The Hessian of a function $f$ of multiple variables $\vec{x}$ is defined as the matrix of second Partial derivatives:

$H[f](\vec{x}) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1 ^2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ . & & . \\ . & & . \\ . & & . \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \cdots &\frac{\partial^2 f}{\partial x_n ^2} \end{bmatrix}$

Definition of Hessian of a function on a Surface

video, defined at a critical point of the function $f$, which turns out when represented as a matrix in the canonical basis equals the standard defintion above