0
$\begingroup$

We can make a variable change if we have some diffeomorphism $\phi$. A usual variable change is polar coordinates. But when $\rho = 0\quad |\phi'| = 0$, so $\phi$ is not a diffeomorphism if our region contains $0$. But anyway we do change and integrate. Why can we?

$\endgroup$
5
  • 1
    $\begingroup$ because the set $\{0\}$ have Lebesgue measure zero $\endgroup$ – Masacroso Jan 16 at 17:09
  • $\begingroup$ @Masacroso We just use additivity of integral, one part is zero, in the other we make the change? $\endgroup$ – Maxim Jan 16 at 17:12
  • $\begingroup$ what you says is a way to see it $\endgroup$ – Masacroso Jan 16 at 17:13
  • $\begingroup$ @Masacroso I just want to know if it's correct. Thank you. (But what are other ways?) $\endgroup$ – Maxim Jan 16 at 17:15
  • $\begingroup$ yes, its correct. Any other way to see it is equivalent to the one you propose, it rely in the fact that integration on sets of zero measure are zero. $\endgroup$ – Masacroso Jan 18 at 7:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.