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Clairaut--Schwarz theorem for continuous second partial derivatives
Statement
If is on an open subset of , then for every pair of coordinate indices.
Facts & Assumptions
Given: A scalar field and coordinate indices .
If exists on a neighbourhood of a point and is continuous at that point, while exists there, then the two values are equal (Peano's mixed-partial theorem from continuity of one mixed partial).
The condition supplies every ordered partial derivative of length at most two, continuously on the open set ( maps and multi-index derivative notation in Euclidean space).
Proof
At an arbitrary point, [L2] supplies on a neighbourhood and continuously there, as well as the reversed partial at the point.
Apply [L1] at an arbitrary point to obtain .
Depends on
Used by
- The curl of a curl is the gradient of the divergence minus the Laplacian Corollary
- The Hessian determinant of each C² holomorphic component is nonpositive; a nondegenerate critical point is a saddle Corollary
- The Hessian of a C² scalar field is symmetric Corollary
- Peano's function has unequal mixed partials at the origin Counterexample
- The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
- Continuous mixed partials of order k are invariant under permutations Theorem
- Every plane harmonic function is locally the real part of a holomorphic function Theorem
- Harmonic conjugates exist on homologically simply connected plane domains Theorem
- If a holomorphic function has C² components, then its derivative is holomorphic Theorem
- The C² real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair Theorem
- The curl of the gradient of a C² function vanishes Theorem
- The divergence of the curl of a C² field vanishes Theorem
- The exterior derivative squares to zero 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)