Let $ c : I \rightarrow \mathbb{R}^3 $ be a space curve with $c(t) = (f(t),g(t),0)$.
if we rotate the image of $c$ around the x-axis, we get the set $R$.
First task: Find a parameterisation of $R$. When is this parameterisation regular?
Second task: Consider $c(t) = (\cos(t), 2 + \sin(t), 0)$ and make a sketch of the rotation around the x-axis.
So the second task was no problem. It is a torus. So I only need help with the first task. My assumption is that the parameterisation is $X :(0,2\cdot \pi] \times I \rightarrow \mathbb{R}^3$ with $X(t,a) =$ $(f(t),g(t)\cos(a),g(t)\sin(a))$.Why? : Because of the rotation I think that we need polar coordinates $ r\cos(a)$ and $r\sin(a)$. But to be honest: I don't know how I can explain the parameterisation above. It was just an ''idea'' . So if the parameterisation is right, I need help with the explanation. And if my parameterisation is false: Can you tell me where's my mistake?
For the second question of first task : $D_t(t,a) = ( f'(t), g'(t)\cos(a),g'(t)\sin(a)) $ and $D_a(t,a) = ( 0, -g(t)\sin(a),g(t)\cos(a)) $ has to be linearly independent. My thoughts here: first of all we need that $f,g$ are differentiable. Moreover $g(t) \neq 0 $. And if $f'(t) = 0$ we need that $ g'(t) \neq 0 $. And if $g'(t) = 0$ we need that $ f'(t) \neq 0 $, right? Do you see more problems?