16.5-bo'lim : Chiziqli integrallar uchun asosiy teorema

1. \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\)ni baholang, bu erda \(f\left( {x,y} \right) = 5x - {y^2} + 10xy + 9\) va \(C\) \(\displaystyle \vec r\left( t \right) = \left\langle {\frac{{2t}}{{{t^2} + 1}},1 - 8t} \right\rangle \) tomonidan \( - 2 \le t \le 0\) bilan berilgan.
2. \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) ni baholang, bu erda \(\displaystyle f\left( {x,y,z} \right) = \frac{{3x - 8y}}{{z - 6}}\) va \(C\) \(\vec r\left( t \right) = 6t\,\vec i + 4\vec j + \left( {9 - {t^3}} \right)\vec k\) tomonidan \( - 1 \le t \le 3\) bilan berilgan.
3. \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) ni baholang, bu erda \(f\left( {x,y} \right) = 20y\cos \left( {x + 3} \right) - y{x^3}\) va \(C\) \(\displaystyle {\left( {x + 3} \right)^2} + \frac{{{{\left( {y - 1} \right)}^2}}}{{16}} = 1\) tomonidan soat yo'nalishi bo'yicha aylantirilgan ellipsning o'ng yarmidir.
4. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni hisoblang, bu erda \(\vec F = 2x\,\vec i + 4y\,\vec j\) va \(C\) soat miliga teskari aylanish bilan 5 radiusning boshiga markazlashgan doiradir. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) yo'ldan mustaqilmi? Agar \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) yo'ldan mustaqil yoki yo'qligini aniqlashning iloji bo'lmasa, nima uchun buni aniq tushuntiring.
5. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni hisoblang, bu erda \(\vec F = y\,\vec i + {x^2}\,\vec j\) va \(C\) soat miliga teskari aylanish bilan 5 radiusning boshida joylashgan doiradir. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) yo'ldan mustaqilmi? Agar \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) yo'ldan mustaqil yoki yo'qligini aniqlashning iloji bo'lmasa, nima uchun buni aniq tushuntiring.
6. \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) ni baholang, bu erda \(f\left( {x,y,z} \right) = z{x^2} + x{\left( {y - 2} \right)^2}\) va \(C\) \(\left( {1,2,0} \right)\) dan \(\left( { - 3,10,9} \right)\) gacha bo'lgan chiziq segmenti va keyin \(\left( { - 3,10,9} \right)\) dan \(\left( {6,0,2} \right)\) gacha bo'lgan chiziq segmenti.
7. \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) ni baholang, bunda \(f\left( {x,y} \right) = 4x + 3x{y^2} - \ln \left( {{x^2} + {y^2}} \right)\) va \(C\) \({x^2} + {y^2} = 1\) ning soat yo‘nalishi bo‘yicha aylantirilgan yuqori yarmi, keyin esa soat miliga teskari aylantirilgan \(\displaystyle {\left( {x - 1} \right)^2} + \frac{{{{\left( {y - 2} \right)}^2}}}{4} = 1\)ning o‘ng yarmi. Quyidagi eskizga qarang.