U matematici poverhne vij integra l ce viznachenij integral kotrij beretsya po poverhni yaka mozhe buti zignutoyu mnozhinoyu v prostori jogo mozhna rozglyadati yak podvijnij integralnij analog linijnogo integralu Z oglyadu na poverhni mozhna integruvati skalyarni polya tobto funkciyi yaki povertayut chisla yak znachennya i vektorni polya tobto funkciyi yaki povertayut vektori yak znachennya Viznachennya poverhnevogo integralu spirayetsya na rozbittya poverhni na mali elementiPoverhnevi integrali mayut zastosuvannya u fizici zokrema v klasichnij teoriyi elektromagnetizmu Zmist 1 Poverhnevi integrali 2 Plosha gladkoyi poverhni 3 Poverhnevi integrali 1 go ta 2 go rodu 3 1 Poverhnevi integrali 1 go rodu 3 2 Poverhnevi integrali 2 go rodu 3 2 1 Obchislennya poverhnevogo integrala 2 go rodu zvedennya do podvijnogo integrala 3 3 Zv yazok mizh poverhnevimi integralami 1 go i 2 go rodu 4 Geometrichni i fizichni zastosuvannya poverhnevogo integrala 4 1 Ob yem tila 4 2 Centr tyazhinnya ta sila prityagannya 5 Div takozh 6 DzherelaPoverhnevi integrali RedaguvatiShmat poverhni S displaystyle S nbsp zadanij u parametrichni formi x x u v displaystyle x x u v nbsp y y u v displaystyle y y u v nbsp z z u v displaystyle z z u v nbsp prichomu u v displaystyle u v nbsp probigayut deyaku oblast G displaystyle Gamma nbsp ploshini nazivayetsya gladkim yaksho rizni pari znachen u v displaystyle u v nbsp dayut rizni tochki S displaystyle S nbsp chastkovi pohidni funkcij x x u v displaystyle x x u v nbsp y y u v displaystyle y y u v nbsp z z u v displaystyle z z u v nbsp neperervni i zavzhdi A 2 B 2 C 2 gt 0 displaystyle A 2 B 2 C 2 gt 0 nbsp deA y u y v z u z v displaystyle A begin vmatrix partial y over partial u amp partial y over partial v partial z over partial u amp partial z over partial v end vmatrix nbsp B z u z v x u x v displaystyle B begin vmatrix partial z over partial u amp partial z over partial v partial x over partial u amp partial x over partial v end vmatrix nbsp C x u x v y u y v displaystyle C begin vmatrix partial x over partial u amp partial x over partial v partial y over partial u amp partial y over partial v end vmatrix nbsp Yaksho poverhnya S displaystyle S nbsp skladayetsya z skinchennogo chisla gladkih kuskiv poverhni to S displaystyle S nbsp nazivayetsya kuskovo gladkoyu Gladka poverhnya S displaystyle S nbsp nazivayetsya dvostoronnoyu yaksho pri obhodi kozhnoyi zamknutoyi krivoyi na S displaystyle S nbsp vihodyachi z bud yakoyi tochki M 0 displaystyle M 0 nbsp na S displaystyle S nbsp povertayemosya v pochatkove polozhennya z napryamom normali vibranim v M 0 displaystyle M 0 nbsp Obidvi storoni dvostoronnoyi poverhni mozhut buti takim chinom oharakterizovani napryamom vidpovidnih normalej Odnostoronnoyu poverhneyu ye napriklad list Mebiusa Usyudi nadali pid poverhneyu rozumiyetsya dvostoronnya poverhnya Plosha gladkoyi poverhni RedaguvatiDokladnishe Plosha poverhniHaj poverhnya S displaystyle S nbsp zadana parametrichno x x u v displaystyle x x u v nbsp y y u v displaystyle y y u v nbsp z z u v displaystyle z z u v nbsp prichomu u displaystyle u nbsp i v displaystyle v nbsp probigayut deyaku oblast G displaystyle Gamma nbsp ploshini u displaystyle u nbsp v displaystyle v nbsp Todi plosha S displaystyle S nbsp poverhni viznachayetsya poverhnevim integralom G E G F 2 d u d v displaystyle iint limits Gamma sqrt EG F 2 mathrm d u mathrm d v nbsp deE x u 2 y u 2 z u 2 displaystyle E left frac partial x partial u right 2 left frac partial y partial u right 2 left frac