16.4-bo'lim: Vektor maydonlarining chiziqli integrallari
Oldingi ikki bo'limda biz funktsiyalarning chiziqli integrallarini ko'rib chiqdik. Ushbu bo'limda biz vektor maydonlarining chiziqli integrallarini baholaymiz. Biz vektor maydonidan boshlaymiz,
\[\vec F\left( {x,y,z} \right) = P\left( {x,y,z} \right)\vec i + Q\left( {x,y,z} \right)\vec j + R\left( {x,y,z} \right)\vec k\]va tomonidan berilgan uch o'lchovli, silliq egri
\[\vec r\left( t \right) = x\left( t \right)\vec i + y\left( t \right)\vec j + z\left( t \right)\vec k\hspace{0.25in}\hspace{0.25in}a \le t \le b\]Vektor maydonining chiziqli integrali
\(\vec F\) ning \(C\) bo'ylab chiziqli integrali is
\[\int\limits_{C}{{\vec F\centerdot d\,\vec r}} = \int_{{\,a}}^{{\,b}}{{\vec F\left( {\vec r\left( t \right)} \right)\centerdot \vec r'\left( t \right)\,dt}}\]Chap tarafdagi integraldagi yozuvga e'tibor bering. Bu haqiqatan ham a nuqta mahsuloti vektor maydoni va differensial haqiqatda vektordir. Bundan tashqari, \(\vec F\left( {\vec r\left( t \right)} \right)\) - bu stenografiya:
\[\vec F\left( {\vec r\left( t \right)} \right) = \vec F\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right)\]Shuningdek, vektor maydonlarining chiziqli integrallarini yoy uzunligiga nisbatan chiziqli integral sifatida quyidagicha yozishimiz mumkin,
\[\int\limits_{C}{{\vec F\centerdot d\,\vec r}} = \int\limits_{C}{{\vec F\centerdot \vec T\,ds}}\]bu yerda \(\vec T\left( t \right)\) birlik tangens vektori va quyidagicha ifodalanadi:
\[\vec T\left( t \right) = \frac{{\vec r'\left( t \right)}}{{\left\| {\vec r'\left( t \right)} \right\|}}\]Agar chiziq integrallarini yoy uzunligi bo'yicha hisoblash bo'yicha bilimlarimizdan foydalansak, bu ikkinchi shakl yuqorida keltirilgan birinchi shaklga ekvivalent ekanligini ko'rishimiz mumkin.
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\,\vec r}} & = \int\limits_{C}{{\vec F\centerdot \vec T\,ds}}\\ & = \int_{{\,a}}^{{\,b}}{{\vec F\left( {\vec r\left( t \right)} \right)\centerdot \frac{{\vec r'\left( t \right)}}{{\left\| {\vec r'\left( t \right)} \right\|}}\,\,\left\| {\vec r'\left( t \right)} \right\|\,dt}}\\ & = \int_{{\,a}}^{{\,b}}{{\vec F\left( {\vec r\left( t \right)} \right)\centerdot \vec r'\left( t \right)\,dt}}\end{align*}\]Umuman olganda, biz ushbu chiziqli integralni hisoblash uchun birinchi shakldan foydalanamiz, chunki undan foydalanish odatda ancha oson. Keling, bir nechta misollarni ko'rib chiqaylik.
Yechim
OK, bizga birinchi navbatda egri chiziq bo'ylab baholangan vektor maydoni kerak.
\[\vec F\left( {\vec r\left( t \right)} \right) = 8{t^2}\left( {{t^2}} \right)\left( {{t^3}} \right)\vec i + 5{t^3}\,\vec j - 4t\left( {{t^2}} \right)\vec k = 8{t^7}\,\vec i + 5{t^3}\,\vec j - 4{t^3}\,\vec k\]Keyinchalik, parametrlashning hosilasi kerak.
\[\vec r'\left( t \right) = \,\vec i + 2t\,\vec j + 3{t^2}\,\vec k\]Va nihoyat, keling, nuqta mahsulotiga g'amxo'rlik qilaylik.
