Hessian matrix

The matrix of second partial derivatives, $\begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{pmatrix}$ , whose determinant is the second derivative test's discriminant $D$ .

The discriminant $D$ used in the second derivative test is really the determinant of the Hessian matrix $\begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{pmatrix}$ , which is symmetric whenever the mixed partials are continuous (by Clairaut's theorem). Its determinant, $D=f_{xx}f_{yy}-f_{xy}^2$ , packages all three second partials into the single number the classification test relies on.

Examples

For $f(x,y)=x^2-y^2$ , the Hessian is $\begin{pmatrix} 2 & 0 \\ 0 & -2\end{pmatrix}$ , with determinant $-4$ , matching the saddle classification at $(0,0)$ .