🧄Laplace Operator Derivation and Verification in Cylindrical Coordinates

The Laplace-Beltrami Operator is defined generally in tensor calculus as the divergence of the gradient. When applied to the cylindrical coordinate system ( $\rho, \phi, z$ ), the formula simplifies significantly because the metric $g_{a b}$ is diagonal, meaning only self-coupled terms ( a=b ) survive. The crucial geometric factor is $\sqrt{g}=\rho$, which determines the scale of the differential volume element. Substituting this factor and the inverse metric components into the general formula reveals the origin of the terms in the final expression: the $\rho$ factor is retained inside the radial derivative $\partial_\rho$ as $\rho g^{\rho \rho} \partial_\rho \Phi$, resulting in the characteristic term $\frac{1}{\rho} \partial_\rho\left(\rho \partial_\rho f\right)$, while for the $\phi$ and z components, $\rho$ is independent of the coordinate being differentiated, leading to simple second derivatives like $\frac{1}{\rho^2} \partial_\phi^2 f$. The successful match between the derived formula and the standard vector analysis expression confirms the consistency of the abstract tensor approach with traditional physics formulas.

Laplace Operator Derivation and Verification in Cylindrical Coordinates | Proof and Derivationviadean on Notion