CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The Hessian of a $C^2$ scalar field is symmetric

Statement

For a $C^2$ scalar field $f$ , $H_f(a)^T=H_f(a)$ at every point $a$ .

Facts & Assumptions

Given: A $C^2$ scalar field $f$ and a point $a$ .

[L1]

The $(i,j)$ entry of the Hessian is $\partial_i\partial_j f(a)$ (The Hessian matrix and critical points of a scalar field).

[L2]

Continuous second partial derivatives commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

technique · direct

1.1

For every $i,j$ , [L1] and [L2] give $(H_f(a))_{ij}=\partial_j\partial_i f(a)=(H_f(a))_{ji}$ .

L1L2

2.1

Entrywise equality with the transpose proves symmetry.

step 1.1algebra∎

Depends on

Used by

The two-variable Hessian determinant test Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 8 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

Mixed partial derivatives (Eremenko) (standard reference, not scraped)

The Hessian of a C2C^2C2 scalar field is symmetric