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

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Continuous mixed partials of order $k$ are invariant under permutations

Statement

Let $k \geq 2$ . If $f \in C^{k} (U)$ for an open $U \subseteq R^{m}$ , then every iterated derivative of $f$ of order $k$ is unchanged by any permutation of its coordinate differentiations.

Facts & Assumptions

Given: A $C^{k}$ scalar field on $U$ and a word of $k$ coordinate indices.

[L1]

Adjacent second coordinate derivatives commute under the $C^{2}$ hypotheses (Clairaut--Schwarz theorem for continuous second partial derivatives).

[L2]

A $C^{k}$ field has every ordered iterated partial derivative through length $k$ , continuously on $U$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Proof

technique · induction

1.1basealgebra

Label the $k$ differentiation positions and count inversions of a permutation of these labels. A permutation with zero inversions is the identity, so it leaves the derivative unchanged.

1.2ih

Assume every reordering with at most $n$ inversions leaves the derivative unchanged.

2.1step 1.2L1L2algebra

A reordering with $n + 1$ inversions has an adjacent inverted pair; exchanging that pair reduces its inversion count by one. If that pair occupies positions $r, r + 1$ in the sequence of differentiation operations, first apply only the operations in positions $1, \dots, r - 1$ and call the resulting field $g$ . Every ordered partial of $g$ through order two is an ordered partial of $f$ of length at most $k$ , hence is continuous by [L2]; thus $g \in C^{2} (U)$ and [L1] swaps precisely the operations in positions $r, r + 1$ . Apply the remaining outer operations in positions $r + 2, \dots, k$ to this equality; their existence is again supplied by [L2].

3.1step 1.1step 1.2step 2.1discharge-induction∎

The induction hypothesis applies after the swap in step 2.1, so the original reordering leaves the derivative unchanged. Induction on inversion number proves the claim for every finite permutation.

Depends on

Dependency tree · 2 levels

Sources

Mixed partial derivatives (Eremenko)

Continuous mixed partials of order kk are invariant under permutations