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 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
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.
Examples in low dimensions
Differential forms in and 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 here is
where the subscripts denote partial derivatives. Therefore the condition for 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 and .
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.