Let $S=S_{1} \cup S_{2}$ be a oriented surface in $\mathbb{R}^{3}$ where $$ S_{1}: x^{2}+y^{2}=4,1 \leq z \leq 2, S_{2}: x^{2}+y^{2} \leq 4, z=2 $$ with an orientation $n$ pointing outward from the origin. Consider a vector field $$ F(x, y, z)=\left(y^{2} \sin (\pi z)+e^{\sqrt{x^{2}+y^{2}+z^{2}-1}}, \cos \left(\pi z e^{y}\right)+\sin \left(\pi \sqrt{x^{2}+y^{2}}\right), \sin \left(e^{x^{2} y^{2} z^{2}}\right)\right) $$ Calculate the flux $$ \iint_{S}(\nabla \times F) \cdot n d \sigma $$
What are the general ideas behind such flux problems? It seems that there should be heavy usage of some flux properties, but I could not match any famous ones (like Stokes' Theorem) into this specific scenario. What could be the potential suggestions specifically?
Let $E=\{(x,y,z)\in\mathbb R^3:x^2+y^2< 4,z\in[1,2]\}$ be the "volume" region defined by the surface of your exercise and note that $E$ is clearly $\mu_3$-measurable and bounded.
Its boundary $\partial E$ can be decomposed as $\Sigma_1\cup\Sigma_2\cup S$, where $\Sigma_1=\{(x,y,z)\in\mathbb R^3:x^2+y^2=4,z\in[1,2)\}$, $\Sigma_2=\{(x,y,z)\in\mathbb R^3:x^2+y^2<4, z=2\}$ and $S=\{(x,y,z)\in\mathbb R^3:x^2+y^2=4, z=2\}$.
Observe that $\Sigma_1$ and $\Sigma_2$ are open sets in $\partial E$ and you can find two $2$-manifold $M_1$ and $M_2$ such that $\overline\Sigma_1\subseteq M_1$ and $\overline\Sigma_2\subseteq M_2$.
$S$ is a compact set contained in a $1$-manifold and $\overline \Sigma_1 \cap\overline \Sigma_2\subseteq S$, then $E$ is a regular domain for the divergence theorem.
The vector field is $\mathcal C^1(\overline E,\mathbb R)$, so using cylindrical coordinates you can evaluate the surface integral you wrote as $\int_E \text{div}(F)d\mu_3$.
The decomposition I did has decomposed the boundary of the domain of integration in a regular ($\Sigma_1$ and $\Sigma_2$) and in a singular part (the compact set $S$ which gives a zero contribute to the integral since $\mu_3(S)=0$) and the flux you compute on the surface corresponds to the sum of the fluxes on the regular parts of the boundary of the set, which means $$\int_{\Sigma_1}F\cdot \vec\nu d\mu_2+\int_{\Sigma_2}F\cdot \vec\nu d\mu_2=\int_{E}\text{div}(F) d\mu_3,$$ where $\vec\nu$ is the outward pointing versor.