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.
The Hessian matrix and critical points of a scalar field
Definition
Let be open and let have second partial derivatives. Its Hessian at is the matrix in the matrix space of The vector space of by matrices over a field, with entrywise operations, using the multi-index notation of maps and multi-index derivative notation in Euclidean space. A point is critical when its gradient is zero, with the gradient convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.
Depends on
Used by
- Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+tfrac12h^TH_f(a)h+o(‖h‖²) Corollary
- The Hessian of a C² scalar field is symmetric Corollary
- A zero Hessian occurs at a strict minimum, a strict maximum, and a saddle Counterexample
- x²+y²(1-x)³ has a unique critical point, a strict local but nonglobal minimum Counterexample
- Local and strict local extrema for scalar fields on Euclidean open sets Definition
- Positive definite, negative definite, semidefinite, and indefinite quadratic forms Definition
- The monkey saddle x³-3xy² has an indefinite higher-order critical point Example
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 78 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Analysis, Convexity, and Optimization (standard reference, not scraped)