Timeline for Proof of Helmholtz-Hodge decomposition, poor man's version
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2019 at 18:12 | vote | accept | Ivica Smolić | ||
| Oct 16, 2019 at 17:30 | answer | added | Willie Wong | timeline score: 2 | |
| Oct 16, 2019 at 17:12 | comment | added | Ivica Smolić | @Deane Yang OK, I see, thanks! Still, I don't know how to properly solve the initial problem (polishing of the proof)... | |
| Oct 16, 2019 at 16:52 | comment | added | Deane Yang | @IvicaSmolić, here you can define distributions on $\overline{\Omega}$ as the set of linear functions on $C^\infty(\overline{\Omega})$. For the purposes of this calculation, you can simply use define $\delta$ as a bounded linear functional on $C^k(\overline{\Omega})$, which is a Banach space with respect to the $C^k$ norm. | |
| Oct 16, 2019 at 15:33 | comment | added | Ivica Smolić | @Deane Yang You'll have to enlighten me about that remark on delta -- in my narrow view, I'm thinking of $\delta$ as a linear functional on $C_c^\infty$ | |
| Oct 16, 2019 at 13:57 | comment | added | Deane Yang | No. Integrating against delta does not require the test function to be compactly supported since delta itself is compactly supported. But, on a closer look, I’m not sure about the switching of the integral and Laplacian. Is that valid? | |
| Oct 16, 2019 at 6:29 | comment | added | Ivica Smolić | @Willie Wong I see, just to do everything in reverse... my concern was with the initial usage of the $\delta$ (as I've commented above), which I wanted to avoid. | |
| Oct 16, 2019 at 6:27 | comment | added | Ivica Smolić | @Deane Yang My concern is due to fact that in that proof (from Wiki page) we start with an integral of $F$ with $\delta$, doesn't that assume that $F$ is a test field? | |
| Oct 16, 2019 at 5:30 | comment | added | Deane Yang | I don’t see any assumption of compact support stated or needed in the Wikipedia article. | |
| Oct 16, 2019 at 1:16 | comment | added | Willie Wong | I am confused: for $F$ in $C^2(\Omega) \cap C^0(\bar{\Omega})$, and $\Omega$ bounded with smooth boundary, the integral expressions defining $U$ and $W$ make sense. Why isn't it sufficient to simply check that $- \mathrm{grad}(U) + \mathrm{curl}(W)$ in fact equals $F$? (This you can do using Gauss-Green carefully I think.) | |
| Oct 15, 2019 at 22:39 | history | edited | Ivica Smolić | CC BY-SA 4.0 |
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| Oct 15, 2019 at 21:47 | history | edited | Ivica Smolić | CC BY-SA 4.0 |
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| Oct 15, 2019 at 21:08 | history | asked | Ivica Smolić | CC BY-SA 4.0 |