Given the surface $S\{(z,y,z)\in \mathbb{R} | x^2+y^2=1, 0\leq z\leq 1\}$ and the vectorial fields $X(x,y,z)=\frac{1}{x^2+y^2}(x,y,0)$ and $Y(x,y,z)=\frac{1}{(x-3)^2+y^2}(x-3,y,0)$ I have to calculate the flux of both vectors fields through $S$
I have calculated the divergence of $X$ and $Y$, and they are both $0$
So $$\int_S divXdV=\int_S divYdV=0$$
However when I calculate $$\int_{\partial S} \langle F·N \rangle dS$$ does not give me $0$. What am I doing wrong?
Your surface is not closed, and so it doesn’t bound a volume to which you can apply the Divergence Theorem. Add on the disks and see what happens.