Closed and exact differential forms

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 differential form "β". Thus, an "exact" form is in the "image" of "d", and a "closed" form is in the "kernel" 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 of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. 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 information using differential methods.
Examples.
A simple example of a form which is closed but not exact is the 1-form formula_1 given by the derivative of argument on the punctured plane Since formula_2 is not actually a function (see the next paragraph) formula_1 is not an exact form. Still, formula_1 has vanishing derivative and is therefore closed.
Note that the argument formula_2 is only defined up to an integer multiple of formula_6 since a single point formula_7 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 formula_7 counterclockwise around the origin and back to the argument increases by Generally, the argument formula_2 changes by
over a counter-clockwise oriented loop 
Even though the argument formula_2 is not technically a function, the different "local" definitions of formula_2 at a point formula_7 differ from one another by constants. Since the derivative at formula_7 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 upshot is that formula_1 is a one-form on formula_16 that is not actually the derivative of any well-defined function We say that formula_1 is not "exact". Explicitly, formula_1 is given as:
which by inspection has derivative zero. Because formula_1 has vanishing derivative, we say that it is "closed".
This form generates the de Rham cohomology group formula_21 meaning that any closed form formula_22 is the sum of an exact form formula_23 and a multiple of where formula_24 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 R2 and R3 were well known in the mathematical physics 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 formula_27 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 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, "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, meaning that it is the derivative (gradient) of a 0-form (smooth scalar field), called the scalar potential. A closed vector field (thought of as a 1-form) is one whose derivative (curl) vanishes, and is called an irrotational vector field.
Thinking of a vector field as a 2-form instead, a closed vector field is one whose derivative (divergence) vanishes, and is called an incompressible flow (sometimes solenoidal vector field). 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 with respect to "X" given by In particular 
Now define
By the fundamental theorem of calculus we have that
With formula_33 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", formula_35 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.
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 formula_39.
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 [0,1] × "U" to "V", set "F""t"("x") = "F"("t","x"). For formula_22 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 
Formulation as cohomology.
When the difference of two closed forms is an exact form, they are said to be cohomologous to each other. That is, if "ζ" and "η" are closed forms, and one can find some "β" such that
then one says that "ζ" and "η" are cohomologous to each other. Exact forms are sometimes said to be cohomologous to zero. The set of all forms cohomologous to a given form (and thus to each other) is called a de Rham cohomology class; the general study of such classes is known as cohomology. It makes no real sense to ask whether a 0-form (smooth function) is exact, since "d" increases degree by 1; but the clues from topology suggest that only the zero function should be called "exact". The cohomology classes are identified with locally constant functions.
Using contracting homotopies similar to the one used in the proof of the Poincaré lemma, it can be shown that de Rham cohomology is homotopy-invariant.
Application in electrodynamics.
In electrodynamics, the case of the magnetic field formula_48 produced by a stationary electrical current is important. There one deals with the vector potential formula_49 of this field. This case corresponds to , and the defining region is the full formula_50. The current-density vector is It corresponds to the current two-form
For the magnetic field formula_52 one has analogous results: it corresponds to the induction two-form and can be derived from the vector potential formula_53, or the corresponding one-form formula_54,
Thereby the vector potential formula_53 corresponds to the potential one-form
The closedness of the magnetic-induction two-form corresponds to the property of the magnetic field that it is source-free: i.e., that there are no magnetic monopoles.
In a special gauge, formula_58, this implies 
This equation is remarkable, because it corresponds completely to a well-known formula for the "electrical" field formula_61, namely for the "electrostatic Coulomb potential" formula_62 of a "charge density" formula_63. At this place one can already guess that
can be "unified" to quantities with six rsp. four nontrivial components, which is the basis of the relativistic invariance of the Maxwell equations.
If the condition of stationarity is left, on the left-hand side of the above-mentioned equation one must add, in the equations for to the three space coordinates, as a fourth variable also the time "t", whereas on the right-hand side, in the so-called "retarded time", must be used, i.e. it is added to the argument of the current-density. Finally, as before, one integrates over the three primed space coordinates. (As usual "c" is the vacuum velocity of light.)