Why is curvature quadratic?

Question

In my textbook, it describes the hessian being the local quadratic of the optimization landscape. The hessian is synonymous with curvature.

I'm wondering why curvature is quadratic?

Answer 1

The Hessian is not synonymous with curvature. The quadratic (involving Hessian) is the lowest order term in a multivariate Taylor series expansion which captures any curvature. Higher order terms than quadratic (Hessian) are not usually used in optimization. But higher order methods can be used in optimization, and sometimes are.

Answer 2

Slope is obtained by first differentiation... like with differential operator $D$. Curvature is obtained by second differentiation, like with the operator $D^2$.

Second order terms and their products like $rt-s^2$ in Hessian vanish for flat planes and the like. It is positive for positive (double or Gauss) curvature surfaces and negative for negative curvature surfaces. Hessian indicates sign of curvature.