Let $\varphi: (-\frac{\pi}{2}, \frac{\pi}{2}) \times \mathbb{R} \to \mathbb{R}^{3}$ with $\varphi(t,s)=(\cos\ t-s\ \sin\ t, \sin\ t + s\ \cos\ t, s)$. I have to find the maximum open set $U$ for which $\varphi$ is "ruled surface". Unfortunately I don't know the exact english term so I'll try to explain, I want that:
- $\varphi \in \mathscr{C}^{\infty}(U)$, and it's true $\forall\ U$
- $\varphi$ is injective, and it's true only if $s=\pm1$
- $\mathrm{rank}(J_{\varphi}(x, y))$ is maximum (i.e 2) $\forall\ (x,y) \in U$, and it's true $\forall U$
This is the definition of "surface" (the literal translation of what I mean would be "simple sheet of surface"). So I can say that $U$ is a "surface" for $U=(-\frac{\pi}{2}, \frac{\pi}{2})\times\left\{\pm1\right\}$ and it's the maximum open set.
Now the problem is: how can I check that this "surface" is ruled too? I know the definition but I have no idea how to apply it.
Hint: In this example, it may help to write $$ \varphi(t, s) = (\cos t, \sin t, 0) + s(-\sin t, \cos t, 1). $$