Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 1 / 2A (Algebra and Geometry)

In the standard spherical coordinate system, $\theta$ is measured from the positive $z$ axis. The radial unit vector in spherical coordinates isThe coordinate forms of the elliptic paraboloid are given in the labelled parts below.

For the other surface, $r^2\cos 2\theta =r^2({\cos }^2\theta -{\sin }^2\theta )$ converts toFor $a>0$, this is a two-sheeted hyperboloid, with two disconnected sheets starting at $z=\pm \sqrt{a}$ and opening along the $z$ axis. For $a<0$, it is a one-sheet hyperboloid:Its waist is the circle $z=0$, $x^2+y^2=|a|$. Both are surfaces of revolution about the $z$ axis and approach the cone $z^2=x^2+y^2$ far from the origin.

The figure shows $a=1$ and $a=-1$; scaling all coordinates by $\sqrt{|a|}$ gives the other magnitudes.

In cylindrical coordinates, $x^2+y^2={\rho }^2$, so the elliptic paraboloid is

In spherical coordinates, $z=r\cos \theta$ and $x^2+y^2=r^2{\sin }^2\theta$, so the elliptic paraboloid has equationAway from the origin this is $r=\cos \theta /{\sin }^2\theta$, with $0<\theta <\pi /2$. The undivided equation also includes the origin, where angular coordinates are not unique.