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CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02
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Second-order Taylor expansion f(a+h)=f(a)+∇f(a)⋅h+12hTHf(a)h+o(∥h∥2)

Statement

For a C2 scalar field near a,

f(a+h)=f(a)+∇f(a)⋅h+12⟨Hf(a)h,h⟩+o(∥h∥2).

Facts & Assumptions

Given: A C2 scalar field near a.

[L1]

The degree-two multi-index Taylor formula has a Peano remainder (Multivariable Taylor formula with o(∥h∥k) remainder).

[L2]

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 ⟨x,y⟩=∑k<nxkyk on Rn.

Proof

technique · direct
1.1

Expand the degree-one and degree-two multi-index sums in [L1].

L1algebra
2.1

The degree-one sum is ∇f(a)⋅h, while symmetry of the repeated second derivatives identifies the degree-two sum with 12⟨Hf(a)h,h⟩.

step 1.1L2algebra∎

Depends on

Used by

Dependency tree · two levels

21 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