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29/05/2017
maTematikuri fizikis gantolebebi
leqcia 8
3.3.4. gausis formulebi. vTqvaT,   R n SemosazRvruli area A k ,h , k  1 , klasis sazRvriT. ganvixiloT ormagi fenis potenciali erTis toli simkvriviT. davamtkicoT, rom
1, Tu   ,
1
E

(3.3.11)
(
x
,

)
dS

 , Tu    ,
x
  x
2
0, Tu    ,
sadac  x gare normalia   -s mimarT.
roca    , maSin E ( x,  ) funqcia harmoniulia  areSi da amitom, rogorc viciT (ix.
Teorema 3.1.1),
E
 ( x, )dS x  0 .

 x
ganvixiloT    SemTxveva (ix. nax. 3.3.2). vTqvaT,
d :   x      da    \ d .
advili dasanaxia, rom Tu   0 sakmaod mcire ricxvia, maSin    sazRvari Sedgeba:
a)   -sgan da S : d sferosgan, roca   ;
b)   -s  1 nawilisgan da d -is  2 nawilisgan, roca   .
cxadia, rom   areSi E ( x,  ) x-is mimarT harmoniuli funqciaa, amitom    sazRvarze
normaliT warmoebulisgan integrali nulis tolia. cal-calke ganvixiloT orive SemTxveva:
a) Tu   , maSin
E
E
  ( x,  )dS x    * ( x, )dS x  0 ,
S

saidanac
E
  ( x,  )dS
x

E
1
( x,  )dS x 
 *
n
S
 
dS x

S
n 1
 1.


sadac  * aris S -is mimarT Siga normali.
b) Tu   , maSin
E
  ( x,  )dS

x

1
saidanac
E
  * ( x,  )dS

x
 0,
2
E
E
1
(3.3.12)
( x ,  )dS x 
dS x .
 *
 n  n 1 

advili dasanaxia (ix. nax. 3.3.3), rom, roca   0 , maSin  2 miiswrafvis naxevar-
  ( x,  )dS

1
sferosken. maSasadame,
x
 
2
2
leqcia 8
maTematikuri fizikis gantolebebi


*


d
2


1
*

S




nax. 3.3.2
lim  dS x 
 0
2
n
2
 n 1 .
amitom, Tu (3.3.12)-Si  -s nulisken mivaswrafebT,
E
E
1
( x,  )dS x  .
 x ( x,  )dS x  lim

 0 
2
x
1
2
nax. 3.3.3
amiT gausis (3.3.11) formula damtkicebulia.
ganvixiloT
U ( x)   f ( ) E ( x,  )d

~
moculobiTi potenciali. vTqvaT,   R n SemosazRvruli area A k ,h , k  1 , klasis sazRvriT.
maSin, radganac
U
E
( x)   f ( )
( x,  )d ,

amitom


U
E
E
( x)dS x   dS  f ( )
( x,  )d   f ( )d 
( x,  )dS x
~ 
~
~ 








2
leqcia 8
maTematikuri fizikis gantolebebi

E
 f ( )d   ( x,  )dS
~

~

x

E
 f ( )d   ( x,  )dS
~
x
,

*
sadac
~
*:   \  ,
~
magram, (3.3.11) formulis Tanaxmad, aq Semavali meore integrali   -ze nulis tolia, xolo
~
pirveli integrali   -ze erTis tolia, amitom
U
 ( x)dS x   f ( )d ,
~


~

romelsac aseve gausis formula ewodeba.
3.3.5. ormagi fenis potenciali. ganvixiloT   R n SemosazRvruli are A k ,h , k  1 , klasis
sazRvriT da ormagi fenis
E
W ( x)    ( )
( x,  )dS

 
potenciali. radgan   da E ( x,  ) harmoniulia yvelgan, sadac   x , amitom W funqcia
harmoniulia R n \  -Si. amasTan, radgan
E
( x,  )  0 , roca x   ,
 
amitom
W ( x)  0 , roca x   .
SemoviRoT aRniSvnebi
  :   da   :  R n \  .
gamovikvlioT W funqciis yofaqceva   sazRvarze da mis maxloblad. ganvixiloT n  2
SemTxveva, maSin ormagi fenis potenciali Semdegi saxiT Caiwereba:

1
(3.3.13)
W ( x) 
 ( )
ln x   dS ,

2 
 
sadac   Caketili martivi wiria uwyveti simrudiT, xolo  uwyveti funqciaa. gausis (3.3.11)
formulis gamoyenebiT mtkicdeba, rom
1
W  x0   x0  W x0 ,
2
(3.3.14)
1

0
0
0
W x    x W x ,
2
sadac
W  x 0 :  lim 0 W ( x) , W  x 0 :  lim 0 W ( x) .
 
