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Применение двойных интегралов к задачам механики и геометрии

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Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey ?.O.

E?eooneee ainoaea?noaaiiue oaoie/aneee oieaa?neoao.

Eaoaae?a aunoae iaoaiaoeee.

?aoa?ao.

I?eiaiaiea aeaieiuo eioaa?aeia e caaea/ai iaoaieee e aaiiao?ee.

Auiieieea: nooaeaioea

a?oiiu OY-97-1

Iaeeinooiiaa N.N.

I?iaa?ee i?aiiaeaaaoaeue

eaoaae?u aunoae
iaoaiaoeee

Naaeuo A.E.

E?eoone 1998.

Niaea?aeaiea.

1.Iau?i oeeeeiae?e/aneiai oaea. Aeaieiie eioaa?ae.

2. Au/eneaiea aeaieiuo eioaa?aeia.

a) i?eia?u.

3.I?eeiaeaiey aeaieiuo eioaa?aeia e caaea/ai iaoaieee.

a) ianna ieineie ieanoeiee ia?aiaiiie ieioiinoe.

a) noaoe/aneea iiiaiou e oeaio? oyaeanoe ieanoeiee.

a) iiiaiou eia?oeee ieanoeiee.

4.Au/eneaiea ieiuaaeae e iau?iia n iiiiuueth aeaieiuo eioaa?aeia.

a) Iau?i.

a) Au/eneaiea ieiuaaee ieineie iaeanoe.

5.Au/eneaiea ieiuaaee iiaa?oiinoe.

a) I?eia?u.

1.Iau?i oeeeeiae?e/aneiai oaea. Aeaieiie eioaa?ae.

Oeeeeiae?e/aneei oaeii iacuaaaony oaei, ia?aie/aiiia ieineinoueth Oxy,
iiaa?oiinoueth, n eioi?ie ethaay i?yiay, ia?aeeaeueiay ine Oz,
ia?anaeaaony ia aieaa /ai a iaeiie oi/ea, e oeeeeiae?e/aneie
iiaa?oiinoueth, ia?acothuay eioi?ie ia?aeeaeueia ine Oz.

.

?en. 1

Iau/ii oaei iiaeii ninoaaeoue ec iaeioi?iai /enea oeeeeiae?e/aneeo oae e
ii?aaeaeeoue eneiiue iauaeo eae noiio iau?iia oeeeeiae?e/aneeo oae,
ninoaaeythueo yoi oaei.

I?aaeaea anaai iaiiiiei aeaa i?eioeeia, ec eioi?uo iu enoiaeei i?e
ii?aaeaeaiee iau?ia oaea:

anee ?acaeoue oaei ia /anoe, oi aai iau?i aoaeao ?aaai noiia iau?iia
anao /anoae;

iau?i i?yiiai oeeeeiae?a, o.a. oeeeeiae?e/aneiai oaea, ia?aie/aiiiai
ieineinoueth, ia?aeeaeueiie ieineinoe Oxy, ?aaai ieiuaaee iniiaaiey,
oiiiaeaiiie ia aunioo oaea.

anthaeo a iaeanoe D.

?en. 2

iu aoaeai o?aaiaaoue, /oiau ia oieueei ieiuaaeue eaaeaeie /anoe/iie
iaeanoe no?aieeanue e ioeth, ii /oiau no?aieeenue e ioeth ana aa
?acia?u. Anee iacaaoue aeeaiao?ii iaeanoe iaeaieueoaa ?annoiyiea iaaeaeo
oi/eaie aa a?aieoeu (Iai?eia?, aeeaiao? i?yiioaieueieea ?aaai aai
aeeaaiiaee, aeeaiao? yeeeina—aai aieueoie ine. Aeey e?oaa i?eaaaeaiiia
ii?aaeaeaiea aeeaiao?a ?aaiineeueii iau/iiio.), oi auneacaiiia
o?aaiaaiea aoaeao icia/aoue, /oi eaaeaeue ec aeeaiao?ia /anoe/iuo
iaeanoae aeieaeai no?aieoueny e ioeth; i?e yoii naie iaeanoe aoaeoo
noyaeaaoueny a oi/eo (Anee ecaanoii oieueei, /oi ieiuaaeue iaeanoe
no?aieony e ioeth, oi yoa iaeanoue iiaeao e ia noyaeaaoueny a oi/eo.
Iai?eia?, ieiuaaeue i?yiioaieueieea n iinoiyiiui iniiaaieai e aunioie,
no?aiyuaeny e ioeth, no?aieony e ioeth, a i?yiioaieueiee noyaeaaaony e
naiaio iniiaaieth, o. a. e io?aceo).

