Open main menu

Differential form

  (Redirected from Differential forms)

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an interval [a, b] in the domain of f:

Similarly, the expression f(x, y, z) dxdy + g(x, y, z) dxdz + h(x, y, z) dydz is a 2-form that has a surface integral over an oriented surface S:

The symbol denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dxdydz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over k-dimensional sets, and is homogeneous of degree k in the coordinate differentials.

The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form, produces a (k + 1)-form. This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem.

The general setting for the study of differential forms is on a differentiable manifold. Differential 1-forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.

Contents

HistoryEdit

Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper.[1]

ConceptEdit

Differential forms provide an approach to multivariable calculus that is independent of coordinates.

IntegrationEdit

A differential k-form can be integrated over a manifold of dimension k. A differential one-form can be thought of as measuring an infinitesimal (oriented) length, or one-dimensional density. A differential two-form can be thought of as measuring an infinitesimal (oriented) area, or two-dimensional density. And so on.

Integration of differential forms is well-defined only on oriented manifolds. An example of a one dimensional manifold is an interval [a, b], and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential one-form f(x) dx over the interval [a, b] (with its natural positive orientation) is

 

which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:

 

This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (b < a), the increment dx is negative. The integrals are negatives of one another because the oriented lengths "dx" have opposite directions.

More generally, an m-form is an oriented density that can be integrated over an m-dimensional oriented manifold. (For example, a 1-form can be integrated over an oriented curve, a 2-form can be integrated over an oriented surface, etc.) If M is an oriented m-dimensional manifold, and M is the same manifold with opposed orientation and ω is an m-form, then one has:

 

These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure μ and integrates over a subset A, without any notion of orientation; one writes   to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details.

Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra. The basic 1-forms are the differentials of the coordinates: dx1, ..., dxn. Each of these represents a covector that measures a small displacement in the corresponding coordinate direction. A general 1-form is a linear combination of these differentials

 

where the   are functions of the coordinates. A differential 1-form is integrated along an oriented curve as a line integral.

The basic two-forms are expressions dxidxj, where i < j. This represents an infinitesimal oriented square parallel to the xixj-plane. A general two-form is a linear combination of these, and it is integrated just like a surface integral.

A fundamental operation defined on differential forms is the exterior product (the symbol is the wedge ). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,

 

because the square whose first side is dx1 and second side is dx2 is to be regarded as having the opposite orientation as the square whose first side is dx2 and whose second side is dx1. The exterior product allows higher dimensional differential forms to be built out of lower-dimensional ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides.

In addition to the exterior product, there is also the exterior derivative operator d. Like the differential of a function, the exterior derivative gives a way of quantifying a differential form's sensitivity to change. In Rn, if ω = f dxa is a k-form, then is a k + 1-form defined by

 

with extension to general k-forms occurring linearly.

This more general approach allows for a more natural coordinate-free approach to integration on manifolds. It also allows for a natural generalization of the fundamental theorem of calculus (see § Stokes' theorem).

Differential calculusEdit

Let U be an open set in Rn. A differential 0-form ("zero-form") is defined to be a smooth function f on U. If v is any vector in Rn, then f has a directional derivative v f, which is another function on U whose value at a point pU is the rate of change (at p) of f in the v direction:

 

(This notion can be extended to the case that v is a vector field on U by evaluating v at the point p in the definition.)

In particular, if v = ej is the jth coordinate vector then v f is the partial derivative of f with respect to the jth coordinate function, i.e., f / ∂xj, where x1, x2, ..., xn are the coordinate functions on U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1, y2, ..., yn are introduced, then

 

The first idea leading to differential forms is the observation that v f (p) is a linear function of v:

 

for any vectors v, w and any real number c. This linear map from Rn to R is denoted dfp and called the derivative of f at p. Thus dfp(v) = ∂v f (p). The object df can be viewed as a function on U, whose value at p is not a real number, but the linear map dfp. This is just the usual Fréchet derivative – an example of a differential 1-form.