 
 
 
  x x
 
 
 
 
  x x
(3.3.14) formulebidan cxadia, rom, roca x  x   , ormagi fenis potenciali wyvetas
ganicdis. aRsaniSnavia, rom ormagi fenis potencialis normaluri warmoebuli am dros wyvetas
ar ganicdis.
(3.3.14)-Si Tu pirvels gamovaklebT meores, miviRebT, rom
W  x0  W  x0   x0 .
0
 
 
 
am formulas naxtomis formula ewodeba.
ganvixiloT dirixles
u( x )  0 , x    ,
3
leqcia 8
maTematikuri fizikis gantolebebi
u  g,
amocana [(+) Seesabameba Siga, xolo (-) _ gare amocanas. am ukanasknel SemTxvevaSi
damatebiT viTxovT amonaxsnis SemosazRvrulobas uasasrulobaSi (roca n  3 , es piroba
usasrulobaSi qrobis pirobiT icvleba)], sadac   wirs uwyveti simrude aqvs da g uwyveti
funqciaa. veZeboT am amocanis amonaxsni ormagi fenis
1

u ( x) 
 ( )
ln x   dS

2  
 
potencialis saxiT. radgan ormagi fenis potenciali harmoniulia  areSi, amitom saWiroa,
Semowmdes mxolod sasazRvro pirobis Sesruleba. maSin, (3.3.14) formulebis Tanaxmad, 
funqciisTvis miiReba
1
u  x 0     x 0   u x 0 
2
gantoleba, romelic Caiwereba e. w. fredholmis meore gvaris Semdegi integraluri gantolebis
saxiT
1
1

(3.3.15)
  x0 
 ( )
ln x 0   dS  g x 0 .

2
2  
 
 
 
integralur gantolebaTa Teoriidan cnobilia, rom (3.3.15) gantoleba amoxsnadia, Tu mis
Sesabamis erTgvarovan gantolebas aqvs mxolod trivialuri amonaxsni (amas fredholmis
alternativa ewodeba). magram dirixles erTgvarovan amocanas, rogorc viciT, mxolod
trivialuri amonaxsni aqvs. erTgvarovan amocanas ki erTgvarovani  g( x)  0 (3.3.15)
gantoleba Seesabameba. e. i., am ukanasknelsac mxolod trivialuri amonaxsni aqvs. es ki imas
niSnavs, rom (3.3.15) araerTgvarovani integraluri gantoleba calsaxad amoxsnadia.
3.3.6. martivi fenis potenciali. ganvixiloT SemosazRvruli   R n are A k ,h , k  1 , klasis
sazRvriT,  funqcia, romelic uwyvetia   -ze, da
V ( x)    ( ) E ( x,  )dS

martivi fenis potenciali.
cxadia, rom R n \   -Si V harmoniuli funqciaa, radgan am SemTxvevaSi x   . amasTan,
Tu n  3 , maSin u( x)  0 , roca x   . Tu n  2 , maSin
V ( x) 
1
2
 ln

x 
1
 ( )dS 
x
2
 ln x  ( )dS ,

saidanac gamomdinareobs, rom V ( x)  0 maSin da mxolod maSin, roca
  ( )dS  0 .

SemovifargloT wina qvepunqtis pirobebSi n  2 SemTxveviT. gausis (3.3.11) formulis
gamoyenebiT mtkicdeba, rom martivi fenis potenciali uwyvetia   sazRvarze gavlisas, xolo
 V ( x,  )
misi
normaluri warmoebuli wyvetas ganicdis, amasTan
 x
0
 V

  0
 x
 V

  0
 x

 0 1
 x   x0   V x0 ,

2
 x0


 
 
 
 0
 x   1  x0   V x0 .

2
 x 0

 
 
 
4
leqcia 8
maTematikuri fizikis gantolebebi
ganvixiloT noimanis
u( x )  0 , x    ,
u
( x)
 x
0
 g ( x0 ) ,
x x 0  
amocana da misi u  C ()  C ( ) amonaxsni veZeboT martivi fenis
1
u( x) 
 ( ) ln x   dS
2 
2
1
potencialis saxiT. radgan martivi fenis potenciali harmoniulia, roca x  R n \   , amitom
saWiroa, Semowmdes mxolod sasazRvro pirobis Sesruleba. maSin  funqciisTvis miiReba
1
1
  ( x0 ) 
2
2


 ln x  

0
 x
 ( )dS  g ( x )
0
0
fredholmis meore gvaris gantoleba.
noimanis amocanis amonaxsni, cxadia, ganisazRvreba mudmivi Sesakrebis sizustiT da
amonaxsnis arsebobisTvis aucilebelia Sesruldes (ix. Teorema 3.1.1)
 g ( x)dS  0

piroba. integralur gantolebaTa fredholmis Teoriaze dayrdnobiT mtkicdeba, rom es piroba
sakmarisicaa amonaxsnis arsebobisTvis.
SevniSnoT, rom zogadi saxis meore rigis elifsuri gantolebebisTvis n damoukidebeli cvladis
SemTxvevaSi SeiZleba, aigos ganzogadebul potencialTa Teoria, romelsac arsebobis
Teoremebis damtkiceba, bunebrivia, integralur gantolebaTa Teoriis gamoyenebaze dayavs.
5