I am trying to evaluate the flux of a cylinder $C$ which is closed at either the top or bottom using Stokes.
The actual integral is $\iint_{\partial S} \mathrm{curl}\ F\ dS$ and using Stokes I can break the surface down into the outer part of the cylinder and either the bottom disk or the top disk and evaluate $\int F \cdot dr$.
However, since the bottom disk has normal down and the top disk has normal up, wouldn't that lead to two different orientations and answers depending on which I use?
PS the line integral of the circular projection of the surface is given to be -2 so the answer is -4, but I am getting $0$ when I use the bottom disk.
Alternatively, with the divergence theorem : $$ \iint_S \mbox{curl } F dS = \underbrace{\iiint_V \mbox{div(curl) } F dV}_{=0} - \iint_{S_1} \mbox{curl } F dS - \iint_{S_2} \mbox{curl } F dS $$ where $S_1$ and $S_2$ denote the top and bottom surfaces of the cylinder.
Use the rest of the information (in particular, the fact that the line integral of the circular projection of the cylinder $=-2$) and Stokes theorem ($\iint_{S_1} \mbox{curl } F dS = \int_{C_1} F dr$) to obtain the $-4$ answer.