Global analysis, analysis on manifolds Differential forms and exterior differential systems Theorem

Poincaré lemma for one-forms on convex sets

Builds on Closed and exact forms One-form Differentiation under the integral sign Curl-free vector field on a convex set is conservative Simply connected domain Curl-free field that is not conservative

Statement

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $\omega = \sum_i f_i \, dx_i$ be a closed one-form 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, 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. 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 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.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

Euclidean space object
Function concept
Function composition concept
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Limit of a function of several variables concept
Differentiable function of several variables concept
Parametrised curve concept
Partial derivative concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Gradient concept
Chain rule in several variables statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Jacobian matrix definition
Differentiation under the integral sign statement
Iterated derivative definition
Chain rule for a composition with a curve statement
Symmetry of second derivatives statement
C^k function characterised by continuous derivatives statement
Differential form definition
Parametrised curve definition
Schwarz's theorem for C^2 functions statement
Vector field definition
Wedge product of forms definition
Closed curve definition
Rectifiable curve definition
Conservative vector field definition
Tangent vector of a curve definition
Curl definition
Divergence definition
Line integral of a scalar field definition
Simply connected domain definition
Line integral of a vector field definition
Regular curve definition
Scalar potential definition
Curl of a gradient is zero statement
One-form definition
Arc length parametrisation definition
Curl-free vector field on a convex set is conservative statement
Line integral as an integral over the parameter interval statement
Exterior derivative definition
Additivity of line integrals under concatenation statement
Reversing a path changes the sign of a line integral statement
Gradient theorem statement
Closed and exact forms definition
Path independence of line integrals statement
Curl-free field that is not conservative counterexample
Poincaré lemma for one-forms on convex sets statement

Tag	`05MC3HND79`
URI	`m2://05MC3HND79`
Aliases	`poincare-lemma`
Kind	statement, theorem
Writing	W0 stub
MSC	58A12
Status	This copy of the site has no build data. Run `m2 status 05MC3HND79`.
Uses	Closed and exact forms, One-form, Differentiation under the integral sign, Curl-free vector field on a convex set is conservative, Simply connected domain, Curl-free field that is not conservative

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