If the symbol $\partial \Omega$ is used to represent the boundary of $\Omega \subset \mathbb{R}^2$, is smooth or differentiable a pre-requisite?

Question

If we can use the symbol $\partial \Omega$ to represent the boundary of $\Omega$, for instance $\Omega$ is in $\mathbb{R}^2$ ($\mathbb{R}^3$), and thus $\partial \Omega$ is a curve (surface), do we require the curve (surface) to be smooth or differentiable as a pre-requisite?

For example, if $\Omega$ is a polygon or some other shapes with non-smooth edges, can we use $\partial \Omega$ to denote the boundary of $\Omega$?

In particular, if we want to do integration over all the edges of a triangle, should we just write one integral for three edges

\begin{align} \int_{\partial T} n ds \end{align}

OR write one integral for each edge and sum them up

\begin{align} \sum_{\text{edge}=1}^3 \int_{\partial T} n ds \end{align} where $n$ is the unit normal of each edge?

And what about the case for higher dimensional manifolds?

Answer 1

The source of the confusion here is that there are several areas of mathematics which use the symbol $\partial$ for the boundary (and assign a slightly different meaning to it).

1. Point-set (aka general) topology. (I personally prefer to refer to this "boundary" as the "frontier".) I do not expect you to be familiar with this area of math since you are just taking a real analysis class, but for a subset $\Omega$ in $R^n$ the topological boundary (the frontier of $\Omega$) is the set of points $x\in R^n$ such that every open ball $B(x,r)$ of radius $r$ centered at $x$ intersects both $\Omega$ and its complement in $R^n$. For example, if $C$ is the Cantor set in $R$ and $\Omega= R - C$ then $\partial \Omega= C$. I do not think you are planning to integrate over this boundary of $\Omega$ (at least, not until you learn the Lebesgue integral). For the item 2 below, we will also need another topological notion, the closure $cl(\Omega)$ of $\Omega$, which would be the union of $\Omega$ and its boundary.

2. The notion of boundary (with the same notation $\partial$) is also used by: (a) differential topology, (b) real analysis, (c) differential geometry,.... In this context, it is expected that the boundary has some reasonably nice structure (Cantor set will not qualify). Such structure is supposed to be the one of a smooth manifold or piecewise-smooth manifold. I do not expect you to know what this would mean either. But, in the case of domains $\Omega\subset R^2$, the assumption is that $\partial \Omega$ (the one defined in the item 1 above) is a smooth curve. Why? because all the three listed areas of math have something in common: They aim to do some form of real analysis on some spaces (curves, surfaces, manifolds...). For instance, for $\partial \Omega$ to be a smooth curve (in the case when $\Omega$ is a domain in $R^2$) one needs some restrictions on $\Omega$. The technical condition is that the closure of $\Omega$ in $R^2$ is a smooth submanifold with boundary whose interior is equal to $\Omega$ itself. In practical terms, for a domain in $R^2$ this means that for every point $a\in \partial \Omega$ there exists some $r>0$ and a choice of rectangular coordinates in $R^2$ (as a practical matter, you may just swap the $x$ and $y$ axes) such that:

i. $\partial \Omega \cap B(a, r)$ equals the intersection of the graph of a smooth (how many derivatives one requires is negotiable, but at least $C^1$) function $y=f(x)$.

ii. Similarly, $cl(\Omega)\cap B(a,r)$ is required to be the intersection of the epi-graph of $f$ $$ \{(x,y): y\ge f(x)\}, $$ with $B(a,r)$.

I suggest you test this definition for $\Omega=B(a,r)$ and check that it applies in this case. The same definition with very little change, goes through in $R^n$ (just $f$ will be a function of $n-1$ variables).

Given this, one defines the orientation on $\partial \Omega$ (for you, a choice of a normal vector, which is traditionally required to point outward of the domain $\Omega$). Given this, one defines integrals over $\partial \Omega$. (It more advanced real analysis/differential topology/differential geometry classes, you learn that you integrate not functions but differential forms, but that's another question.)

One can make sense of all this in the setting of "piecewise-smooth" boundaries, but I will skip this part. In short, if you just want to integrate, then go ahead and integrate over each "smooth piece" of the boundary and add up the results, this is what an integral in this setting would mean. A formal definition of the boundary and a justification will be given (or not) in a more advanced class.