I keep reading that a smooth curve, say r(t) = (x(t),y(t)), is a curve such that r'(t) is continuous and r'(t) \neq 0.
I understand the "r'(t) is continuous" bit - so there are no "kinks" or anything in the curve (hence, smooth), but I don't understand why we require r'(t) \neq 0. Would anyone be able to explain this part to me please?
There's no such restriction. Perhaps you misread or misunderstood?
On page 1070 of Stewart's Calculus (6th Ed, section on line integrals) for instance: "Start with a plane curve given by r(t) = x(t)i + y(t)j and assume that this curve is smooth (ie, r' is continuous and r'(t) \neq 0)"
Oh I'm sorry, I see we're talking about a parametric curve. I'm not entirely sure about that, but my intuition says that if r'(t)=0 and we interpret t as time, then the curve "stops" at a particular point. I'm not sure why that is bad though.
I'll let someone else more familiar with parametric curve smoothness answer. (Lipssealed)
Yep, I just can't figure out why that is "bad" :)
Thanks for the reply though!