Let ${\bf F}(x,y,z)=\begin{bmatrix}f(z)\\f(z)\\f(z)\end{bmatrix}$ be a smooth vector field that depends on the z-coordinate. Let $R$ be the cilinder in $\mathbb R^3$ bounded by $S=\{(x,y,z):x^2+y^2\le 1\text{ and } -1\le z\le 1\}$, and the closed disks $S_\pm=\{x^2+y^2\le 1, z=\pm1\}$.
Let $\vec N$ be the outward pointing unit normal on $R$. How can I prove $\iint_S{\bf F}\bullet N dS$=0? And is it still true if ${\bf F}$ only depends on $x$?