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 -simplex"? Is it the same as oriented affine -simplex defined in the previous page?

Second, if is in open set , that is, its range , how to guarantee that is also in ? If not, integral may not be meaningful at all.

Three, the proof deals only with the relation of Jacobians of and , what about the coefficient function? In page 254, the integral is defined as or , where is the parameter domain, but in general , how can we say that the two integrals differ only in the sign?

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