Note: the same question was asked here - rudin's principles of mathematical analysis 10.31 , but I am not able to understand the answer in above post, so I'm re-asking it.
In Section 10.31 of "Baby" Rudin, he defines positively oriented boundary of subsets of $\mathbb{R}^n$.
Let $\sigma_0$ be the identity mapping with domain $Q^n$. Let $T$ be a 1-1 mapping of $Q^n$ into $\mathbb{R}^n$, of class $C''$, whose Jacobian is positive in the interior of $Q^n$. Let $E=T(Q^n )$. Then, we define the positively oriented boundary of the set $E$ to be the $(n-1)$-chain $\partial T = T(\partial \sigma_0 )$.
But, to prove that this definition is well-defined, we need to prove the following, which Rudin mentioned (that it is true) without proof.
If $E=T_1 (Q^n ) = T_2 (Q^n )$ (where $T_1$ and $T_2$ satisfies the condition listed above), $\partial T_1 = \partial T_2 $, so that $\int_{\partial T_1} \omega = \int_{\partial T_2} \omega$ for all $(n-1)$-form $\omega$.
How can I prove this statement? (Plus, if I understood correctly, I think the answer in rudin's principles of mathematical analysis 10.31 is saying that $\partial T_1$ and $\partial T_2 $ has same image (of $Q^n$), but it is not enough to prove what we want)
Below are some of the definitions Rudin is using.
$C'$(or $C''$)-mapping with compact domain (since we define differentiation in open sets only)
To say that $f$ is a $C'$-mapping (or $C''$-mapping) of a compact set $D \subset \mathbb{R}^k$ into $\mathbb{R}^n$ means that there is a $C'$-mapping (or $C''$-mapping) $g$ of an open set $W \subset \mathbb{R}^k$ into $\mathbb{R}^n$ such that $D \subset W$ and such that $g(x)=f(x)$ for all $x \in D$.
$k$-surface
Suppose $E \subset \mathbb{R}^n$ is open. A $k$-surface in $E$ is a $C'$-mapping $\Phi$ from a compact set $D \subset \mathbb{R}^k $ into $E$. Here, we call $D$ a parameter domain of $\Phi$.
$k$-forms
$k$-form in $E$ is a function $\omega = \sum a_{i_1 \cdots i_k}(x ) dx_{i_1} \wedge \cdots \wedge dx_{i_k}$, which assigns to each $k$-surface $\Phi$ in $E$ a number $\int_{\Phi} \omega = \int_{D} \sum a_{i_1 \cdots i_k}(\Phi (u) ) \frac{\partial (x_{i_1} , \cdots , x_{i_k} )}{\partial (u_1 , \cdots , u_k)} du$ where $D$ is parameter domain of $\Phi$.
affine simplexes
Define $Q^k \subset \mathbb{R}^k$ to be a set with points $u=(u_1 , \cdots , u_k )$ with $u_1 + \cdots + u_k \le 1$ and $u_i \ge 0$. oriented affine $k$-simplex $\sigma = [p_0 , p_1 , \cdots , p_k ]$ is a $k$-surface in $\mathbb{R}^k$ with parameter domain $Q^k$ which is given by affine mapping $\sigma (\alpha_1 e_1 + \cdots + \alpha_k e_k )=p_0 + \sum_{i=1}^{k} \alpha_i (p_i -p_0 )$.
affine chains
affine $k$-chain $\Gamma$ in an open set $E \subset \mathbb{R}^n$ is a collection of finitely many oriented affine $k$-simplexes $\sigma_1 , \cdots, \sigma_r$ in $E$. We use the notation $\Gamma = \sigma_1 + \cdots + \sigma_r$, and we define $\int_{\Gamma} \omega = \sum_{i=1}^{r} \int_{\sigma_i} \omega$.
Boundary of affine simplexes
$boundary$ of the oriented affien $k$-simplex $\sigma = [p_0 , \cdots , p_k ]$ is defined to be the affine $(k-1)$-chain $\partial \sigma = \sum_{j=0}^{k} (-1)^j [p_0 , \cdots, p_{j-1} , p_{j+1} , \cdots , p_k ]$
Differentiable simplexes and chains
Let $T$ be a $C''$-mapping of an open set $E \subset \mathbb{R}^n$ into an open set $V \subset \mathbb{R}^m$. If $\sigma$ is an oriented affine $k$-simplex in $E$, then $\Phi = T \circ \sigma $(or $T \sigma $) is a $k$-surface in $V$, with parameter domain $Q^k$. We call $\Phi$ an oriented $k$-simplex of class $C''$. And finite collection $\Psi$ of oriented $k$-simplexes $\Phi_1 , \cdots, \Phi_r $ of class $C''$ in $V$ is called a $k$-chain of class $C''$ in $V$. We also define $\int_{\Psi}\omega = \sum_{i=1}^{r} \int_{\Phi_i}\omega$.
Plus, if $\Gamma = \sum \sigma_i$ is affine $k$-chain, $T \circ \Gamma$ is defined to be a $k$-chain $\sum T \sigma_i$.
- boundary of differentiable simplex and chains
If $\Gamma = \sum \sigma_i $ is affine $k$-chain, $\partial \Gamma = \sum \partial \sigma_i $. If $\Phi = T \circ \sigma $ is oriented $k$-simplex of class $C''$, we define $\partial \Phi = T(\partial \sigma)$. If $\Psi = \sum \Phi_i$ is $k$-chain of class $C''$, we define $\partial \Psi = \sum \partial \Phi_i $.
Thank you for reading my long question. I spent lot of time proving the main statement(the second yellow box), but I couldn't get any meaningful result. Hope to see a satisfying answer here. Any short ideas are also welcome. Thanks again.