partial z partial u right 2 nbsp F x u x v y u y v z u z v displaystyle F partial x over partial u partial x over partial v partial y over partial u partial y over partial v partial z over partial u partial z over partial v nbsp G x v 2 y v 2 z v 2 displaystyle G left frac partial x partial v right 2 left frac partial y partial v right 2 left frac partial z partial v right 2 nbsp Pidintegralnij viraz d S E G F 2 d u d v displaystyle mathrm d S sqrt EG F 2 mathrm d u mathrm d v nbsp nazivayetsya elementom poverhni Yaksho S displaystyle S nbsp zadana yavno rivnyannyam z ϕ x y displaystyle z phi x y nbsp prichomu x y displaystyle x y nbsp probigayut oblast S displaystyle S nbsp proyekciyu oblasti S displaystyle S nbsp na ploshinu x 0 y displaystyle x0y nbsp to S S 1 p 2 q 2 d x d y displaystyle S iint limits S prime sqrt 1 p 2 q 2 mathrm d x mathrm d y nbsp dep z x displaystyle p partial z over partial x nbsp q z y displaystyle q partial z over partial y nbsp Poverhnevi integrali 1 go ta 2 go rodu RedaguvatiPoverhnevi integrali 1 go rodu Redaguvati nbsp Ris 1Viznachennya poverhnevogo integralu 1 go rodu Nehaj deyaka funkciya f x y z displaystyle f x y z nbsp viznachena i obmezhena na gladkij poverhni S displaystyle S nbsp Haj Z displaystyle Z nbsp poznachaye deyake rozbittya S displaystyle S nbsp na skinchennu kilkist elementarnih poverhon S i displaystyle S i nbsp i 1 2 i z ploshami D S i displaystyle Delta S i nbsp D Z displaystyle Delta Z nbsp ye najbilshim diametrom elementarnih poverhon S i displaystyle S i nbsp i M i x i y i z i displaystyle M i x i y i z i nbsp dovilna tochka na vidpovidnij elementarnij poverhni Ris 1 Chislo S Z i 1 N f x i y i z i D S i displaystyle S Z sum i 1 N f x i y i z i Delta S i nbsp nazivayetsya integralnoyu sumoyu sho vidpovidaye rozbittyu Z displaystyle Z nbsp Yaksho isnuye chislo I displaystyle I nbsp z takoyu vlastivistyu dlya kozhnogo ϵ gt 0 displaystyle epsilon gt 0 nbsp znajdetsya takeD ϵ gt 0 displaystyle Delta epsilon gt 0 nbsp sho dlya kozhnogo rozbittya Z displaystyle Z nbsp z D Z lt D displaystyle Delta Z lt Delta nbsp nezalezhno vid viboru tochok M i displaystyle M i nbsp S Z I lt D displaystyle S Z I lt Delta nbsp to I displaystyle I nbsp nazivayetsya poverhnevim integralom 1 go rodu vid f x y z displaystyle f x y z nbsp po poverhni S displaystyle S nbsp i zapisuyetsya I S f x y z d s displaystyle I iint limits S f x y z mathrm d s nbsp Dlya okremogo vipadku pidintegralnogo virazu f x y z 1 displaystyle f x y z equiv 1 nbsp chislo I displaystyle I nbsp daye ploshu S displaystyle S nbsp poverhni S displaystyle S nbsp Obchislennya zvedennya do podvijnogo integrala yaksho poverhnya zadana parametrichno x x u v displaystyle x x u v nbsp y y u v displaystyle y y u v nbsp z z u v displaystyle z z u v nbsp prichomu u displaystyle u nbsp ta v displaystyle v nbsp probigayut oblast G displaystyle Gamma nbsp ploshini u displaystyle u nbsp v displaystyle v nbsp I S f x y z d s G f x u v y u v z u v E G F 2 d u d v displaystyle I iint limits S f x y z mathrm d s iint limits Gamma f x u v y u v z u v sqrt EG F 2 mathrm d u mathrm d v nbsp Yaksho poverhnya zadana yavno rivnyannyam z ϕ x y displaystyle z phi x y nbsp prichomu x y displaystyle x y nbsp probigayut oblast S displaystyle S nbsp to I S f x y z d s S f x y ϕ x y 1 p 2 q 2 d x d y displaystyle I iint limits S f x y z mathrm d s iint limits S f x y phi x y sqrt 1 p 2 q 2 mathrm d x mathrm d y nbsp Analogichni formuli virni yaksho S displaystyle S nbsp predstavlena rivnyannyami vidu x