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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 = F¢, where (~ is diagonal. Using the transformation X = fl/4~), where f = det(F), the result V2¢ = 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.

Author

I.J. Clark

Publisher

Elsevier Science; Elsevier ; Elsevier Ltd.; Elsevier BV (ISSN 0893-9659)

Published

1999

Language

EN