I am reading "Analysis on Manifolds" by James R. Munkres.
Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. The set $Y=\alpha(A)$, together with the map $\alpha$, constitute what is called parametrized-manifold, of dimension $k$. We denote this parametrized-manifold by $Y_\alpha$; and we define the ($k$-dimensional) volume of $Y_\alpha$ by the equation $$v(Y_\alpha)=\int_A V(D\alpha),$$ provided the integral exists.
Let us give a plausibility argument to justify this definition of volume. Suppose $A$ is the interior of a rectangle $Q$ in $\mathbb{R}^k$, and suppose $\alpha:A\to\mathbb{R}^n$ can be extended to be of class $C^r$ in a neighborhood of $Q$. Let $Y=\alpha(A)$.
Let $P$ be a partition of $Q$. Consider one of the subrectangles $$R=[a_1,a_1+h_1]\times\cdots\times [a_k,a_k+h_k]$$ determined by $P$. Now $R$ is mapped by $\alpha$ onto a "curved rectangle" contained in $Y$. The edge of $R$ having endpoints $a$ and $a+h_ie_i$ is mapped by $\alpha$ into a curve in $\mathbb{R}^n$; the vector joining the initial point of this curve to the final point is the vector $$\alpha(a+h_ie_i)-\alpha(a).$$ A first-order approximation to this vector is, as we know, the vector $$v_i=D\alpha(a)\cdot h_ie_i=\left(\frac{\partial \alpha}{\partial x_i}\right)\cdot h_i.$$
$\cdots$
The author wrote as follows:
suppose $\alpha:A\to\mathbb{R}^n$ can be extended to be of class $C^r$ in a neighborhood of $Q$.
I think we don't need to extend $\alpha$ to a neighborhood of $Q$.
$A$ is an open rectangle in $\mathbb{R}^k$ and $a\in A$.
The author calculated $D\alpha(a)$ and $\alpha$ is a map of $C^r$ in $A$.
Why did the author need to extend $\alpha$ to a neighborhood of $Q$?
Munkres wants to consider a division of a fixed cube $Q$ into rectangles, but the boundaries of those rectangles must intersect the boundary of the cube we start with. To get around this (for instance, if we want to talk about the value of $\alpha$ at some of the boundary points), we could just start from a slightly smaller subcube $Q_0\subset \overline{Q_0}\subset Q$ and discuss what $\alpha$ does to a partition of this smaller subcube $Q_0$. The alternative, which is what Munkres does, is to assume that $\alpha$ extends to a neighborhood of the closure of $Q$.
This is all done for the purpose of motivating the definition of integrals on parametrized manifolds, so we should pick whichever one we feel comfortable with.