LineIntegralsVectorFields - Amaliy mashqlar | MathModel.uz

16.4-bo'lim: Vektor maydonlarining chiziqli integrallari

$ \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
$ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang, bunda $\vec F\left( {x,y} \right) = \left( {x + y} \right)\,\vec i + \left( {1 - x} \right)\vec j$ va $C$ $\displaystyle\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1$ ning 4-banddagi qismidir.^th soat miliga teskari aylanish bilan kvadrant. Yechim
$ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang, bu erda $\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {{x^2} - 4} \right)\vec j$ va $C$ $y = {\left( {x - 1} \right)^2}$ ning $x = 0$ dan $x = 3$ gacha bo'lgan qismidir. Yechim
$ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ni baholang, bu erda $\vec F\left( {x,y,z} \right) = {{\bf{e}}^{2x}}\,\vec i + z\left( {y + 1} \right)\vec j + {z^3}\,\vec k$ va $C$ $0 \le t \le 2$ uchun $\vec r\left( t \right) = {t^3}\,\vec i + \left( {1 - 3t} \right)\vec j + {{\bf{e}}^t}\,\vec k$ tomonidan berilgan. Yechim
$ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang, bunda $\vec F\left( {x,y} \right) = 3y\,\vec i + \left( {{x^2} - y} \right)\vec j$ va $C$ aylananing yuqori yarmi radius 1 ning boshiga markazlashgan, soat miliga teskari aylanuvchi va $y = {x^2} - 1$ ning $x = - 1$ dan $x = 1$ gacha bo‘lgan qismi. Quyidagi eskizga qarang.
$The curve $C$. It is a closed curve made up of the upper half of the circle of radius 1 centered at the origin together with the portion of the parabola $y = {x^2} - 1$ from $x = - 1$ to $x = 1$, which joins the two ends of the semicircle at $\left( { - 1,0} \right)$ and $\left( {1,0} \right)$. The result is a rounded, slightly pear shaped closed curve and the arrows show that it is traveled in the counter clockwise direction.$
Yechim
$ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang, bunda $\vec F\left( {x,y} \right) = xy\,\vec i + \left( {1 + 3y} \right)\vec j$ va $C$ $\left( {0, - 4} \right)$ dan $\left( { - 2, - 4} \right)$ gacha bo‘lgan chiziq segmenti, so‘ngra $y = - {x^2}$ ning $x = - 2$ dan $x = 2$ ga bo‘lgan qismi, bu o‘z navbatida MABLOCK dan $\left( {2, - 4} \right)$4 ga bo‘lgan segment. Quyidagi eskizga qarang.
$The curve $C$. The arrows show that it starts at $\left( {0, - 4} \right)$ and runs left along a horizontal line to $\left( { - 2, - 4} \right)$. From there it follows the parabola $y = - {x^2}$, up to the origin and back down to $\left( {2, - 4} \right)$, and then runs up and to the right along a straight line to $\left( {5,1} \right)$.$
Yechim
Quyidagi egri chiziqlarning har biri uchun $\vec F\left( {x,y} \right) = \left( {6x - 2y} \right)\,\vec i + {x^2}\vec j$ ni $ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang.
1. $C$ - $\left( {6, - 3} \right)$ 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.
2. $C$ - $\left( {6, - 3} \right)$ dan $\left( {6,3} \right)$ gacha bo'lgan chiziq segmenti.
Yechim
Quyidagi egri chiziqlarning har biri uchun $\vec F\left( {x,y} \right) = 3\,\vec i + \left( {xy - 2x} \right)\vec j$ ni $ \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$ ni baholang.
1. $C$ - soat miliga teskari aylanish bilan 4 radiusning boshiga markazlashgan doiraning yuqori yarmi.
2. $C$ - soat yo'nalishi bo'yicha aylanish bilan 4 radiusning boshida joylashgan aylananing yuqori yarmi.
Yechim