PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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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.

Smoothness of a section is equivalent to smooth local components

Statement

Let $E \to M$ be a smooth vector bundle, let $U \subseteq M$ be open, and let $(s_{1}, \dots, s_{r})$ be a local frame on $U$ . A local section $σ$ on $U$ is smooth if and only if there are smooth functions $f_{1}, \dots, f_{r} : U \to R$ with

$σ (p) = \sum_{i = 1}^{r} f_{i} (p) s_{i} (p) (p \in U) .$

Facts & Assumptions

Given: A local frame $(s_{1}, \dots, s_{r})$ on $U$ and a local section $σ : U \to E ∣_{U}$ .

[L1]

A local frame determines a local trivialization, and conversely (Local frames and local trivializations are equivalent data).

[L2]

Smoothness is local on the source (Smoothness is local on the source).

Proof

technique · direct

1.1

By [L1], the chosen frame gives a bundle chart $Φ : E ∣_{U} \to U \times R^{r}$ in which $s_{i} (p)$ corresponds to the $i$ -th standard basis vector. Therefore $Φ (σ (p)) = (p, f_{1} (p), \dots, f_{r} (p))$ exactly when $σ (p) = \sum_{i} f_{i} (p) s_{i} (p)$ .

L1given

2.1

In this chart, $σ$ is smooth exactly when the coordinate map $p \mapsto (f_{1} (p), \dots, f_{r} (p))$ is smooth. Equivalently, each component $f_{i}$ is smooth, and the criterion is local on $U$ by [L2].

L2step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)