# Variable change in multiple integrals: polar coordinates at zero.

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?

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