Since any vector v is a linear combination vjej of its components, df is uniquely determined by dfp(ej) for each j and each pU, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2, ..., xn are themselves functions on U, and so define differential 1-forms dx1, dx2, ..., dxn. Let f = xi. Since xi / ∂xj = δji, the Kronecker delta function, it follows that

 

 

 

 

 

(*)

The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined "pointwise", so that

 

Applying both sides to ej, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

More generally, for any smooth functions gi and hi on U, we define the differential 1-form α = ∑i gi dhi pointwise by

 

for each pU. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as

 

for some smooth functions fi on U.

The second idea leading to differential forms arises from the following question: given a differential 1-form α on U, when does there exist a function f on U such that α = df? The above expansion reduces this question to the search for a function f whose partial derivatives f / ∂xi are equal to n given functions fi. For n > 1, such a function does not always exist: any smooth function f satisfies

 

so it will be impossible to find such an f unless

 

for all i and j.

The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product on differential 1-forms, the exterior product, so that these equations can be combined into a single condition

 

where is defined so that:

 

This is an example of a differential 2-form. This 2-form is called the exterior derivative of α = ∑n
j=1
fj dxj
. It is given by

 

To summarize: = 0 is a necessary condition for the existence of a function f with α = df.

Differential 0-forms, 1-forms, and 2-forms are special cases of differential forms. For each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as

 

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.

Differential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential (k + 1)-form called the exterior derivative of α.

Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any smooth manifold M. One way to do this is cover M with coordinate charts and define a differential k-form on M to be a family of differential k-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.

Intrinsic definitionsEdit

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

 

By the universal property of exterior powers, this is equivalently an alternating multilinear map

 

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.

The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping

 

For a tensor at a point p,

 

where Sk is the symmetric group on k elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding

 

This map exhibits β as a totally antisymmetric covariant tensor field of rank k. The differential forms on M are in one-to-one correspondence with such tensor fields.

OperationsEdit

As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.

Exterior productEdit

The exterior product of a k-form α and an l-form β is a (k + l)-form denoted αβ. At each point p of the manifold M, the forms α and β are elements of an exterior power of the tangent space at p. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).

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

 

This description is useful for explicit computations. For example, if k = l = 1, then αβ is the 2-form whose value at a point p is the alternating bilinear form defined by

 

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

 
 

It is skew commutative (also known as graded commutative), meaning that it satisfies a variant of anticommutativity that depends on the degrees of the forms: if α is a k-form and β is an l-form, then

 

Riemannian manifoldEdit

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 structuresEdit

On a pseudo-Riemannian manifold, 1-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.

Firstly, each (co)tangent space generates a Clifford algebra, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the metric. This algebra is distinct from the exterior algebra of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anti-commutative ("quantum") deformations of the exterior algebra. They are studied in geometric algebra.

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 complexEdit

One important property of the exterior derivative is that d2 = 0. This means that the exterior derivative defines a cochain complex:

 

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  .

PullbackEdit

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

 

The pullback of a form can also be written in coordinates. Assume that x1, ..., xm are coordinates on M, that y1, ..., yn are coordinates on N, and that these coordinate systems are related by the formulas yi = fi(x1, ..., xm) for all i. Locally on N, ω can be written as

 

where, for each choice of i1, ..., ik, ωi1⋅⋅⋅ik is a real-valued function of y1, ..., yn. Using the linearity of pullback and its compatibility with exterior product, the pullback of ω has the formula

 

Each exterior derivative dfi can be expanded in terms of dx1, ..., dxm. The resulting k-form can be written using Jacobian matrices:

 

Here,   denotes the determinant of the matrix whose entries are  ,  .

IntegrationEdit

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 spaceEdit

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 chainsEdit

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

 

where

 

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:

 

This approach to defining integration does not assign a direct meaning to integration over the whole manifold M. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly triangulated in an essentially unique way, and the integral over M may be defined to be the integral over the chain determined by a triangulation.

