For Stoke's Theorem we are tasked with picking the boundary of a surface. What happens if there'smore than one boundary?
e.g. in
$$\int_{S}(\nabla \times {\textbf{F}})\cdot dS$$
where ${\textbf{F}} = (y,z,x^2y^2)$ and $S$ is the surface given by $z=x^2 + y^2$ and $0\leq z \leq 5$.
The surface looks like it has a boundary at $z=0$ with the circle $x^2 + y^2 = 1$ and also the same boundary except at $z=5$. How do we pick the correct one?
What if instead of the boundary $0 \leq z \leq 5$, we have $0 \leq z \leq 4x+3y+5$ or something like that? (There will also be a boundary with a slanted plane).