DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 rank of a derivative and constant-rank Euclidean maps

Definition

Let $m, n \geq 1$ , let $U \subseteq R^{m}$ be open, and let $f : U \to R^{n}$ be $C^{1}$ ( $C^{k}$ Euclidean maps and diffeomorphisms). The rank of $f$ at $a \in U$ is ${rank}_{a} f : = rank D f (a),$ where rank is the dimension of the image (Rank and nullity of a linear map with finite-dimensional domain). In the standard bases, this is also the rank of the Jacobian matrix $J f (a)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

For $S \subseteq U$ and $0 \leq r \leq \min {m, n}$ , the map $f$ has constant rank $r$ on $S$ when $rank D f (x) = r$ for every $x \in S$ . On the empty set this condition is vacuous, so it may hold for more than one $r$ ; every assertion that needs a determined rank will assume $S$ is nonempty or specify $r$ .

The rank of a derivative and constant-rank Euclidean maps

Definition

Depends on

Dependency tree · 2 levels

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