15.8-bo'lim: O'zgaruvchilarning o'zgarishi
1-3-masalalar uchun har bir transformatsiyaning Yakobianini hisoblang.
- \(x = 4u - 3{v^2}\hspace{0.25in}y = {u^2} - 6v\) Yechim
- \(x = {u^2}{v^3}\hspace{0.25in}y = 4 - 2\sqrt u \) Yechim
- \(\displaystyle x = \frac{v}{u}\hspace{0.25in}\hspace{0.25in}y = {u^2} - 4{v^2}\) Yechim
- Agar \(R\) \(\displaystyle \frac{{{x^2}}}{4} + \frac{{{y^2}}}{{36}} = 1\) ichidagi mintaqa bo'lsa, mintaqani aniqlang, biz \(x = 2u\), \(y = 6v\)ni \(R\) ga o'zgartiramiz. Yechim
- Agar \(R\) \(\left( {1,0} \right)\), \(\left( {4,3} \right)\), \(\left( {1,6} \right)\) va \(\left( { - 2,3} \right)\) cho'qqilari bilan parallelogramm bo'lsa, mintaqani aniqlaydi, biz \(\displaystyle x = \frac{1}{2}\left( {v - u} \right)\), \(\displaystyle y = \frac{1}{2}\left( {v + u} \right)\)ni \(R\) ga o'zgartiramiz. Yechim
- Agar \(R\) \(xy = 1\), \(xy = 3\), \(y = 2\) va \(y = 6\) bilan chegaralangan mintaqa bo'lsa, biz \(\displaystyle x = \frac{v}{{6u}}\), \(y = 2u\)ni \(R\) ga o'zgartiramiz. Yechim
- \(\displaystyle \iint\limits_{R}{{x{y^3}\,dA}}\) ni baholang, bunda \(R\) \(xy = 1\), \(xy = 3\), \(y = 2\) va \(y = 6\) bilan chegaralangan mintaqadir, \(\displaystyle x = \frac{v}{{6u}}\), \(y = 2u\) transformatsiyasi yordamida. Yechim
- \(\displaystyle \iint\limits_{R}{{6x - 3y\,dA}}\)ni baholang, bunda \(R\) \(\left( {2,0} \right)\), \(\left( {5,3} \right)\), \(\left( {6,7} \right)\) va \(\left( {3,4} \right)\) cho‘qqilari bilan parallelogramma bo‘lib, \(\displaystyle x = \frac{1}{3}\left( {v - u} \right)\), \(\displaystyle y = \frac{1}{3}\left( {4v - u} \right)\)ni \(R\) ga o‘zgartirish orqali. Yechim
- \(\displaystyle x = \frac{1}{2}\left( {u - v} \right)\), \(\displaystyle y = \frac{1}{4}\left( {3u + v + 12} \right)\)ni \(R\) ga o‘zgartirish yordamida \(R\) uchburchak bo‘lgan \(\left( {0,3} \right)\), \(\left( {4,1} \right)\) va \(\left( {2,6} \right)\) uchburchakni baholang. Yechim
- 8-masalada qo‘llanilgan transformatsiyani chiqaring. Yechim
- \(\left( {1,0} \right)\), \(\left( {6,0} \right)\) va \(\left( {3,8} \right)\) uchli uchburchakni \(uv\) tizimining boshida yuzaga keladigan to'g'ri burchakli to'g'ri burchakli uchburchakka aylantiradigan transformatsiyani oling. Yechim