The term differential form (often called alternating differential form ) goes back to the mathematician Élie Cartan Joseph. Differential forms are a fundamental concept of differential geometry. They allow a coordinate- independent integration on general oriented differentiable manifolds.

  • 3.1 Definition
  • 3.2 Features
  • 3.3 coordinate representation of the exterior derivative
  • 3.4 Example
  • 4.1 Inner Product
  • 4.2 Repatriation ( pullback ) of differential forms
  • 4.3 Dual shape and star - operator
  • 5.1 Exact and closed forms
  • 5.2 The de Rham cohomology
  • 5.3 The lemma of Poincaré
  • 5.4 An example from electrodynamics
  • 6.1 Orientation
  • 6.2 Integral of differential forms
  • 6.3 Stokes' theorem


It should be an open subset

  • Of
  • Or more generally of a differentiable submanifold of
  • Or generally a differentiable manifold.

In each of these cases there is

  • The concept of differentiable function on the space of infinitely differentiable functions on is denoted by;
  • The notion of tangent space in a point
  • The concept of directional derivative of a tangent vector and a differentiable function
  • The concept of differentiable vector field on the space of vector fields on be denoted by.

The dual space of the tangent space is called the cotangent space.

Differential forms


A differential form of degree k or short - form is a smooth cut in the -th exterior power of the cotangent bundle of math notation:

This means that each point is mapped to the tangent an alternating multi- linear form; in such a way that smooth vector field function

Smooth ( that is, infinitely differentiable ) is.

Alternatively, one can interpret a form as an alternate, smooth multilinear map. This means that maps vector fields to a function, so that



Alternative by resorting to tensor fields: A form is an alternating covariant tensor field of stage

Space of differential forms

The amount of the k- shapes form a vector space and is referred to. Furthermore, it is

For finite-dimensional manifolds, this sum is finite, since it is for the vector space of the zero vector space. The amount is an algebra with the outer product as multiplication and thus again a vector space. From a topological point of view, this space is also a sheaf.

One can interpret as an element of the external potency; As a result, the outer product is defined (i.e., the product in the outer algebra) pictures

Where by

Is defined pointwise.

This product is graded - commutative, it is

In this case, refers to the degree of k i.e. a shape, it is. That is, the product of two types of odd degree anticommutative is, in all other combinations, the product is commutative.


  • Smooth functions are 0- forms.
  • Pfaffian forms are 1- forms.

Coordinate representation

It is an n-dimensional differentiable manifold. Next is a local coordinate system ( a map). so is

A base of the outer algebra over the cotangent space of, ie

Each differential form has a unique representation on all cards

With suitable differentiable functions on the cards transition areas, the differential form is also well defined. In this view one can readily see that the zero differential form is the only shape.

Exterior derivative

The outer discharge is an operator that assigns a differential form of a differential form. Considering on the amount of the differential shapes, ie the amount of the smooth functions as the outer discharge of the corresponding derivative for usual functions.


The exterior derivative of a form is inductively using the Lie derivative and the Cartan formula

Defined; there is a vector field, the Lie derivative and the establishment of

For example, a 1-form, it is



For vector fields; it denotes the Lie bracket.

The general formula is

Here means the hook in that the corresponding argument is omitted.


The exterior derivative has the following properties:

  • The exterior derivative is a Antiderivation. That is, nonlinear, and the Leibniz rule
  • Be then matches the exterior derivative with the total differential.
  • The exterior derivative respected restrictions. It was open and Then is called the exterior derivative, therefore, also a local operator.

These four properties characterize the exterior derivative completely. That is, one can derive from these properties, the above sum formula. If we add to the exterior derivative, so it is preferable to calculating with the characteristics of the derivation and avoids the above formula.

Coordinate representation of the exterior derivative

Be a point on the manifold. The exterior derivative has at this point, the representation

To express the resulting expressions back by the standard basis, the identities are




  • For valid
  • For n = 3 the coefficients of the differential form form in the similar procedure (rotation ) vector red vector analysis.

Further operations on differential forms

Inner Product

Be a smooth vector field. The inner product is a linear map


Is given. That is, the inner product is a K- form of a (k -1) from shape by the mold is evaluated in a fixed vector field. This illustration is an analogue of the Tensorverjüngung on the space of differential forms. Therefore, this operation is also sometimes called its contraction.

The inner product is a Antiderivation. That is, for and the Leibniz rule

Also applies to the inner product

Transport back ( pullback ) of differential forms

If a differentiable map between differentiable manifolds, then, for the means of retrieved form defined as follows:

It is induced by f Figure of the derivatives, also known as "push -forward ". With the other operations, the retraction is compatible, the following applies:

( pedantically written on the left, on the right side, however ) and

For all

In particular, induces a map

The reversal of the opposite direction of the arrow is observed ( " pull-back ", " cohomology " instead of " homology" ).

Besides, the "pull -back" operations, by differential shapes but can be referred to as I. W. " trivial ."

Dual form and star - operator

Are considered external forms in an n- dimensional space, in which an inner product ( metric ) defined so that an orthonormal basis of the space can be formed. The dual to the outer form of degree k, in this n-dimensional space is a form of ( nk )-form

Both sides were written in oriented form. Formally, the dual form is called by application of the ( Hodge ) * operator. Especially for differential forms in three dimensional Euclidean space we have:

Dx with the 1- form, dy dz It was considered that the sequence here oriented (y, z ), ( x, y) and (z, x) is (cyclic Verstauschungen in (x, y, z) ).

