dg.differential geometry - Equivalent definitions of Gaussian curvature

It seems to me that the most difficult step is learning and understanding the intrinsic definition of Gauss curvature. The wikipedia page, http://en.wikipedia.org/wiki/Gaussian_curvature, provides a few different versions.

My favorite way to describe the extrinsic definition is the following: To calculate the Gauss curvature at a point $p$ on the surface: Move the surface so that $p$ is at the origin. Rotate the surface so that the $xy$-plane is tangent to the surface at the origin. The surface near the origin is now given as a graph $z = f(x,y)$, where the partials $ partial_x f $ and $partial_y f$ vanish at the origin. The second fundamental form of the surface at $p$ is given by the Hessian $partial^2 f(0)$, and the Gauss curvature at $p$ is the determinant of the Hessian.

Proving the theorem egregium is now pretty straightforward, especially if you remember that you don't need exact formulas for anything but only second order approximations at the origin.