I am reading Smooth Manifolds by John M.Lee. In chapter 16 he defines the integral of an n-form $\omega$ over a domain of integration $D$ in $\mathbb{R}^n$.
My question:
Unless a domain of integration is in itself a smooth manifold (which I don't think is true), I don't understand the meaning of the phrase "let $\omega$ be a n-form on $\overline{D}$".
My reasoning:
A n-form is a multilinear map on the (product of) tangent space. The latter is defined only if the notion of a smooth function is defined. However, unless $\overline{D}$ is a smooth manifold I don't understand what is meant by this.
Further:
let's say the phrase just means that is a restriction of a n-form on $\mathbb{R}^n$ to $\overline{D}$. In that case, I don't understand the necessity of including the boundary. That is, if $D$ is a domain of integration than any integral should not be changed if it is computed over $D$ instead of $\overline{D}$. So why bother with the definition on the boundary at all?
Thanks for any help and explanations!
Note that what I actually wrote was "let $\omega$ be a (continuous) $n$-form on $\overline D$. Any such form can be written as $\omega = f\, dx^1\wedge \cdots \wedge dx^n$ for some continuous function $f\colon \overline D\to \mathbb R$." What that means is that $\omega$ is a continuous section of the restricted bundle $\Lambda^nT^*\mathbb R^n|_{\overline D}$, that is, a continuous map from $\overline D$ to $\Lambda^nT^*\mathbb R^n$ such that $\omega(x) \in \Lambda^n(T_x^*\mathbb R^n)$ for each $x\in \overline D$.
The reason for insisting that $\omega$ be defined on $\overline D$ rather than just on $D$ is to ensure that it's bounded. Otherwise, the integral might not be finite.