12.12-bo'lim : Silindrsimon koordinatalar
3. Dekart koordinatalarida yozilgan quyidagi tenglamani Silindr koordinatalaridagi tenglamaga aylantiring.
\[{x^3} + 2{x^2} - 6z = 4 - 2{y^2}\]Yechim
Bu erda, albatta, ko'p narsa qilish mumkin emas. Biz qilishimiz kerak bo'lgan yagona narsa quyidagi \(x\) va \(y\) qutbli konvertatsiya formulalarini iloji bo'lsa (va agar) tenglamaga kiritishdir.
\[x = r\cos \theta \hspace{0.5in}y = r\sin \theta \hspace{0.5in}{r^2} = {x^2} + {y^2}\]2-qadam
Biroq, avval biz biroz qayta yozamiz.
\[{x^3} + 2{x^2} + 2{y^2} - 6z = 4\hspace{0.5in} \to \hspace{0.5in}{x^3} + 2\left( {{x^2} + {y^2}} \right) - 6z = 4\]3-qadam
Endi tenglamani silindrsimon koordinatalarga aylantirish uchun 1-bosqichdagi formulalardan foydalanamiz.
\[{\left( {r\cos \theta } \right)^3} + 2\left( {{r^2}} \right) - 6z = 4\hspace{0.5in} \to \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{{r^3}{{\cos }^3}\theta + 2{r^2} - 6z = 4}}\]