# 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.

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