16.4-bo'lim : Vektor maydonlarining chiziqli integrallari
7. Quyidagi egri chiziqlarning har biri uchun \(\vec F\left( {x,y} \right) = \left( {6x - 2y} \right)\,\vec i + {x^2}\vec j\) bo'lgan \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang.
- \(C\) - \(\left( {6, - 3} \right)\) dan \(\left( {0,0} \right)\) gacha bo'lgan chiziq segmenti va undan keyin \(\left( {0,0} \right)\) dan \(\left( {6,3} \right)\) gacha bo'lgan chiziq segmenti.
- \(C\) - \(\left( {6, - 3} \right)\) dan \(\left( {6,3} \right)\) gacha bo'lgan chiziq segmenti.
Yechim
Keling, muammoning ushbu qismi uchun egri chiziqning qisqacha chizmasidan boshlaylik.
Mana bu egri chiziqlarning har biri uchun parametrlash.
\({C_1}:\,\,\vec r\left( t \right) = \left( {1 - t} \right)\left\langle {6, - 3} \right\rangle + t\left\langle {0,0} \right\rangle = \left\langle {6 - 6t, - 3 + 3t} \right\rangle \hspace{0.25in}\,\,\,\,\,\,\,0 \le t \le 1\)\({C_2}:\,\,\vec r\left( t \right) = \left\langle {t,\frac{1}{2}t} \right\rangle \hspace{0.25in}\,\,\,\,\,\,\,0 \le t \le 6\)
\({C_2}\) uchun parametrlarni olish uchun chiziq tenglamasidan foydalandik, chunki u ishlash uchun biroz chiroyliroq shakl berdi. Biz buni \({C_1}\) bilan amalga oshira olmadik, chunki egri chiziqning belgilangan yo‘nalishi kamayuvchi \(x\) yo‘nalishida edi va integral chegaralari kichikroq qiymatdan kattaroq qiymatgacha bo‘lishi kerak.
2-qadam
Mana bu egri chiziqlarning har biri uchun chiziqli integrali.
\[\begin{align*}\int\limits_{{{C_1}}}{{\vec F\centerdot d\vec r}} & = \int_{0}^{1}{{\left\langle {6\left( {6 - 6t} \right) - 2\left( { - 3 + 3t} \right),{{\left( {6 - 6t} \right)}^2}} \right\rangle \centerdot \left\langle { - 6,3} \right\rangle \,dt}}\\ & = \int_{0}^{1}{{252t - 252 + 3{{\left( {6 - 6t} \right)}^2}\,dt}}\\ & = \left. {\left( {126{t^2} - 252t - \frac{1}{6}{{\left( {6 - 6t} \right)}^3}} \right)} \right|_0^1 = \underline { - 90} \end{align*}\] \[\begin{align*}\int\limits_{{{C_2}}}{{\vec F\centerdot d\vec r}} & = \int_{0}^{6}{{\left\langle {6\left( t \right) - 2\left( {\frac{1}{2}t} \right),{{\left( t \right)}^2}} \right\rangle \centerdot \left\langle {1,\frac{1}{2}} \right\rangle \,dt}}\\ & = \int_{0}^{6}{{5t + \frac{1}{2}{t^2}\,dt}} = \left. {\left( {\frac{5}{2}{t^2} + \frac{1}{6}{t^3}} \right)} \right|_0^6 = \underline {126} \end{align*}\]3-qadam
Bu qism uchun chiziqli integrali shunday bo'ladi:
\[\int\limits_{C}{{\vec F\centerdot d\vec r}} = \left( { - 90} \right) + \left( {126} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{36}}\]b \(C\) - \(\left( {6, - 3} \right)\) dan \(\left( {6,3} \right)\) gacha bo'lgan chiziq segmenti.
Yechim
Keling, muammoning ushbu qismi uchun egri chiziqning qisqacha chizmasidan boshlaylik.
Shunday qilib, bizda bu qismda \(\left( {6, - 3} \right)\) dan \(\left( {6,3} \right)\) ga o'tadigan boshqa egri chiziq mavjud. Ushbu egri chiziq birinchi qismdagi egri chiziq bilan bir xil boshlang'ich va tugash nuqtasiga ega bo'lishiga qaramay, chiziqli integrali bir xil qiymatga ega bo'lishini kutish uchun hech qanday sabab yo'q. Shuning uchun biz ishni ko'rib chiqishimiz va chiziqli integraldan nimani olishimizni ko'rishimiz kerak.
Biz egri chiziqni parametrlashimiz kerak, shuning uchun bu haqda g'amxo'rlik qilaylik.
\(C:\,\,\vec r\left( t \right) = \left\langle {6,t} \right\rangle \hspace{0.25in}\,\,\,\,\,\,\, - 3 \le t \le 3\)2-qadam
Endi biz faqat chiziq integralini hisoblashimiz kerak.
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\vec r}} & = \int_{{ - 3}}^{3}{{\left\langle {6\left( 6 \right) - 2\left( t \right),{{\left( 6 \right)}^2}} \right\rangle \centerdot \left\langle {0,1} \right\rangle \,dt}}\\ & = \int_{{ - 3}}^{3}{{36\,dt}} = \left. {\left( {36t} \right)} \right|_{ - 3}^3 = \require{bbox} \bbox[2pt,border:1px solid black]{{216}}\end{align*}\]Shunday qilib, ushbu qismning boshida ta'kidlanganidek, egri chiziqning bir xil boshlang'ich va tugash nuqtalariga qaramay, chiziqli integralning qiymati birinchi qismdagi chiziqli integralning qiymati bilan bir xil emas edi. E'tibor bering, bir xil boshlang'ich va tugash nuqtalari bo'lgan ikkita chiziqli integral bir xil qiymatga ega bo'lishi mumkin, ammo biz bunday bo'lishini kuta olmaymiz va shuning uchun ishni bajarishimiz kerak.