Timeline for Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 11 at 20:24 | vote | accept | Isaac | ||
| Oct 8 at 21:25 | comment | added | Isaac | @OlegEroshkin Anyway, now I review some of the materials on differential geometry, I can see that my comment above was an absurd one. I just deleted it. Thank you for enlightening me on this. | |
| Oct 8 at 21:04 | comment | added | Isaac | @OlegEroshkin Well, I am aware of the definitions. And de Rham cohomology doesn't seem "critically" relevant for my question in the first place, as it only pertains to smooth functions. As you can see in my answer above, it is more about appearance of jump discontinuity (and the Dirac delta distribution) that causes the problem.. | |
| Oct 8 at 19:28 | comment | added | Oleg Eroshkin | Sorry Isaac, but you need to review some basic things. Every exact form is closed but not every closed is exact. That failure is measured by de Rham cohomology. The form $d\theta$ is closed but not exact. That's the relation. | |
| Oct 8 at 16:44 | comment | added | Neal | It is not a problem, just a connection with de Rham cohomology. | |
| Oct 8 at 16:33 | comment | added | Isaac | What exactly is the problem of $d \theta$ not being exact? I am sorry that I have forgotten most of the details I learned before. | |
| Oct 8 at 16:29 | comment | added | Neal | One connection is that you can obtain $F$ by integrating the closed one-form $d\theta$ from a base point identified with $0$, and the fact the total integral is not zero means that the one-form is not exact. | |
| Oct 8 at 16:08 | history | answered | Isaac | CC BY-SA 4.0 |