I am having problems getting my head around this problem:
Evaluate the surface integral $$\int_S \vec F\bullet d\vec s$$ where $\vec F=x \vec i-y \vec j +z \vec k$ and where the surface S is the cylinder defined by $x^2+y^2\le 4$ and $0\le z \le 1$. Verify your answer using the Divergence Theorem.
The thing I am confused about is do we integrate over the ends of the cylinder and if so how? I think that we should since the divergence theorem requires a closed surface, but if I am right how do we do the surface integral since the normal changes direction (with a non-continuous derivative) between the top and bottom and the curved edges and therefore I cannot see how split the surface integral up into the three sections (top, bottom and curved side), how do we do this?
The divergence theorem
$$\int_{\Omega} \nabla \cdot \mathbf{F} \, d\mathbf{x}=\int_{\partial \Omega}\mathbf{F} \cdot \mathbf{n} \, dS$$
is applicable when the closed surface $\partial \Omega$ is piecewise smooth.
The discontinuity of the normal vector $\mathbf{n}$ is restricted to a set of measure $0$ and does not affect the integration. Just integrate over the top and bottom disks with $\mathbf{n} = \pm \mathbf{e_z}$ and over the side using $\mathbf{n} = \mathbf{e_r} = \cos \theta \mathbf{e_x} + \sin \theta \mathbf{e_y}.$