Laplacian Operator in Spherical Coordinates, an Alternative Derivation

An alternative method for obtaining the Laplacian operator ∇2 in the spherical coordinate system from the Cartesian coordinates is described. The procedure consists of three steps: (1) The transformation from plane Cartesian coordinates to plane polar coordinates is accomplished by a simple exercise in the theory of complex variables. (2) The transition to cylindrical coordinates is made by utilizing the result of step (1). (3) The transformation from cylindrical to spherical coordinates follows by a second application of the result in step (1) and the evaluation of a first derivative. The great gain over the brute-force method is that the calculation of the second-order derivative is bypassed, thus avoiding most of the chain rule labor.