I read somewhere that this measure can be equivalent described as "stress", which is interesting; I think that refers to mechanical engineering. So by these terms "curvature" and "stress", one can interpret this intuition (and connect it with the Laplacian's definition as the divergence of a gradient) to measure the relative differences of the rate of change compared to its surroundings' average rate of change.
keenan
@emmaloool Yeah, there's a lot of crossover between geometry and physics... and terms often get tossed around with reckless abandon! I can think of one specific case where there is a relationship between the Laplacian and mechanical stress: for a thin plate, a common bending energy is the integral of the square of the mean curvature $H$, i.e., $\int_\Omega H^2\ dA$, where $\Omega \subset \mathbb{R}^2$ is the domain describing the plate in its undeformed state. Also, if $f: \Omega \to \mathbb{R}^3$ describes the deformed configuration of the plate, then there is a relationship between the Laplacian and mean curvature: $\Delta f = 2HN$, where $N$ is the unit normal of the deformed plate. So, you can write the bending energy for a thin plate as $\int_\Omega |\Delta f|^2\ dA$, or in other words as just the $L^2$ norm $||\Delta f||^2$. Coming back to the original intuition for the Laplacian, this makes some kind of sense: if the position of a thin plate deviates from the average position in a local neighborhood, then it cannot possibly be flat; it must be bending (which causes mechanical stress).
Took me a long time to realize that the link should have an O and not 0, so here's the link for others: https://www.youtube.com/watch?v=oEq9ROl9UmklI
I read somewhere that this measure can be equivalent described as "stress", which is interesting; I think that refers to mechanical engineering. So by these terms "curvature" and "stress", one can interpret this intuition (and connect it with the Laplacian's definition as the divergence of a gradient) to measure the relative differences of the rate of change compared to its surroundings' average rate of change.
@emmaloool Yeah, there's a lot of crossover between geometry and physics... and terms often get tossed around with reckless abandon! I can think of one specific case where there is a relationship between the Laplacian and mechanical stress: for a thin plate, a common bending energy is the integral of the square of the mean curvature $H$, i.e., $\int_\Omega H^2\ dA$, where $\Omega \subset \mathbb{R}^2$ is the domain describing the plate in its undeformed state. Also, if $f: \Omega \to \mathbb{R}^3$ describes the deformed configuration of the plate, then there is a relationship between the Laplacian and mean curvature: $\Delta f = 2HN$, where $N$ is the unit normal of the deformed plate. So, you can write the bending energy for a thin plate as $\int_\Omega |\Delta f|^2\ dA$, or in other words as just the $L^2$ norm $||\Delta f||^2$. Coming back to the original intuition for the Laplacian, this makes some kind of sense: if the position of a thin plate deviates from the average position in a local neighborhood, then it cannot possibly be flat; it must be bending (which causes mechanical stress).