll,HM

CBOHCTBaM

(49.4)

HOPMhI

,n;JIH

KOHe'IHOMepHliIX

IIpOCTpaHCTB

fJiJ

.

lIep-

n

BhIe

me

,n;Ba

yCJIOBHH

H3

(56.1)

Hym,n;aIOTCH

B

rrOHCHeHMII.

,n:eJIo

B

TOM,

'ITO

BO

MHorIIX

3ap;a'IaX

H3MeHeHHe

3Ha'IeHlIH

<PYHKD;HH h

(1:')

B

KaKOH-

JIII60

O,n;HOH

TO'IKe

(HJIH

B pH,n;e OT,n;eJIbHhIX

TO'IeK)

MomeT

HHKaK

He

OTpamaThCH

Ha

pe3YJIhTaTe

Tex

orrepaD;HH,

ROTophIe

npOH3Bo,n;HTCH

C

aTOH

<pyHKD;HeH

B

xo,n;e

pellieHIIH

3a,n;a'IH.

no

aTOH

npH'IHHe

<PYHKD;HH

h

(1:'),

KOTophIe

JIMllih

B

OT,n;eJIhHliIX

TO'IKaX

OTJIH'IHliI

01'

HyJIH,

MoryT

BeCTH

ce6H

B

KOHe'IHOM

C'IeTe

TaK

me,

KaK

H <PYHKD;MH,

Tomp;ecTBeHHo

paBHliIe

HyJIIO.

IIHa'Ie

rOBopH,

,rJ;JIH

p;aHHoro

Kpyra

3ap;a'I

aTH

<PYHKD;HH

OTJIH'IaIOTCH.oT

HyJIH

HecYUfecmeeHHo.

HarrpHMep,

eCJIH

pe'Ib

H,n;eT 0

BhI-

'IHCJIeHMH

Koa<p<pHD;HeHTOB

Wyphe

<PYHKD;HH h

(1:')

t(3

In

=

1

-

~

h ('t)

sin

•

t

a

2~n

(T

-

ta)

t _ t

dt

(3

a

(n = 1, 2,

...

), (56.2)

TO

HHTerpaJIliI

(56.2)

6y,n;yT

rrOJIy'IaThCH

paBH1>IMH

HyJIIO

KaK

,n;JIH

<:PYHK-

D;HH h

(1:')

=0,

TaK

H ,n;JIH <PYHRD;HH

h(l)

(1:'),

OTJIH'IHOH

01'

HyJIH

JIIIllih

B

KOHe'IHOM

HJIH

C'IeTHOM

'IIICJIe

TO'IeK.

CJIep;OBaTeJIhHO,

,n;JIH

3a,n;a'I1I

o

BliI'IIICJIeHIIH

IIHTerpaJIOB

(56.2)

<pyHKD;uH

h(l)

(1:')

HecYUfecmeeHHO

OTJIH'IaeTCH

01'

HyJIH.

A:J;laJIOrH'IHbIM

06pa30M

06CTOHT

,n;eJIO H B

CJIyqae

C

HOpMOH.

,n:eJIO B

TOM,

'ITO

HOpMa

p [h] BBO,n;HTCH 061>I'IHO C D;eJIhIO

06-

CJIymIITh

HeKOTOp1>IH

Rpyr

3a,n;a'I.

nOaTOMY

OKa3liIBaeTCH

BamHhIM,

'IT06liI

HopMa

OTJIII'laJIaCh

01'

HyJIH

JIIIllih

Ha

Tex

<pyHKD;HHX

h

(1:'),

KO-

TopliIe

,n;JIH

,n;aHHoro

KJIaCCa

3a,n;a'I

CYlll,eCTBeHHO

OTJIH'IaIOTCH

01'

Tom-

,n;eCTBeHHoro

HyJIH.