ps y z displaystyle x psi y z nbsp chi y x x z displaystyle y chi x z nbsp Poverhnevi integrali 2 go rodu Redaguvati nbsp Ris 2Oriyentaciya dvostoronnoyi nezamknutoyi poverhni vibirayetsya pevna storona poverhni S displaystyle S nbsp na kozhnij zamknutij krivij na S displaystyle S nbsp viznachayetsya dodatnij napryam obhodu tak sho vin razom z normallyu vibranoyi storoni utvoryuvav pravu trijku vektoriv Nehaj v tochkah poverhni S displaystyle S nbsp roztashovanoyi odnoznachno nad ploshinoyu x y displaystyle x y nbsp i zadanoyu yavno rivnyannyam z ϕ x y displaystyle z phi x y nbsp viznachena obmezhena funkciyeyu f x y z displaystyle f x y z nbsp Nehaj Z displaystyle Z nbsp ye rozbittya poverhni S displaystyle S nbsp na skinchennu kilkist elementarnih poverhon S i displaystyle S i nbsp i 1 2 n displaystyle i 1 2 n nbsp D Z displaystyle Delta Z nbsp najbilshij diametr elementarnih poverhon M i x i y i z i displaystyle M i x i y i z i nbsp dovilna tochka vibrana na elementarnij poverhni S i displaystyle S i nbsp Yaksho vibrana pevna storona poverhni i tim samim oriyentaciya po nij to napryam obhodu mezhi kozhnoyi elementarnoyi poverhni S i displaystyle S i nbsp viznachaye napryam obhodu v ploshini x y displaystyle x y nbsp bilya kordonu proyekciyi S i displaystyle S i nbsp Plosha D S i displaystyle Delta S i nbsp ciyeyi proyekciyi beretsya iz znakom yaksho mezha proyekciyi S i displaystyle S i nbsp prohoditsya v dodatnomu napryami inakshe iz znakom Ris 2 Chislo S Z i 1 N f x i y i z i D S i displaystyle S Z sum i 1 N f x i y i z i Delta S i nbsp nazivayetsya integralnoyu sumoyu sho vidpovidaye rozbittyu Z displaystyle Z nbsp Na protivagu utvorennyu integralnih sum poverhnevih integraliv 1 go rodu tut f M i displaystyle f M i nbsp mnozhitsya ne na ploshu D S i displaystyle Delta S i nbsp elementarnij poverhni S i displaystyle S i nbsp a na vzyatu iz znakom plosha D S i displaystyle Delta S i nbsp proyekciyi S i displaystyle S i nbsp poverhni S i displaystyle S i nbsp na ploshinu x y displaystyle x y nbsp Yaksho isnuye chislo I displaystyle I nbsp z takoyu vlastivistyu dlya kozhnogo ϵ gt 0 displaystyle epsilon gt 0 nbsp znajdetsya take D ϵ gt 0 displaystyle Delta epsilon gt 0 nbsp sho dlya kozhnogo rozbittya Z displaystyle Z nbsp z D Z lt D displaystyle Delta Z lt Delta nbsp nezalezhno vid viboru tochok M i displaystyle M i nbsp zavzhdi S Z I lt ϵ displaystyle S Z I lt epsilon nbsp to I displaystyle I nbsp nazivayut poverhnevim integralom 2 go rodu vid f x y z displaystyle f x y z nbsp za vibranoyu storonoyu S displaystyle S nbsp i pishut S f x y z d x d y displaystyle iint limits S f x y z mathrm d x mathrm d y nbsp Yaksho S displaystyle S nbsp ne maye vzayemno odnoznachnoyi proyekciyi na ploshinu x y displaystyle x y nbsp ale yiyi mozhna rozbiti na skinchennu kilkist poverhon dlya kozhnoyi z yakih isnuye taka proyekciya to poverhnevij integral po S displaystyle S nbsp viznachayetsya yak suma integraliv po okremih poverhnyah Yaksho S displaystyle S nbsp maye odnoznachnu proyekciyu na ploshinu y z displaystyle y z nbsp abo x z displaystyle x z nbsp to mozhna viznachiti analogichno dva inshih poverhnevih integrala 2 go rodu S f x y z d y d z displaystyle iint limits S f x y z mathrm d y mathrm d z nbsp ta S f x y z d z d x displaystyle iint limits S f x y z mathrm d z mathrm d x nbsp de u vidpovidnih integralnih sumah stoyat ploshi proyekcij S i displaystyle S i nbsp na ploshinu y z displaystyle y z nbsp abo x z displaystyle x z nbsp Nareshti dlya troh funkcij P x y z displaystyle