Timeline for Reference request: A multidimensional generalization of the fundamental theorem of calculus
Current License: CC BY-SA 4.0
17 events
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| Jan 14, 2021 at 3:46 | comment | added | Iosif Pinelis | @WillieWong : Thank you for the reference. | |
| Jan 14, 2021 at 2:25 | comment | added | Willie Wong | Taking the limit as $u \to (-\infty, -\infty, -\infty, \ldots, -\infty)$ this also gives you a proof of the $W^{1,n}(\mathbb{R}^n) \hookrightarrow L^\infty$ Sobolev inequality. A variant of it is how I argued Lemmata 3 and 11 of this paper. | |
| Jan 13, 2021 at 22:16 | comment | added | Fedor Petrov | @IosifPinelis well, Stokes theorem for the box (and the derivative in the coordinate direction) is nothing but one-dimensional fundamental theorem of calculus integrated against all other coordinates. | |
| Jan 13, 2021 at 21:54 | comment | added | Iosif Pinelis | @FedorPetrov : But then, one can also say that it is just the one-dimensional fundamental theorem of calculus applied $p$ times, which is of course true. | |
| Jan 13, 2021 at 21:38 | comment | added | Iosif Pinelis | @FedorPetrov : Yes, I guess that is why I did not see a relation with Stokes' theorem; the key was your "applied $p$ times". | |
| Jan 13, 2021 at 21:11 | comment | added | Fedor Petrov | @IosifPinelis I do not know, but I would not expect this: in Stokes theorem the dimensions in LHS and RHS differ by 1, here they differ by $p$. | |
| Jan 13, 2021 at 20:49 | comment | added | Iosif Pinelis | @FedorPetrov : Thank you for your comment. So, it's Green's formula applied $p$ times. Can one get this from a Stokes formula applied just once? | |
| Jan 13, 2021 at 18:58 | history | edited | Abdelmalek Abdesselam |
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| Jan 13, 2021 at 18:35 | answer | added | Abdelmalek Abdesselam | timeline score: 7 | |
| Jan 13, 2021 at 16:57 | comment | added | Fedor Petrov | @Iosif Stokes theorem relies the integral of $D_ig$ over, say, the box and the integral of $g$ over it's boundary. This should be Stokes theorem applied $p$ times. | |
| Jan 13, 2021 at 16:52 | answer | added | Zach Teitler | timeline score: 5 | |
| Jan 13, 2021 at 16:30 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 12, 2021 at 22:57 | comment | added | Iosif Pinelis | @Malkoun : I think it does; cf. arxiv.org/abs/1705.09159 -- where the statement posted here is presented as Lemma 5.1. | |
| Jan 12, 2021 at 22:48 | comment | added | Malkoun | It probably generalizes to polytopes (I know this is not what you are asking, but it is just a comment). | |
| Jan 12, 2021 at 21:57 | comment | added | Iosif Pinelis | @RiversMcForge : I don't see a relation with Stokes' theorem. | |
| Jan 12, 2021 at 21:54 | comment | added | Rivers McForge | Is this not Stokes’ Theorem? | |
| Jan 12, 2021 at 21:41 | history | asked | Iosif Pinelis | CC BY-SA 4.0 |