I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$.
If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is regular, $\gamma(t)\neq 0$, and so $S$ is differentiable with $S'(t)=|\gamma'(t)|$.
My question is, why do we need $\gamma$ to be regular? My understanding is, there are smooth curves that aren't regular -- e.g. $f(t)=(t^3,t^2,0)$-- and to use the fundamental theorem of calculus here, we only need that $\gamma$ is continuous. So why do we need the extra condition that $\gamma\neq 0$? Why wouldn't just $C^1$ curves do?
Also, I may be to early in my studies of elementary differential geometry, but I'm having a hard time seeing why smooth regular curves are so important as opposed to just smooth ones. If regularity really is needed in the above scenario, I guess that's one reason, but are there others?
The meaning of regularity here is
$$\gamma(t) \neq 0$$
You may ask, what will happen if $\gamma(t)= 0$ ? It refers to re-parametrization concept. What if I re-parametrize $\gamma(t)$ in a way that new curve goes back and forward sometimes on the same curve?
$\gamma(t) \neq 0$ guarantees that the curve never turns back and overlays on itself. And re-parametrization has only the same direction or opposite direction but not both. So the length is not calculated two times.