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

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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

1.1F2F3F4given

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 $\sum_{a = 0}^{k} (- 1)^{a} \partial_{j_{a}} ω_{J ∖ j_{a}}$ .

2.1F2F4step 1.1algebra∎

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.

Depends on

Used by

The closed angular form on the punctured plane is not exact Counterexample
Bigraded complex forms and the Dolbeault operators Definition
Curl and divergence encoded by the exterior derivative Example
Curvature two-form of a connection on a trivial plane bundle 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
Every smooth form is either closed or exact False statement
The de rham cohomology class of a form is defined without closedness False statement
The poincare lemma says every closed form is globally exact False statement
Averaged pullback Whitney moments and uniform near identity Lemma
Form calculus extends locally across a manifold boundary Lemma
Stokes for complex forms on a bounded C1 Euclidean domain Lemma
Stokes theorem for the standard simplex Lemma
The de Rham complex and pullback extend to manifolds with boundary Lemma
Coordinate formula and well-definedness of divergence Proposition
Curvature two-form structure equation Theorem
De rham homotopy formula on a product Theorem
Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
The exterior derivative commutes with pullback Theorem
The exterior derivative is a graded derivation Theorem
The exterior derivative squares to zero Theorem
Zero th de rham cohomology is locally constant functions Theorem

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)