Timeline for How does one motivates the method of separation of variables when teaching PDE's?
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14 events
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| Nov 16, 2010 at 21:27 | comment | added | j.c. | Thanks to Denis Serre and Willie Wong for the suggestion. Miller's book is apparently out of print, but available on his website here ima.umn.edu/~miller/separationofvariables.html | |
| Nov 16, 2010 at 14:27 | comment | added | J. M. isn't a mathematician | @Willie: I got the vibe (so no offense taken ;) ), just clarifying my position. | |
| Nov 16, 2010 at 14:15 | comment | added | Willie Wong | @J.M.: please note the smiley face! My statement was made in jest and with tongue firmly in cheek. | |
| Nov 16, 2010 at 14:09 | comment | added | J. M. isn't a mathematician | @Willie: I'm not saying it's "bad"; more of it's slightly more intricate to solve nonseparable PDEs. The "sigh of relief" in finding that your PDE is separable is in the theme of "exploit any structure you find in your problem", to which I strongly adhere. | |
| Nov 16, 2010 at 13:54 | answer | added | Bob Terrell | timeline score: 2 | |
| Nov 16, 2010 at 12:19 | comment | added | Willie Wong | +1 @Denis Serre's suggestion. | |
| Nov 16, 2010 at 12:18 | comment | added | Willie Wong | @J.M.: as someone who studies PDEs for a living, I strongly object to your repetition of the "ODE good, PDE bad" mantra. :) @jc: separation of variables for the Laplace-Beltrami operator (and also of the Hamilton-Jacobi flow) is intimately tied to the number of symmetries of the underlying (pseudo-)Riemannian metric. For general operators a sufficient condition for separation of variables in $n$ dimensions is the existence of $n$ non-vanishing, mutually commuting vector fields that commute with your operator. Then you just integrate the holonomic vector fields to get a coordinate system. | |
| Nov 16, 2010 at 12:03 | comment | added | Denis Serre | @jc. Look at Willard Jr Miller. Symmetry and separation of variables. Encyclopedia of Mathematics and its Applications, Vol. 4. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. | |
| Nov 16, 2010 at 8:59 | answer | added | Ryan Reich | timeline score: 5 | |
| Nov 16, 2010 at 7:03 | answer | added | Denis Serre | timeline score: 12 | |
| Nov 16, 2010 at 6:48 | answer | added | john mangual | timeline score: 4 | |
| Nov 16, 2010 at 6:42 | comment | added | j.c. | If I may make a side request: I've always wanted a conceptual explanation of how to think about which coordinate systems admit separation of variables (for the 3D Laplacian operator, say)... references I have found have always seemed a bit opaque and only managed to convince me further that the geometry of quadrics is very special (?). | |
| Nov 16, 2010 at 6:24 | comment | added | J. M. isn't a mathematician | Because it's easier to solve ODEs than PDEs. Finding out that the solution of a PDE in n independent variables can be turned into the solution of n ODEs. should make you heave a sigh of relief at least. | |
| Nov 16, 2010 at 6:03 | history | asked | Yuhao Huang | CC BY-SA 2.5 |