I propose to you this problem.
Let $\Sigma = \{ (x,y,z) \in \mathbb{R}^3 : 4z = \sqrt{x^2+y^2} - 1, 0 \leq z \leq \frac{1}{2} \}$. Sigma is oriented such that the normal exiting from it forms an obtuse angle with the z-axis.
I want to find the flux of the curl of $F(x,y,z) = e^{x^2+y^2}(2yz,3z^2,\frac{x}{2} + z^3 \}$. I do not want to directly calculate the curl of F, instead I want to use Stokes Theorem. By drawing, I see that I have a cone cut by two planes, thus I have two borders. The flux of the curl is then the sum with appropriate signs of the line integral of these two borders.
Here I get confused because the integral of the upper circumference needs a minus sign, while the other does not. I do not understand how to choose the appropriate sign, to me they should have the same.
Our teacher told us to imagine a man walking on the border, with the head oriented towards the normal exiting from the surface. If the man has on his left the surface then I need a plus sign, otherwise a minus, but I still struggle to understand. Any tips?
Let's flatten your surface into the plane (and flip it over - forgive me for not checking carefully which side you wanted when I made the picture). Then cut it so we have a simply connected surface:
Note that if we follow its border, taking the outside counter-clockwise, then we end up circling the inner circle clockwise instead. By the right-hand rule, the flux in this case is up, coming out of the screen towards you. A person traveling this border in the indicated direction, with their head in the direction of the flux, whould find that the region is always on their left, no matter where they are along the border, inside circle or outside.
While I have separated the two sides of the cut here for clarity in how the border must be followed, they actually are the same curve. We follow this curve twice, but in opposite directions, meaning that any integral along the curve will be added in, then subtracted back out. The resulting integral will will have the same value as if we just integrated along the inside and outside borders, without any cut at all - but only when we take the inside border in the opposite direction as the outside border.
If we take the insider border in the same direction from the outside border, then we jump from one side of the cut to the other. Our hypothetical traveler suddenly switches from traveling with the region on his left to traveling with it on his right. It is a discontinuity in how we are treating the region.
Stoke's theorem always requires traversing the border with the region on the left, never the right.