CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Second-order Taylor expansion $f(a+h)=f(a)+\nabla f(a)\cdot h+\tfrac12h^TH_f(a)h+o(\|h\|^2)$

Statement

For a $C^2$ scalar field near $a$ ,

$f(a+h)=f(a)+\nabla f(a)\cdot h+\frac12\langle H_f(a)h,h\rangle+o(\|h\|^2).$

Facts & Assumptions

Given: A $C^2$ 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 $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ .

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 $\nabla f(a)\cdot h$ , while symmetry of the repeated second derivatives identifies the degree-two sum with $\tfrac12\langle H_f(a)h,h\rangle$ .

step 1.1L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 94 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MAT237 notes: Taylor's theorem in several variables (standard reference, not scraped)

Second-order Taylor expansion f(a+h)=f(a)+∇f(a)⋅h+12hTHf(a)h+o(∥h∥2)f(a+h)=f(a)+\nabla f(a)\cdot h+\tfrac12h^TH_f(a)h+o(\|h\|^2)f(a+h)=f(a)+∇f(a)⋅h+21​hTHf​(a)h+o(∥h∥2)