IITaK,

IIepBliIe

,n;Ba

yCJIOBHH

B (56.1)

6y,n;eM

nQHH-

MaTh

CJIe,n;ylOlll,IIM

06pa30M:

HopMa

p

[h]

,n;OJImHa

6liITh

BeJIH'IHHOH

nOJIOmIITeJIhHOH

P;JIH

Bcex

Tex

<PYHKD;IIH h

(1:'),

OTJIH'IHe

KOTOphIX

01'

Tom,n;ecTBeHHoro

HyJIH

KaK-JIH60

CRa3liIBaeTCH

Ha

pe3YJIhTaTax

orrepa-

D;IIH,

BCTpe'IalOIIJ;HXCH

B

paCCMaTpHBaeMOM

KJIaCCe

3a,n;a'I;

B

rrpOTHB-

HOM

CJIyqae

<PYHKD;HIO

Ha,n;JIemIIT

C'IIITaTh

OTJIH'IHOH

01'

HyJIH

HeCYlll,eCT-

BeHHO,

a

ee

HOpMy

IIOJIaraTh

paBHOH

HyJIIO

•

3aMeTIIM

elll,e,

.'ITO

CJIOBa

«CYlll,eCTBeHHO»

H (<HeCYlll,eCTBeHHO»

MhI

6y,n;eM

B TeKC.Te

06hI'IHO

OIIyC-

KaTh,

HO

BCHKIIH

pa3,

KaK

<PYHKD;IIH h

(1:')

6y,n;eT

cpaBHIIBaThCH

C

HY-

JIeM,

CJIep;yeT

HMeTh

B

BII,n;y,

'ITO

pe'Ih

IIp;eT

0

cpaBHeHHH

HMeHHO

IIO

cYUfecmey.

}\oHKpeTHhIe

rrpaBIIJIa

,n;JIH

BliI'IHCJIeHIIH

HOpM1>I,

paccMoTpeHHhle

B § 49

,n;JIH

KOHe'IHOMepHhIx

BeKTopoB,

CBH3aHhI

C

orrepaD;IIeH

B3HTJa:H

MaKcHMyMa

HJIM

C

orrepaD;HeH

CJIOmeHHH

Mo,n;yJIeH

KOMrrOHeHT

hI.

-

=c-

-"

'G

,.~-,

tIC

,-.....-..,'

.•

g-r

...... .L

••

3

.

_.~

..

~_

......

:.

p

~:

.:_

..

":'.

',.:'''::.':

::

.....

:

:.:

";.

:,.:,

...

:"

..

;~:.;,:

"::':;

;':'-:':.;::.

.

:.'

':,'.

':::

:':::'-':":':'1';&7','

l"".::'

.....

,

D.

~~

...

:r

HOPMUPOBAHHbIE

IJPOCTPAHCTBA

«I>YHRIJ;u:B

[rJI.

13

416

§

56]

HOPM:bI B «I>YHRIJ;HOHAJIbHDIX

IIPOCTPAHCTBAX

417

ABilJIorHlJBhIM

oopaaOM

CTPOJlTCJI OOLIlJBO H

BOpMhI

<PYBKIJ;HH h

(-r)

,KOTO-

pLIe

MhI

MO;JWM

TpaKTOBaTb

KaK

BeKTopbI

h C

oeCKOBelJHLIM

'IHe-

Ha

MHomeCTBe

TOlJeK •

HYJIeBoi!

MepbI,

a B

OCTaJIbHbIX

CJIYlJaJIX :31'01'

JIOM

KOMnOH~HT

h

(-r)

(t

l1

<'

-r

<'

t

s

).

IIpH

aTOM eCTeCTBeHHbIM

oopaaOM

HHTe1'paJI

JIBJIJIeTCJI

BeJIHlJHHoi!

nOJIOmHTeJIbHoi!,

CnpaBeAJIHBOCTh

onepaIJ;HJ1

CJIOmeHHJI

no

HHAeKCY

i

aaMeHHeTCJI

OnepaIJ;HeH

HHTe1'pHpo-

TpeTbe1'o

yCJIOBHJI

CJIeAyeT

H3

HHTe1'paJIbHoro

HepaBeHcTBa

MHHROBCRO_

1'0 ([14'1,

CTp,

497):

BaHHK

no

nepeMeHHOH

-r.

3AeCb,

BnpOlJeM,

BoaHHKaeT

aaTpYAHeHHe.

OnepaIJ;HJI

BaJlTHJI

MaKCHMYMa

HJIH

OnepaIJ;HJI

HHTerpHpOBaHHJI:no

't

MomeT

OhITb

BHnOJIHeHa

He

co

BCJlKOH

<PYHKIJ;HeH h

(-r).

