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

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 View the MathML source, where View the MathML source is diagonal. Using the transformation View the MathML source, where f = det(F), the result View the MathML source 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.