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Multivariable Taylor formula with a Lagrange remainder along a line segment

Statement

Let $k\in\mathbb N$ , let $U\subseteq\mathbb R^m$ be open and convex, $a,a+h\in U$ , and $f\in C^{k+1}(U)$ . Write $\iota:\mathbb N\to\mathbb R$ for the canonical-natural map of The multivariable Taylor polynomial in multi-index notation. Then some $\theta\in(0,1)$ satisfies

$f(a+h)=T_kf(a;h)+\sum_{|\alpha|=k+1}\frac{D^\alpha f(a+\theta h)}{\iota(\alpha!)}h^\alpha.$

Facts & Assumptions

Given: The hypotheses of the statement.

[L1]

Convexity keeps the segment $a+th$ in $U$ for $0\le t\le1$ (A convex subset of $\mathbb{R}^m$ contains every line segment between two of its points).

[L2]

On an open interval containing $[0,1]$ , the derivatives of $t\mapsto f(a+th)$ have the multi-index expansion through order $k+1$ (Repeated derivatives along a line expand by the multinomial formula).

[L3]

By The Lagrange and Cauchy forms of Taylor's remainder, if a one-variable function has derivatives through order $k+1$ on $[0,1]$ with the required endpoint continuity, then some $\theta\in(0,1)$ satisfies

$g(1)=\sum_{j=0}^{k}\frac{g^{(j)}(0)}{\iota(j!)}+\frac{g^{(k+1)}(\theta)}{\iota((k+1)!)}$

by the Lagrange remainder formula.

Proof

technique · direct

1.1

Put $I:=\{t\in\mathbb R:a+th\in U\}$ and $g(t):=f(a+th)$ . The set $I$ is open, contains $[0,1]$ by [L1], and is an interval because $U$ is convex. Hence [L2] shows that $g$ has the derivatives through order $k+1$ required by [L3].

L1L2

2.1

Apply [L3] to $g$ between $0$ and $1$ .

step 1.1L3choose

3.1

Substitute the formula of [L2] for $g^{(j)}(0)$ and $g^{(k+1)}(\theta)$ ; the degree- $k$ part is the definition of $T_kf(a;h)$ .

step 2.1L2algebra∎

Depends on

Used by

Multivariable Taylor formula with o(‖h‖ᵏ) remainder Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 80 results over 17 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)