IIoaTOMY

npH

[

~

J

11

I

h(l)

Cr)

+h(2)

(t)

/P

dT

]@['SS]@

Ihill

(t)

IP

dT

< +

onpeAeJIeHHH

TOH

HJIH

HHOH

HOpMLI

P [h] AJIJI <PYHKIJ;HH h

(-r)

Ha

1 1

.Ai'

{h

(-r)}

npH

nOMOm;H

ynOMJlHYThIX

OnepaIJ;HH

npHxOAHTCJI

orpaa:wIH-

11

BaTbCJI

JIHillb

TaKHMH

JIHHeHHbIMH

npOCl'paHCTBaMH

.Ai'

aTHX

<PYHKIJ;Hii:,

l{l

1/

AJIJI

KOl'OpLIX

HeOOXOAHMhIe

OnepaIJ;HH

BhInOJIHHMhI.

BCJIeACTBHe

aT01'O

TeOpHJI

HOpMHpoBaHHhIx

<PYHKIJ;HOHaJIbHhIX

npocTpaHcTB

TeCHO CBJI-

+

[S

Ih(2)

(T)

/P

dT]

p

(56.4)

11

aaHa

C

TeopHeH

UHme<JpupoeaHU.fL

<PYHKIJ;HH h

(-r).

'

npH

p = 2.

lJeTBepToe

YCJIOBHe BbIllOJIHJIeTCJI C OlJeBHAHOCTbIO.

IIpo-.

IIpHBeAeM

HeROTophIe

npHMephI

onpeAeJIeHHH

HOpM

p[h],

C KOTO-

CTpaHcTBO

<PYHKIJ;Hi! h

(or)

C

HOpMOH

(56.3)

6YAeM

Ha3bIBaTb

rrpocTpaH_

CTBOM,2(2)CI

I1

•

'/3]

HJIH,

ROpoqe,

npoCTpaHCTBOM

2(2).

.

phIMH

OYAeM

BCTpelJaTbCJI

B

AaJIbHeHilleM.

B CJIeAYIOm;HX

HHme

npHMe-

pax

56.1-56.4

pelJb

nOHAeT

0

CKaJIJlpHhIX

<PYHKIJ;HJlX.

1°.

PaccMoTpHM

MHomeCTBO

Bcex

<PYHKIJ;HH h

(-r)

(tat

<,-r

<'

t

s

),

HH-

2°.

HOpMa,

onpeAeJIeHHaJI

AJIJI

n-MepHo1'o

BeRTopa

paBeHcTBOM

(49.6),

0606m;aeTCJI

AJIJI <PYHRIJ;HH h

(.)

TaR;

Te1'paJI

01'

KBaApaTa

KOl'OphIX

cym;eCl'ByeT

H JlBJIJleTCJI

KOHe'lHbIM.

3TH

<PYHKIJ;HH

oopaaYIOT

JIHHeHHOe

npOCTpaHCTBO,B

KOTOpOM

HOPMY

Mom-

'S

HO

onpeAeJIHTb

paBeHCTBOM

IS

p [hI =

~

/ h

(T)

I

dT.

(56.5)

11

Ia.

,II

P [h] =

[

J

('

h

2

(T:)

dT:

J '

.

(56.3)

llPH

aTOM

MbI

AOJImHbI

paCCMaTpHBaTb

JIHHeHHoe

rrpOCTpaHCTBO

JY"

.

TaRHX

<PYHKIJ;HH h

(T)

(t

l1

< •<

t/3),

AJIJI

ROTOpbIX

cym;eCTByeT

H

HopMa

(56.3)

JlBJIJleTCJI, OlJeBHAHO, OOOOm;eHHeM

eBKJIHAOBOH

HOp-

1

11

O1'paHHqeH

HHTerpaJI

01'

HXMOAYJIJI.

RaR

H B

rrpep;bIP;ym;eM

IIPHMepe,

MhI

n-MepH01'O

BeKTopa

h =

{hd

(CM. <POPMYJIY

(49.2)

CTp.

395).MHTe1'-

HHTe1'paJI

(56.5) rrOHHMaeTCJI 3p;eCb B CMbICJIe

HHTe1'paJIa

JIe6era

•.

paJI

(56.3)

nOHHMaeTCJI

3AeCb

B CMbICJIe

HHTe1'paJIa

JIeoe1'a

(CM. §

55,

MO",HO,

'aR

H B apHMepe 1°,

apOB

epHTh,

'To

BeH""HHa (56.5)

CTp.

