🎬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 ( ) 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 ( ), 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.
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