DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Dependency tree · 2 levels

Sources

MAT237 notes: Taylor's theorem in several variables