P x y z nbsp Q x y z displaystyle Q x y z nbsp R x y z displaystyle R x y z nbsp viznachenih na S displaystyle S nbsp ci integrali mozhna dodati i viznachiti zagalnishij poverhnevij integral drugogo rodu S P d y d z Q d z d x R d x d y S P d y d z S Q d z d x S R d x d y displaystyle iint limits S P mathrm d y mathrm d z Q mathrm d z mathrm d x R mathrm d x mathrm d y iint limits S P mathrm d y mathrm d z iint limits S Q mathrm d z mathrm d x iint limits S R mathrm d x mathrm d y nbsp Obchislennya poverhnevogo integrala 2 go rodu zvedennya do podvijnogo integrala Redaguvati 1 Nehaj poverhnya S displaystyle S nbsp maye yavne predstavlennya z ϕ x y displaystyle z phi x y nbsp prichomu x y displaystyle x y nbsp zminyuyutsya v oblasti S displaystyle S nbsp Todi poverhnevij integral po tij storoni S displaystyle S nbsp dlya yakoyi kut mizh normallyu i vissyu z displaystyle z nbsp ye gostrim obchislyuyetsya tak S f x y z d x d y S f x y ϕ x y displaystyle iint limits S f x y z mathrm d x mathrm d y iint limits S f x y phi x y nbsp Yaksho vibrana insha storona poverhni to S f x y z d x d y S f x y ϕ x y displaystyle iint limits S f x y z mathrm d x mathrm d y iint limits S f x y phi x y nbsp Analogichni formuli vihodyat dlya inshih integraliv S f x y z d y d z S f ps y z y z displaystyle iint limits S f x y z mathrm d y mathrm d z iint limits S f psi y z y z nbsp de S displaystyle S nbsp zadana rivnyannyam x ps y z displaystyle x psi y z nbsp S displaystyle S nbsp proyekciya S displaystyle S nbsp na ploshinu y z displaystyle y z nbsp a poverhnevij integral beretsya po tij storoni normal do yakoyi utvoryuye z vissyu x displaystyle x nbsp gostrij kut Tak samo S f x y z d z d x S f x x z x y d z d x displaystyle iint limits S f x y z mathrm d z mathrm d x iint limits S f x chi z x y mathrm d z mathrm d x nbsp de S displaystyle S nbsp zadana rivnyannyam y x z x displaystyle y chi z x nbsp S displaystyle S nbsp proyekciya S displaystyle S nbsp na ploshinu x z displaystyle x z nbsp a poverhnevij integral beretsya po tij storoni normal do yakoyi skladaye z vissyu u gostrij kut 2 Yaksho poverhnya S displaystyle S nbsp zadana v parametrichnij formi x x u v displaystyle x x u v nbsp y y u v displaystyle y y u v nbsp z z u v displaystyle z z u v nbsp to S f x y z d x d y G f x u v y u v z u v C d u d v displaystyle iint limits S f x y z mathrm d x mathrm d y pm iint limits Gamma f x u v y u v z u v C mathrm d u mathrm d v nbsp S f x y z d y d z G f x u v y u v z u v A d u d v displaystyle iint limits S f x y z mathrm d y mathrm d z pm iint limits Gamma f x u v y u v z u v A mathrm d u mathrm d v nbsp S f x y z d z d x G f x u v y u v z u v B d u d v displaystyle iint limits S f x y z mathrm d z mathrm d x pm iint limits Gamma f x u v y u v z u v B mathrm d u mathrm d v nbsp de A y z u v displaystyle A partial y z over partial u v nbsp B z x u v displaystyle B partial z x over partial u v nbsp C x y u v displaystyle C partial x y over partial u v nbsp divis rivnyannya ugori dodatnij znak pered integralom sprava vikoristovuyetsya todi koli oriyentaciya oblasti G displaystyle Gamma nbsp ploshini u v displaystyle u v nbsp vidpovidaye oriyentaciyi vibranoyi storoni Dlya sumi troh integraliv otrimuyemo S P d y d z Q d z d x R d x d y G P A Q B R C d u d v displaystyle iint limits S P mathrm d y mathrm d z Q mathrm d z mathrm d x R mathrm d x mathrm d y pm iint limits Gamma PA QB RC mathrm d u mathrm d v nbsp Zv yazok mizh poverhnevimi integralami 1 go i 2 go rodu Redaguvati Yaksho a