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

Abstract

In a curvilinear coordinate system with metric tensor G, the Laplace-Beltrami operator ▿ 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 G ̌ , where G ̌ is diagonal. Using the transformation χ = f 1 4 ψ , where f = det( F), the result ▿ 2ψ = f − 1 4 (▿ 0 2χ+K 0χ+U 0χ) is obtained, where ▿ 0 2 is the Laplacian in a “straightened-out” coordinate system, perturbed by differential and multiplication operators K 0 and U 0. This allows the investigation of partial differential equations in complicated geometries by perturbation methods in simpler geometries. An illustrative example is given.