Alphabeta MathHow 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.
Local and strict local extrema for scalar fields on Euclidean open sets
Definition
Let be open, , and . The point is a local minimum when some Euclidean neighbourhood of satisfies for every ; it is a strict local minimum when the inequality is strict for . Local and strict local maxima reverse these inequalities. Euclidean neighbourhoods use The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement. A critical point is as in The Hessian matrix and critical points of a scalar field.
Depends on
Used by
- Every local minimum of a convex function on an open Euclidean convex set is global Corollary
- A smooth flat refinement has a strict minimum on every line through the origin but no local extremum Counterexample
- Peano's surface has a strict minimum on every line through the origin but no local extremum Counterexample
- Two constraints on a sphere-plane circle, where one multiplier solution is only a local maximum Example
- A constrained local extremum annihilates every velocity of a differentiable parametrization Theorem
- Fermat's theorem: an interior differentiable local extremum has zero gradient Theorem
- Lagrange multipliers for a regular vector-valued level-set constraint Theorem
- The multivariable second-derivative test by definiteness of the Hessian Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Analysis, Convexity, and Optimization (standard reference, not scraped)