Английская Википедия:Differential form

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Шаблон:Short description

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, 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 Шаблон:Math is an example of a [[1-form|Шаблон:Math-form]], and can be integrated over an interval Шаблон:Math contained in the domain of Шаблон:Math:

<math>\int_a^b f(x)\,dx.</math>

Similarly, the expression Шаблон:Math is a Шаблон:Math-form that can be integrated over a surface Шаблон:Math:

<math>\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz).</math>

The symbol Шаблон:Math denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a Шаблон:Math-form Шаблон:Math represents a volume element that can be integrated over a region of space. In general, a Шаблон:Mvar-form is an object that may be integrated over a Шаблон:Mvar-dimensional manifold, and is homogeneous of degree Шаблон:Mvar in the coordinate differentials <math>dx, dy, \ldots.</math> On an Шаблон:Math-dimensional manifold, the top-dimensional form (Шаблон:Math-form) is called a volume form.

The differential forms form an alternating algebra. This implies that <math>dy\wedge dx = -dx\wedge dy</math> and <math>dx\wedge dx=0.</math> This alternating property reflects the orientation of the domain of integration.

The exterior derivative is an operation on differential forms that, given a Шаблон:Math-form <math>\varphi</math>, produces a Шаблон:Math-form <math>d\varphi.</math> This operation extends the differential of a function (a function can be considered as a Шаблон:Math-form, and its differential is <math>df(x)=f'(x)dx</math>). This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.

Differential Шаблон:Math-forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and Шаблон:Math-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.

History

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] Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).

Concept

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

Integration and orientation

A differential Шаблон:Mvar-form can be integrated over an oriented manifold of dimension Шаблон:Mvar. A differential Шаблон:Math-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential Шаблон:Math-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.

Integration of differential forms is well-defined only on oriented manifolds. An example of a 1-dimensional manifold is an interval Шаблон:Math, and intervals can be given an orientation: they are positively oriented if Шаблон:Math, and negatively oriented otherwise. If Шаблон:Math then the integral of the differential Шаблон:Math-form Шаблон:Math over the interval Шаблон:Math (with its natural positive orientation) is

<math>\int_a^b f(x) \,dx</math>

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

<math>\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx.</math>

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 (Шаблон:Math), the increment Шаблон:Math is negative in the direction of integration.

More generally, an Шаблон:Mvar-form is an oriented density that can be integrated over an Шаблон:Mvar-dimensional oriented manifold. (For example, a Шаблон:Math-form can be integrated over an oriented curve, a Шаблон:Math-form can be integrated over an oriented surface, etc.) If Шаблон:Mvar is an oriented Шаблон:Mvar-dimensional manifold, and Шаблон:Math is the same manifold with opposite orientation and Шаблон:Mvar is an Шаблон:Mvar-form, then one has:

<math>\int_M \omega = - \int_{M'} \omega \,.</math>

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 Шаблон:Mvar with respect to a measure Шаблон:Mvar and integrates over a subset Шаблон:Mvar, without any notion of orientation; one writes <math display="inline">\int_A f\,d\mu = \int_{[a,b]} f\,d\mu</math> to indicate integration over a subset Шаблон:Mvar. 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 differentials of a set of coordinates, Шаблон:Math, ..., Шаблон:Math can be used as a basis for all Шаблон:Math-forms. Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general Шаблон:Math-form is a linear combination of these differentials at every point on the manifold:

<math>f_1\,dx^1+\cdots+f_n\,dx^n ,</math>

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

The expressions Шаблон:Math, where Шаблон:Math can be used as a basis at every point on the manifold for all Шаблон:Math-forms. This may be thought of as an infinitesimal oriented square parallel to the Шаблон:Math–Шаблон:Math-plane. A general Шаблон:Math-form is a linear combination of these at every point on the manifold: Шаблон:Nowrap 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 Шаблон:Math). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,

<math>dx^1\wedge dx^2=-dx^2\wedge dx^1</math>

because the square whose first side is Шаблон:Math and second side is Шаблон:Math is to be regarded as having the opposite orientation as the square whose first side is Шаблон:Math and whose second side is Шаблон:Math. This is why we only need to sum over expressions Шаблон:Math, with Шаблон:Math; for example: Шаблон:Math. The exterior product allows higher-degree differential forms to be built out of lower-degree 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. Alternating also implies that Шаблон:Math, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, Шаблон:Math if any two of the indices Шаблон:Math, ..., Шаблон:Math are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.

Multi-index notation

A common notation for the wedge product of elementary Шаблон:Mvar-forms is so called multi-index notation: in an Шаблон:Mvar-dimensional context, for Шаблон:Nowrap we define Шаблон:Nowrap[2] Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length Шаблон:Mvar, in a space of dimension Шаблон:Mvar, denoted Шаблон:Nowrap Then locally (wherever the coordinates apply), <math>\{dx^I\}_{I \in \mathcal{J}_{k,n}}</math> spans the space of differential Шаблон:Mvar-forms in a manifold Шаблон:Mvar of dimension Шаблон:Mvar, when viewed as a module over the ring Шаблон:Math of smooth functions on Шаблон:Mvar. By calculating the size of <math>\mathcal{J}_{k,n}</math> combinatorially, the module of Шаблон:Mvar-forms on an Шаблон:Mvar-dimensional manifold, and in general space of Шаблон:Mvar-covectors on an Шаблон:Mvar-dimensional vector space, is Шаблон:Mvar choose Шаблон:Mvar: Шаблон:Nowrap This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.