):

E iouneaieth i?aaeaea iiaeiaiuo noii aeey ooieoeee aeaoo ia?aiaiiuo
i?eaiaeyo naiua ?aciiia?aciua caaea/e, a ia oieueei caaea/a ia iauaia.

e ninoaaei noiio

(*)

, – ieiuaaeue /anoe/iie iaeanoe.

a iaeanoe D, niioaaonoaothuae aeaiiiio ?acaeaieth yoie iaeanoe ia n
/anoe/iuo iaeanoae.

ii iaeanoe D iacuaaaony i?aaeae, e eioi?iio no?aieony n-y
eioaa?aeueiay noiia (*) i?e no?aieaiee e ioeth iaeaieueoaai aeeaiao?a
/anoe/iuo iaeanoae.

Caienuaaaony yoi oae:

– yeaiaioii ieiuaaee, iaeanoue D – iaeanoueth eioaa?e?iaaiey,
iaeiiaoe, ia?aiaiiua x e o iacuaathony ia?aiaiiuie eioaa?e?iaaiey.

, acyoui ii iaeanoe, yaeythuaeny iniiaaieai oeeeeiae?e/aneiai oaea:

.

Aiaeiae/ii oai?aia nouanoaiaaiey iaueiiaaiiiai eioaa?aea eiaao ianoi
neaaeothuay oai?aia.

Oai?aia nouanoaiaaiey aeaieiiai eioaa?aea.

e io auai?a a ieo oi/ae Pi.

Aeaieiie eioaa?ae, ?acoiaaony, i?aaenoaaeyao niaie /enei, caaenyuaa
oieueei io iiaeuioaa?aeueiie ooieoeee e iaeanoe eioaa?e?iaaiey e aiana
ia caaenyuaa io iaicia/aiee ia?aiaiiuo eioaa?e?iaaiey, oae /oi,
iai?eia?,

.

Aeaeaa iu oaaaeeiny a oii, /oi au/eneaiea aeaieiiai eioaa?aea iiaeao
auoue i?iecaaaeaii iin?aaenoaii aeaoo iaueiiaaiiuo eioaa?e?iaaiee.

2.Au/eneaiea aeaieiuo eioaa?aeia.

o.a. yeaiaio ieiuaaee a aeaea?oiauo eii?aeeiaoao yaeyaony
i?iecaaaeaieai aeeooa?aioeeaeia iacaaeneiuo ia?aiaiiuo. Iu eiaai

. (*)

. Iaiiiiei, /oi iu oaea caieiaeenue caaea/ae ia iau?ia oaea, eiaaea
?anniao?eaaee i?eiaiaiey ii?aaeae?iiiai eioaa?aea e caaea/ai aaiiao?ee e
iieo/eee oi?ioeo

(**)

?en.3

– o?aaiaiey ieineinoae, ia?aie/eaathueo oaei. I?eiaiei oaia?ue yoo
oi?ioeo e au/eneaieth aeaieiiai eioaa?aea

I?aaeiieiaeei nia/aea, /oi iaeanoue eioaa?e?iaaiey D oaeiaeaoai?yao
neaaeothuaio oneiaeth: ethaay i?yiay, ia?aeeaeueiay ine Ox eee Oy,
ia?anaeaao a?aieoeo iaeanoe ia aieaa /ai a aeaoo oi/eao.
Niioaaonoaothuaa oeeeeiae?e/aneia oaei ecia?aaeaii ia ?en.3

Iaeanoue D caeeth/ei aioo?ue i?yiioaieueieea

noi?iiu eioi?iai eanathony a?aieoeu iaeanoe a oi/eao A, A, N, A.
Eioa?aae [a, b] yaeyaony i?oiaiiaeueiie i?iaeoeeae iaeanoe D ia inue Io,
a eioa?aae [c, d] – i?oiaiiaeueiie i?iaeoeeae iaeanoe D ia inue Oy. Ia
?en.5 iaeanoue D iieacaia a ieineinoe Ioo.

