In mathematics, the Poincaré lemma gives a sufficient condition for a
closed differential form
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (); and an exact form is a differential form, ''α'', that is the exterior derivative of another di ...
to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R
''n'' is exact for ''p'' with . The lemma was introduced by
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
in 1886.
Informal Discussion
Especially in
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, the Poincaré lemma also says that every closed 1-form on a
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
open subset in
is exact.
In simpler terms, it means that if a differential form is closed in a region that can be shrunk to a point, then it can be written as the derivative of another form; i.e. if dα = 0 on a simplely connected region, we can always find α = dβ; therefore we have d(dβ) = 0, expressed simply as d
2 = 0. This concept is used in mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a 2-form) and you take its boundary (a 1-form, a curve), then the boundary of that boundary (a 0-form, a point) is an empty set.
In electromagnetism, magnetic fields can be described using a vector potential, and the Poincaré lemma helps in finding such potentials when the magnetic field is "well-behaved" (i.e., when the magnetic field is not due to a monopole),
Gauss's law for magnetism states that the total magnetic flux through a closed surface is always zero, which implies that magnetic monopoles, if they exist, are not isolated but must be accompanied by other magnetic charges.
In the language of
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
, the Poincaré lemma says that the ''k''-th
de Rham cohomology group of a
contractible open subset of a manifold ''M'' (e.g.,
) vanishes for
. In particular, it implies that the de Rham complex yields a
resolution of the
constant sheaf
In mathematics, the constant sheaf on a topological space X associated to a set (mathematics), set A is a Sheaf (mathematics), sheaf of sets on X whose stalk (sheaf), stalks are all equal to A. It is denoted by \underline or A_X. The constant presh ...
on ''M''. The
singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma ''does not follow'' from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the
de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration.
The Poincaré lemma is also a special case of the
homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
Proofs
A standard proof of the Poincaré lemma uses the homotopy invariance formula (cf. see the proofs below as well as
Integration along fibers#Example). The local form of the homotopy operator is described in and the connection of the lemma with the
Maurer-Cartan form is explained in .
Direct proof
The Poincaré lemma can be proved by means of
integration along fibers.
(This approach is a straightforward generalization of constructing a primitive function by means of integration in calculus.)
We shall prove the lemma for an open subset
that is
star-shaped or a cone over