For a scalar-valued function $f$ of three variables $x,y,z$, the [[Jacobian#Jacobian Vector|Jacobian vector]] is defined as follows: $ f(x,y,z) \mapsto J\Big[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\Big] $ To compute the Hessian $\mathbf H$ , we differentiate each component of the Jacobian vector w.r.t. $x,y,z$ in turn. The first row of $\mathbf H$ shows $J$ differentiated w.r.t. $x$. The proceeding differentiated vectors w.r.t. $y$ and $z$ are stacked below. $ \mathbf H= \begin{bmatrix} \frac{\partial^2f}{\partial x^2} & \frac{\partial^2f}{\partial x \partial y} & \frac{\partial^2f}{\partial x \partial z} \\[4pt] \frac{\partial^2f}{\partial y \partial x} & \frac{\partial^2f}{\partial y^2} & \frac{\partial^2f}{\partial y \partial z} \\[4pt] \frac{\partial^2f}{\partial z\partial x} & \frac{\partial^2f}{\partial z\partial y} & \frac{\partial^2f}{\partial z^2} \end{bmatrix} $ **Shape:** The Hessian is a square matrix that contains all second-order partial derivatives of $f$. **Symmetry:** If $f$ is twice differentiable, then the order of the differentiation does not matter. This implies that the Hessian is symmetric around its diagonal. $ \frac{\partial ^2 f}{\partial x \,\partial y}= \frac{\partial ^2 f}{\partial y \,\partial x}$ When we have identified a point with zero slope $J=[0, \dots ,0]$, the Hessian tells us if that point is a minimum, maximum or saddle point. This is because second-order-derivatives measure the [[Identify Convex and Concave Functions|convexity or concavity]]. **Determinant:** In a $2 \times 2$ matrix the determinant is $(ad-bc)$. However, in bigger matrices the computation is more complex. $ \det = (ad-bc) \impliedby \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ Possible scenarios: - If determinant of the Hessian $\det >0$, the respective $[x,y,z]$ point is a minimum OR maximum - If first element of the Hessian $f_{xx}>0$, the respective $[x,y,z]$ point is a minimum. - If determinant of the Hessian $\det <0$, the respective $[x,y,z]$ point is a saddle point. > [!note:] > The determinant of a Hessian is also called the Discriminant.