Answer to Question #106868 in Analytic Geometry for khushi

Question

a) Express the following surfaces in spherical coordinates
i) xz = 3
ii)x^2+y^2-z^2=1

Answer 1

$1)\ xz=3$

$x=r\sin{\theta}\cos{\phi} \\ z=r\cos{\theta}\\ xz= r\sin{\theta}\cos{\phi}\ r\cos{\theta}=r^2\sin{\theta}\cos{\theta}\cos{\phi}=\frac{1}{2}r^2\sin{2\theta}\cos{\phi}=3\\ r^2\sin{2\theta}\cos{\phi}=6, \ r>0,\ \theta\in[0,\pi], \ \phi\in[0,2\pi]$

$2)\ x^2+y^2-z^2=1\\ x=r\sin{\theta}\cos{\phi} \\ y=r\sin{\theta}\sin{\phi} \\ z=r\cos{\theta}\\ x^2+y^2-z^2=r^2\sin^2{\theta}\cos^2{\phi}+r^2\sin^2{\theta}\sin^2{\phi}-r^2\cos^2{\theta}=\\ =r^2\sin^2{\theta}(\cos^2{\phi}+\sin^2{\phi})-r^2\cos^2{\theta}=r^2\sin^2{\theta}-r^2\cos^2{\theta}=\\ =r^2(\sin^2{\theta}-\cos^2{\theta})=-r^2\cos{2\theta}=1\\ r^2\cos{2\theta}=-1,\ r>0,\ \theta\in[0,\pi], \ \phi\in[0,2\pi]$