According to Hessian matrix, It describes the **local curvature** of a function.

AFAIK, for one-variable function $f(x)$, its local curvature is $$\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}},$$ and its Hessian matrix is $$\mathcal{Hess}(f) = [f''],$$ right? And here is my problem, I think the local curvature is not just described by its Hessian matrix, because $f'$ also has its role in it, doesn't it?

And furthermore, for 2-variable function $f(x,y)$, its Hessian matrix is $$\mathcal{Hess}(f) = \left[ \begin{array}{cc} f_{xx}'' & f_{xy}'' \\ f_{xy}'' & f_{yy}'' \end{array} \right].$$ How does it relate to the local curvature of $f(x,y)$?