Oi/eaie A e C a?aieoea ?acaeaaaony ia aeaa eeiee: ABC e AEC, eaaeaeay ec
eioi?uo ia?anaeaaony n ethaie i?yiie, ia?aeeaeueiie ine Oy, a iaeiie
oi/ea. Iiyoiio, eo o?aaiaiey iiaeii caienaoue a oi?ia, ?ac?aoaiiie
ioiineoaeueii y:

(ABC),

(AEC).

Aiaeiae/ii oi/eaie A e A a?aieoea ?acaeaaaony ia eeiee AAA e ANA,
o?aaiaiey eioi?uo iiaeii caienaoue oae:

(BAE),

(BCE).

?en.5

.

Neaaeiaaoaeueii, eioaa?ae

aeaao au?aaeaiea aeey ieiuaaee ieineiai na/aiey PMNR. ssnii, /oi
aaee/eia yoiai eioaa?aea caaeneo io aua?aiiiai cia/aiey o; ae?oaeie
neiaaie, ieiuaaeue ?anniao?eaaaiiai iiia?a/iiai na/aiey yaeyaony
iaeioi?ie ooieoeeae io o, iu iaicia/ei aa /a?ac S(o):

. Iniiauaaynue ia ooi/iaiiii aaiiao?e/aneii niunea aeaieiiai eioaa?aea,
iao?oaeii aeieacaoue, ia /ai iu ia aoaeai inoaiaaeeaaoueny, /oi
iieo/athuayny oi?ioea aeey au/eneaiey aeaieiiai eioaa?aea aoaeao aa?ia
aeey ethauo ooieoeee.

Caiaiyy a yoie oi?ioea S(x) a? au?aaeaieai, ieii/aoaeueii iieo/ei

eee a aieaa oaeiaiie oi?ia

(A)

I?aaeaeu aioo?aiiaai eioaa?aea ia?aiaiiua; iie oeacuaatho a?aieoeu
eciaiaiey ia?aiaiiie eioaa?e?iaaiey o i?e iinoiyiiii cia/aiee aoi?iai
a?aoiaioa o. I?aaeaeu aiaoiaai eioaa?aea iinoiyiiu; iie oeacuaatho
a?aieoeu, a eioi?uo iiaeao eciaiyoueny a?aoiaio o.

, aaea o i?e eioaa?e?iaaiee n/eoaaony aaee/eiie iinoiyiiie. Eioaa?e?oy
caoai Q(o) a i?aaeaeao eciaiaiey o, o. a. io c aei d, iu i?eaeai ei
aoi?iio au?aaeaieth aeey aeaieiiai eioaa?aea

(A)

Caeanue eioaa?e?iaaiea niaa?oaaony nia/aea ii o, a iioii ii o.

.Oi?ioeu (A) e (A) iieacuaatho, /oi au/eneaiea aeaieiiai eioaa?aea
naiaeeony e iineaaeiaaoaeueiiio au/eneaieth aeaoo iaueiiaaiiuo
ii?aaeaeaiiuo eioaa?aeia; ioaeii oieueei iiiieoue, /oi ai aioo?aiiai
eioaa?aea iaeia ec ia?aiaiiuo i?eieiaaony i?e eioaa?e?iaaiee ca
iinoiyiioth. Aeey e?aoeinoe i?aaua /anoe oi?ioe (A) e (A) iacuaatho
iiaoi?iuie (eee aeaoe?aoiuie) eioaa?aeaie, a nai i?ioeann ?annoaiiaee
i?aaeaeia eioaa?e?iaaiey – i?eaaaeaieai aeaieiiai eioaa?aea e
iiaoi?iiio.

Oi?ioeu i?eaaaeaiey aeaieiiai eioaa?aea e iiaoi?iiio i?eia?aoatho
iniaaiii i?inoie aeae, eiaaea iaeanoue D yaeyaony i?yiioaieueieeii ni
noi?iiaie, ia?aeeaeueiuie inyi eii?aeeiao (?en.6). A yoii neo/aa
noaiiayony iinoiyiiuie i?aaeaeu ia oieueei aiaoiaai, ii e aioo?aiiaai
eioaa?aeia:

A ae?oaeo neo/ayo aeey naaaeaiey aeaieiiai eioaa?aea e iiaoi?iiio
iaiaoiaeeii i?aaeaea anaai iino?ieoue iaeanoue eioaa?e?iaaiey; eo/oa
anaai ecia?aceoue yoo iaeanoue i?yii a ieineinoe Ioo, eae yoi naeaeaii
ia ?en. Caoai ioaeii onoaiiaeoue ii?yaeie eioaa?e?iaaiey, o. a.
iaiaoeoue, ii eaeie ia?aiaiiie aoaeao i?iecaiaeeoueny aioo?aiiaa
eioaa?e?iaaiea, a ii eaeie – aiaoiaa, e ?annoaaeoue i?aaeaeu
eioaa?e?iaaiey.

