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Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$

Statement

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

$f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} ⟨ H_{f} (a) h, h ⟩ + o (∥ h ∥^{2}) .$

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 $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[L3]

A $C^{2}$ scalar field on an open neighborhood has equal mixed second partial derivatives: $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

technique · direct

1.1L1algebra

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

2.1step 1.1L2L3algebra∎

The degree-one sum is $\nabla f (a) \cdot h$ , while symmetry of the repeated second derivatives identifies the degree-two sum with $\frac{1}{2} ⟨ H_{f} (a) h, h ⟩$ .

Depends on

Dependency tree · 2 levels

Sources

MAT237 notes: Taylor's theorem in several variables