Hessian Determinant
Text The Hessian determinant of a function f(x,y) is defined as
H(x,y) = f_{xx}(x,y)f_{yy}(x,y)  f_{xy}(x,y)f_{yx}(x,y).
The Second Partials Test states that if a function f(x,y) has continuous second partials and f_{x}(x_{0},y_{0}) = 0 and f_{y}(x_{0},y_{0}) = 0, then
1. H > 0 and f_{xx}(x_{0},y_{0}) > 0 implies (x_{0},y_{0}) is a local minimum;
2. H > 0 and f_{xx}(x_{0},y_{0}) < 0 implies (x_{0},y_{0}) is a local maximum;
3. H < 0 implies (x_{0},y_{0}) is a saddle point;
4. H = 0 then the test is inconclusive.
Demos
Hessian determinant
 
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 (bowlshaped) and a region with a negative hessian will have a shape resembling that of a hyperbolic paraboloid (saddleshaped). 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.

Hessian determinant
 
This example shows how the sign of the Hessian affects the shape of a region of a surface.

Exercises 1. For each of the following functions, find the sign of the Hessian determinant at (x,y) = (0,0) as a function of c. Verify your results using the demo.
 f(x,y) = x^{2} + cy^{2}
 f(x,y) = x^{2} + cxy
 f(x,y) = sin(x) + c cos(y)
2. Is it possible to have a surface with two or more distinct "saddles" and no "bowls"? Why or why not?
3. How could you describe a mountain range using Hessian determinants?