Iiyniei ia i?eia?ao, eae i?iecaiaeeony ?annoaiiaea i?aaeaeia
eioaa?e?iaaiey.

a) I?eia?u.

anee iaeanoue D- o?aoaieueiee,

?en. 6. ?en. 7.

ia?aie/aiiue i?yiuie y=0, y=x e o=a (?en.7). Anee eioaa?e?iaaoue nia/aea
ii o, a iioii ii o, oi aioo?aiiaa eioaa?e?iaaiea i?iecaiaeeony io eeiee
o=0 aei eeiee o=o, a aiaoiaa – io oi/ee o=0 aei oi/ee o=a. Iiyoiio

Iaiyy ii?yaeie eioaa?e?iaaiey, iieo/ei

anee iaeanoue D ia?aie/aia eeieyie o=0, o=o2 e o+o=2.

Iaeanoue D, a oaeaea eii?aeeiaou e?aeieo aa oi/ae iieacaiu ia ?en. 158.
Aeae iaeanoe oeacuaaao ia oi, /oi oaeiaiaa eioaa?e?iaaoue nia/aea ii x,
a iioii ii y:

Anee eciaiei ii?yaeie eioaa?e?iaaiey, oi ?acoeueoao oaea ia oaeanony
caienaoue a aeaea iaeiiai iiaoi?iiai eioaa?aea, oae eae eeiey OBA eiaao
ia ?aciuo o/anoeao ?aciua o?aaiaiey.

?en.8

?acaeaay iaeanoue D ia aeaa : OBC e CBA, iieo/ei

Yoio i?eia? iieacuaaao, eae aaaeii n naiiai ia/aea i?iaeoiaoue ii?yaeie
eioaa?e?iaaiey.

Oi?ioeu (A) e (A) naaaeaiey aeaieiiai eioaa?aea e iiaoi?iiio
ni?aaaaeeeau e aeey neo/ay iaeanoae aieaa iauaai aeaea. Oae, oi?ioea (A)
i?eiaieia e iaeanoe, oeacaiiie ia ?en.9, a oi?ioea (A) – e iaeanoe,
ecia?aaeaiiie ia ?en.10. A neo/aa iaeanoe au? aieaa iauaai aeaea
(?en.11) aeaieiie eioaa?ae neaaeoao ?acaeoue ia noiio eioaa?aeia ii
aieaa i?inoui iaeanoyi, a caoai eaaeaeue ec ieo naiaeeoue ioaeaeueii e
iiaoi?iiio, iieuecoynue oi?ioeaie (A) e (A).

?anniio?ei oaia?ue ianeieueei i?eia?ia, naycaiiuo n au/eneaieai aeaieiuo
eioaa?aeia.

I?eia?u. 1) Iaeae?i aeaieiie eioaa?ae io ooieoeee

Aaiiao?e/anee I au?aaeaao iau?i /aou??ooaieueiie i?eciu

.

Aicuei?i iiaoi?iue eioaa?ae nia/aea ii y, caoai ii x:

Oi aea naiia iieo/ei, eioaa?e?oy nia/aea ii x, a caoai ii y:

2) Au/eneei aeaieiie eioaa?ae

ii iaeanoe D, ia?aie/aiiie eeieyie y=x e y=x2. Iaeanoue D

ecia?aaeaia ia ?en.13. Eioaa?e?oy nia/aea ii y, a iioii ii x,

iieo/aai

I?aaeeueiinoue ?acoeueoaoa iiaeii i?iaa?eoue, eciaiea ii?yaeie
eioaa?e?iaaiey :

e ieineinoueth z=0 (?en.14,a).

n eeieae ia?ana/aiey oeeeeiae?a z=4-y2 e ieineinoe z=0, o.a. n i?yiie
y=2 (?en. 14, a). Aaeaeo neiiao?ee oaea ioiineoaeueii ieineinoe Oyz
au/eneyai iieiaeio eneiiiai iau?ia :

eoa.aae.

e ieineinoueth Oxy.

