Hessian matrix

The Hessian matrix of a twice-differentiable function $f (x, y)$ is the matrix of its second-order partial derivatives. It generalizes the second derivative test from single-variable calculus to multiple dimensions. The Hessian, denoted as $H_{f}$ , is given by:

H_{f} (x, y) = (\begin{matrix} f_{x x} (x, y) & f_{x y} (x, y) \\ f_{y x} (x, y) & f_{y y} (x, y) \end{matrix})

where $f_{x x} = \frac{\partial^{2} f}{\partial x^{2}}$ , $f_{x y} = \frac{\partial^{2} f}{\partial y \partial x}$ , and so on. By Clairaut's theorem, if the second partial derivatives are continuous, then $f_{x y} = f_{y x}$ , making the Hessian a symmetric matrix.

Second Derivative Test Criterion

To classify a critical point $(a, b)$ (where $\nabla f (a, b) = 0$ ), we evaluate the determinant of the Hessian, $D$ , at that point.

D (a, b) = det (H_{f} (a, b)) = f_{x x} (a, b) f_{y y} (a, b) - [f_{x y} (a, b)]^{2}

The classification is as follows:

Local Minimum: If $D (a, b) > 0$ and $f_{x x} (a, b) > 0$ . The surface is shaped like a bowl opening upwards.
Local Maximum: If $D (a, b) > 0$ and $f_{x x} (a, b) < 0$ . The surface is shaped like a dome.
Saddle Point: If $D (a, b) < 0$ . The surface curves up in one direction and down in another, like a horse's saddle or a Pringles chip.
Inconclusive: If $D (a, b) = 0$ , the test fails, and further analysis is needed.

Relation to the shape operator

The Hessian matrix provides a powerful link between calculus and differential geometry, specifically through its connection to the curvature of the surface $z = f (x, y)$ .

At a critical point $(a, b)$ , the tangent plane to the surface is horizontal. At this specific point, the Hessian matrix is a representation of the shape operator (or Weingarten map). The shape operator describes how the surface curves away from its tangent plane.

Principal Curvatures: The eigenvalues of the Hessian at a critical point, $λ_{1}$ and $λ_{2}$ , are the principal curvatures of the surface. These values represent the maximum and minimum normal curvature at that point.
Gaussian Curvature (K): The determinant of the Hessian is the Gaussian curvature, $K = λ_{1} λ_{2}$ .
- If $K > 0$ ( $D > 0$ ), the principal curvatures have the same sign. The surface is locally shaped like a dome (elliptic paraboloid) or a bowl, corresponding to a maximum or minimum.
- If $K < 0$ ( $D < 0$ ), the principal curvatures have opposite signs. The surface is locally shaped like a saddle (hyperbolic paraboloid), corresponding to a saddle point.
Mean Curvature (H): Half the trace of the Hessian, $\frac{1}{2} (f_{x x} + f_{y y})$ , gives the mean curvature, $H = \frac{λ_{1} + λ_{2}}{2}$ . The sign of $f_{x x}$ (and $f_{y y}$ ) when $D > 0$ determines if the surface is curving up or down on average. This is related to the Laplacian operator.