displaystyle alpha nbsp b displaystyle beta nbsp g displaystyle gamma nbsp kuti normali do vibranoyi storoni poverhni z osyami x y displaystyle x y nbsp i z displaystyle z nbsp to S P d y d z Q d z d x R d x d y S P cos a Q cos b R cos g d S displaystyle iint limits S P mathrm d y mathrm d z Q mathrm d z mathrm d x R mathrm d x mathrm d y pm iint limits S P cos alpha Q cos beta R cos gamma mathrm d S nbsp tobto poverhnevij integral 2 go rodu sho stoyit zliva peretvoritsya v poverhnevij integral 1 go rodu sho stoyit sprava nbsp Ris 3Poverhnevij integral S P d y d z Q d z d x R d x d y displaystyle iint limits S P mathrm d y mathrm d z Q mathrm d z mathrm d x R mathrm d x mathrm d y nbsp maye dlya riznih nezamknutih poverhon S 1 displaystyle S 1 nbsp i S 2 displaystyle S 2 nbsp z odniyeyu i tiyeyu zh graniceyu C displaystyle C nbsp u zagalnomu vipadku rizni znachennya Ris 3 tobto vin v zagalnomu vipadku ne obertayetsya v nul na zamknutij poverhni analogichno zalezhnosti vid shlyahu krivolinijnogo integrala Yaksho funkciyi P Q R P x Q y R z displaystyle P Q R partial P over partial x partial Q over partial y partial R over partial z nbsp neperervni v odnozv yaznij prostorovij oblasti V displaystyle V nbsp tobto v oblasti yaka razom z kozhnoyu zamknutoyu poverhneyu mistit takozh i oblast obmezhenu ciyeyu poverhneyu to poverhnevij integral po vsyakij zamknutij poverhni S displaystyle S nbsp v V displaystyle V nbsp obertayetsya v nul todi i tilki todi koli P x Q y R z 0 displaystyle partial P over partial x partial Q over partial y partial R over partial z 0 nbsp Geometrichni i fizichni zastosuvannya poverhnevogo integrala RedaguvatiOb yem tila Redaguvati Ob yem V displaystyle V nbsp tila V displaystyle V nbsp obmezhenogo kuskovo gladkimi poverhnyami S displaystyle S nbsp mozhna riznimi sposobami obchisliti yak poverhnevij integral drugogo rodu V S z d x d y displaystyle V iint S z dx dy nbsp chi V S x d y d z displaystyle V iint S x dy dz nbsp chi V S y d z d x displaystyle V iint S y dz dx nbsp abo V 1 3 S x d y d z y d z d x z d x d y displaystyle V 1 over 3 iint S x dy dz y dz dx z dx dy nbsp pri comu integrali slid brati po zovnishnij storoni poverhni S displaystyle S nbsp Centr tyazhinnya ta sila prityagannya Redaguvati Yaksho poverhnya S displaystyle S nbsp pokrita masoyu z poverhnevoyu gustinoyu d x y z displaystyle delta x y z nbsp to povna masa poverhni S displaystyle S nbsp dorivnyuye M S d x y z d S displaystyle M iint S delta x y z dS nbsp koordinati 3 h z displaystyle xi eta zeta nbsp centru tyazhinnya dorivnyuyut 3 1 M S x d x y z d S displaystyle xi 1 over M iint S x delta x y z dS nbsp h 1 M S y d x y z d S displaystyle eta 1 over M iint S y delta x y z dS nbsp z 1 M S z d x y z d S displaystyle zeta 1 over M iint S z delta x y z dS nbsp komponenti sili prityagannya F displaystyle F nbsp cogo rozpodilu masi sho diye na materialnu tochku M x 0 y 0 z 0 displaystyle M x 0 y 0 z 0 nbsp odinichnoyi masi dorivnyuyut F x g S x x 0 r 3 d S displaystyle F x gamma iint S x x 0 over r 3 dS nbsp F y g S y y 0 r 3 d S displaystyle F y gamma iint S y y 0 over r 3 dS nbsp F z g S z z 0 r 3 d S displaystyle F z gamma iint S z z 0 over r 3 dS nbsp g c o n s t displaystyle gamma const nbsp Div takozh Redaguvati nbsp Portal Matematika Integralne chislennya Dzherela RedaguvatiBronshtejn I N Semendyaev K A Spravochnik po matematike dlya inzhenerov i uchashihsya vtuzov M Nauka 1980 976 s il nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Poverhnevij integral amp oldid 40291048
