I am asked to calculate $\begin{gather*} \iint_S \nabla\times\vec{A} \end{gather*}$,
$\vec{A}=(z,x,y)$ and S the surface defined by $x^2 + y^2 + z^2= 1, ~~~~0\leq z \leq1/2.$
This is meant to compare the flux and the circulation of $\vec{A}$ over the boundary. We have been taught Stokes' theorem at a very basic level, since we aren't taking pure maths. Most examples involve simple surfaces that seem to 'close', like a paraboloid $z=x^2 +y^2$, or a pyramid and its base.
In all of them, we only used one curve along which to calculate the circulation, but apparently here two are needed: the level sets $z=0$ and $z=1/2$ of the sphere. I don't understand why, since one of the curves could be used to bound the surface. What am I missing?
I found in this page http://www.math.toronto.edu/courses/mat237y1/20189/notes/Chapter5/S5.6.html the definition for the boundary points. So it makes sense in this case... but what if we don't know what is $S_0$? I feel like there's some sort of simplification there. What is a true definition for boundary curves of surfaces?
Also, it doesn't explain why the direction of circulation of both curves must be opposite in order to calculate the flux over the surface. One could interpret it as "subtracting" the flux between $1/2\leq z \leq 1$ and $ 0 \leq z\leq 1/2$, but again it looks a bit fishy.
This post Stokes' theorem on surface with boundary being a union of closed, simple, piecewise smooth curves seems related, but it's using precisely the aforementioned approach.
To start with, the sphere all by itself, as defined by the equation $x^2+y^2+z^2=1$ has no boundary at all.
Let's restrict the sphere to the region where $z \ge 0$. Then indeed, as you say, the resulting surface has just one curve on its boundary, namely the curve where the sphere intersects the plane $z=0$, which forms a circle $x^2+y^2=1$ in the $z=0$ plane. Let me denote that circle $C_0$. Visually, imagine slicing the sphere along the $z=0$ plane, throwing away everything below that, and keeping what's above it.
If on the other hand we could restrict the sphere to the region where $z \le 1/2$. Then again the resulting surface has just one curve on its boundary namely the curve where the sphere intersects the plane $z=1/2$, which forms a circle $x^2 + y^2 + (1/2)^2 = 1$ in that plane; we can simplify that to $x^2+y^2 = 3/4$ in the $z=1/2$ plane. Let me denote that circle $C_{1/2}$. Visually, imaging slicing the sphere along the $z=1/2$ plane, throwing away everything above that plane, and keeping what's below.
But when you form a surface by restricting the sphere to the region where $0 \le z \le 1/2$, then the resulting surface has two boundary circles: $C_0$ and $C_{1/2}$. Imagine slicing the sphere simultaneously along the $z=0$ and $z=1/2$ planes, throwing away what's below $z=0$ and also throwing away above $z=1/2$, and keeping what's between. Using just one of those curves will not get you the whole boundary.