TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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 local coordinate formula for the exterior derivative

Statement

Let $(U, x^{1}, \dots, x^{n})$ be a smooth chart on a smooth manifold and $ω$ a smooth $k$ -form on $U$ , with $k \geq 0$ . Summing over increasing $k$ -tuples $I$ , and writing $d x^{I} = d x^{i_{1}} \land \dots \land d x^{i_{k}}$ , if $ω = \sum_{I} ω_{I} d x^{I}$ , then $d ω = \sum_{I} d ω_{I} \land d x^{I} .$

Facts & Assumptions

Given: The chart and smooth form in the statement; for k=0 the empty wedge is 1.

[F2]

The exterior derivative is given by the invariant vector-field formula (The exterior derivative by the invariant vector-field formula).

[F3]

Coordinate vector fields commute (Coordinate vector fields commute).

[F4]

The increasing coordinate wedges give a unique expansion of each smooth differential form (Local coordinate expression for a differential form).

Proof

technique · direct

1.1

Evaluate the invariant formula on coordinate vector fields. By [F3] their brackets vanish. On an increasing $(k + 1)$ -tuple $J = (j_{0}, \dots, j_{k})$ the resulting value is $\sum_{a = 0}^{k} (- 1)^{a} \partial_{j_{a}} ω_{J ∖ j_{a}}$ .

F2F3F4given

2.1

For a function $f$ , [F2] in degree zero gives $d f (\partial_{j}) = \partial_{j} f$ , hence $d f = \sum_{j} \partial_{j} f d x^{j}$ . Evaluating $\sum_{I} d ω_{I} \land d x^{I}$ on $J$ gives exactly the alternating sum in step 1.1. Uniqueness in [F4] proves the formula. For $k \geq n$ both sides vanish in degree $k + 1$ ; for $k = 0$ it is the function formula just established.

F2F4step 1.1algebra∎

Depends on

Used by

Curl and divergence encoded by the exterior derivative Example
Exterior derivatives of coordinate one-forms Example
Lie derivative of an area form and planar divergence Example
The angular one-form on the punctured plane is closed Example
The Euclidean area form is closed Example
Every closed differential form is globally exact False statement
The exterior derivative commutes with pullback Theorem
The exterior derivative is a graded derivation Theorem
The exterior derivative squares to zero Theorem

Dependency tree · two levels

11 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, 2nd ed. (standard reference, not scraped)