Timeline for How do I prove the change of variables into polar coordinates using measure theory?
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Oct 28 at 23:29 | vote | accept | user1591353 | ||
| Oct 26 at 7:42 | answer | added | peek-a-boo | timeline score: 2 | |
| Oct 26 at 3:25 | history | edited | Rócherz | CC BY-SA 4.0 | added 20 characters in body |
| Oct 26 at 0:12 | comment | added | user1591353 | @peek-a-boo Thank you for the reference; I'll try going through the proof. "it is kind of surprising that Axler wrote an entire book on real analysis without proving the change of variables theorem" I've searched for relevant keywords and didn't find this result but maybe it comes later and is called something else. | |
| Oct 25 at 20:00 | comment | added | peek-a-boo | … especially since he does prove the very important theorems of Lebesgue differentiation and Radon-Nikodym. Rudin does an excellent job showing the utility of these two theorems in C7 of RCA. Anyway, for now I think you’re better off consulting Folland (elementary but annoying/tedious proof of change of variables). Of course, there are the usual handwavy arguments about drawing inscribed sectors anc comparing them to triangles (but there are several loose ends to be tied up that I think it’s better to just prove the general case precisely (in the spirit of Folland)) | |
| Oct 25 at 19:58 | comment | added | peek-a-boo | another type of proof (for the general case) is given in Folland’s real analysis book (section 2.6 IIRC), and this is based on the more typical idea of approximating how the transformation $g$ distorts the measures from the domain and target; it boils down to approximating $g$ near a point $a$ by the derivative $Dg_a$ (i.e $g(a+h)\approx g(a)+ Dg_a(h)$, then using translation-invariance, and that linear maps distort measure by absolute value of determinant). Anyway, it is kind of surprising that Axler wrote an entire book on real analysis without proving the change of variables theorem… | |
| Oct 25 at 19:31 | history | edited | RobPratt | edited tags | |
| Oct 25 at 18:20 | comment | added | user1591353 | @peek-a-boo Do you think there might be a way to get this special two-dimensional case without Radon-Nikodym? I haven't seen this theorem yet. I'm working my way through this book and it only comes later. The change of variables formula however is used in the proof to show the volume of the unit ball in an earlier chapter (page 141) so I wanted to justify it using the results I've seen so far. In any case thank you for the relevant link. | |
| Oct 25 at 18:03 | history | edited | user1591353 | CC BY-SA 4.0 | added 1 character in body |
| S Oct 25 at 17:21 | history | suggested | user683 | CC BY-SA 4.0 | spaced out equations for readability, modified punctuation |
| Oct 25 at 17:03 | review | Suggested edits | |||
| S Oct 25 at 17:21 | |||||
| Oct 25 at 16:00 | comment | added | peek-a-boo | See here for the change of variables in $\Bbb{R}^n$. | |
| Oct 25 at 14:34 | history | asked | user1591353 | CC BY-SA 4.0 |