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:

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

Second Derivative Test Criterion

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

The classification is as follows:

1. Local Minimum: If $D(a,b)>0$ and $f_{xx}(a,b)>0$. The surface is shaped like a bowl opening upwards.
2. Local Maximum: If $D(a,b)>0$ and $f_{xx}(a,b)<0$. The surface is shaped like a dome.
3. 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.
4. 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, ${\lambda }_1$ and ${\lambda }_2$, 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, $K={\lambda }_1{\lambda }_2$.
  • 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, $\frac{1}{2}(f_{xx}+f_{yy})$, gives the mean curvature, $H=\frac{{\lambda }_1+{\lambda }_2}{2}$. The sign of $f_{xx}$ (and $f_{yy}$) when $D>0$ determines if the surface is curving up or down on average. This is related to the Laplacian operator.

Relation to normal curvature

Moreover, for a direction $v$ in the domain $(x,y)$, $v^{T}Hv$ is the normal curvature in disguise.

For the actual normal curvature ${\kappa }_n(v)$ we need to make two corrections, to account for the chosen "bad parametrization" of the surface:

Where:

• correction 1 ($\sqrt{1+|\mathrm{\nabla }u{|}^2}$): Accounts for the fact that the surface is tilted relative to the vertical axis
• correction 2 ($\parallel v{\parallel }_{\text{I}}$): Accounts for the fact that distances on the surface differ from distances in the domain

If you:

1. Reescale time so you're moving at unit speed on the surface (removes correction 2)
2. Rotates the normal direction properly (removes correction 1)

Then $v^{T}Hv$ steps forward and reveals itself as the normal curvature.