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A simplifying transformation for the Laplace-Beltrami operator in curvilinear coordinates

✍ Scribed by I.J. Clark

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
292 KB
Volume
12
Category
Article
ISSN
0893-9659

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✦ Synopsis

In a curvilinear coordinate system with metric tensor G, the Laplace-Beltrami operator V 2 expresses the Laplacian in terms of partial derivatives with respect to the coordinates. This paper describes a simplifying transformation, useful in curvilinear coordinate systems with a nondiagonal G, where the mixed partial derivative terms are problematic. G is expressed as the matrix multiple G = FΒ’, where (~ is diagonal. Using the transformation X = fl/4~), where f = det(F), the result V2Β’ = f-1/4(V~x+Kox+Uox) is obtained, where V~ is the Laplacian in a "straightened-out" coordinate system, perturbed by differential and multiplication operators K0 and Uo. This allows the investigation of partial differential equations in complicated geometries by perturbation methods in simpler geometries. An illustrative example is given.

πŸ“œ SIMILAR VOLUMES

On the absolutely continuous spectrum of
On the absolutely continuous spectrum of the Laplace–Beltrami operator acting on p -forms for a class of warped product metrics
✍ Francesca Antoci πŸ“‚ Article πŸ“… 2006 πŸ› John Wiley and Sons 🌐 English βš– 270 KB πŸ‘ 1 views

## Abstract We explicitely compute the absolutely continuous spectrum of the Laplace–Beltrami operator for __p__ ‐forms for the class of warped product metrics __dΟƒ__ ^2^ = __y__ ^2__a__^ __dy__ ^2^ + __y__ ^2__b__^ __dΞΈ__ ^2^, where __y__ is a boundary defining function on the unit ball __B__ (0,