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Second-order Taylor expansion
Statement
For a scalar field near ,
Facts & Assumptions
Given: A scalar field near .
The degree-two multi-index Taylor formula has a Peano remainder (Multivariable Taylor formula with remainder).
Gradient, Hessian, and Euclidean inner-product notation are defined in The Hessian matrix and critical points of a scalar field and The Euclidean inner product on .
A scalar field on an open neighborhood has equal mixed second partial derivatives: (Clairaut--Schwarz theorem for continuous second partial derivatives).
Proof
Expand the degree-one and degree-two multi-index sums in [L1].
The degree-one sum is , while symmetry of the repeated second derivatives identifies the degree-two sum with .
Depends on
Used by
- A second-order Taylor polynomial computed from gradient and Hessian data Example
- A regular level set realizes every compatible second-order jet Lemma
- A strictly pseudoconvex boundary point has a local holomorphic separator Lemma
- Peak functions at strongly pseudoconvex boundary points, by a dbar correction Lemma
- Stationary phase with a compactly supported amplitude Lemma
- The Hessian is negative semidefinite at an interior local maximum Lemma
- Continuation trajectories are compact up to breaking Theorem
- Continuous ball-mean-value functions are harmonic Theorem
- Dynkin formula for bounded Brownian stopping Theorem
- Full-memory Bfgs is locally superlinear when the Hessian is Lipschitz Theorem
- Laplace asymptotic under a unique nondegenerate interior maximum Theorem
- MFCQ identifies the linearized and Bouligand tangent cones Theorem
- Multidimensional Ito formula for Brownian-driven processes Theorem
- One-dimensional Ito formula Theorem
- Second-order correction reduces the active constraint residual to cubic order Theorem
- Second-order necessity on the critical cone under LICQ Theorem
- Strict second-order positivity on the critical cone gives a strict local minimum Theorem
- The multivariable second-derivative test by definiteness of the Hessian Theorem
Dependency tree · two levels
23 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
- MAT237 notes: Taylor's theorem in several variables (standard reference, not scraped)