413).

OAHaKo,

KaK

yme

OTMe'laJIOCb

BhIille,

B

AaJIbHeHmeM

npH

YP;OBJIeTBOPJIeT YCJIOBHHM (56.1),

npep;'bJlBJIJIeMbIM

R

HOpMe.

B

qacT-

lJTeHlm

MaTepHaJIa,

OTHOCJlm;erOCJI K

aaAalJaM

00

ynpaBJIeHHH

H 0

Ha-

ROCTH,TpeTbe

YCJIOBHe

H3

(56.1) BbInOJIRJIeTCJI BCJlep;cTBHe

RepaBeHCTBa

OJIIOAeHHH,

Ha

aTO OOCTOJlTeJIbCTBO

MomHO

He

oopam;aTb

BHHMaHHJI.

(56.4)

UpH

P =

1.

O,eB_O,

HOpMa

(56.5)

HMeeT

CMMCH

UHOU1a'H,

. ,l];JIJI

Bcex

KOHKpeTHhIX

<PYHKIJ;HH h

(-r),

KOTophIe

BCTpeTJlTCJI B

aTHX

o1'paHHlJeHHOH

RpHBOH

h

~

k

~.),

OCbIO • H

rrpJIMbIMH

• = t , • = til.

rx

aaAalJaX,

HHTe1'paJI

(56.3)

MomHO

nOHHMaTb

KaK

OOhI1JHhIH

HHTe1'paJI

<P_~Hii

h (T)

PaCCMoTpemlOe

UPOCTpaHcTBO

",..".,

'"'

C

HOPMOii

PHMaHa.

IIOHJlTHe

HHTe1'paJIa

JIeoe1'a

BamHO

JIHillb

AJIJI CTpo1'01'O

(56.5) Haa""'aroT

06""'"0

UPOcTpaHcTBOM

2[,.,

',J

HHH,

ROpo,e,

UPOCT-

paRcTBOM

2.

OoocHOBaHHJI

Tex

TeopeM

H3

<PYHKIJ;HOHaJIbH01'O

aHaJIH3a,

KOTophIe

OYAYl'

c<popMymlpoBaHbI

B

AaJIbHeHilleM

B

§§

58-61.

AHaJIOrH'IHOe

3°.

B §

49

'HH

.-MepHoro BeRTopa h = (h;) 6MHa yRaaaHa

HOp-

aaMe'laHHe

CJIeAyeT

HMeTb

B

BHAY

H B

CJIYlJae

HOpMhI

p [h],

papcMoT-

Ma

(49.5).

ECJIH

pa,CcMaTpHBaTb

<PYHRIl;HIO h

(.)

RaR

BeRTOp

h C

6ecRo-

Hc

peHHoH

HHme,

B

npHMepe

2°.

He'HMM ' 40M

ROMuoneHT

h (T),

TO

HOpMy (49.5)

MO",HO

eCTe".BeHHO.

MomHO

npOBepHTb,

'ITO

BeJIHlJHHa

(56.3)

YAOBJIeTBOpJleT

BceM

Tpe-

0606_'b

'HH

mmemlOro

apoCTpaHcTBa",.. <PYH""Bii h (T), Heape

PM'-

60BaHHJlM

(56.1),

npelrhHBJIJleMhIM

K

HopMe.

IIpH

aTOM <PYHKIJ;HH h

(-r)

HbIx

Ra

OTpe3Re

[t

l1

,

t/3

J,

CJIeP;YIOIIJ;HM

06pa30M:

HaAJIemHT

C'IHTaTb

cym;ooTBeBHO

OTJIH'IHhIMH

01'

HYJIH

TorAa

H TOJIbKO

To1'Aa,

KorAa

h

(-r)

=1=

0

Ha

MBO»teCTBe

@

HeHYJIeBOH

MephI.

(To

me

ca-

p

(h)

=,'max

Ih

(.)

/

rrpH

t

l1

< •< t{l. (56.6)-

Moe

aaMelJaHHe

OTHOCHTCH H K

p.

2°.)