The * symbol is intended to emphasize the fact that this an inner product in the space of shapes is given on an underlying space M, as can be written for two k- forms and as a volume form and the integral

Returns a real number. The dual additive indicates that the two-fold application of a K- shape is obtained again, the K - form - apart from the sign, it must be considered separately. More precisely, for a k- form in an n- dimensional space whose metric has the signature s (s = 1 in Euclidean space, s = -1 in Minkowski space ):

It was shown above, as in the 3- dimensional Euclidean space under external derivative of a 1- form, the 2- form results with the components of the rotation vector of the vector analysis as coefficients. This red vector you can use the * operator now formally write directly as a 1- form. Similar to the * operator to the " translation " of the above- formulated set of Stokes vector analysis is used in the form.

The relativistic Maxwell equations of electrodynamics on a four-dimensional space - time manifold M ( with metric and the determinant of the metric g, and of course the signature of Minkowski space is present, for example, for according to the previously given definition of the pseudo - length ) are denominated, for example, using these symbols:

( the so-called Bianchi identity ) and

Expressed with the electromagnetic field tensor as a 2- form

For example, with the z- component of the vector of the magnetic induction and the current (written as a 3- form)

Here, the Antisymmetrisierungs symbol is ( Levi- Civita symbol ), and the semicolon stands for the covariant derivative. As usual on duplicate indices are summed ( Einstein summation convention ) and there are natural units used ( the speed of light c replaced by 1). By using the * operator, we can write the second set of four Maxwell equations, alternatively, with a 1 - form for the current. From the Maxwell equations one can see that obeying and in electrodynamics completely different equations, so the duality is not a symmetry of the theory. The reason is that the duality reversed electric and magnetic fields in electrodynamics are known but no magnetic monopoles. The free Maxwell equations are obtained for the other hand, have dual symmetry.

The features that the transition to dual thereby resulting in that the electric dynamics of non Euclidean space but the Minkowski space is based, have already been indicated in a previous section and used here.

De Rham cohomology

From the graded algebra can be constructed together with the exterior derivative, a Kokettenkomplex. This is a cohomology is then defined by the usual methods of homological algebra. Georges de Rham showed that this coincides named after him cohomology theory with the singular cohomology. To define the De Rham cohomology, the terms of the exact and the closed differential form are defined first:

Exact and closed forms

One form is called closed if and only if; it is accurate, if there is a shape so that applies. Due to the exact formula, each mold is closed. Note that unity in contrast to exactness is a local property: Is an open cover of such a form if and only closed when the restriction of to is closed for each.

The de Rham cohomology

The factor space

Ie -th de Rham cohomology group - containing information on the global topological structure of

The lemma of Poincaré

More generally, the statement of the lemma for said contractible open subsets U of the proof is constructive, that is, there are explicit examples constructed, which is very important for applications. Note that consists of the locally constant functions, as there is no exact forms 0 by definition. So it is for each

Is closed and exact, it follows

The same applies if is exact and closed. Thus, there is induced pictures

An example from electrodynamics

In electrodynamics implies the lemma of Poincaré that for every pair of electromagnetic fields that can be combined into a two-stage alternating differential form in a four-dimensional so-called Minkowski space, a single-stage vector potential form with exists, a so-called " four-potential " (see also: four-vector ).

Also current and charge densities may be combined to form a four-vector and a corresponding 3- form.

Satisfies the field strength form is equivalent to the first two Maxwell 's equations. The third and fourth of Maxwell's equations yield ( in appropriate units )

Wherein the dual to form (see below).

The potential form is unique only up to an additive additive: and give the same to a calibration shape satisfying, but is otherwise arbitrary. You can use this additional so-called gauge freedom to pointwise to fulfill additional constraints. In electrodynamics one calls, for example, that for all the additional so-called Lorenz condition ( Lorenz gauge ) should apply ( in the four components of this condition is easy). This " gauge fixing " is finally obtained as the unique solution of all four Maxwell's equations, the so-called " retarded potential"

In the transition to the dual is important to note that one does not have to do it with but with a correspondingly different metric. Wherein the invariant transformations Lorentz line element wherein the differential of the operating time, and the summation convention was used. Co-and contravariant four-vector components now differ. But It is true that

Integration theory


Is so called one - form which vanishes at any point, an orientation on with such a form is called oriented. An orientation defined orientations of the tangential and Kotangentialräume: A base of Kotangentialraums at one point was positively oriented if

Applies with a positive number; a basis of the tangent space at a point is positively oriented if


Two orientations are called equivalent if they differ only by a factor everywhere positive; this condition is equivalent to the fact that they define on each tangential or cotangent same orientation.

If connected, then there is either exactly two or no equivalence classes.

Orientable if there exists an orientation of.

Integral of differential forms

It was back and we assume to be an orientation selected. Then there is a canonical integral

For forms is an open subset, a positively oriented basis and

So true

The integral on the right side is the usual Lebesgue integral in

It follows from the transform set that this definition is invariant under change of coordinates.

Stokes' theorem

Is a compact oriented -dimensional differentiable manifold with boundary and you know with the induced orientation, then for every shape

This sentence is a far-reaching generalization of the fundamental theorem of differential and integral calculus.

Is closed, that is, is considered as follows for any precise form i.e. in the relationship:

To illustrate the above property of we often use the formulation with a loop integral:

The integral provides an illustration

If connected, then this map is an isomorphism. You come back with it (see above) to the De Rham cohomology.

Complex differential forms

In the theory of complex differential forms introduced here the calculus is transferred to complex manifolds. This works largely analogous to the definition of the forms described here. However, here are analogous to the complex numbers, the complex spaces of differential forms in two rooms ( real) differential forms

Disassembled. Then, the space is the space of the (p, q ) forms. On these spaces we can define two new derivations analogous to the exterior derivative. These are called Dolbeault and Dolbeault cross - operator, and analogous to de Rham cohomology one can form a cohomology again using the Dolbeault - cross operator. This is called Dolbeault cohomology.