CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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 $(H_{f} (a))_{i j} = \partial_{j} \partial_{i} f (a) = (H_{f} (a))_{j i}$ .

L1L2

2.1

Entrywise equality with the transpose proves symmetry.

step 1.1algebra∎

Depends on

Used by

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)

The Hessian of a C2C2 scalar field is symmetric