DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

How statement and proof provenance work

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The multivariable Taylor polynomial in multi-index notation

Definition

For a natural $k$ , a map $f:U\subseteq\mathbb R^m\to\mathbb R$ with the derivatives $D^\alpha f(a)$ for $|\alpha|\le k$ , and with $\iota:\mathbb N\to\mathbb R$ the canonical embedding of The canonical natural $\iota(n) = n \cdot 1_F$ of a field, the Taylor polynomial of degree at most $k$ at $a$ is

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

The multi-index conventions are those of $C^k$ maps and multi-index derivative notation in Euclidean space, and the displayed sum is the real finite sum of Finite sums and finite products, by recursion. When $m=1$ , this agrees with the polynomial of Taylor polynomials and their remainders.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 18 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)