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:
Throughout this article, the symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms.
Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz 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.
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.
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 dxi ∧ dxj, where i < j. This represents an infinitesimal oriented square parallel to the xi–xj-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 dω 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).
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 p ∈ U 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 p ∈ U, 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 p ∈ U. 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 dα of α = ∑n
j=1 fj dxj. It is given by
To summarize: dα = 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. (As assumed below, one can restrict the sum to the case 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 dα 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.
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
The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M).
For example, a differential 1-form α assigns to each point p ∈ M 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.
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.
The exterior product of a k-form α and an l-form β is a (k + l)-form denoted α ∧ β. 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 an equivalence relation).
The antisymmetry inherent in the exterior algebra means that when α ∧ β is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is viewed as a subspace of the tensor algebra, the tensor product α ⊗ β 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 α ∧ β 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: for instance, if α, β, and γ are any differential forms, 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
On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, vector and covector fields can be identified (the metric is a fiber-wise isomorphism of the tangent space and the cotangent space), and additional operations can thus be defined, such as the Hodge star operator and codifferential (degree −1), which 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, and consider the (noncommutative) algebra of differential operators they generate, which 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:
One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. The existence of pullback homomorphisms in de Rham cohomology depends on the pullback of differential forms.
Differential forms can be moved from one manifold to another using a smooth map. If f : M → N is smooth and ω is a smooth k-form on N, then there is a differential form f∗ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.
To define the pullback, recall that the differential of f is a map f∗ : TM → TN. Fix a differential k-form ω on N. For a point p of M and tangent vectors v1, ..., vk to M at p, the pullback of ω is defined by the formula
More abstractly, if ω is viewed as a section of the cotangent bundle T∗N of N (which thus means that ω is a 1-form), then f∗ω is the section of T∗M defined as the composite map
Pullback respects all of the basic operations on forms:
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. Then, 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 , .
A differential k-form can be integrated over an oriented k-dimensional manifold. More generally, it can be integrated over k-dimensional chains. 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, etc. Simply, a chain parametrizes a domain of integration as a collection of cells (images of cubes or other domains D) that are patched together; to integrate, one pulls back the form on each cell of the chain to a form on the cube (or other domain) and integrates there, which is just integration of a function on Rk, as the pulled back form is simply a multiple of the volume form du1 ∧ ... ∧ duk. For example, given a path γ(t) : [0, 1] → R2, integrating a form on the path is simply pulling back the form to a function on [0, 1] (properly, to a form f(t) dt) and integrating the function on the interval.
for u in the parameter domain D. Then (Rudin 1976) defines the integral of the differential form over S as
where the integral on the right-hand side is the standard Riemann (or Lebesgue) integral over D, and
is the determinant of the Jacobian. The Jacobian exists because S is differentiable.
More generally, a k-form can be integrated over a p-dimensional submanifold, for p ≤ k, to obtain a (k − p)-form. This comes up, for example, in defining the pushforward of a differential form by a smooth map f : M → N by attempting to integrate over the fibers of f.
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 M − N 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.
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, 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.
There is in general no meaningful way to integrate k-forms over subsets for k < n because there is no consistent way to orient k-dimensional subsets; geometrically, a k-dimensional subset can be turned around in place, reversing any orientation but yielding the same subset. 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.
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.
Applications in physicsEdit
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.)
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, A ∧ A = 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.
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