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 ( = 0), 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, α = for some differential form β of one-lesser degree than α. The form β is called a "potential form" or "primitive" for α. Since d2 = 0, β is not unique, but can be modified by the addition of the differential of a two-step-lower-order form.

Because d2 = 0, any exact form is automatically 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

Vector field corresponding to dθ.

A simple example of a form which is closed but not exact is the 1-form given by the derivative of argument on the punctured plane . [note 1] 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 "". [note 2]

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.