From: Stephen Montgomery-Smith Subject: Re: Solving Laplacian on Sphere Date: Sun, 20 Feb 2000 07:59:10 GMT Newsgroups: sci.math.research,sci.math Summary: [missing] Thanks to all who answered. The answer was K(x,y) = constant (1/|x-y| - 1/|y||x-y*|) where y* = y/|y|^2 is the 'reflection' of y through the sphere. It was very helpful to me to know this. Thank you very very much. Stephen Montgomery-Smith wrote: > > Suppose that I wish to solve the equation for f given g: > > Laplacian f = g > > where Laplacian f = sum d^f/dxi^2, and f,g:sphere->R, where > sphere = {(x1,...,xn) : sum xi^2 <= 1}, > and I require that f = 0 on the boundary of the sphere. > > Then I can see that f would be given by an equation like: > > f(x) = integral_sphere K(x,y) g(y) dy. > > I would like an explicit formula in the 3 dimensional case. > -- Stephen Montgomery-Smith stephen@math.missouri.edu 307 Math Science Building stephen@showme.missouri.edu Department of Mathematics stephen@missouri.edu University of Missouri-Columbia Columbia, MO 65211 USA Phone (573) 882 4540 Fax (573) 882 1869 http://www.math.missouri.edu/~stephen ============================================================================== From: "Jonathan Barker" Subject: Re: Solving Laplacian on Sphere Date: Sun, 20 Feb 2000 11:11:04 -0000 Newsgroups: sci.math.research,sci.math Stephen I think I'm right in saying that the function K you seek is the Green's function for your problem. Look up "Green's function" and also "Method of images" for details. This is a well known problem and the solution is well documented in many textbooks. I found a brief treatment which is mostly 2d (but contains some 3d work) on the web. Doubtless there are more if you search. The URL is http://www.mathphysics.com/pde/green/g17.html Hope this helps Jonathan -- Jonathan Barker Research Fellow Dept of Biochemistry and Molecular Biology University College London ucapjab@ucl.ac.uk (work) jandk@easynet.co.uk (home) Stephen Montgomery-Smith wrote in message news:38AC22DF.B0DA4206@math.missouri.edu... > Suppose that I wish to solve the equation for f given g: > > Laplacian f = g > > where Laplacian f = sum d^f/dxi^2, and f,g:sphere->R, where > sphere = {(x1,...,xn) : sum xi^2 <= 1}, > and I require that f = 0 on the boundary of the sphere. > > Then I can see that f would be given by an equation like: > > f(x) = integral_sphere K(x,y) g(y) dy. > > I would like an explicit formula in the 3 dimensional case. > ============================================================================== From: george.ivey@gallaudet.edu (G.E. Ivey) Subject: Re: The Laplacian .. Date: 12 May 2000 10:43:39 -0400 Newsgroups: sci.math Doug McKean wrote: > I've been reading how the Laplacian can be applied to >Maxwell's equation to get to the wave equation. This >is used to describe how moving charge in a wire causes >radiation. Specifically, accelerated charge. >Question - What if any is some physical basis of the >Laplacian? What's going on there? >I use the derivative to find rates, >the curl to figure out solenoidal action in a field, >divergence to figure out ... well ... a divergent field, >the Laplacian to figure out ... how to change something >into a wave equation??? >What is it in applied terms that the Laplacian is >doing to a function? >And why when you read older engineering books, there >seems to be in some areas a real effort to transform >functions by way of the Laplacian? Like it was some >holy Grail in analysis. "Ah, yes, now we've finally >gotten something by applying the Laplacian ..." >I can mathematically do it, I've seen loads of >explanations about how to do it, but what is it? As has been said above, the Laplacian is the simplest differential operator that is invariant under "rigid motions" (translation, rotation, etc.) and so shows up in all forms of physics. As for the second part of your query, I think you are confusing "the Laplacian" with the "Laplace Transform"- a completely different thing.