while I was studying Vector Calculus with Apostol book, I read this part:
"In a completely rigorous treatment of Green’s theorem it would be necessary to describe analytically what it means to traverse a closed curve in the “counterclockwise direction.”
and I'd like to know where to find more information about this, or if anyone has a precise definition of an analitycal description of moving along a curve in the counterclockwise direction, thanks.
I've never seriously considered this question before you asked it, so these are my thoughts which may be overcomplicating things.
You could perhaps think about it in terms of a parameterization for the simple, closed curve $C$. For instance, if your curve is parameterized by $r(t) = \langle x(t), y(t) \rangle$ and your parameterization traverses the curve in the "counterclockwise direction'' then your unit normal vector is given by $$N(t) = \frac{T'(t)}{||T'(t)||}$$ where $T(t)$ is the unit tangent vector, $$T(t) = \frac{r'(t)}{||r'(t)||}$$ The vector $N$ should point toward the region enclosed by $C$ (you know the curve divides the plane into an interior and exterior region by the Jordan Curve Theorem, which is never a thing I like to rely on but hey it works). So perhaps your analytic way of determining "counterclockwise direction'' would be to say that at any time $t$, there is some $c$ which is sufficiently small for which $$ r(t) + cN(t)$$ lies in the region enclosed by $C$.