## Closed and exact differential forms |

In **closed form** is a *α* whose *dα* = 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

For an exact form *α*, *α* = *dβ* 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 *d*^{2} = 0, any exact form is necessarily closed. The question of whether *every* closed form is exact depends on the

- examples
- examples in low dimensions
- poincaré lemma
- formulation as cohomology
- application in electrodynamics
- notes
- footnotes
- references

A simple example of a form which is closed but not exact is the 1-form ^{[note 1]} given by the derivative of

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