🎬the Laplace-Beltrami Operator measures the local curvature of the field by detecting peaks and troug

The dynamic visualization effectively proves that the Laplace-Beltrami Operator ( $\nabla^2 \Phi$ ) is the mathematical measure of a scalar field's local curvature. By tracking the sign of the Laplacian as the probe moves over the Gaussian peak, the demo clearly shows that a negative Laplacian signals a peak (maximum), where the field curves downward. Conversely, a positive Laplacian signals a trough or shoulder (concave-up region). Crucially, the Laplacian approaches zero precisely at the inflection points ( $r=\sqrt{2}$ ), where the field transitions from one curvature type to the other, visually confirming the operator's fundamental role in detecting maxima, minima, and inflection points regardless of the underlying coordinate system.

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