15.8-bo'lim : O'zgaruvchilarni almashtirish
1-4-masalalar uchun har bir transformatsiyaning yakobiyini hisoblang.
- \(x = 4{u^2}v\hspace{0.25in}y = 6v - 7u\)
- \(x = \sqrt u \hspace{0.25in}\hspace{0.25in}y = 10u + v\)
- \(\displaystyle x = {v^3}u\hspace{0.25in}\hspace{0.25in}y = \frac{{{u^2}}}{v}\)
- \(x = {{\bf{e}}^u}\cos v\hspace{0.25in}y = {{\bf{e}}^u}\sin v\)
- Agar \(R\) \(\displaystyle \frac{{{x^2}}}{{25}} + 49{y^2} = 1\) ichidagi mintaqa bo'lsa, mintaqani aniqlang, biz \(x = 5u\), \(\displaystyle y = \frac{1}{7}v\)ni \(R\) ga o'zgartiramiz.
- Agar \(R\) uchburchak bo'lsa, \(\left( {2,0} \right)\), \(\left( {6,4} \right)\) va \(\left( {1,4} \right)\) mintaqani aniqlasa, biz \(\displaystyle x = \frac{1}{5}\left( {u - v} \right)\), \(\displaystyle y = \frac{1}{5}\left( {u + 4v} \right)\)ni \(R\) ga o'zgartiramiz.
- Agar \(R\) \(\left( {0,0} \right)\), \(\left( {4,2} \right)\), \(\left( {0,4} \right)\) va \(\left( { - 4,2} \right)\) cho'qqilari bilan parallelogramm bo'lsa, mintaqani aniqlaydi, biz \(x = u - v\), \(\displaystyle y = \frac{1}{2}\left( {u + v} \right)\)ni \(R\) ga o'zgartiramiz.
- Agar \(R\) \(0 \le x \le 3\) tomonidan belgilangan kvadrat bo'lsa va \(0 \le y \le 3\) mintaqani aniqlasa, biz \(x = 3u\), \(y = v\left( {2 + {u^2}} \right)\)ni \(R\) ga o'zgartiramiz.
- Agar \(R\) \(\left( {1,1} \right)\), \(\left( {5,3} \right)\), \(\left( {8,8} \right)\) va \(\left( {4,6} \right)\) cho'qqilari bilan parallelogramm bo'lsa, mintaqani aniqlaydi, biz \(\displaystyle x = \frac{6}{7}\left( {u - v} \right)\), \(\displaystyle y = \frac{1}{7}\left( {10u - 3v} \right)\)ni \(R\) ga o'zgartiramiz.
- Agar \(R\) \(xy = 4\), \(xy = 10\), \(y = x\) va \(y = 6x\) bilan chegaralangan mintaqa bo'lsa, biz \(\displaystyle x = 2\sqrt {\frac{u}{v}} \), \(y = 4\sqrt {uv} \)ni \(R\) ga o'zgartirishni qo'llaymiz.
- \(\displaystyle \iint\limits_{R}{{{x^2}{y^4}\,dA}}\) ni baholang, bunda \(R\) \(xy = 4\), \(xy = 10\), \(y = x\) va \(y = 6x\) bilan chegaralangan mintaqadir, \(\displaystyle x = 2\sqrt {\frac{u}{v}} \), \(y = 4\sqrt {uv} \) transformatsiyasi yordamida.
- \(\displaystyle \iint\limits_{R}{{1 - y\,dA}}\) ni baholang, bunda \(R\) uchburchak \(\left( {0,4} \right)\), \(\left( {1,1} \right)\) va \(\left( {2,5} \right)\) bo‘lib, \(\displaystyle x = \frac{1}{7}\left( {7 + u - v} \right)\), \(\displaystyle y = \frac{1}{7}\left( {7 + 4u + 3v} \right)\) \(R\) ga o‘zgartiriladi.
- \(\displaystyle \iint\limits_{R}{{121x\,dA}}\) ni baholang, bu erda \(R\) \(\left( {0,0} \right)\), \(\left( {6,2} \right)\), \(\left( {7,6} \right)\) va \(\left( {1,4} \right)\) cho'qqilari bilan parallelogramma bo'lib, \(\displaystyle x = \frac{1}{{11}}\left( {v - 3u} \right)\), \(\displaystyle y = \frac{1}{{11}}\left( {4v - u} \right)\)ni \(R\) ga o'zgartiring.
- \(\displaystyle \iint\limits_{R}{{\frac{{15y}}{x}\,dA}}\) ni baholang, bunda \(R\) \(xy = 2\), \(xy = 6\), \(y = 4\) va \(y = 10\) bilan chegaralangan mintaqa bo‘lib, \(x = v\), \(\displaystyle y = \frac{{2u}}{{3v}}\) transformatsiyasi yordamida.
- \(\displaystyle x = \frac{1}{4}\left( {u - v} \right)\), \(\displaystyle y = \frac{1}{2}\left( {u + v} \right)\)ni \(R\) ga o‘zgartirish yordamida \(\displaystyle \iint\limits_{R}{{2y - 8x\,dA}}\)ni baholang.
- \(\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) ellipsini birlik doirasiga aylantiradigan transformatsiyani chiqaring.
- 12-masalada qo‘llanilgan transformatsiyani chiqaring.
- 13-masalada qo‘llanilgan transformatsiyani chiqaring.
- \(\left( {4,1} \right)\), \(\left( {7,4} \right)\), \(\left( {6,8} \right)\) va \(\left( {3,5} \right)\) cho'qqilari bilan parallelogrammani \(uv\) tizimida to'rtburchakga aylantiradigan transformatsiyani oling.
- \(\left( {4,1} \right)\), \(\left( {7,4} \right)\), \(\left( {6,8} \right)\) va \(\left( {3,5} \right)\) cho'qqilari bilan parallelogrammani \(uv\) tizimining boshida joylashgan bir burchagi bo'lgan to'rtburchakka aylantiradigan transformatsiyani chiqaring.