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Concept
Closed and exact differential forms
Summary
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (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 of d, and a closed form is in the kernel of d.
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, 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.
A simple example of a form that is closed but not exact is the 1-form given by the derivative of argument on the punctured plane . 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 "".
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