Exact form

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 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 = dβ 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.

The simplest example of a form which is closed but not exact is the 1-form "dθ" (quotes because it is not the derivative of a globally defined function), defined on the punctured plane ${\displaystyle \mathbf {R} ^{2}\setminus \{0\},}$ which is locally given as the derivative of the argument—note that argument is locally but not globally defined, since a loop around the origin increases (or decreases, depending on direction) the argument by 2π, which corresponds to the integral:

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