What is the boundary of a surface?

Question

This might be a really basic question, but for Stoke's Theorem, I always see the use of the word "boundary" without any explanation and the boundary is pretty intuitive, but how would I determine the boundary for more complicated shapes (shapes that perhaps might be too difficult to sketch)?
Is there a sort of basic definition of a boundary (preferably in simple terms, since I'm not very proficient at Calculus)?

Answer 1

Intuitively, the difference between an interior point of a surface and a boundary point of a 2-D surface is whether a neighborhood surrounding that point looks like $\mathbb{R}^2$ or the upper half space $$ \mathbb{H}^2=\{(x,y)\in\mathbb{R}^2:y\geq0\}. $$

One way to think of this is that in an interior point, I may move in any "cardinal direction," i.e. North, South, East, West, or any direction in between, while staying within my surface. However, on a boundary point, it looks as if I am standing on the $y$-axis in $\mathbb{H}^2$, so I cannot move south; I can only move East, West, or North. This is perhaps not the most formal definition, but it is how I picture it in my head.

Answer 2

Suppose a surface $S$ of $\Bbb R^3$ parametrized by $\mathbf Σ :D\subset \Bbb R^2\to \Bbb R^3:\mathbf Σ (D)=S$. From a purely topological point of view the boundary of $S$ is the whole surface, because for every point of $S$ every open sphere centered at it intersects $S$ and $\Bbb R^3 \setminus S$. However the usual "$\partial S$" of Stoke 's Theorem is about the "geometric boundary", namely the set $\mathbf Σ(\partial D)$, i.e. the map of parametric region 's boundary via $\mathbf Σ$. For example, if $S$ is the unitary upper semisphere and $D$ is the unitary closed disc, then the geometric boundary of $S$ is the unitary circle, for $\mathbf Σ(\partial D)=\partial D$.