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The Hessian of a scalar field is symmetric
Statement
For a scalar field , at every point .
Facts & Assumptions
Given: A scalar field and a point .
The entry of the Hessian is (The Hessian matrix and critical points of a scalar field).
Continuous second partial derivatives commute (Clairaut--Schwarz theorem for continuous second partial derivatives).
Proof
For every , [L1] and [L2] give .
Entrywise equality with the transpose proves symmetry.
Depends on
Used by
- Adjoint hypergradient with a measured linear-solve residual Algorithm
- Damped Newton method Algorithm
- Finite-budget adaptive regularization using cubic models Algorithm
- Finite-budget index-one eigenvector-following Newton step Algorithm
- The two-variable Hessian determinant test Corollary
- Validated ordered eigenvalue enclosures for an index-one Hessian Definition
- A curved double well has an exact circular minimum-energy path Example
- Distance hessian and laplacian in space forms Example
- Cubic regularization bounds consecutive unsuccessful trials Lemma
- Paired force evaluations give a second-order Hessian-vector estimate Lemma
- An exact index-one eigenvector-following step is the Newton correction at a saddle Proposition
- A C² function is convex exactly when its Hessian is positive semidefinite Theorem
- A gap-controlled dimer mode gives a local saddle-translation error recurrence Theorem
- A separated dimer mode gives a local inexact saddle error bound Theorem
- Adaptive cubic regularization reaches first-order accuracy in O(ε⁻³ᐟ²) evaluations Theorem
- Damped Newton reaches its local quadratic phase in finitely many continued updates Theorem
- Newton's method has local quadratic convergence under a Lipschitz Hessian Theorem
- Residual bounds for exact and approximate-state adjoint hypergradients Theorem
- Validated Hessian enclosures and a small gradient residual certify a nearby index-one saddle Theorem
Dependency tree · two levels
6 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
- Mixed partial derivatives (Eremenko) (standard reference, not scraped)