12.13-bo'lim : Sferik koordinatalar
4. Dekart koordinatalarida yozilgan quyidagi tenglamani Sferik koordinatadagi tenglamaga aylantiring.
\[{x^2} + {y^2} = 4x + z - 2\]Yechim
Bu erda qilishimiz kerak bo'lgan yagona narsa quyidagi konvertatsiya formulalarini tenglamaga kiritish va biroz soddalashtirishdir.
\[x = \rho \sin \varphi \cos \theta \hspace{0.5in}y = \rho \sin \varphi \sin \theta \hspace{0.5in}z = \rho \cos \varphi \]2-qadam
Konvertatsiya formulasini ulang,
\[{\left( {\rho \sin \varphi \cos \theta } \right)^2} + {\left( {\rho \sin \varphi \sin \theta } \right)^2} = 4\left( {\rho \sin \varphi \cos \theta } \right) + \rho \cos \varphi - 2\]Birinchi ikkita atamani quyidagicha soddalashtirish mumkin.
\[\begin{align*}{\rho ^2}{\sin ^2}\varphi {\cos ^2}\theta + {\rho ^2}{\sin ^2}\varphi {\sin ^2}\theta & = 4\rho \sin \varphi \cos \theta + \rho \cos \varphi - 2\\ {\rho ^2}{\sin ^2}\varphi \left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) & = 4\rho \sin \varphi \cos \theta + \rho \cos \varphi - 2\\ {\rho ^2}{\sin ^2}\varphi & = 4\rho \sin \varphi \cos \theta + \rho \cos \varphi - 2\end{align*}\]