DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $f : U \subseteq R^{m} \to R$ with the derivatives $D^{α} f (a)$ for $∣ α ∣ \leq k$ , and with $ι : N \to R$ the canonical embedding of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, the Taylor polynomial of degree at most $k$ at $a$ is

$T_{k} f (a; h) : = \sum_{∣ α ∣ \leq k} \frac{D^{α} f (a)}{ι (α!)} h^{α} .$

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 · two levels

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Sources

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