Differential form

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with extension to general k-forms occurring linearly.


Differential calculus[edit]
















for some smooth functions fi on U.


so it will be impossible to find such an f unless

for all i and j.


where is defined so that:

fj dxj
. It is given by



for a collection of functions fi1i2⋅⋅⋅ik. Antisymmetry, which was already present for 2-forms, makes it possible to restrict the sum to those sets of indices for which i1 < i2 < ... < ik−1 < ik.



Intrinsic definitions[edit]

Let M be a smooth manifold. A smooth differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M).

The definition of a differential form may be restated as follows. At any point pM, a k-form β defines an element

where TpM is the tangent space to M at p and Tp*M is its dual space. This space is naturally isomorphic to the fiber at p of the dual bundle of the kth exterior power of the tangent bundle of M. That is, β is also a linear functional


Consequently a differential k-form may be evaluated against any k-tuple of tangent vectors to the same point p of M. For example, a differential 1-form α assigns to each point pM a linear functional αp on TpM. In the presence of an inner product on TpM (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vector Xp. Differential 1-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics.


For a tensor at a point p,





Exterior product[edit]


The antisymmetry inherent in the exterior algebra means that when αβ is viewed as a multilinear functional, it is alternating. However, when the exterior algebra embedded a subspace of the tensor algebra by means of the alternation map, the tensor product αβ is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is


for v, w ∈ TpM.

The exterior product is bilinear: If α, β, and γ are any differential forms, and if f is any smooth function, then


Riemannian manifold[edit]

On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism of the tangent and cotangent spaces. This makes it possible to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as the Hodge star operator and the codifferential , which has degree −1 and is adjoint to the exterior differential d.

Vector field structures[edit]



Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of differential operators they generate is the Weyl algebra and is a noncommutative ("quantum") deformation of the symmetric algebra in the vector fields.

Exterior differential complex[edit]


This complex is called the de Rham complex, and its cohomology is by definition the de Rham cohomology of M. By the Poincaré lemma, the de Rham complex is locally exact except at Ω0(M). The kernel at Ω0(M) is the space of locally constant functions on M. Therefore the complex is a resolution of the constant sheaf , which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of .


Suppose that f : MN is smooth. The differential of f is a smooth map df : TMTN between the tangent bundles of M and N. This map is also denoted f and called the pushforward. For any point pM and any vTpM, there is a well-defined pushforward vector f(v) in Tf(p)N. However, the same is not true of a vector field. If f is not injective, say because qN has two or more preimages, then the vector field may determine two or more distinct vectors in TqN. If f is not surjective, then will be a point qN at which f does not determine any tangent vector at all. Since a vector field on N determines, by definition, a unique tangent vector at every point of N, the pushforward of a vector field does not always exist.

By contrast, it is always possible to pull back a differential form. A differential form on N may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential df : TMTN defines a linear functional on each tangent space of M and therefore a differential form on M. The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

Formally, let f : MN be smooth, and let ω be a smooth k-form on N. Then there is a differential form fω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f. To define the pullback, fix a point p of M and tangent vectors v1, ..., vk to M at p. The pullback of ω is defined by the formula

There are several more abstract ways to view this definition. If ω is a 1-form on N, then it may be viewed as a a section of the cotangent bundle TN of N. Using to denote a dual map, the dual to the differential of f is (df) : TNTM. The pullback of ω may be defined to be the composite

This is a section of the cotangent bundle of M and hence a differential 1-form on M. In full generality, let denote the kth exterior power of the dual map to the differential. Then the pullback of a k-form ω is the composite

Another abstract way to view the pullback comes from viewing a k-form ω as a linear functional on tangent spaces. From this point of view, ω is a morphism of vector bundles

where N × R is the trivial rank one bundle on N. The composite map

defines a linear functional on each tangent space of M, and therefore it factors through the trivial bundle M × R. The vector bundle morphism defined in this way is fω.

Pullback respects all of the basic operations on forms. If ω and η are forms and c is a real number, then






A differential k-form can be integrated over an oriented k-dimensional manifold. When the k-form is defined on an n-dimensional manifold with n > k, then the k-form can be integrated over oriented k-dimensional submanifolds. If k = 0, this is just evaluation of a function at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.

Integration on Euclidean space[edit]

Let U be an open subset of Rn. Give Rn its standard orientation and U the restriction of that orientation. Every smooth n-form ω on U has the form

for some smooth function f : RnR. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of ω to be the integral of f:

Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, dx1dx2 must be the negative of the integral of dx2dx1. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity.

