Let $f : \mathbb{R}^N \to \mathbb{R}$ be a D4364: Real function such that

(i)	\begin{equation} f \in \mathcal{C}^2(\mathbb{R}^N \to \mathbb{R}) \end{equation}

The Hessian of $f$ is the D5655: Real matrix function \begin{equation} \mathbb{R}^N \to \mathbb{R}^{N \times N}, \quad x \mapsto \begin{bmatrix} \partial_1 \partial_1 f (x) & \partial_1 \partial_2 f (x) & \cdots & \partial_1 \partial_N f (x) \\ \partial_2 \partial_1 f (x) & \partial_2 \partial_2 f (x) & \cdots & \partial_2 \partial_N f (x) \\ \vdots & \vdots & \ddots & \vdots \\ \partial_N \partial_1 f (x) & \partial_N \partial_2 f (x) & \cdots & \partial_N \partial_N f (x) \end{bmatrix} \end{equation}