📢The General Coordinate Laplacian Formula

The general expression for the Laplace operator on a scalar field in curvilinear coordinates is derived to be the formula, which is established by starting with the definition of the Laplacian as the divergence of the gradient, and then utilizing the crucial tensor identity, which links the contracted Christoffel symbols to the partial derivative of the local volume factor. The identity allows the two components of the divergence (the partial derivative and the Christoffel symbol term) to be combined via the reverse product rule, demonstrating how the local volume factor is necessary to properly account for the expansion or contraction of the coordinate grid lines in the generalized space.

Derivation of the Laplacian Operator in General Curvilinear Coordinates | Proof and Derivationviadean on Notion