In
mathematics, especially
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
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 differential form ''β''. Thus, an ''exact'' form is in the ''
image'' of ''d'', and a ''closed'' form is in the ''
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
'' of ''d''.
For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''.
Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of the domain of interest. On a
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
domain, every closed form is exact by the
Poincaré lemma 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 ...
. More general questions of this kind on an arbitrary
differentiable manifold are the subject of
de Rham cohomology, which allows one to obtain purely
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
information using differential methods.
Examples
A simple example of a form which is closed but not exact is the 1-form
[This is an abuse of notation. The argument is not a well-defined function, and is not the differential of any zero-form. The discussion that follows elaborates on this.] given by the derivative of
argument on the
punctured plane
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also funda ...
Since
is not actually a function (see the next paragraph)
is not an exact form. Still,
has vanishing derivative and is therefore closed.
Note that the argument
is only defined up to an integer multiple of
since a single point
can be assigned different arguments etc. We can assign arguments in a locally consistent manner around but not in a globally consistent manner. This is because if we trace a loop from
counterclockwise around the origin and back to the argument increases by Generally, the argument
changes by
:
over a counter-clockwise oriented loop
Even though the argument
is not technically a function, the different ''local'' definitions of
at a point
differ from one another by constants. Since the derivative at
only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative
[The article covering spaces has more information on the mathematics of functions that are only locally well-defined.]
The upshot is that
is a one-form on
that is not actually the derivative of any well-defined function We say that
is not ''exact''. Explicitly,
is given as:
:
which by inspection has derivative zero. Because
has vanishing derivative, we say that it is ''closed''.
This form generates the de Rham cohomology group
meaning that any closed form
is the sum of an exact form
and a multiple of where
accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a
potential function) being the derivative of a globally defined function.
Examples in low dimensions
Differential forms in R
2 and R
3 were well known in the
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element , so that it is the 1-forms
:
that are of real interest. The formula for the
exterior derivative ''d'' here is
:
where the subscripts denote
partial derivatives. Therefore the condition for
to be ''closed'' is
:
In this case if is a function then
:
The implication from 'exact' to 'closed' is then a consequence of the
symmetry of second derivatives, with respect to ''x'' and ''y''.
The
gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
asserts that a 1-form is exact if and only if the line integral of the form depends only on the endpoints of the curve, or equivalently,
if the integral around any smooth closed curve is zero.
Vector field analogies
On a
Riemannian manifold, or more generally a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, ''k''-forms correspond to ''k''-vector fields (by
duality via the metric), so there is a notion of a vector field corresponding to a closed or exact form.
In 3 dimensions, an exact vector field (thought of as a 1-form) is called a
conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
, meaning that it is the derivative (
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
) of a 0-form (smooth scalar field), called the
scalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
. A closed vector field (thought of as a 1-form) is one whose derivative (
curl) vanishes, and is called an
irrotational vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
.
Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative (
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
) vanishes, and is called an
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. A ...
(sometimes
solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
). The term incompressible is used because a non-zero divergence corresponds to the presence of sources and sinks in analogy with a fluid.
The concepts of conservative and incompressible vector fields generalize to ''n'' dimensions, because gradient and divergence generalize to ''n'' dimensions; curl is defined only in three dimensions, thus the concept of irrotational vector field does not generalize in this way.
Poincaré lemma
The Poincaré lemma states that if ''B'' is an open ball in R
''n'', any smooth closed ''p''-form ''ω'' defined on ''B'' is exact, for any integer ''p'' with .
Translating if necessary, it can be assumed that the ball ''B'' has centre 0. Let ''α''
''s'' be the flow on R
''n'' defined by . For it carries ''B'' into itself and induces an action on functions and differential forms.
The derivative of the flow is the vector field ''X'' defined on functions ''f'' by : it is the ''radial vector field'' . The derivative of the flow on forms defines the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
with respect to ''X'' given by In particular
:
Now define
:
By the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
we have that
:
With
being the
interior multiplication or contraction by the vector field ''X'',
Cartan's formula states that
:
Using the fact that ''d'' commutes with ''L''
''X'',
and ''h'', we get:
:
Setting
:
leads to the identity
:
It now follows that if ''ω'' is closed, i. e. , then , so that ''ω'' is exact and the Poincaré lemma is proved.
(In the language of
homological algebra, ''g'' is a "contracting homotopy".)
The same method applies to any open set in R
''n'' that is
star-shaped about 0, i.e. any open set containing 0 and invariant under ''α''
''t'' for
.
Another standard proof of the Poincaré lemma uses the homotopy invariance formula and can be found in , , and . The local form of the homotopy operator is described in and the connection of the lemma with the
Maurer-Cartan form is explained in .
This formulation can be phrased in terms of
homotopies between open domains ''U'' in ''R''
''m'' and ''V'' in ''R''
''n''. If ''F''(''t'',''x'') is a homotopy from
,1× ''U'' to ''V'', set ''F''
''t''(''x'') = ''F''(''t'',''x''). For
a ''p''-form on ''V'', define
:
Then
:
Example: In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.
If is a closed 1-form on , then . If then and . Set
:
so that . Then must satisfy and . The right hand side here is independent of ''x'' since its partial derivative with respect to ''x'' is 0. So
:
and hence
:
Similarly, if then with . Thus a solution is given by and
:
Alternative notions
Related notions to the Poincaré lemma can be proven in other contexts.
On
complex manifolds, the use of the
Dolbeault operators
and
for
complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifol ...
s, which refine the exterior derivative by the formula
, lead to the notion of
-closed and
-exact differential forms. The local exactness result for such closed forms is known as the
Dolbeault–Grothendieck lemma (or
-Poincaré lemma). Importantly, the geometry of the domain on which a
-closed differential form is
-exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
may be applied) and there exist counterexamples to the lemma even on contractible domains.
[For counterexamples on contractible domains which have non-vanishing first Dolbeault cohomology, see the post https://mathoverflow.net/a/59554.] The
-Poincaré lemma holds in more generality for
pseudoconvex domains.
Using both the Poincaré lemma and the
-Poincaré lemma, a refined
local -Poincaré lemma can be proven, which is valid on domains upon which both the aforementioned lemmas are applicable. This lemma states that
-closed complex differential forms are actually locally
-exact (rather than just
or
-exact, as implied by the above lemmas).
Basic derivation for a 1-form
Let
be an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
star domain
In geometry, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lie ...
and let
be a vantage point of
, so for all
the
line segment