``Grin formulasi va uning tadbiqlari ``

Ikkinchi tur egri chiziqli integrallarning xossalari

^1._Agar ^P^{x, y}_fun^k_siya ^AB _y^o_{y b}^o_’y^l_ab ⁱⁿ_te^g_ra^l_laⁿ_u^vc_{hi b}^o’l_sa,

^{kPx, y dx k} ^{Px,y dx} AB AB

tenglik o’rinli.

P

P
^{2 .}_{Agar 1} ^{x, y}_{va 2} ^{x, y}_f^u_nksiyalar ^AB _y^o_y ^bo_’y^l_{ab i}ⁿ_tegr^a_l^l_a^nuv_chi bo’lsa,

^{Px, yP x, ydx }^{P x,y dx}^{P x,ydx} AB AB AB

tenglik o’rinli.

^{3 .}(Additivlik xossasi)Agar ^AB yoy biror ^C nuqta orqali^AC va ^CB yoylarga _a^j_rati^l_{gan b}^o’_lⁱ_b,^P^{x, y}_f^u_n^k_sⁱ_ya^AC _va ^CB _y^o_y^l_arning ^h_ar ^bi_{ri bo’y}^l_ab integrallanuvchi bo’lsa,

^{Px, y dx }^{Px, y dx}^{Px, y dx} AB AC CB

tenglik o’rinli bo’ladi.

^{4 .Agar} ^{Px, ydx egri chiziqli integral ham mavjud bo’lsa,} AB

^{Px, ydx }^{Px, ydx} AB BA

22
tenglik o’rinli bo’ladi.

^{5 .}Agar funksiya L yopiq kontur bo’ylab integrallanuvchi bo’lsa, u holda

^{Px, y dx} L

U holda egri chiziqli integralning qiymati L konturdagi qaysi nuqtani boshlang’ich

nuqta (bu nuqta ham bo’ladi) deb olinishiga bog’liq emas.


A
Isboti. ^A va ₁ lar teng bo’lmagan ixtiyoriy nuqtalar bo’lsin.

( 3-chizma)

A nuqtani boshlang’ich nuqta deb, egri chiziqli integralni ko’rsatilgan yo’nalish

bo’yicha hisoblasak



A

1 1
^{Px, ydx }^{Px, ydx}^{Px, y dx} AmA nA AmA ₁nA

tenglikka ega bo’lamiz.

Agar nuqtani boshlang’ich nuqta deb hisoblasak, u holda




1 1 1
^{Px, ydx }^{Px, ydx}^{Px, y dx} AnA mA AnA A mA

tenglikka ega bo’lamiz.

Bu tengliklarning o’ng tomonlari bir xil qo’shiluvchilardan iborat.Shuning

uchun chap tomonlari ham teng bo’ladi.Demak, xossa isbotlandi.

Grin formulasini keltirib chiqarish

23



d




Ikkinchi tur egri chiziqli integrallar ham asosan Riman integrallariga



keltirilib hisoblanadi:

^{f x, yx }^{f t,t}' ^tdt AB 






d
^{f x, yy }^{f t,t}' ^tdt AB 




d




^{Px, yxQx, ydy }^{Pt,t}' ^{tQt,t}' ^tdt AB 

(0.15)

(0.16)

(0.17)

^{Xususan AB egri chiziq y y}^x^{a x b}^{tenglama bilan aniqlangan bo’lib, y y}^x^funksiya ^a,b^{da hosilaga ega va uzluksiz b}_o^{’lsa (1.15)va (1.17) formulalar}

b
quyidagi

^{f x, ydx }^{f x, yxdx} AB a

(0.18)

b






_^P^{x, y dxQ}^{x, y}^{dy }_^P^{x, y}^x^Q^{x, y}^x^y' ^x_^dx AB a

ko’rinishga keladi.

d
_S^h_u^ningdek_, ^AB _e^gr_i ^c_hⁱ_ziq ^{x x}^y^{c y d}^teng_lama ^bi_l^a_{n a}ⁿ_i^q_l^a_n^ga_n ^bo’lib_,^x^y^fu_nks^iy_a ^c,d_or^a_l^iq_{da h}^o_sⁱ_la^g_{a e}^g_{a va uzl}^uks_{iz b}^o’l_{sa (}^1.1₆₎ ^v_{a (}^1.₁⁷₎ tenglamalar quyidagi

^{f x, ydy }^{f xy, y dy} AB c


d



 
_^P^{x, y dxQ}^{x, y}^{dy }_^P^x^y^{, y}^x' ^y^Q^x^y^{, y}_^dy AB c

(0.19)

(0.20)

ko’rinishga keladi.

24

^AgarP^{x, y}^{va Q}^{x, y}^{funksiyalar uchun}



Q P x y

^{Shart bajarilsa, u holda P}^{x, y}^dxQ^{x, y}^{dyifoda biror U} ^{x, y}^{funksiyaning to’la}

differensiali bo’ladi va

^{Px, ydxQx, ydy} AB

integral integrallash yo’liga bog’liq bo’lmaydi, faqat A va B nuqtalarning

berilishi bilan birqiymatli aniqlanadi.To’la differensial bo’yicha funksiyaning o’zi


_y

x




^{U x, y}^{Pt, y0 dt }^{Qx,t dt C} x₀ y₀

formula bo’yicha topiladi.

Ikki karrali va egri chiziqli integrallarni bog’lovchi

D ^P^{x, y}^dxQ^{x, y}^{dy }D _x ^_{y }^dxdy _(0.21)

formula Grin formulasi deyiladi.Grin formulasidan D sohaning yuzasini hisoblash

uchun ushbu

^{S }^{xdy ,} D
^{S }^ydx D

(0.22)

^{S }₂ ^{xdy ydx} _(0.23)

formulalar kelib chiqadi.