Poincaré lemma

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The Poincaré lemma is a mathematical theorem and was named after the French mathematician Henri Poincaré .

Exact and closed differential forms

  • A differential form of degree is called closed if applies. The outer derivation denotes .
  • A differential form of degree is called exact if there is a differential form such that it holds. The form is called a potential form of

The potential form is not clearly defined, but only "except for re-calibration" (see below).

Because of this , every exact differential form is also closed. The Poincaré lemma specifies conditions under which the opposite statement also applies. In the proof, there is also a generalization of the lemma: An exact part can be split off from every differential form “by construction”.


The Poincaré lemma says that every closed differential form defined on a star-shaped open set is exact.

The statement can also be formulated more abstractly as follows: For a star-shaped open set , the -th De Rham cohomology vanishes for all :

In the three-dimensional special case, the Poincaré lemma, translated into the language of vector analysis , says that an eddy-free vector field defined on a simply connected area as a gradient of a potential field ( ), a source-free vector field on a convex area through rotation of a vector potential ( ), and a scalar field density ("source density") can be represented as the divergence of a vector field ( ).

Proof (constructive)

Be the point around which is star-shaped. The Poincaré Lemma gives an explicit form, and although having the formula: any form unity can not necessary provided that a form assign, from which the desired potential shape results in unity: This associated shape can by itself define the following figure:


(The roof symbol in the -th column on the right-hand side means that the corresponding differential is omitted.)

Now one shows directly that the following identity applies: what formally corresponds to the product rule of differentiation and splits the properties represented by into two parts, of which the second has the property sought.

Because of the prerequisite and because of this, this initially applies without restricting the generality even without the foremost of the right-hand side, namely because the requirement only considers the form at the zero point, so that, as with the total differential, a function from up to so-called Calibration transformations (see below) can also be deduced.

This leaves only the last term of the above identity, and the required statement follows: with

The given identity also generalizes Poincaré's lemma by breaking down any differential form into an inexact (“anholonomic”) and an exact (“holonomic”) part (the bracketed names correspond to the so-called constraining forces in analytical mechanics ). At the same time, it corresponds to the decomposition of any vector field into a vortex part and a source part.

In the language of homological algebra , a contracting homotopy , e.g. B. contracted to the central point of the star-shaped area considered here .


What is so defined is not the only form whose external differential is. All others differ from one another by the differential of a -form: If and are two such -forms, then there exists a -form such that it holds.

The addition is also known as calibration transformation or re-calibration of .

Application in electrodynamics

The case of a magnetic field generated by a stationary current is known from electrodynamics , with the so-called vector potential . This case corresponds , the star-shaped area being the . The vector of the current density is and corresponds to the current shape. The same applies to the magnetic field : it corresponds to the magnetic flux shape and can be derived from the vector potential:, or . The vector potential corresponds to the potential form. The closeness of the magnetic flux form corresponds to the absence of sources of the magnetic field

Using the Coulomb calibration or as appropriate, the following then applies to i = 1,2,3

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

In this equation u. a. remarkable that it fully corresponds to a well-known formula for the electric field , the Coulomb potential of a given charge distribution with density . At this point it is already assumed that

  • and or
  • and as well
  • and

can be summarized and that the relativistic invariance of Maxwell's electrodynamics results from it, see also electrodynamics .

If you give up the condition of stationarity , the time argument must be added to the space coordinates on the left side of the above equation , while on the right side the so-called "retarded time" must be added. As before, it is integrated using the three spatial coordinates . After all, the speed of light is in a vacuum .

Application in continuum mechanics

In continuum mechanics , the lemma is applied to tensors . B. is needed for the establishment of the compatibility conditions . The starting point is the lemma in the formulation:


The operator "grad" forms the gradient , the vectors are the standard basis of the Cartesian coordinate system with coordinates and Einstein's summation convention was applied, according to which indices occurring twice in a product, here k, are to be summed from one to three, which is also the case in The following should be practiced.

Let us now be given a tensor field whose row vectors are combined with the dyadic product “ ” to form the tensor. The rotation of the transposed tensor vanishes

so that every row vector is rotation-free. Then there is a scalar field for every row vector , the gradient of which it is:

because the gradient of the vector is formed according to:

Thus the second form of the lemma applies:


If the trace of the tensor also disappears, then the vector field is free of divergence:

In this case, the Kronecker delta δ ij is used to calculate :

and the tensor is skew symmetric :

Inside is the permutation symbol . The third form of the lemma follows:


In the literature, a rotation operator is also used, which directly forms the rotation of the row vectors:

With this operator:


  • Otto Forster : Analysis. Volume 3: Integral calculus in R n with applications. 4th edition. Vieweg + Teubner, Braunschweig et al. 2007, ISBN 978-3-528-37252-1 .
  • John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
  • C. Truesdell: Solid Mechanics II in S. Flügge (Ed.): Handbook of Physics , Volume VIa / 2. Springer-Verlag, 1972, ISBN 3-540-05535-5 , ISBN 0-387-05535-5 .