Describe a 2-D surface by the total/average/maximum amount of curvature

I want to describe a function by the amount of curvature or variability it has. For example the surface

f(x,y)=sin(4πx)cos(7π2y)f(x,y) = \sin(4 \pi x) \cos \left(\frac{7 \pi}{2} y \right)

has more variability than

g(x,y)=sin(πx)cos(π2y)g(x,y) = \sin(\pi x) \cos \left(\frac{\pi}{2} y \right)

on the same region because f(x,y)f(x,y) has a greater frequency. I would like to quantify this amount of variability for these functions and others (h(x,y)=x2y2\left(h(x,y) = x^2 y^2 \right., or j(x,y)=exp(x)exp(y))\left. j(x,y) = \exp(x) \exp(y) \right).




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1 Answer


In Dynamics, time rate of change of acceleration is at times referred to as jerk. I do not know about corresponding name of rate of change of curvature or its next derivative with spatial independent variable. In mechanics of beams these are proportional to shear forces and rate of load but still no name is given / known to me.

In surface theory you can perhaps, if needed device your own suitable differential definition like dH/dl dH/dl on average curvature for a specific direction ll or dK/dA dK/dA as Gauss curvature change per unit area defined by chosen parameters.