,l];eHcTBHTeJIbHO,

nepBhIe

ABa

yCJIOBHJI BhInOJIHJIIOTCJI BCJIeACTBHe

1'01'0,

lJTO

HHTerpaJI

(56.3),

nOHH-

BeJIHqHRa

(56.6) YP;OBJIeTBOPJIeT

BceM

YCJIOBHHM (56.1),

orrpep;eJIJI_

IOm;HM

HOpMy

P [hI.

IIpoBepRa

aTHX

YCJIOBHH

He

COCTaBJIJIeT

60JIbilloro-

MaeMhIH

B CMLICJIe

HHTe1'paJIa

JIe6e1'a,

oopam;ae1'CJI

B

HYJIb

TOJIbKO

r·

TPYP;a, H

MbI

Ra

ReH

OC1:aHaBJIHBaTbC:JI

He

Uyp;eM.

3aMeTHM,

'ITO

P;JIJI

r'

TorAa,

K01',!J;a JIHOO h

(-r)

= 0, JIHOO h

(-r)

OTJIHlJaeTCJI 01'

HYJIJI

JIHillb

.u;aHRo1'O

rrpHMepa

RenpepbIBHaJI

<PYRRIl;HH k

(or)

cym;ecTBeRRO

OTJJHQRa

~'

;

%1

<:

H. H.

RpaCOBCKHlt

.-..:.....~~,.

'

--~-----'-'-~~-

()

' , • • • • : "

...

"

":"

• -" • "

'.

'

.•••

' : ,

:.

.!

' •

::

',.

••

',

.

.':

',:

••

~

••

','

••

'"

