Teorema Grina vstanovlyuye zv yazok mizh krivolinijnim integralom po zamknutomu konturu C displaystyle C i podvijnim integralom po oblasti D displaystyle D obmezhenij cim konturom Faktichno cya teorema ye okremim vipadkom zagalnishoyi teoremi Stoksa Teorema nazvana na chest anglijskogo matematika Dzhordzha Grina Zmist 1 Formulyuvannya 1 1 Dovedennya 2 Zv yazok z formuloyu Ostrogradskogo 3 Div takozh 4 DzherelaFormulyuvannya Redaguvati nbsp D displaystyle D nbsp oblast obmezhena zamknutoyu krivoyu C displaystyle C nbsp Nehaj C displaystyle C nbsp dodatno oriyentovana kuskovo gladka zamknuta kriva na ploshini a D displaystyle D nbsp oblast obmezhena krivoyu C displaystyle C nbsp Yaksho funkciyi P P x y displaystyle P P x y nbsp Q Q x y displaystyle Q Q x y nbsp viznacheni v oblasti D displaystyle D nbsp i mayut neperervni chastkovi pohidni P y displaystyle frac partial P partial y nbsp Q x displaystyle frac partial Q partial x nbsp to C P d x Q d y D Q x P y d x d y displaystyle oint C P dx Q dy iint limits D left frac partial Q partial x frac partial P partial y right dx dy nbsp Na simvoli integrala chasto malyuyut kolo shob pidkresliti sho kriva C displaystyle C nbsp zamknena Dovedennya Redaguvati Nehaj oblast D displaystyle D nbsp krivolinijna trapeciya oblast pravilna v napryamku O Y displaystyle OY nbsp D x y a x b y 1 x y y 2 x displaystyle D x y a leq x leq b y 1 x leq y leq y 2 x nbsp Dlya krivoyi C displaystyle C nbsp sho obmezhuye oblast D displaystyle D nbsp zadamo napryamok obhodu za godinnikovoyu strilkoyu Todi D P y d x d y a b d x y 1 x y y 2 x y P y d y a b P x y 2 x P x y 1 x d x displaystyle iint limits D frac partial P partial y dx dy int limits a b dx int limits y 1 x y y 2 x y frac partial P partial y dy int limits a b P x y 2 x P x y 1 x dx nbsp a b P x y 2 x d x a b P x y 1 x d x 1 displaystyle int limits a b P x y 2 x dx int limits a b P x y 1 x dx quad 1 nbsp Pomitimo sho obidva oderzhani integrali mozhna zaminiti krivolinijnimi integralami C 1 P x y d x C 1 P x y d x a b P x y 1 x d x 2 displaystyle int limits C 1 P x y dx int limits C 1 P x y dx int limits a b P x y 1 x dx quad 2 nbsp C 3 P x y d x a b P x y 2 x d x 3 displaystyle int limits C 3 P x y dx int limits a b P x y 2 x dx quad 3 nbsp Integral po C 1 displaystyle C 1 nbsp beretsya zi znakom minus oskilki zgidno z oriyentaciyeyu konturu C displaystyle C nbsp napryamok obhodu danoyi chastini vid b displaystyle b nbsp do a displaystyle a nbsp Krivolinijni integrali po C 2 displaystyle C 2 nbsp i C 4 displaystyle C 4 nbsp dorivnyuvatimut nulyu oskilki x const displaystyle x operatorname const nbsp C 2 P x y d x 0 4 displaystyle int limits C 2 P x y dx 0 quad 4 nbsp C 4 P x y d x 0 5 displaystyle int limits C 4 P x y dx 0 quad 5 nbsp Zaminimo v 1 integrali zgidno z 2 i 3 a takozh dodamo 4 i 5 sho rivni nulyu i ne vplivayut na znachennya virazu D P y d x d y C 1 P x y d x C 3 P x y d x C 2 P x y d x C 4 P x y d x displaystyle iint limits D frac partial P partial y dx dy int limits C 1 P x y dx int limits C 3 P x y dx int limits C 2 P x y dx int limits C 4 P x y dx nbsp Oskilki obhid za godinnikovoyu strilkoyu za pravoyi oriyentaciyi ploshini ye vid yemnim napryamkom to suma integraliv v pravij chastini ye krivolinijnim integralom po zamknutij krivij C displaystyle C nbsp u vid yemnomu napryamku D P y d x d y C P x y d x 6 displaystyle iint limits D frac partial P partial y dx dy int limits C P x y dx quad 6 nbsp Analogichno dovoditsya formula D Q x d x d y C Q x y d y 7 displaystyle iint limits D frac partial Q partial x dx dy int limits C Q x y dy quad 7 nbsp yaksho za oblast D displaystyle D nbsp vzyati oblast pravilnu v napryamku O X displaystyle OX nbsp Vidnimayuchi 6 z 7 oderzhimo C P d x Q d y D Q x P y d x d y displaystyle int limits C P dx Q dy iint limits D left frac partial Q partial x frac partial P partial y right dx dy nbsp Zv yazok z formuloyu Ostrogradskogo RedaguvatiRozglyadayuchi dvovimirne vektorne pole teorema Grina rivnoznachna dvovimirnomu vipadku formuli Ostrogradskogo D F d A C F n d s displaystyle iint D left nabla cdot mathbf F right dA oint C mathbf F cdot mathbf hat n ds nbsp de F displaystyle nabla cdot mathbf F nbsp ce divergenciya dvovimirnogo vektornogo polya F displaystyle mathbf F nbsp a n displaystyle mathbf hat n nbsp ce normal na granici sho vkazuye nazovni Sho pobachiti ce rozglyanemo odinichnu normal n displaystyle mathbf hat n nbsp u pravij chastini rivnosti Oskilki v teoremi Grina d r d x d y displaystyle d mathbf r dx dy nbsp ce vektor napryamlenij vzdovzh dotichnoyi do krivoyi i kriva C dodatno oriyentovana tobto proti godinnikovoyi strilki kriva vzdovzh mezhi zovnishnya normal ce vektor napryamlenij 90 pravoruch vid cogo mozhna obrati d y d x displaystyle dy dx nbsp Cej vektor zavdovzhki d x 2 d y 2 d s displaystyle sqrt dx 2 dy 2 ds nbsp Tomu d y d x n d s displaystyle dy dx mathbf hat n ds nbsp Otzhe C L d x M d y C M L d y d x C M L n d s displaystyle oint C L dx M dy oint C M L cdot dy dx oint C M L cdot mathbf hat n ds nbsp Div takozh RedaguvatiDiskretna teorema GrinaDzherela RedaguvatiFihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1966 T 3 656 s ros Otrimano z https uk wikipedia org w index php title Teorema Grina amp oldid 40290994