Integration over chains[edit]

Let M be an n-manifold and ω an n-form on M. First, assume that there is a parametrization of M by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism

where DRn. Give M the orientation induced by φ. Then (Rudin 1976) defines the integral of ω over M to be the integral of φω over D. In coordinates, this has the following expression. Fix a chart on M with coordinates x1, ..., xn. Then

Suppose that φ is defined by

Then the integral may be written in coordinates as


is the determinant of the Jacobian. The Jacobian exists because φ is differentiable.

In general, an n-manifold cannot be parametrized by an open subset of Rn. But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of k-dimensional subsets for k < n, and this makes it possible to define integrals of k-forms. To make this precise, it is convenient to fix a standard domain D in Rk, usually a cube or a simplex. A k-chain is a formal sum of smooth embeddings DM. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a k-dimensional submanifold of M. If the chain is

then the integral of a k-form ω over c is defined to be the sum of the integrals over the terms of c:


Integration using partitions of unity[edit]

There is another approach, expounded in (Dieudonne 1972), which does directly assign a meaning to integration over M, but this approach requires fixing an orientation of M. The integral of an n-form ω on an n-dimensional manifold is defined by working in charts. Suppose first that ω is supported on a single positively oriented chart. On this chart, it may be pulled back to an n-form on an open subset of Rn. Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of ω is independent of the chosen chart. In the general case, use a partition of unity to write ω as a sum of n-forms, each of which is supported in a single positively oriented chart, and define the integral of ω to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate k-forms on oriented k-dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before. For example, given a path γ(t) : [0, 1] → R2, integrating a 1-form on the path is simply pulling back the form to a form f(t) dt on [0, 1], and this integral is the integral of the function f(t) on the interval.

Integration along fibers[edit]

Fubini's theorem states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well. Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let M and N be two orientable manifolds of pure dimensions m and n, respectively. Suppose that f : MN is a surjective submersion. This implies that each fiber f−1(y) is (mn)-dimensional and that, around each point of M, there is a chart on which f looks like the projection from a product onto one of its factors. Fix xM and set y = f(x). Suppose that

and that ηy does not vanish. Following (Dieudonne 1972), there is a unique

which may be thought of as the fibral part of ωx with respect to ηy. More precisely, define j : f−1(y) → M to be the inclusion. Then σx is defined by the property that,


is any (mn)-covector for which

The form σx may also be notated ωx / ηy.

Moreover, for fixed y, σx varies smoothly with respect to x. That is, suppose that

is a smooth section of the projection map; we say that ω is a smooth differential m-form on M along f−1(y). Then there is a smooth differential (mn)-form σ on f−1(y) such that, at each xf−1(y),

This form is denoted ω / ηy. The same construction works if ω is an m-form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber f−1(y) is orientable. In particular, a choice of orientation forms on M and N defines an orientation of every fiber of f.

The analog of Fubini's theorem is as follows. As before, M and N are two orientable manifolds of pure dimensions m and n, and f : MN is a surjective submersion. Fix orientations of M and N, and give each fiber of f the induced orientation. Let θ be an m-form on M, and let ζ be an n-form on N that is almost everywhere positive with respect to the orientation of N. Then, for almost every yN, the form θ / ζy is a well-defined integrable mn form on f−1(y). Moreover, there is an integrable n-form on N defined by

Denote this form by

Then (Dieudonne 1972) proves the generalized Fubini formula

It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let α be a compactly supported (mn + k)-form on M. Then there is a k-form γ on N which is the result of integrating α along the fibers of f. The form α is defined by specifying, at each yN, how α pairs against each k-vector v at y, and the value of that pairing is an integral over f−1(y) that depends only on α, v, and the orientations of M and N. More precisely, at each yN, there is an isomorphism

defined by the interior product

If xf−1(y), then a k-vector v at y determines an (mk)-covector at x by pullback:

Each of these covectors has an exterior product against α, so there is an (mn)-form βv on M along f−1(y) defined by

This form depends on the orientation of N but not the choice of ζ. Then the k-form γ is uniquely defined by the property

and γ is smooth (Dieudonne 1972). This form also denoted α and called the integral of α along the fibers of f. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the projection formula (Dieudonne 1972). If λ is any -form on N, then

Stokes' theorem[edit]





Relation with measures[edit]




Even in the presence of an orientation, there is in general no meaningful way to integrate k-forms over subsets for k < n because there is no consistent way to use the ambient orientation to orient k-dimensional subsets. Geometrically, a k-dimensional subset can be turned around in place, yielding the same subset with the reserve orientation; for example, the horizontal axis in a plane can be rotated by a half-circle. Compare the Gram determinant of a set of k vectors in an n-dimensional space, which, unlike the determinant of n vectors, is always positive, corresponding to a squared number. An orientation of a k-submanifold is therefore extra data not derivable from the ambient manifold.




Applications in physics[edit]




The current 3-form is







Applications in geometric measure theory[edit]


See also[edit]




External links[edit]