How to find the closed path such that $\int\limits_\Gamma F\cdot dl\ne0$ If there is some vector field $F(x,y)=(F^1(x,y),F^2(x,y))$ such that $\frac{\partial F^2}{\partial x}-\frac{\partial F^1}{\partial y}\ne0$ then by a theorem there does not exist a scalar field $f$ such that $F$ is the gradient of $f$. And another theorem says that there exists a closed smooth curve $\Gamma$ such that $$\int\limits_\Gamma F\cdot dl\ne0$$
For example: $F(x,y)=(3x^2y,x^2)$ The curl is $\nabla\wedge F(x,y)= 2x-3x^2\ne0$ and if we integrate $F$ along the following path, the result will not be zero:
$$\Gamma_1=\{(t,t^2)\ :\ t\in[0,1]\}, \ \Gamma_2=\{(2-t,2-t)\ :\ t\in[1,2]\}$$
$$\Gamma:=\Gamma_1\cup\Gamma_2$$
In fact: $\int\limits_\Gamma F\cdot dl=\int\limits_0^1(3t^4,t^2)\cdot(1,2t)dt+\int\limits_1^2(3(2-t)^3,(2-t)^2)dt=\frac{1}{60}\ne0$
My question is how to find the adequate path? And is this related to the residue theorem, like, does the path have to go around some singularity?
 A: First, note that if $F$ has any singularities, its domain is not simply connected, and the theorem stated does not hold for every closed loop containing the singularity, even if $F$ has zero curl.
So, let's assume that $F$ has no singularities in its domain. If $F$ has continuous first partials, Green's Theorem states that the integral of $F$ along a closed path in $\mathbb{R}^2$ is equal to the integral of the (scalar) curl of $F$ over the interior of the path. Then, simply find an open set in the domain where the curl is entirely positive or negative, and integrate along its boundary. Such a set necessarily exists because the first partials are continuous, so the curl is continuous and not everywhere zero. In your example, any path with $x\notin \{0,\frac{2}{3}\}$ for all $t$ would do the trick.
I'm not sure if there's an analytic way to do it for $F$ without continuous partials. One might guess that if such a closed curve exists, then there may be a circle which does it, in which case you can integrate around a circle with variable center and radius, then choose values which give a non-zero integral. However, I'm not sure if this will always work.
