PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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.

Smoothness of a tensor field is equivalent to smooth coordinate components

Statement

A type $(r, s)$ tensor field is smooth if and only if, in every smooth chart, its coordinate component functions are smooth.

Facts & Assumptions

Given: A type $(r, s)$ tensor field $T$ on a smooth manifold $M$ .

[F1]

A smooth tensor field is a smooth section of the tensor bundle (A smooth tensor field).

[L1]

The tensor bundle is a smooth vector bundle with the standard tensor-coordinate trivializations (Tensor transition laws define a smooth vector bundle).

[L2]

Smoothness of a section is equivalent to smoothness of its local component functions (Smoothness of a section is equivalent to smooth local components).

Proof

technique · direct

1.1

By [F1] and [L1], a tensor field is a section of a smooth vector bundle whose local bundle coordinates are exactly the tensor coefficients relative to the chart bases $\partial / \partial x^{i}$ and $d x^{i}$ .

F1L1given

2.1

Applying [L2] to those trivializations shows that the section is smooth exactly when each local coefficient function is smooth.

L1L2step 1.1

3.1

Therefore smoothness of a tensor field is equivalent to smoothness of its coordinate components.

step 2.1∎

Depends on

Used by

Tensor components do not transform as independent scalar functions False statement
Differential forms form a graded commutative algebra Proposition
Tensor products and contractions of smooth tensor fields are smooth Proposition

Dependency tree · two levels

12 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

Will J. Merry, Differential Geometry (standard reference, not scraped)