Chapter 12 - Vector-Valued Functions - 12.1 Introduction To Vector-Valued Functions - Exercises Set 12.1 - Page 846: 39

See proof

Let us put $x = t\sin t,$ $y = t\cos t,$ $z = t^2.$ Then we get $x^2 + y^2 = (t\sin t)^2 + (t\cos t)^2 = t^2(\sin^2 t + \cos^2 t) = t^2.$ Thus, any point of the parametric curve lies on the given paraboloid. Therefore, the entire curve lies on the paraboloid.