The Poincare Lemma states that on a contractible open subset of Euclidean space, or more generally on any open set that is star-shaped about a point, every closed differential form, one whose exterior derivative vanishes, is also exact, meaning it is itself the exterior derivative of some other form. Named for Henri Poincare, it is a foundational local result underlying de Rham cohomology, showing that the cohomology it measures is entirely a global obstruction, absent on any sufficiently simple region.
Facts
StatementEvery closed p-form on an open ball in R^n is exact for p with 1 <= p <= n. 1 Connections
Sources
1. Poincare lemma (Wikipedia)
Introduction, statement
Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n.
Introduction, history
The lemma was introduced by Henri Poincaré in 1886.
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