:.,~.

~~~~~~~'""c~;;~"'~.~.

'.'~"~

.....

::=:·:.··:::=.;c'·:::::'::~.·::':'::~·':-'->:i'·'::

.....

~·.=:

JilL.

418

HOPMHPOBAHHbIE rrpOCTPAHCTBA

<I>YHHUHn

ern.

13

01' HyJIH B

TOM

R

TOJIbKO

TOM

CJIy'Iae, Horp;a

OHa

He

paBHa Tomp;eCT-

BeHHoMy HyJIIO.

HopMa,

orrpep;eJIeHHaH

paBeHCTBOM

(56.6),

HMeeT

npocToH

reOMeTpHlJeCHHH

CMIiICJI:

OHa

rrpep;CTaBJIHeT C060H MaKCHMaJIb-

Hoe

OTKJIOHeHRe

rpaqJHRa

cPYHKIJ;HR

h

(-r)

01'

OCH

1'.

TlpocTpaHCTBO HenpepliIBHIiIX

cPYHKIJ;HH

h (1'), sap;aHHliIX Ha OTpes-

Re [ta,

t{3l

C

HOpMOH

(56.6),

HaSIiIBaIOT

rrpOCTpaHCTBOM

<ifJ[t,,,,

til]

HJIH

npOCTO rrpOCTpaHCTBOM

<ifJ.

4°.

B rrpep;blp;ym;eM rrpHMepe

HopMa

(56.6),

BBep;eHHaH paHee B

~

49

P;JIH

KOHe'IHOMepHoro rrpOCTpaHCTBa, 6liIJIa 0606m;eHa

P;JIH

npo~

'CTpaHCTBa JII'

cPYHKIJ;RH

h ('i), HerrpepbIBHliIx

Ha

OTpeSKe [ta, til). Op;Ha-

EO

B xop;e pemeHRH sap;a'I

06

yrrpaBJIeHHR R Ha6JIIOp;eHRH

MIiI

6yp;eM

I3CTpe'IaTbCH C

cPYHKIJ;HHMH

h

(.)

60JIee

CJIOmHOH

rrpRpop;liI, 'IeM Herrpe-

JlbIBHb18

cPyHHIJ;RR.

TloaToMy

meJIaTeJIbHO RMeTb 0606m;eHHe HOpMbl

{56.6)

P;JIH

<PYHHIJ;HH

h

(.)

60JIee

mlipOKoro

KJIaCCa, 'IeM

KJIaCC

<PYHK-

IJ;HH,

HerrpepliIBHIiIX Ha OTpeSKe

[tot,

til). Tlpemp;e 'IeM

Cp;eJIaTb

TaHoe

-o606IIJ;eHHe,

p;ap;HM

OP;HO

BcrrOMOraTeJIbHOe orrpep;eJIeHHe.

Byp;eM rOBopRTb, 'ITO

cPYHHIJ;KH

h

(.)

CYUfeCm6el-tliO

OepaliU"ielia

lia

<Jmpea"e

t

a

< • <

tp"

eCJIH

cym;ecTByeT MHomeCTBO @ TO'IeK • HyJIe-

.BoB:

MepbI TaKoe, 'ITO rrpH

Bcex

1'RS OTpeSKa [ta,

t{3l

BHe

nora

MHomeCT-

Ba

@

<PYHKIJ;HH

h

(.)

OrpaHH'IeHa,

T.

e.

eCJIH

rrpH

Bcex

•

HS

[ta,

t{3l,

.

KpOME',

MOiKeT 61iITb, •

RS

@,

BliIrrOJIHeHO HepaBeHCTBO

!I

Ih

(.)

I< v,

(56.7)

rp;e v - HeKOTopoe

p;OCTaTO'IHO

60JIbmOe 'IHCJIO. MHomeCTBO

<PYHK-

IJ;HB:

h

(.),

HHTerplIpyeMliIx

II

cym;ecTBeHHO OrpaHH'IeHHIiIX Ha OTpeSKe

:I

[t

a

,

til

l,

06pasyeT

JIHHeHHOe

rrpOCTpaHCTBO JII'. BBep;eM Terrepb B

aTOM

npOCTpaHCTBe

HopMy

p [hl, 0606m;aIOm;yIO HOpMy

(56.6)

CJle;O;YIOm;IIM

·o6pasoM. BliI6poCHM

HS

OTpeSKa

[t

oc

,

t{3l

KaKOe-JIII60

MHOiKeCTBO

@

TO'18K

• HyJIeBOH MepliI

II

Ha

OCTaBmeMCH

MHomeCTBe

HaHp;eM

BepXHIOIO

rpaHb

BeJIH'IRHbI I h

(.)

I.

01'0

6yp;eT BeJIR'IHHa 0' (@),

saBHCHIIJ;aH

01'

@.

ITOCTpOHM

TaHRe BeJIR'IIIHbI

0'

(@)

P;JIH

Bcex

BOSMomHIiIX

MHOiKeCTB

@

HyJIeBOH MepliI

HS

OTpeSKa

[toc,

t(3).

HRmHIOIO

rpaHb

Bcex

rrOJIy'IeHHIiIX

''lUCeJI

(j

(@)

rrpHMeM

sa

HOpMy

cPYHKIJ;HH

h

(1')

p [hl =

inf

sup

Ih

(1')

I.

(56.8)

@

~

MOiImo

rrpOBepRTb, 'ITO BeJUI'IHHa

(56.8)

yp;OBJIeTBOpHeT BceM

Tpe60BaHHHM

(56.1),

rrpep;'bHBJIHeMbIM K HopMe. fOBOpHT

([23*

J,

I

T.

3, CTp. 74), 'ITO HopMa p [hl, BBep;eHHaH

