TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

How 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.

Clairaut--Schwarz theorem for continuous second partial derivatives

Statement

If $f$ is $C^{2}$ on an open subset of $R^{m}$ , then $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ for every pair of coordinate indices.

Facts & Assumptions

Given: A $C^{2}$ scalar field and coordinate indices $i, j < m$ .

[L1]

If $f_{x y}$ exists on a neighbourhood of a point and is continuous at that point, while $f_{y x}$ exists there, then the two values are equal (Peano's mixed-partial theorem from continuity of one mixed partial).

[L2]

The $C^{2}$ condition supplies every ordered partial derivative of length at most two, continuously on the open set ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Proof

technique · direct

1.1L2given

At an arbitrary point, [L2] supplies $\partial_{i} \partial_{j} f$ on a neighbourhood and continuously there, as well as the reversed partial $\partial_{j} \partial_{i} f$ at the point.

2.1step 1.1L1∎

Apply [L1] at an arbitrary point to obtain $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ .

Depends on

Dependency tree · 2 levels

Sources

Mixed partial derivatives (Eremenko)