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Vektornyi analiz

Vektornyi analiz - razdel matematiki, v kotorom izuchayutsya skalyarnye i vektornye polya i razlichnye operacii s nimi. Skalyarnoe pole sopostavlyaet kazhdoi tochke (3-mernogo) prostranstva nekotoroe (deistvitel'noe) chislo ( $\varphi=\varphi(r)$ , a vektornoe pole - nekotoryi vektor (a=a(r)). Esli tochka zadaetsya svoimi dekartovymi koordinatami, $r=\{x_1, x_2, x_3\}$ a vektor - svoimi komponentami $a=\{a_1, a_2, a_3\}$ , to gradient skalyarnogo polya, divergenciya i rotor vektornogo polya vyrazhayutsya formulami:
$(\mathrm{grad}\ \varphi)_i={\displaystyle{\partial \varphi} \over \displaystyle{\partial x_i}}$ , $\mathrm{div}\ a={\displaystyle{\partial a_{1}} \over \displaystyle{\partial x_{1}}}+{\displaystyle{\partial a_{2}} \over \displaystyle{\partial x_{2}}}+{\displaystyle{\partial a_{3}} \over \displaystyle{\partial x_{3}}}$

$\mathrm{rot}\ a=\{{\displaystyle{\partial a_3} \over \displaystyle{\partial x_2}}-{\displaystyle{\partial a_2} \over \displaystyle{\partial x_3}},{\displaystyle{\partial a_1} \over \displaystyle{\partial x_3}}-{\displaystyle{\partial a_3} \over \displaystyle{\partial x_1}},{\displaystyle{\partial a_2} \over \displaystyle{\partial x_1}}-{\displaystyle{\partial a_1} \over \displaystyle{\partial x_2}} \}$

Gradient, divergenciyu i rotor udobno vyrazhat' s pomosh'yu simvolicheskogo vektora $\nabla$ (nabla), komponentami kotorogo yavlyayutsya operatory differencirovaniya po koordinatam, $\nabla = \{{\displaystyle{\partial} \over \displaystyle{\partial x_1}},{\displaystyle{\partial} \over \displaystyle{\partial x_2}}, {\displaystyle{\partial} \over \displaystyle{\partial x_3}} \}$ Deistvuya etim simvolicheskim vektorom na skalyarnye i vektornye polya po pravilam vektornoi algebry, poluchim:
$\mathrm{grad}\ \varphi =\nabla \varphi$ , $\mathrm{div}\ a = (\nabla a)$ , $\mathrm{rot}\ a = [\nabla a]$

Skalyarnyi kvadrat vektora u predstavlyaet soboi Laplasa operator, ili laplasian, kotoryi oboznachaetsya $\Delta$ :
$\Delta=\nabla^2={\displaystyle{\partial ^{2}} \over \displaystyle{\partial x_1^2}}+{\displaystyle{\partial ^2} \over \displaystyle{\partial x_2^2}}+{\displaystyle{\partial ^2} \over \displaystyle{\partial x_3^2}}$

Formal'noe primenenie pravil vektornoi algebry k vektoru $\nabla$ privodit k ryadu sootnoshenii mezhdu gradientom, divergenciei i rotorom, naprimer,
$[\nabla(\nabla\varphi)]=0$ , ili $\mathrm{rot}\ \mathrm{grad}\ \varphi=0$ ;
$(\nabla[\nabla\alpha])=0$ , ili $\mathrm{rot}\ \mathrm{grad}\ \alpha=0$ ;
$[\nabla[\nabla\alpha]]=\nabla(\nabla\alpha)-\nabla^2 a$ ,
ili $\mathrm{rot}\ \mathrm{rot}\ a=\mathrm{grad}\ \mathrm{div}\ a - \Delta a$
Pri takogo roda formal'nyh preobrazovaniyah neobhodimo sledit', chtoby differencial'nyi operator $\nabla$ v okonchatel'nom vyrazhenii stoyal sleva ot toi funkciii, na kotoruyu on deistvuet. Esli operator $\nabla$ deistvuet na proizvedenie dvuh funkcii, to po pravilu Leibnica (pravilo differencirovaniya proizvedeniya) mozhno zapisat' rezul'tat v vide summy dvuh chlenov:
$\nabla(\varphi \psi)=\varphi \nabla \psi+\psi \nabla \varphi$ ,
ili
$\mathrm{grad}\ (\varphi \psi)=\varphi \ \mathrm{grad}\ \psi+\psi \ \mathrm{grad}\ \varphi$ ,
Sochetaya pravilo Leibnica s pravilami vektornoi algebry, mozhno poluchat' sootnosheniya takogo tipa:
$(\nabla (a\varphi))=\varphi(\nabla a)+(a \nabla \varphi)$
ili
$\mathrm{div}\ (a \varphi))=\varphi \ \mathrm{div}\ a + a \ \mathrm{grad}\ \varphi$
V sluchae bolee slozhnyh algebraicheskih vykladok pa promezhutochnyh etapah sleduet otmechat' strelkoi tu funkciyu, na kotoruyu deistvuet operator $\nabla$ , ne zabotyas' o poryadke sledovaniya operatora i funkcii, i lish' na poslednem etape vozvrashat'sya k obychnomu poryadku:
$[\nabla (a \varphi)]=[\nabla\check{a}\varphi]+[\nabla a \check{\varphi}]=\varphi [\nabla a]-[a \nabla \varphi]$
ili
$\mathrm{rot}\ (a \varphi)=\varphi \ \mathrm{rot}\ a -[a \ \mathrm{grad} \ \varphi]$
Takim obrazom, poluchaem:
$\mathrm{div}\ [ab]=b \ \mathrm{rot}\ a-a \ \mathrm{rot}\ b$ ,
$rot[ab]=a \ \mathrm{div}\ b - b \ \mathrm{div}\ a + (b \nabla) a - (a \nabla)b$ ,
$\mathrm{grad}\ (ab)=[a \ \mathrm{rot}\ b] + [b \ \mathrm{rot}\ a] + (b \nabla) a + (a \nabla)b$
Vse osnovnye differencial'nye operacii vektornogo analiza imeyut opredelennyi smysl, poetomu znacheniya vyrazhenii $\mathrm{grad}\ \varphi$ , div a, rota ne zavisyat ot vybora sistemy koordinat. Vse sootnosheniya mezhdu differencial'nymi vyrazheniyami takzhe nosyat invariantnyi harakter.

