Let $K$ be the surface of an infinite cone with circular cross section, vertex at the origin and axis lying along the positive $z$-axis. If the angle between the $z$-axis and the surface of the cone is $α$, find expressions for $K$ in terms of Cartesian, spherical and cylindrical polar coordinates.
Solution:
In terms of spherical polars, $K = \{(ρ,ϕ,θ) : 0 \leq ρ < ∞,ϕ = α,0 \leq θ < 2π\}$. By simple trigonometry, $tanα = r/z = (\sqrt{x^2 + y^2})/z$. In terms of Cartesian coodinates then, $K = \{(x,y,z) : x^2 +y^2 = z \tan α\}$ and, in terms of cylindrical polars, $K = \{(r,θ,z) : r = z tanα,0 \leq θ < 2π\}$.
For the Cartesian, shouldn't it say $(z \tan α)^2$ instead, since it is the radius?
