16.4-bo'lim : Vektor maydonlarining chiziqli integrallari
1. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {3x - 6y} \right)\vec j\) va \(C\) \(\left( {3,7} \right)\) dan \(\left( {0,12} \right)\) gacha bo'lgan chiziq segmentidir.
Yechim
Bu erda ko'rsatilgan muammo bayonotida ko'rsatilgan yo'nalish bilan \(C\) ning tezkor eskizi.
2-qadam
Keyinchalik, egri chiziqni parametrlashimiz kerak.
\[\vec r\left( t \right) = \left( {1 - t} \right)\left\langle {3,7} \right\rangle + t\left\langle {0,12} \right\rangle = \left\langle {3 - 3t,7 + 5t} \right\rangle \,\,\,\,\,\,\,\,0 \le t \le 1\]3-qadam
Ushbu chiziqli integralni baholash uchun bizga vektor maydonining nuqta mahsuloti (egri chiziq bo'ylab baholanadi) va parametrlashning hosilasi kerak bo'ladi.
Mana egri chiziq bo'ylab baholangan vektor maydoni (i.e. vektor maydoniga parametrlashdan \(x\) va \(y\) ni ulang).
\[\vec F\left( {\vec r\left( t \right)} \right) = {\left( {7 + 5t} \right)^2}\,\vec i + \left( {3\left( {3 - 3t} \right) - 6\left( {7 + 5t} \right)} \right)\vec j = {\left( {7 + 5t} \right)^2}\,\vec i + \left( { - 33 - 39t} \right)\vec j\]Parametrlashning hosilasi,
\[\vec r'\left( t \right) = \left\langle { - 3,5} \right\rangle \]Nihoyat, vektor maydonining nuqta mahsuloti va parametrlashning hosilasi.
\[\vec F\left( {\vec r\left( t \right)} \right)\centerdot \vec r'\left( t \right) = - 3{\left( {7 + 5t} \right)^2} - 5\left( {33 + 39t} \right)\]4-qadam
Endi biz qilishimiz kerak bo'lgan narsa - bu integralni baholash.
\[\begin{align*}\int\limits_{C}{{\vec F\centerdot d\vec r}} & = \int_{0}^{1}{{ - 3{{\left( {7 + 5t} \right)}^2} - 5\left( {33 + 39t} \right)\,dt}}\\ & = \left. {\left[ { - \frac{1}{5}{{\left( {7 + 5t} \right)}^3} - 165t - \frac{{195}}{2}{t^2}} \right]} \right|_0^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{1079}}{2}}}\end{align*}\]