Any reparametrization of a regular curve is regular

Question

In A.Pressley's book, there is a proposition "Any reparametrization of a regular curve is regular".

In its proof, the author used the Chain Rule to the equation (Φ ο Ψ)(t) = t, where Φ is the reparametriztion map and Ψ its inverse,
and he concluded the differential of Φ is not zero.

But, I doubt the proof because to apply the Chain Rule, the two functions should be check whether they are differentiable or not.
For example, if we take the reparametriztion curve β(t)=(cost^3, sont^3) of γ(t)=(cost, sint), t ∈ (-1, 1), the β is singular at t=0. But γ is obiously regular.

I ask you that is Pressley's proof correct?????

Answer 1

A re-parametrization is by definition a smooth map with nowhere vanishing derivative. A regular curve is by definition a smooth curve with nowhere vanishing derivative. Therefore, a re-parametrization of a regular curve is regular by the chain rule.

In particular, your example isn't a counterexample because the derivative of $\beta$ vanishes at $0$ (as you note) and so $\beta$ isn't a re-parametrization (by definition). (Moreover, the range of $\beta$ is $\mathbb{R}^2$; the domain and codomain of a re-parametrization are required to be open intervals so this is another reason $\beta$ isn't a re-parametrization.)

I also recommend you to see my answer here for more details:

How (and why) would I reparameterize a curve in terms of arclength?

I hope this helps!