🧄Derivation of the Laplacian Operator in General Curvilinear Coordinates

The general expression for the Laplace operator ( $\nabla^2 \phi$ ) on a scalar field $\phi$ in curvilinear coordinates is derived to be $\nabla^2 \phi=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} g^{a b} \partial_b \phi\right)$. This formula is established by starting with the definition of the Laplacian as the divergence of the gradient, $\nabla \cdot(\nabla \phi)$, and then utilizing the crucial tensor identity $\Gamma_{a b}^b=\partial_a \ln (\sqrt{g})$, which links the contracted Christoffel symbols to the partial derivative of the local volume factor ( $\sqrt{g}$ ). 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 $\sqrt{g}$ 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