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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?

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  • $\begingroup$ When it says $\dot{r}(t)$ is continuous, what exactly does it mean? $\endgroup$ – user160738 Dec 23 '16 at 13:13
  • $\begingroup$ I guess the standard $\epsilon-\delta$ definition $\endgroup$ – Euler_Salter Dec 23 '16 at 13:23
  • $\begingroup$ Doesn't it give some kind of examples of this "smooth" function? So in those example he varifies $\dot{r}(t):\mathbb{R}\to\mathbb{R}^n$ is continous directly using $\epsilon-\delta$? $\endgroup$ – user160738 Dec 23 '16 at 13:26
  • $\begingroup$ "A helix is a smooth curve, for example." this is one example he gives. No I don't think he gives any $\epsilon-\delta$ proof in this section(as it would rather be Calculus 1 stuff). However it normally deals with standard definitions $\endgroup$ – Euler_Salter Dec 23 '16 at 13:28
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    $\begingroup$ The definitions aren't equivalent. Requiring that the derivative vanishes nowhere ensures a) that the curve has a tangent everywhere [well, locally, the curve may intersect itself, then you have no tangent as in tangent space of a submanifold at (transversal) intersection points, but if you consider the curve restricted to a small interval around one of the pertinent parameter values, it has], and b) you can reparametrise the curve by arclength (and still get a smooth curve). For some applications these things are good to have, for others it's not important. $\endgroup$ – Daniel Fischer Dec 23 '16 at 13:35

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