TITLE: What does smooth curve mean?
QUESTION [17 upvotes]: In this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?

REPLY [4 votes]: I stumbled upon this old question and I'd like to add something: there is a difference of perspective on smoothness depending whether you look at the geometric object or its parametrization.
Look at the standard example: the real cusp. It is a curve in the real plane parametrized $f:t\to (t^2,t^3)$. Of course, the mapping $f$ is smooth (of any order), and the graph of $f$ is a smooth manifold in $\mathbb{R}^3$, but its  image is singular: it is the zero set $x^3=y^2$. It is "worse than a corner"!
So you need to be always clear what you want: do you need only differentiability of the parametrization or do you want the image to be a differentiable manifold (typically in such a case you would assume that the derivative of $f$ does not vanish).