Evaluate the surface integral $ \int_{S}\int \vec{F} \cdot \vec{n}\, dS,$ with the vector field $ \vec{F} = zx\vec{i} + xy\vec{j} + yz\vec{k} \ $. $S$ is the closed surface composed of a portion of the cylinder $ x^2 + y^2 = R^2 $ that lies in the first octant, and portions of the planes $ x=0, y =0, z = 0\,\,\text{and}\,\, z = H $. $\vec{n}$ is the outward unit normal vector.
Attempt: I said $S$ consisted of the five surfaces $ S_1, S_2, S_3, S_4 $ and $ S_5$ $S_1 $ being the portion of the cylinder, $S_2$ being where the plane $ z=0$ cuts the cylinder and similarly, $ S_3, S_4 ,S_5 $ being where the planes $ x = 0, y = 0 $ and $z = H $ cut the cylinder.
For $ S_2 $, the normal vector points in the -k direction. so the required integral over $S_2$ is: $ \int_{0}^{R} \int_{0}^{\sqrt{R^2-x^2}} -yz\,dy\,dx $ Am I correct? I think for the surface $ S_5$ the only thing that would change in the above would be that the unit normal vector points in the positive k direction?
I need some guidance on how to set up the integrals for the rest of the surfaces. I tried $ \int_{0}^{R} \int_{0}^{H} -xy\,dz\,dx $ for the y = 0 plane intersection with the cylinder, but I am not sure if this is correct.
Any advice on how to tackle the remaining surfaces would be very helpful. Many thanks.