paBeHCTBOM

(56.8),

HB-

.JIHeTCH

UCmUlilibl.M

Ma"CUMYMOM

UIlU

Ma"CUMYMOM

no

CYUfecm6Y

P;JIH

<PYHKIJ;HH

Ih

(.)

IHa OTpeSKe

[tQt,

tftJ.

01'01'

HCTHHHliIH MaKcHMyM

060-

I

.<3Ha'laIOT

em;e

iii

TaK:

ill'

i

.1

P

[h]

=

vrai

max

Ih

('t')

I.

(56.9)

tQt~~~t{3

II

f

r

o

§ 561

HOPMM

II

<I>YHHUHOHAJIbHbIX npOCTPAHCTBAX

419

OUHcaHHliIH

P;OBOJIbHO

c,rromHliIH rrYTb orrpep;eJIeHHH HOpMIiI

(56.8)

BIiISIiIBaeTCH TeOpeTH'IeCKHMII

coo6pameHIIHMII.

Ha

npaKTIIHe

me

MbI

6y,n;eM

CTaJIKIIBaTbCH JIHmb C TaKHMH

1>YHKIJ;IIHMH

h (1') (t

Qt

<.

< tft)

p;aHHoro

npocTpaHcTBa

JII',

P;JIH

KOTOpliIX HopMa

p[h]

(56.8)

Haxo-

P;HTCH

cpaBHHTeJIbHO JIerKO. TlYCTb,

HanplIMep,

Tpe6yeTcH

onpep;e-

JIIITb HOpMy

(56.8)

cPYHKIJ;HH

h(l)

(.)

(pHC.

56.1):

I

--r,

O=tQt<1'<1;

h(l)

(1') =

(1'-1),

1

<1'<2;

+

2,

-r

= t{3= 2.

HeTpyP;Ho yCTaHoBHTb, 'ITO

p

[h(l)l

=

vrai

max

I

h(l)

(-r) I=

O";;;~~

I1pocTpaHcTBo

<PYHKJ:WH,

P;JIH

KOTOpliIX

HaSIiIBaTb rrpocTpaHCTBoM

-At[ta.

t{3]

(IiIJI.Iil,

KOpO'Ie, .At)l).

5°.

PaCCMOTpHM

eIIJ;e

OP;UH

BamHbIH

cnoco6

onpeAeJIeHHH HOpMIiI, HBJIH-

IOIn;eHcH o606m;eHHeM

eBKJIUp;OBOH

HOpMIiI

n-MepHoro

rrpocTpaHcTBa

fliJ

•

n

IIMeHHo,

paccMoTplIM,

KaR

onpe-

p;eJIHeTCH

HopMa

.n;JIH

aJIeMeHTOB

cPYHKIJ;UoHaJIbHIiIX rrpocTpaHcTB,

.IilMeHyeMbIx

eUIlb6e

pm06blJrtU

(CM.,

Hanp.lilMep,

[14*1, CTp.

83).

OnHmeM

3p;eCb

BKpaTIJ;e HaH60JIee

xapaKTep-

Hoe

CBO.lilCTBO

aTHX

npOCTpaHCTB.

I1Y.CTb

HMeeTCH

JIHHeHHoe npOCTpaH-

CTBO

JII'[ta.

t{3J

BeIIJ;ecTBeHHliIx 'IUCJIO-

lim

I

h(l)

(1') I = 1.

~-+l-O

BBe.n;eHa

HopMa (56.8),6YP;eM

Ii"

'('f')

2r-

.

2

_II

....

,

't

PIlC.

56.1.

BIiIX

cPYHRuUH

{h

(t)}, sap;aHHliIx Ha oTpesKe

[tah),

I1pep;nOJIOiImM,

'IT()

Kamp;oH rrape

cPYHKuHH

h

y

(t), h

s

(t)

HS

JII'[t

•

tf!]

corrOCTaBJIeHO

no

on-

ex

pep;eJIeHHoMY rrpaBUJIY

6eUfecm6elilioe

'lUCIlO

(hyh

),

npH'IeM

BliInOJIHHIOT-

CH

YCJIOBIiIH:

s

(h

y

(t),

hs

(t»

= (h

s

(t),

h

y

(t»;

(56.1Oa)

('Alh

y

(t) +

'A

2

h&

(t),

hr.

(t»

=

'AI

(h

y

(t),

h"

(t»

+

'A

2

(h

s (t), h"

(t»,

(56.106)

1)

BcIO.n;y

B OCHOBHOM

TeRCTe

RHHrH,

'lTo6M

He

aarpOMO}R.n;aTb

TeI~CT

JUlillHlIMD

1'epMIIHaMH,

Mhl

UOaBOJIHM

ce6e

CJIe)l;YIO~YIO

BOJIhHOCTb:

paccMaTpHBaJI

1meMeHTbl

h (or)

upocTpaHcTBa

.Jt.

6y.n;eM

rOBopHTb

UpOCTO

05

OrpaHII'leHHbIX

<PYHRIl;HJIX H

o

MaRCHMYMe

HX

MO.n;YJIJI H

6y.n;eM

UlicaTb

rrpoC1'o

sup

I h

('r)

I,

rrOHHMaR

rro.n;

3THH

BCJIRHH

paa

cooTBeTcTBYIO~lIe

CBoHcTBa

no

cYUjeC1nIJy,

T.

e.

B

CMhlCJIe

(56.9).

1/

2

15

H.

RpaCOBCRHR

Красовский Н.Н. Теория управления движением

Подождите немного. Документ загружается.