I am revising line integrals. Thomas's Calculus, 14th Edition, says the following:
As I understand it, a smooth curve is a curve that is defined by a smooth function, and a smooth function is defined as a function that has derivatives of all orders defined everywhere in its domain.
What I'm confused about is where $\mathbf{v} = \dfrac{ d\mathbf{r} }{ dt }$ never being zero came from? I don't see how $C$ being a smooth curve demands this?
I would greatly appreciate it if people could please take the time to clarify this.
EDIT:
Best Answer
If $v$ is zero at some point, then this means that the 'velocity' of the curve is $=0$ In such a point the image of the parametrization may have all kinds of singularities, which -- in a geometric context -- is not desired.
As a simple example you may consider the curve
$$c(t) = (t^5, |t|^{5/2})$$
which is (quite obviously) $C^1$, but which is a parametrization of the graph of $y= \sqrt{|x|}$, which has a cusp at $t=0$. Of course this kind of example can be given in any dimension.
With the help functions like $\exp(-\frac{1}{|x|^2})$ it's possible to create parametriziations of such curves which are actually smooth.
The requirement $v\neq 0$ prevents the curve from having geometric singularities, for this reason one calls a curve with this property regular.
Edit (in response to a question in a comment): if you want to define line integrals then it is not really necessary to assume that the curves are regular. With this assumption the definition is, however, much easier and straightforward. Just assuming that there is a smooth parametrization, is, for example, not sufficient.
The example I gave is still a 'nice' curve and would allow do define a line integral along the curve without much of a problem. One can write down more nasty ones so that the parametrization is still smooth, but the curve no longer has finite length or has other nasty features like fractal behavior. This does not automatically mean that the definitions of length and line integral don't make any sense anymore, it may just be more difficult to give a consistent defintion.
The case of corners is often the first generalization which is introduced as admissible, since that class of curves is often more convenient in applications.