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Continuous mixed partials of order are invariant under permutations
Statement
Let . If for an open , then every iterated derivative of of order is unchanged by any permutation of its coordinate differentiations.
Facts & Assumptions
Given: A scalar field on and a word of coordinate indices.
Adjacent second coordinate derivatives commute under the hypotheses (Clairaut--Schwarz theorem for continuous second partial derivatives).
A field has every ordered iterated partial derivative through length , continuously on ( maps and multi-index derivative notation in Euclidean space).
Proof
Label the 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.
Assume every reordering with at most inversions leaves the derivative unchanged.
A reordering with inversions has an adjacent inverted pair; exchanging that pair reduces its inversion count by one. If that pair occupies positions in the sequence of differentiation operations, first apply only the operations in positions and call the resulting field . Every ordered partial of through order two is an ordered partial of of length at most , hence is continuous by [L2]; thus and [L1] swaps precisely the operations in positions . Apply the remaining outer operations in positions to this equality; their existence is again supplied by [L2].
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
Used by
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- Basic operations are continuous on Schwartz space Theorem
- Directional Jacobian and Hessian differences retain Taylor and evaluation-noise errors Theorem
- Distributional differentiation is continuous and commutes Theorem
- Forcing residuals determine local inexact-Newton rates Theorem
- Hölder data give a classical Newtonian solution Theorem
- Schwartz space is Fréchet Theorem
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- The closure Hölder spaces are Banach spaces Theorem
Dependency tree · two levels
8 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)