I was reading through Paul's Online Math Notes and I bumped into the following definition for a smooth curve here:
A smooth curve is any curve for which $\dot{\vec{r}}(t)$ is continuous and $\dot{\vec{r}}(t)\neq 0$ for any $t$ except possibly at the endpoints.
This looks to me very different from the usual definition of a smooth curve, i.e. the function is "sufficiently" differentiable and continuous for our purposes. (For example check out Walfram Alpha, Wikipedia and a previous question on MSE.
Are these definitions equivalent then? Is it maybe a mistake from Paul's notes?
I was thinking maybe he is referring to a different object, although I am pretty sure he is introducing smooth curves in that section of Calculus 2 in order to be able to talk about line integrals and in general the Integral Theorems of vector calculus (Stoke's, Green's, Divergence, etc) further on, so this should be the same object as in the links above.
I tried to think about how they could be equivalent, however I could not see why a smooth curve should have non-vanishing derivative except at the end points, I am pretty sure I saw loads of exercises where the curve was smooth and it certainly attained a maximum or a minimum not in the boundaries and therefore the derivative was zero there..
Can you help me clarify this?