ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Linear combination
Linear map
Linear form
Distribution
Distributional derivative
Weak derivative
Real matrix function derivative
Euclidean real function derivative
Strong directional derivative
Strong partial derivative
Strongly partially differentiable function
Everyway strongly partially differentiable function
Definition D5622
Hessian real matrix function
Formulation 3
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}