Caaeaiiia oaei i?aaenoaaeyao niaie naaiaio yeeeioe/aneiai

ia?aaieieaea, ?aniieiaeaiiue iaae ieineinoueth Ioo (?en.15). Ia?aaieieae
ia?anaeaaony n ieineinoueth Ioo ii yeeeino

o. a. ii /aoaa?oe yeeeina. Eioaa?e?oy nia/aea ii o, caoai ii o, iieo/ei

aea?o

3.I?eeiaeaiey aeaieiuo eioaa?aeia e caaea/ai

iaoaieee.

a) Ianna ieineie ieanoeiee ia?aiaiiie ieioiinoe.

?anniio?ei oiieoth ieanoeieo, ?aniieiaeaiioth ia ieineinoe Ioo e
caieiathuoth iaeanoue D. Oieueio yoie ieanoeiee n/eoaai ianoieueei
iaeie, /oi eciaiaieai ieioiinoe ii oieueia aa iiaeii i?aiaa?a/ue.

Iiaa?oiinoiie ieioiinoueth oaeie ieanoeiee a aeaiiie oi/ea iacuaaaony
i?aaeae ioiioaiey iannu ieiuaaeee e aa ieiuaaee i?e oneiaee, /oi
ieiuaaeea noyaeaaaony e aeaiiie oi/ea.

Ii?aaeaeaiiay oaeei ia?acii iiaa?oiinoiay ieioiinoue aoaeao caaenaoue
oieueei io iieiaeaiey aeaiiie oi/ee, o. a. yaeyoueny ooieoeeae aa
eii?aeeiao:

a aua?aiiie oi/ea. Ninoaaei i?eaeeaeaiiia au?aaeaiea aeey iannu
ieanoeiee a aeaea eioaa?aeueiie noiiu

(*)

e eaaeaeay /anoe/iay iaeanoue noyaeaaaony e oi/ea. Oiaaea

a) Noaoe/aneea iiiaiou e oeaio? oyaeanoe ieanoeiee.

iannu niioaaonoaothueo /anoe/iuo iaeanoae e iaeaeai noaoe/aneea iiiaiou
iieo/aiiie nenoaiu iaoa?eaeueiuo oi/ae :

Ia?aoiaey e i?aaeaeo i?e iau/iuo oneiaeyo e caiaiyy eioaa?aeueiua noiiu
eioaa?aeaie, iieo/ei

Iaoiaeei eii?aeeiaou oeaio?a oyaeanoe :

oi oi?ioeu oi?iuathony :

aaea S – ieiuaaeue ieanoeiee.

a) Iiiaiou eia?oeee ieanoeiee.

Iiiaioii eia?oeee iaoa?eaeueiie oi/ee ? n iannie m ioiineoaeueii
eaeie-eeai ine iacuaaaony i?iecaaaeaiea iannu ia eaaae?ao ?annoiyiey
oi/ee ? io yoie ine.

:

.

A iaoaieea /anoi ?anniao?eaatho iiey?iue iiiaio eia?oeee oi/ee, ?aaiue
i?iecaaaeaieth iannu oi/ee ia eaaae?ao aa ?annoiyiey aei aeaiiie oi/ee –
iiethna. Iiey?iue iiiaio eia?oeee ieanoeiee ioiineoaeueii ia/aea
eii?aeeiao aoaeao ?aaai

4. Au/eneaiea ieiuaaeae e iau?iia n iiiiuueth aeaieiuo eioaa?aeia.

a) Iau?i.

ii iaeanoe D :

I?eia? 1. Au/eneeoue iauai oaea, ia?aie/aiiiai iiaa?oiinoyie x=0, o=0,
o+o+z=1, z=0 (?en. 17).

?en.17
?en.18

eoa. aaeeieoe.

(?en.18).

Iiyoiio iau?i V ?aaai ?aciinoe aeaoo aeaieiuo eioaa?aeia :

eee

(1)

– ethaua iai?a?uaiua ooieoeee, oaeiaeaoai?ythuea niioiioaieth

. I?aaeiieiaeei, /oi iaeanoe D1 e D2 oaeiau, /oi aeaieiua eioaa?aeu ii
yoei iaeanoyi nouanoaotho. Oiaaea eioaa?ae ii iaeanoe D1 aoaeao
iieiaeeoaeai e aoaeao ?aaai iauaio oaea, eaaeauaai auoa ieineinoe Ioo.
Eioaa?ae ii D2 aoaeao io?eoeaoaeai e ii aaniethoiie aaee/eia ?aaai
iauaio oaea, eaaeauaai ieaea ieineinoe Ioo, Neaaeiaaoaeueii, eioaa?ae ii
D aoaeao au?aaeaoue ?aciinoue niioaaonoaothueo iauaiia.

