I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text.
"Let T be 1-1 mapping of $Q^n$ into $R^n$, of class $\mathscr{C}$ '', whose Jacobian is positive ( at least in the interior of $Q^n$). Let E=$T(Q^n)$. By the inverse function theorem, E is the closure of an open subset of $R^n$. We define the positively oriented boundary of the set E to be the (n-1)-chain $\partial$T = $\partial\sigma_o$
If E=$T_1(Q^n)$=$T_2(Q^n)$, and if both $T_1$ and $T_2$ have positive Jacobians, is it true that $\partial$$T_1$ = $\partial$$T_2$? " "The answer is yes, but we shall omit the proof"
I tried to prove this particular statement, but I could not. Please can anyone help me with this issue???
Consider the inverse function theorem applied to the diffeomorphism $f=T_2^{-1}\circ T_1$. Because local diffeomorphisms are open maps, $f$ maps $\partial Q^n$ into $\partial Q^n$, as does $f^{-1}$.