I have to use the Stokes' Theorem to evaluate the exterior flow of the vectorial field $$\vec{F}(x,y,z)=(xy^{2},x^{2}y,y)$$ throught the border of the solid bounded by the cylinder $x^{2}+y^{2}=1$ and the planes $z=1$ and $z=-1$.
My question is: Stokes use curl. So should I find a vectorial field $G$ such that $$curl G=F$$ and then apply Stokes on it?
And since Stokes use line integral, what is the curve $\gamma$ that I'm supposed to use? I thought that $$\gamma=\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$$ where $\gamma_{1}$ and $\gamma_{2}$ are the curves described by the bases of the cylinder ($x^{2}+y^{2}=1$) and $\gamma_{3}$ like $\gamma_{4}$ are the curves given by the height of the cylinder, ie, from $z=-1$ to $z=1$, but I'm really not sure about it.
Let $C$ be the described cylinder.
I'm going to assume by "exterior flow" you mean the integral:
$$\int_{\partial C}\mathbf F \cdot \hat{\mathbf n} \, \mathrm dS$$
By the divergence theorem form of Stokes's Theorem, this is equal to:
$$\int_C \nabla\cdot\mathbf F \, \mathrm dV$$
$$=\int_C(y^2 + x^2+0)\, \mathrm dV$$
As $C$ is a cylinder, you should convert to cylindrical coordinates.