Clairaut--Schwarz theorem for continuous second partial derivatives

Depends on

• Peano's mixed-partial theorem from continuity of one mixed partial
• $C^k$ maps and multi-index derivative notation in Euclidean space

Used by

• Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²) Corollary
• 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
• Wave energy need not be conserved through an open boundary Counterexample
• The Bergman metric form on a bounded domain Definition
• A radial three-dimensional wave reduces to one dimension Example
• Cartan hadamard for hyperbolic space Example
• Great circles as round-sphere geodesics Example
• Factorisation of the one-dimensional wave operator Lemma
• General solution of the one-dimensional wave equation Lemma
• Heat-ball representation formula Lemma
• Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel Lemma
• Spatial and time derivatives pass through heat convolution for positive time Lemma
• The complex Hessian of a C² function dominates that of a minorant at a common minimum Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
• The local wave-energy conservation law Lemma
• The radial recursion between dimensions n and n+2 Lemma
• Uniform Hessian bounds control gradient changes and Taylor remainders Lemma
• Characteristics and the Riccati equation for the spatial derivative Proposition
• Coordinate formula for the Lie bracket Proposition
• Coordinate vector fields commute Proposition
• Round sphere model geometry Proposition
• 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
• Full-memory Bfgs is locally superlinear when the Hessian is Lipschitz Theorem
• Harmonic conjugates exist on homologically simply connected plane domains Theorem
• If a holomorphic function has C² components, then its derivative is holomorphic Theorem
• Instantaneous smoothing of the Lp heat flow Theorem
• Kirchhoff's formula in three dimensions Theorem
• The Bergman metric is positive definite on bounded domains and biholomorphically invariant 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
• The odd-dimensional wave formula by iterated spherical means 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)