Surface Stability Estimator in IR^3

[Grufey]

Hello,

I'm solving a problem in optics, and I found the next question:

Let $$f:\mathbb{R}^2\rightarrow\mathbb{R}$$ be a function, such as $$(x,y)\rightarrow(z)$$, which is a surface in $$IR^3$$.I need an estimator of the stability of that surface or any indicator of the inclination of the surface in a point. In other words: if in IR, the derivative indicates the variation of the function or the stability, I require another parameter that measures the variation or stability of a surface in IR^3My first thought was to use something similar to the Jacobian, but this is a matrix, not a number. I need a number. Also, I thought that the determinant of the Hessian matrix could be good enough, but I'm unsure as to what the mathematical meaning of that operation really is.

 

[fresh_42]

I would use the Jacobian and a matrix norm. Of course you could as well compute the curvature.