DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Hessian matrix and critical points of a scalar field

Definition

Let $U \subseteq R^{m}$ be open and let $f : U \to R$ have second partial derivatives. Its Hessian at $a \in U$ is the matrix $H_{f} (a) = (\partial_{i} \partial_{j} f (a))_{i, j < m}$ in the matrix space of The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations, using the multi-index notation of $C^{k}$ maps and multi-index derivative notation in Euclidean space. A point $a$ is critical when its gradient $\nabla f (a)$ is zero, with the gradient convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.

Definition references

The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

$C^{k}$ maps and multi-index derivative notation in Euclidean space

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

Depends on

Dependency tree · 2 levels

Sources

Analysis, Convexity, and Optimization