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.

Laplacian as Divergence of Gradient:

The Laplace operator ( $\nabla^2 \phi$ ) acting on a scalar field $\phi$ is fundamentally defined as the divergence of the gradient ( $\nabla \cdot(\nabla \phi)$ ). This is the starting point for its coordinate-free definition.
General Coordinate Expression:

In arbitrary curvilinear coordinates, the Laplace operator is expressed by the concise and powerful formula:

$$ \nabla^2 \phi=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} g^{a b} \partial_b \phi\right) $$

This formula is valid in any coordinate system (orthogonal or not), including Cartesian, cylindrical, spherical, and general Riemannian spaces.

The Role of the Metric Determinant ( $\sqrt{g}$ ):

The term $\sqrt{ g }$ (the square root of the metric determinant) represents the Jacobian of the coordinate transformation and accounts for the local volume element in the curvilinear space. Its presence ensures the divergence operation correctly accounts for the expansion and contraction of the coordinate grid lines.

Connection to Christoffel Symbols:

The identity $\Gamma_{a b}^b=\partial_a \ln (\sqrt{g})$ is crucial. It shows that the contracted Christoffel symbols, which generally relate to the "curvature" or "non-flatness" of the coordinate system, are directly linked to the change in the local volume element. This link allows the complex terms involving Christoffel symbols in the covariant derivative (divergence) to be neatly reexpressed using the partial derivatives of $\sqrt{ g }$.
Covariant Derivative Implication:

The derivation works by showing that the terms from the partial derivative and the Christoffel symbols in the divergence formula,

$$ \nabla^2 \phi=\partial_a\left(V^a\right)+\Gamma_{a b}^a V^b $$

can be combined using the reverse product rule because the Christoffel term is exactly what's needed to complete the total partial derivative of the weighted vector field $\sqrt{g} V^a$.

$$ \sqrt{g}(\nabla \cdot V )=\partial_a\left(\sqrt{g} V^a\right) $$

This is the core identity for the divergence of a contravariant vector in general coordinates.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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