Integration using partitions of unityEdit

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 fibersEdit

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,

 

where

 

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' theoremEdit

The fundamental relationship between the exterior derivative and integration is given by the Stokes' theorem: If ω is an (n − 1)-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then

 

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If ω is a closed k-form and M and N are k-chains that are homologous (such that MN is the boundary of a (k + 1)-chain W), then  , since the difference is the integral  .

For example, if ω = df is the derivative of a potential function on the plane or Rn, then the integral of ω over a path from a to b does not depend on the choice of path (the integral is f(b) − f(a)), since different paths with given endpoints are homotopic, hence homologous (a weaker condition). This case is called the gradient theorem, and generalizes the fundamental theorem of calculus. This path independence is very useful in contour integration.

This theorem also underlies the duality between de Rham cohomology and the homology of chains.

Relation with measuresEdit

On a general differentiable manifold (without additional structure), differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1-form dx over the interval [0, 1]. Assuming the usual distance (and thus measure) on the real line, this integral is either 1 or −1, depending on orientation:  , while  . By contrast, the integral of the measure |dx| on the interval is unambiguously 1 (formally, the integral of the constant function 1 with respect to this measure is 1). Similarly, under a change of coordinates a differential n-form changes by the Jacobian determinant J, while a measure changes by the absolute value of the Jacobian determinant, |J|, which further reflects the issue of orientation. For example, under the map x ↦ −x on the line, the differential form dx pulls back to dx; orientation has reversed; while the Lebesgue measure, which here we denote |dx|, pulls back to |dx|; it does not change.

In the presence of the additional data of an orientation, it is possible to integrate n-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, [M]. Formally, in the presence of an orientation, one may identify n-forms with densities on a manifold; densities in turn define a measure, and thus can be integrated (Folland 1999, Section 11.4, pp. 361–362).

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate n-forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, n-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no volume forms on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate n-forms. One can instead identify densities with top-dimensional pseudoforms.

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.

On a Riemannian manifold, one may define a k-dimensional Hausdorff measure for any k (integer or real), which may be integrated over k-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over k-dimensional subsets, providing a measure-theoretic analog to integration of k-forms. The n-dimensional Hausdorff measure yields a density, as above.

CurrentsEdit

The differential form analog of a distribution or generalized function is called a current. The space of k-currents on M is the dual space to an appropriate space of differential k-forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.

Applications in physicsEdit

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of electromagnetism, the Faraday 2-form, or electromagnetic field strength, is

 

where the fab are formed from the electromagnetic fields   and  ; e.g., f12 = Ez/c, f23 = −Bz, or equivalent definitions.

This form is a special case of the curvature form on the U(1) principal bundle on which both electromagnetism and general gauge theories may be described. The connection form for the principal bundle is the vector potential, typically denoted by A, when represented in some gauge. One then has

 

The current 3-form is

 

where ja are the four components of the current density. (Here it is a matter of convention to write Fab instead of fab, i.e. to use capital letters, and to write Ja instead of ja. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the International Union of Pure and Applied Physics, the magnetic polarization vector is called   since several decades, and by some publishers J, i.e. the same name is used for different quantities.)

Using the above-mentioned definitions, Maxwell's equations can be written very compactly in geometrized units as

 

where   denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.

The 2-form  , which is dual to the Faraday form, is also called Maxwell 2-form.

Electromagnetism is an example of a U(1) gauge theory. Here the Lie group is U(1), the one-dimensional unitary group, which is in particular abelian. There are gauge theories, such as Yang–Mills theory, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A. The Yang–Mills field F is then defined by

 

In the abelian case, such as electromagnetism, AA = 0, but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of A and F, owing to the structure equations of the gauge group.

Applications in geometric measure theoryEdit

Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2-forms. A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in Gromov's inequality for complex projective space in systolic geometry.

See alsoEdit

NotesEdit

  1. ^ Cartan, Élie (1899), "Sur certaines expressions différentielles et le problème de Pfaff", Annales Scientifiques de l'École Normale Supérieure: 239–332 

ReferencesEdit

External linksEdit