The role of the boxed condition $z'(t)\neq 0$ is to avoid going back, isn't it?
2026-02-27 21:52:49.1772229169
Is the role of the boxed condition $z'(t)\neq 0$ to avoid going back?
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No. It is so that the velocity is never $0$, which implies that we can parametrize the curve by the arc length. It also implies that $z\bigl([a,b]\bigr)$ has no “corners”, which corresponds to the idea of a smooth curve.