---
tag: 05MC3HND79
kind: statement
role: theorem
name: Poincaré lemma for one-forms on convex sets
aliases: [poincare-lemma]
msc: [58A12]
uses: [0F1AVGX26N, 0MTENFBW3Q, 0AR06W1E1B, 0B22P010D6, 075C629XTH, 0GRM3BS0DD]
sources:
  - url: https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma
    locator: Direct proof
  - url: https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma
    locator: Simply connected case
---

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $\omega = \sum_i f_i \, dx_i$ be a [[closed|0F1AVGX26N]] [[one-form|0MTENFBW3Q]] on $U$ with $C^1$ coefficients.
Then $\omega$ is exact. Explicitly, for a fixed $p \in U$, the function
$$g(x) = \int_0^1 \sum_{i=1}^n f_i\big( p + t (x - p) \big) \, (x_i - p_i) \, dt$$
satisfies $dg = \omega$ on $U$.

## Remarks

Proof: [[differentiate under the integral sign|0AR06W1E1B]], and use $\partial_j f_i = \partial_i f_j$. The integrand of $\partial_j g$ becomes $\frac{d}{dt} \big( t \, f_j(p + t (x - p)) \big)$, which integrates to $f_j(x)$.
In $\mathbb{R}^3$, this is the theorem that a [[curl-free field on a convex set is conservative|0B22P010D6]].
The lemma also holds for closed $k$-forms with $k \ge 1$ on star-shaped open sets. For one-forms, it holds on [[simply connected|075C629XTH]] open sets.
Some condition on $U$ is needed: the angle form on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. See also the [[vortex field|0GRM3BS0DD]].

## Explanations

### L1 human

On a convex set, a one-form that passes the local test for being a derivative really is a derivative.
The proof builds the function by integrating the form along straight segments from one base point.