🎬the square root of the determinant of the metric tensor unifies the Divergence of the Gradient in Cu

The visualization explicitly demonstrates the physical necessity of the $\sqrt{g}$ factor in the generalized Laplace (divergence) operator. In polar (curvilinear) coordinates, a volume element defined by fixed coordinate steps ( $\Delta r , \Delta \theta$ ) translates into a physical area that grows proportionally to the radial position $r$ (where $\sqrt{g}=r$ ). Conversely, in Cartesian coordinates, the physical area remains constant ( $\sqrt{g}=1$ ). Therefore, the term $\frac{1}{\sqrt{g}}$ in the formula $\nabla^2 \phi=\frac{1}{\sqrt{g}} \partial_a\left(\sqrt{g} g^{a b} \partial_b \phi\right)$ serves as the essential normalization factor that correctly converts the derivative measured in coordinate space into a measure of flux per unit physical volume (divergence) that is independent of the coordinate system chosen.

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