On Sat, 1 Jun 2013 17:36:00 -0700 (PDT), firstname.lastname@example.org wrote:
>On Sunday, June 2, 2013 1:28:06 AM UTC+1, peps...@gmail.com wrote: >> I'm a bit stuck on Rudin's proof of Change of Variables on page 252. (Yes, I'm aware it's proved in dozens of other texts, but I'm trying to follow this particular text). Two lines down from (31), Rudin appeals to the inverse function theorem. However, the inverse function theorem assumes the Jacobian is invertible, whereas the assumption here is only that the Jacobian is non-zero. >> >> >> >> Therefore the argument that the integrand on the right has compact support doesn't seem valid. >> > >Sorry, I think I can answer my own question. The Jacobian (as defined by Rudin, anyway) means the determinant.
As far as I know the Jacobian always means the determinant. Or the absolute value of the determinant. Who defines it otherwise? (In what seems to me to be very standard terminology, the matrix in question is the _derivative_.)
>So assuming the Jacobian is non-zero guarantees invertibility -- question cancelled. > >Thanks. > >Paul Epstein