a) Au/eneaiea ieiuaaee ieineie iaeanoe.

ii iaeanoe D, oi yoa noiia aoaeao ?aaia ieiuaaee S,

i?e ethaii niiniaa ?acaeaiey. Ia?aoiaey e i?aaeaeo a i?aaie /anoe
?aaainoaa, iieo/ei

Anee iaeanoue D i?aaeeueiay , oi ieiuaaeue au?aceony aeaoe?aoiui
eioaa?aeii

I?iecaiaey eioaa?e?iaaiea a neiaeao, eiaai, i/aaeaeii,

?en.19

Iu iieo/eee aeaa oi/ee ia?ana/aiey

Neaaeiaaoaeueii, eneiiay ieiuaaeue

5. Au/eneaiea ieiuaaee iiaa?oiinoe.

iai?a?uaia e eiaao iai?a?uaiua /anoiua i?iecaiaeiua. Iaicia/ei
i?iaeoeeth eeiee A ia ieineinoue Oxy /a?ac L. Iaeanoue ia ieineinoe Oxy,
ia?aie/aiioth eeieae L, iaicia/ei D.

*a?ac oi/eo Mi i?iaaae?i eanaoaeueioth ieineinoue e iiaa?oiinoe.
O?aaiaiea a? i?eiao aeae

(1)

– no?aieony e ioeth, iu aoaeai iacuaaoue ieiuaaeueth iiaa?oiinoe, o. a.
ii ii?aaeaeaieth iieiaeei

(2)

oaie iaaeaeo eanaoaeueiie ieineinoueth e ieineinoueth Ioo.

?en.20
?en.21

Ia iniiaaiee ecaanoiie oi?ioeu aiaeeoe/aneie aaiiao?ee iiaeii iaienaoue
(?en.21)

eee

(3)

anoue a oi aea a?aiy oaie iaaeaeo inueth Oz e ia?iaiaeeeoey?ii e
ieineinoe (1). Iiyoiio ia iniiaaiee o?aaiaiey (1) e oi?ioeu
aiaeeoe/aneie aaiiao?ee eiaai

Neaaeiaaoaeueii,

Iiaenoaaeyy yoi au?aaeaiea a oi?ioeo (2), iieo/ei

oi ieii/aoaeueii iieo/aai

(4)

oi niioaaonoaothuea oi?ioeu aeey au/eneaiey iiaa?oiinoe eiatho aeae

(3’)

(3’’)

aaea D’ e D’’ – iaeanoe ia ieineinoyo Oyz e Oxz, a eioi?ua i?iaeoe?oaony
aeaiiay iiaa?oiinoue.

a) I?eia?u.

noa?u

(?en.22). A yoii neo/aa

Neaaeiaaoaeueii, iiaeuioaa?aeueiay ooieoeey i?eiao aeae

. Oaeei ia?acii, ia iniiaaiee oi?ioeu (4) aoaeai eiaoue

Neaaeiaaoaeueii,

?en.22 ?en.23

; iiyoiio

Nienie eniieueciaaiiie eeoa?aoo?u.

A.O. Aa?iaio ,E.A. A?aiaiiae/

E?aoeee eo?n iaoaiaoe/aneiai aiaeeca aeey aoocia: O/aaiia iiniaea
aeey aoocia: – I.: Iaoea, Aeaaiay ?aaeaeoeey oeceei-iaoaiaoe/aneie
eeoa?aoo?u , 1971 a.,736n.

I.N. Ieneoiia

Aeeooa?aioeeaeueiia e eioaa?aeueiia en/eneaiey aeey aoocia, Oii 2:

O/aaiia iiniaea aeey aoocia.-13-a ecae. -I. :Iaoea, Aeaaiay
?aaeaeoeey oeceei-iaoaiaoe/aneie eeoa?aoo?u, 1985.-560n.

A.N. Oeia/?a

Aunoay iaoaiaoeea: O/aaiia iiniaea aeey aoocia: –
I: Iaoea,

Aeaaiay ?aaeaeoeey oeceei-iaoaiaoe/aneie eeoa?aoo?u.

?en. 9 ?en.10
?en.11

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