Identify the Following Parametric Surfaces

Question

1. $r(u,v) = ui+(u\cos v)j+(u\sin v)k$
2. $r(u,v) = u\ cos(v)i+u\ sin(v)j+u^2k$
3. $r(u,v) = ui+vj+(2u-3v)k$
4. $r(u,v) = vi+\cos vj+\sin vk$

My Guess:

1. Plane
2. Circular Cylinder
3. Cone
4. Circular Paraboloid

Can someone please check my guess, and correct me if I am wrong.

Answer 1

$1)$ $x^2 = y^2 + z^2$: A cone.

$2)$ $z = x^2+y^2$: A Circular Paraboloid.

$3)$ $z = 2x-3y$: A plane.

$4)$ $y^2+z^2 = 1$: A Circular Cylinder.