\[\vec F\left( {\vec r\left( t \right)} \right)\centerdot \vec r'\left( t \right) = 8{t^7} + 10{t^4} - 12{t^5}\]U holda chiziqli integrali,
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\,\vec r}} & = \int_{{\,0}}^{{\,1}}{{8{t^7} + 10{t^4} - 12{t^5}\,dt}}\\ & = \left. {\left( {{t^8} + 2{t^5} - 2{t^6}} \right)} \right|_0^1\\ & = 1\end{align*}\]Yechim
Bizga birinchi navbatda chiziq segmentini parametrlashtirish kerak bo'ladi. Biz birinchi navbatda chiziq segmentlarining parametrlarini qanday olishni ko'rdik bo'limi chiziqli integrallar. So'nggi bir necha bo'limda biz uning ikki o'lchovli versiyasidan foydalandik. Bu erda chiziq uchun parametrlash.
\[\begin{align*}\vec r\left( t \right) & = \left( {1 - t} \right)\left\langle { - 1,2,0} \right\rangle + t\left\langle {3,0,1} \right\rangle \\ & = \left\langle {4t - 1,2 - 2t,t} \right\rangle ,\hspace{1.0in} 0 \le t \le 1\end{align*}\]Shunday qilib, egri chiziq bo'ylab baholangan vektor maydonini olaylik.
\[\begin{align*}\vec F\left( {\vec r\left( t \right)} \right) & = \left( {4t - 1} \right)\left( t \right)\,\vec i - \left( {2 - 2t} \right)\left( t \right)\,\vec k\\ & = \left( {4{t^2} - t} \right)\vec i - \left( {2t - 2{t^2}} \right)\vec k\end{align*}\]Endi bizga parametrlashning hosilasi kerak.
\[\vec r'\left( t \right) = \left\langle {4, - 2,1} \right\rangle \]Keyin nuqta mahsuloti,
\[\vec F\left( {\vec r\left( t \right)} \right)\centerdot \vec r'\left( t \right) = 4\left( {4{t^2} - t} \right) - \left( {2t - 2{t^2}} \right) = 18{t^2} - 6t\]Chiziqli integralga aylanadi,
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\,\vec r}}& = \int_{{\,0}}^{{\,1}}{{18{t^2} - 6t\,dt}}\\ & = \left. {\left( {6{t^3} - 3{t^2}} \right)} \right|_0^1\\ & = 3\end{align*}\]Vektor maydonlarining chiziqli integrallari va \(x\), \(y\) va \(z\) ga nisbatan chiziqli integrallar o'rtasida yaxshi munosabatga ega bo'lish uchun umumiy shulardan birini bajarib, ushbu bo'limni yopamiz.
\(\vec F\left( {x,y,z} \right) = P\,\vec i + Q\,\vec j + R\,\vec k\) vektor maydonini va \(\vec r\left( t \right) = x\left( t \right)\vec i + y\left( t \right)\vec j + z\left( t \right)\vec k\), \(a \le t \le b\) tomonidan parametrlangan \(C\) egri chizig'ini hisobga olgan holda chiziqli integrali,
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\,\vec r}} & = \int_{{\,a}}^{{\,b}}{{\left( {P\,\vec i + Q\,\vec j + R\,\vec k} \right)\centerdot \left( {x'\,\vec i + y'\,\vec j + z'\,\vec k} \right)\,dt}}\\ & = \int_{{\,a}}^{{\,b}}{{Px' + Qy' + Rz'\,dt}}\\ & = \int_{{\,a}}^{{\,b}}{{Px'\,dt}} + \int_{{\,a}}^{{\,b}}{{Qy'\,dt}} + \int_{{\,a}}^{{\,b}}{{Rz'\,dt}}\\ & = \int\limits_{C}{{P\,dx}} + \int\limits_{C}{{Q\,dy}} + \int\limits_{C}{{R\,dz}}\\ & = \int\limits_{C}{{P\,dx}} + Q\,dy + R\,dz\end{align*}\]Shunday qilib, biz buni ko'ramiz,
Komponent shaklidagi vektor maydonining chiziqli integrali
E'tibor bering, bu bizga vektor maydonlarining chiziqli integrallarini baholashning yana bir usulini beradi.
Bu vektor maydonlarining chiziqli integrallari bilan yo'l yo'nalishini teskari o'zgartirish haqida ham quyidagilarni aytishga imkon beradi.
Egri chiziq yo'nalishini teskari aylantirish
Bu \(x\), \(y\) va/yoki \(z\) ga nisbatan chiziqli integrallar uchun to‘g‘ri ekanligini va vektor maydonlarining chiziqli integrallari \(x\), \(y\) va \(y\) va \(y\) ga nisbatan chiziqli integrallar nuqtai nazaridan aniqlanishi mumkinligini bilsak, bu ma’lum ma’noga ega bo‘lishi kerak.