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

AbstractIn 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 = FǦ, where Ǧ is diagonal. Using the transformation χ = f14ψ, where f = det(F), the result ▿2ψ = f−14(▿02χ+K0χ+U0χ) is obtained, where ▿02 is the Laplacian in a “straightened-out” coordinate system, perturbed by differential and multiplication operators K0 and U0. This allows the investigation of partial differential equations in complicated geometries by perturbation methods in simpler geometries. An illustrative example is given