Closed and exact differential forms

The Poincaré Lemma is a set of mathematics and was named after the French mathematician Henri Poincaré.

Exact and closed differential forms

• A differential form of degree is called closed if applies. It denotes the exterior derivative.
• In addition, a differential form of the degree is exactly when there is a differential form so that the following applies. The form referred to as the potential of the form of

Statement

It states that in any star-shaped open set of a differentiable manifold the -th de Rham cohomology vanishes for all, ie:

In other words, In each of the star-shaped open set each closed differential form is exact.

In the three-dimensional special case of the Poincaré Lemma states in the language of vector analysis convicted that a defined on a simply connected domain irrotational vector field - for example, the electrostatic field - as the gradient of a potential field ( ), a source- free vector field - for example, the magnetic induction - by rotating a vector potential (), and a scalar field as a divergence of a vector field can be represented ().

Remark

The Poincaré Lemma is also such a form explicitly, with the following formula: For any k- forms, can be an associated " potential form " defined by the following (k -1 )-form -valued mapping:

, with

Now one shows directly that:

Because of the requirement that is already the required statement, with

The thus defined is not the only form whose outer differential is. But all other differ at most by a differential form of one another: and two of such forms, there exists a form such that

Applies (see gauge invariance ).

In the language of homological algebra is a contracting homotopy.

Application in electrodynamics

The case of a magnetic field generated by a stationary current in the electrical dynamics known with the so-called vector-potential corresponding to this case, wherein the star-shaped area is. The vector of the current density and corresponds to the current form for the magnetic field applies analogously: it corresponds to the magnetic flux shape and can be derived from the vector potential :, or. This corresponds to the vector potential, the potential form, the unity of the magnetic flux shape corresponds to the source of freedom of the magnetic field

Using the Coulomb gauge, then for i = 1,2,3

It is a natural constant, the so-called magnetic field constant.

In this equation is i.a. remarkable that it fully complies with a known formula for the electric field, the Coulomb potential of a given charge distribution with the density. It is assumed at this point already that

• And or

• And and

• And

Can be summarized and that the relativistic invariance of Maxwell's electrodynamics follows that, see electrodynamics.

If one gives the condition of stationarity, the time argument to be added on the left side of the above equation with the spatial coordinates, while on the right side in the so-called " sustained release time" to complete. It is in this case as previously integrated over the three spatial coordinates. Finally, the speed of light in vacuum.