V prilozheniyah chasto vstrechayutsya potok vektora cherez zadannuyu poverhnost' i integral ot nego vdol' zadannoi krivoi:
$\int_{S}a \ dS=\int_{S}a_n \ dS=\int_{s}(a_1 \ dx_2 \ dx_3+a_2 \ dx_3 \ dx_1+a_3 \ dx_1 \ dx_2)$ ,
$\int_{L}a \ dr=\int_{L}a_{\tau} \ dl=\int_{L}(a_1 \ dx_1+a_2 \ dx_2+a_3 \ dx_3)$
Zdes' $a_n=(an)$ - proekciya vektora a na normal' k poverhnosti v dannoi tochke, $a_{\tau}=(a\tau)$ -proekciya ego na edinichnyi vektor $\tau$ , kasatel'nyi k krivoi, dS - element ploshadi poverhnosti, dl - element dliny krivoi. Pust' a - raspredelenie skorostei dvizhusheisya zhidkosti, togda pervyi integral raven ob'emu zhidkosti, peresekayushei dannuyu poverhnost' v edinicu vremeni. Esli a - silovoe pole, to vtoroi integral raven rabote, sovershaemoi pri peremeshenii probnogo tela vdol' dannoi krivoi. V sluchae zamknutoi krivoi takoi integral nazyvaetsya cirkulyaciei vektornogo polya.

Eti integraly figuriruyut v osnovnyh teoremah vektornoi algebry - Gaussa - Ostrogradskogo formule i Stoksa formule:
$\oint_{\partial V}a_n \ dS=\oint_{V}\mathrm{div}\ a \ dV$ , $\oint_{\partial S}a \ dr=\oint_{S}(\mathrm{rot}\ a)_{n} \ dS$ .
Zdes' $\partial V$ - poverhnost', yavlyayushayasya granicei oblasti V, a $\partial S$ - krivaya, ogranichivayushaya poverhnost' S. Kruzhki na znachkah integralov oznachayut, chto integrirovanie vedetsya po zamknutoi poverhnosti i zamknutoi krivoi. Polozhitel'noe napravlenie normali k poverhnosti S dolzhno byt' orientirovano otnositel'no napravleniya obhoda kontura $\partial S$ tak zhe, kak polozhitel'noe napravlenie osi x₃ - otnositel'no polozhitel'nogo napravleniya vrasheniya v ploskosti x₁, x₂. Polagaya v formule Gaussa-Ostrogradskogo $a=\psi \mathrm{grad}\ \varphi$ , poluchim vazhnuyu teoremu Grina $\oint_{\partial V}\psi (\mathrm{grad}\ \varphi)_n dS= \int_{V}\{(\psi\Delta\varphi +(\mathrm{grad}\ \psi \ \mathrm{grad}\ \varphi)\} \ dV$
Ee sledstviem yavlyaetsya formula
$\oint_{\partial V}(\psi \ \mathrm{grad}\ _n \ \varphi - \varphi \ \mathrm{grad}\ _n \psi) \ dS= \int_{V}(\psi\Delta\varphi - \varphi \Delta \psi) \ dV$
Drugie integral'nye teoremy mozhno poluchit' kak sledstviya uzhe sformulirovannyh:
$\oint_{\partial S} \varphi \ dr= \oint_{S} [n \ \mathrm{grad}\ \varphi] \ dS$ ,
$\oint_{\partial V} \varphi \ n \ dS= \oint_{V} \mathrm{grad}\ \ \varphi \ dV$
$\oint_{\partial V} [na] dS= \oint_{V} \mathrm{rot}\ a \ dV$
Ponyatiya vektornogo analiza, opredelennye vyshe dlya evklidova prostranstva, mozhno obobshit' na rimanovo prostranstvo i drugie mnogoobraziya. Differencial'nye operacii privodyat k ponyatiyu kovariantnoi proizvodnoi, integral'nye teoremy formuliruyutsya na yazyke differencial'nyh form.

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