I have some the smooth curve $\gamma(t) = (0, a(t), b(t))$ around the $z$ axis. I have to prove that $\phi(t, u) = (a(t)\sin(u), b(t)\cos(u), g(t)$ is a $t \in [x, y], u \in [0, 2\pi]$ is a parameterization of the surface, defined by the curve.
My approach was to first proof that this is indeed a parameterization via showing that $\phi$ is $c^1$ and, then showing that the partial derivatives are linearly independent. Done.
Now, I do not know how to continue, so that I show that $\phi$ is a parameterization of $\gamma$.
My idea was to find another parameterization and compute the areas with the surface area with the 2 parameterizations. However, I cannot "guess" another one. And I cannot know if both are actually wrong.
I believe that there must be a straightforward approach, which I fail to find in the textbooks.