16.4-bo'lim : Vektor maydonlarining chiziqli integrallari

1. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bunda \(\vec F\left( {x,y} \right) = 2{x^2}\,\vec i + \left( {{y^2} - 1} \right)\vec j\) va \(C\) \(\displaystyle \frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{9} = 1\) ning 1-banddagi qismidir.^st, 4^th va 3^rd soat yo'nalishi bo'yicha yo'naltirilgan kvadrant.
2. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y} \right) = xy\,\vec i + \left( {4x - 2y} \right)\vec j\) va \(C\) \(\left( {4, - 3} \right)\) dan \(\left( {7,0} \right)\) gacha bo'lgan chiziq segmentidir.
3. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y} \right) = \left( {{x^3} - y} \right)\,\vec i + \left( {{x^2} + 7x} \right)\vec j\) va \(C\) \(y = {x^3} + 2\) ning \(x = - 1\) dan \(x = 2\) gacha bo'lgan qismidir.
4. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\)ni baholang, bu erda \(\vec F\left( {x,y} \right) = xy\,\vec i + \left( {1 + {x^2}} \right)\vec j\) va \(C\) \( - 2 \le t \le 0\) uchun \(\vec r\left( t \right) = {{\bf{e}}^{6t}}\,\vec i + \left( {4 - {{\bf{e}}^{2t}}} \right)\vec j\) tomonidan berilgan.
5. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\)ni baholang, bunda \(\vec F\left( {x,y,z} \right) = \left( {3x - 3y} \right)\,\vec i + \left( {{y^3} - 10} \right)\vec j + y\,z\,\vec k\) va \(C\) \(\left( {1,4, - 2} \right)\)dan \(\left( {3,4,6} \right)\)gacha bo'lgan chiziq segmentidir.
6. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y,z} \right) = \left( {x + z} \right)\,\vec i + {y^3}\vec j + \left( {1 - x} \right)\,\vec k\) va \(C\) \( - \pi \le t \le 2\pi \) uchun \(\vec r\left( t \right) = \cos \left( {2t} \right)\,\vec i - t\,\vec j + \sin \left( {2t} \right)\,\vec k\) tomonidan berilgan \(y\) o'qidagi spiralning qismidir.
7. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y} \right) = {x^2}\,\vec i + \left( {{y^2} - x} \right)\vec j\) va \(C\) \(\left( {2,4} \right)\) dan \(\left( {0,4} \right)\) gacha bo'lgan chiziq segmenti va keyin \(\left( {0,4} \right)\) dan \(\left( {3, - 1} \right)\) gacha bo'lgan chiziq segmenti.
8. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bu erda \(\vec F\left( {x,y} \right) = xy\,\vec i - 3\,\vec j\) va \(C\) \(\displaystyle {x^2} + \frac{{{y^2}}}{16} = 1\) ning 2 qismidagi qismidir.^nd soat yo'nalishi bo'yicha aylantirilgan kvadrant, keyin \(\left( {0,4} \right)\) dan \(\left( {4, - 2} \right)\) gacha bo'lgan chiziq segmenti. Quyidagi eskizga qarang.
   
9. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang, bunda \(\vec F\left( {x,y} \right) = x{y^2}\,\vec i + \left( {2y + 3x} \right)\vec j\) va \(C\) \(x = {y^2} - 1\) ning \(y = - 2\) dan \(y = 2\) gacha bo‘lgan qismi, so‘ngra \(\left( {3,2} \right)\) dan \(\left( {0,0} \right)\) gacha bo‘lgan chiziq segmenti, o‘z navbatida MABLOCK5 dan MABLOCK52 ga bo‘lgan segment. Quyidagi eskizga qarang.
   
10. Quyidagi egri chiziqlarning har biri uchun \(\vec F\left( {x,y} \right) = \left( {1 - {y^2}} \right)\,\vec i - x\,\vec j\) ni \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang.
    1. \(C\) - soat miliga teskari aylanish bilan radius 1ning boshida joylashgan aylananing yuqori yarmi.
    2. \(C\) \(\displaystyle {x^2} + \frac{{{y^2}}}{{36}}=1\) ning pastki yarmi soat yo'nalishi bo'yicha aylanadi.
11. Quyidagi egri chiziqlarning har biri uchun \(\vec F\left( {x,y} \right) = \left( {{x^2} + y + 2} \right)\,\vec i + x\,y\,\vec j\) ni \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang.
    1. \(C\) - \(y = {x^2} - 2\) ning \(x = - 3\) dan \(x = 3\) gacha bo'lgan qismi.
    2. \(C\) - \(\left( { - 3,5} \right)\) dan \(\left( {3,5} \right)\) gacha bo'lgan chiziq segmenti.
12. Quyidagi egri chiziqlarning har biri uchun \(\vec F\left( {x,y} \right) = {y^2}\,\vec i + \left( {1 - 3x} \right)\,\vec j\) ni \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang.
    1. \(C\) - \(\left( {1,4} \right)\) dan \(\left( { - 2,3} \right)\) gacha bo'lgan chiziq segmenti.
    2. \(C\) - \(\left( { - 2,3} \right)\) dan \(\left( {1,4} \right)\) gacha bo'lgan chiziq segmenti.
13. Quyidagi egri chiziqlarning har biri uchun \(\vec F\left( {x,y} \right) = - 2x\,\vec i + \left( {x + 2y} \right)\vec j\) ni \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholang.
    1. \(C\) - \(\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{4} = 1\) ning 1-chi qismi^st soat miliga teskari aylanish bilan kvadrant.
    2. \(C\) - \(\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{4} = 1\) ning 1-dagi qismi^st soat yo'nalishi bo'yicha aylanish bilan kvadrant.