Closed and exact differential forms

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.

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.

[note 1] given by the derivative of argument on the punctured plane

We can assign arguments in a locally consistent manner around

is not technically a function, the different local definitions of

generates the de Rham cohomology group

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.

is a function then The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to

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.

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.

The term incompressible is used because a non-zero divergence corresponds to the presence of sources and sinks in analogy with a fluid.

The concepts of conservative and incompressible vector fields generalize to n dimensions, because gradient and divergence generalize to n dimensions; curl is defined only in three dimensions, thus the concept of irrotational vector field does not generalize in this way.

The Poincaré lemma states that if B is an open ball in Rn, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n.[1] More generally, the lemma states that on a contractible open subset of a manifold (e.g.,

It makes no real sense to ask whether a 0-form (smooth function) is exact, since d increases degree by 1; but the clues from topology suggest that only the zero function should be called "exact".

The cohomology classes are identified with locally constant functions.

Using contracting homotopies similar to the one used in the proof of the Poincaré lemma, it can be shown that de Rham cohomology is homotopy-invariant.

The first law of thermodynamics can be stated as follows: In any process that results in an infinitesimal change of state where the internal energy of the system changes by an amount

in every circumstance, or in mathematical terms that, the differential form

Caratheodory's theorem[3] further states that there exists an integrating denominator

produced by a stationary electrical current is important.

This case corresponds to k = 2, and the defining region is the full

It corresponds to the current two-form For the magnetic field

one has analogous results: it corresponds to the induction two-form

This equation is remarkable, because it corresponds completely to a well-known formula for the electrical field

At this place one can already guess that can be unified to quantities with six rsp.

four nontrivial components, which is the basis of the relativistic invariance of the Maxwell equations.

, to the three space coordinates, as a fourth variable also the time t, whereas on the right-hand side, in

Finally, as before, one integrates over the three primed space coordinates.