In this demo, we color the surface according to the value of the Hessian at every point. The Hessian exists at every point and serves two purposes. One purpose is to tell us about the shape of a surface. A region of a surface where the Hessian is positive will have a shape resembling that an elliptic paraboloid (bowl-shaped) and a region with a negative hessian will have a shape resembling that of a hyperbolic paraboloid (saddle-shaped). A Hessian of 0 means the shape is neither like that of an elliptic paraboloid nor like that of a hyperbolic paraboloid.

The other purpose of a Hessian is to tell us what kind of a critical point we have if the first partial derivatives are equal to 0. If a given critical point is located in a neighborhood colored in orange, then the Hessian at this critical point is positive, and it has to be a maximum or minimum. If the critical point is in a green neighborhood, then the Hessian is negative, so the critical point is a saddle. If the critical point is in a white neighborhood, then the Hessian is zero and doesn't give any conclusive information.