I am reviewing old exam questions, and I am currently doubting everything that I have worked on for the past 3 weeks. The question is as follows [Solution is below, translated from Swedish]
The surface $S$ in $\mathbb{R}^3$ is given as the intersection of the cylinder $x^2 + y^2 ≤ 1$ and the surface defined by $x^2 + z^2\leq 1$ with $z\geq 0$. The vector field $F = (0, 0, x^2 + y^2)$.
(a) Give a parametrization of the surface $S$.
(b) Determine $$\iint\limits_{S} F \cdot N \, dS$$ where N is the unit normal field on $S$ with positive $z$-coordinate. (3)
Now here is where I am getting confused:
I am assuming that S is the union of $S_1$ and $S_2$, why are we adding $S_1$ and $S_2$ to the union only to then apply Gauss's divergence theorem?
Why is the normal vector for $S_1$ directed downwards and not upwards?
why would you parametrise the surface like that and not in cylindrical coordinates where you take $z$ between $0$ and $1$?
I appreciate any help!
Therefore $S\cup S_1\cup S_2$ is a closed surface and we may apply Gauss's divergence theorem.
$S_1=\{(x,y,0): x^2+y^2\leq 1\}$ is the bottom disc in the plane $z=0$. Since the orientation of $S\cup S_1\cup S_2$ is outwards (in order to apply Gauss's divergence theorem), it follows that the normal vector along $S_1$ is $(0,0,-1)$.
In the above solution we only need the parametrization of $S_1$ which is a disc in the plane $z=0$. Along $S_2$, which is a part of the vertical cylinder $x^2+y^2=1$, we know that the normal vector $N_2$ is horizontal and the vector field $F$ is parallel to the $z$-axis. Hence $N_2$ and $F$ are orthogonal which implies that their scalar product is zero at each point of $S_2$. We may conclude that the flux through $S_2$ is zero.