This is Theorem 10.27 in Page 267-268 of Rudin's "Principles of Mathematical Analysis", as the picture below shows.

I have three questions:

First, what is "oriented rectilinear $\displaystyle k$-simplex"? Is it the same as oriented affine $\displaystyle k$-simplex defined in the previous page?

Second, if $\displaystyle \sigma$ is in open set $\displaystyle E$, that is, its range$\displaystyle \subseteq E$, how to guarantee that $\displaystyle \bar\sigma$ is also in $\displaystyle E$? If not, integral $\displaystyle \int\nolimits_{\bar\sigma} \omega$ may not be meaningful at all.

Three, the proof deals only with the relation of Jacobians of $\displaystyle \sigma$ and $\displaystyle \bar\sigma$, what about the coefficient function? In page 254, the integral is defined as $\displaystyle \int\nolimits_D \sum a_{i_1...i_k}(\sigma$ or $\displaystyle \bar\sigma(\mathbf u))J_{i_1...i_k}d\bf u$, where $\displaystyle D$ is the parameter domain, but in general $\displaystyle \sigma(\mathbf u)\ne\bar\sigma(\mathbf u)$, how can we say that the two integrals differ only in the sign?

If any other information is needed, please reply too. Thanks!