# 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.

{% embed url="<https://youtu.be/kk6mETYIgfs>" %}

{% embed url="<https://viadean.notion.site/Laplace-Operator-Derivation-and-Verification-in-Cylindrical-Coordinates-2811ae7b9a3280dcbc76ec5ffc085e0c?source=copy_link>" %}