Timeline for Frobenius Condition for a specific first order pde
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2014 at 9:25 | history | edited | Thomas Richard | CC BY-SA 3.0 |
added 104 characters in body
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| Dec 3, 2014 at 9:18 | comment | added | Thomas Richard | My bad. This case is integrable actually. | |
| Dec 3, 2014 at 5:35 | comment | added | Ali | :your example simply does not work as one can clearly see that the inner products DO vanish everywhere actually as I also mentioned above. Anyways one can indeed find many examples where the system has no solution. Thanks for your comment | |
| Dec 2, 2014 at 9:16 | history | edited | Thomas Richard | CC BY-SA 3.0 |
Added the fact that the solution needs to be non constant.
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| Dec 2, 2014 at 9:14 | comment | added | Thomas Richard | Ok, I did the computation: if $u$ solves the problem, so does any $f(u)$ for $f$ a smooth function from $\mathbb{R}$ to $\mathbb{R}$. So I'll take $u(x,y,z)=z^2-y^2$ instead of yours. Then $\nabla\phi_1=\partial_x$ and $\nabla\phi_2=yz\partial_x+xz\partial_y+xy\partial_z$, while $\nabla u= 2z\partial_z-2y\partial_y$. On can check that the inner products between the gradients don't vanish everywhere. | |
| Nov 29, 2014 at 7:26 | comment | added | Thomas Richard | I'll have a look but I'm skeptical. | |
| Nov 29, 2014 at 1:35 | comment | added | Ali | It is clear that u solves the aforementioned system of odes in the question in euclidean space | |
| Nov 29, 2014 at 1:34 | comment | added | Ali | I am not sure what I am missing here...I agree with the condition you have mentioned but I am not sure the example works.. to make this clear take $ u(x_1,x_2,x_3) = e^{\frac{x_3^2-x_2^2}{2}}$ | |
| Nov 29, 2014 at 1:13 | vote | accept | Ali | ||
| Nov 29, 2014 at 1:32 | |||||
| Nov 28, 2014 at 16:17 | vote | accept | Ali | ||
| Nov 29, 2014 at 1:10 | |||||
| Nov 28, 2014 at 9:39 | history | answered | Thomas Richard | CC BY-SA 3.0 |