Gaussi-Ostrogradski teoreem

Gaussi-Ostrogradski teoreem (ka Gaussi-Ostrogradski valem, Gaussi valem) on valem matemaatilises analüüsis, mis seob omavahel vektorvälja voo läbi pinna ja antud pinna sisese tensorvälja käitumise.

Matemaatiline definitsioon muuda

Kui piirkonna $\Omega$ rajapind $\Sigma$ on tükiti sile ja funktsioonid $X,Y,Z$ ning nende osatuletised $X_{x},Y_{y},Z_{z}$ on pidevad piirkonnas $\Omega$ , siis

$\iint \limits _{\Sigma }\mathbf {Fn} \,d\sigma =\iiint \limits _{\Omega }\mathrm {div} \,\mathbf {F} \,dV,$

kus pindintegraal on võetud üle $\Sigma$ välise pinnapoole.^[1]

Ajalugu ja nimetus muuda

Teoreemi avastas esimesena 1762. aastal Lagrange^[2] ja hiljem iseseisvalt 1813. aastal Gauss.^[3] Ostrogradski oli esimene, kes esitas teoreemi tõestuse üldise juhu jaoks aastal 1826.

Vaata ka muuda

Stokesi valem

Viited muuda

↑ Ivar Tammeraid (2005). Matemaatiline analüüs II. Tallinn: TTÜ kirjastus. Lk 217.
↑ In his 1762 paper on sound, Lagrange treats a special case of the divergence theorem: Lagrange (1762) "Nouvelles recherches sur la nature et la propagation du son" (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 – 172. This article is reprinted as: "Nouvelles recherches sur la nature et la propagation du son" in: J.A. Serret, ed., Oeuvres de Lagrange, (Paris, France: Gauthier-Villars, 1867), vol. 1, pages 151–316; on pages 263–265, Lagrange transforms triple integrals into double integrals using integration by parts.
↑ C. F. Gauss (1813) "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata," Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378; Gauss considered a special case of the theorem; see the 4th, 5th, and 6th pages of his article.

[1] Ivar Tammeraid (2005). Matemaatiline analüüs II. Tallinn: TTÜ kirjastus. Lk 217.

[2] In his 1762 paper on sound, Lagrange treats a special case of the divergence theorem: Lagrange (1762) "Nouvelles recherches sur la nature et la propagation du son" (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 – 172. This article is reprinted as: "Nouvelles recherches sur la nature et la propagation du son" in: J.A. Serret, ed., Oeuvres de Lagrange, (Paris, France: Gauthier-Villars, 1867), vol. 1, pages 151–316; on pages 263–265, Lagrange transforms triple integrals into double integrals using integration by parts.

[3] C. F. Gauss (1813) "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata," Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378; Gauss considered a special case of the theorem; see the 4th, 5th, and 6th pages of his article.

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