The exterior derivative

In addition to the exterior product, there is also the exterior derivative operator Шаблон:Math. The exterior derivative of a differential form is a generalization of the differential of a function, in the sense that the exterior derivative of Шаблон:Math is exactly the differential of Шаблон:Mvar. When generalized to higher forms, if Шаблон:Math is a simple Шаблон:Mvar-form, then its exterior derivative Шаблон:Math is a Шаблон:Math-form defined by taking the differential of the coefficient functions:

<math>d\omega = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i \wedge dx^I.</math>

with extension to general Шаблон:Mvar-forms through linearity: if Шаблон:Nowrap a_I \, dx^I \in \Omega^k(M)</math>,}} then its exterior derivative is

<math>d\tau = \sum_{I \in \mathcal{J}_{k,n}}\left(\sum_{j=1}^n \frac{\partial a_I}{\partial x^j} \, dx^j\right)\wedge dx^I \in \Omega^{k+1}(M)</math>

In Шаблон:Math, with the Hodge star operator, the exterior derivative corresponds to gradient, curl, and divergence, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.

The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in differential geometry, differential topology, and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an antiderivation of degree 1 on the exterior algebra of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on manifolds. It also allows for a natural generalization of the fundamental theorem of calculus, called the (generalized) Stokes' theorem, which is a central result in the theory of integration on manifolds.

Differential calculus

Let Шаблон:Mvar be an open set in Шаблон:Math. A differential Шаблон:Math-form ("zero-form") is defined to be a smooth function Шаблон:Mvar on Шаблон:Mvar – the set of which is denoted Шаблон:Math. If Шаблон:Math is any vector in Шаблон:Math, then Шаблон:Math has a directional derivative Шаблон:Math, which is another function on Шаблон:Mvar whose value at a point Шаблон:Math is the rate of change (at Шаблон:Mvar) of Шаблон:Mvar in the Шаблон:Math direction:

<math> (\partial_\mathbf{v} f)(p) = \left. \frac{d}{dt} f(p+t\mathbf{v})\right|_{t=0} .</math>

(This notion can be extended pointwise to the case that Шаблон:Math is a vector field on Шаблон:Mvar by evaluating Шаблон:Math at the point Шаблон:Mvar in the definition.)

In particular, if Шаблон:Math is the Шаблон:Mvarth coordinate vector then Шаблон:Math is the partial derivative of Шаблон:Mvar with respect to the Шаблон:Mvarth coordinate vector, i.e., Шаблон:Math, where Шаблон:Math, Шаблон:Math, ..., Шаблон:Math are the coordinate vectors in Шаблон:Mvar. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates Шаблон:Math, Шаблон:Math, ..., Шаблон:Math are introduced, then

<math>\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i} .</math>

The first idea leading to differential forms is the observation that Шаблон:Math is a linear function of Шаблон:Math:

<math>\begin{align}
 (\partial_{\mathbf{v} + \mathbf{w}} f)(p) &= (\partial_\mathbf{v} f)(p) + (\partial_\mathbf{w} f)(p) \\
   (\partial_{c \mathbf{v}} f)(p) &= c (\partial_\mathbf{v} f)(p)

\end{align}</math>

for any vectors Шаблон:Math, Шаблон:Math and any real number Шаблон:Mvar. At each point p, this linear map from Шаблон:Math to Шаблон:Math is denoted Шаблон:Math and called the derivative or differential of Шаблон:Mvar at Шаблон:Mvar. Thus Шаблон:Math. Extended over the whole set, the object Шаблон:Math can be viewed as a function that takes a vector field on Шаблон:Mvar, and returns a real-valued function whose value at each point is the derivative along the vector field of the function Шаблон:Mvar. Note that at each Шаблон:Mvar, the differential Шаблон:Math is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential [[1-form|Шаблон:Math-form]].

Since any vector Шаблон:Math is a linear combination Шаблон:Math of its components, Шаблон:Math is uniquely determined by Шаблон:Math for each Шаблон:Math and each Шаблон:Math, which are just the partial derivatives of Шаблон:Mvar on Шаблон:Mvar. Thus Шаблон:Math provides a way of encoding the partial derivatives of Шаблон:Mvar. It can be decoded by noticing that the coordinates Шаблон:Math, Шаблон:Math, ..., Шаблон:Math are themselves functions on Шаблон:Mvar, and so define differential Шаблон:Math-forms Шаблон:Math, Шаблон:Math, ..., Шаблон:Math. Let Шаблон:Math. Since Шаблон:Math, the Kronecker delta function, it follows that

Шаблон:NumBlk

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

<math>df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p .</math>

Applying both sides to Шаблон:Math, the result on each side is the Шаблон:Mvarth partial derivative of Шаблон:Mvar at Шаблон:Mvar. Since Шаблон:Mvar and Шаблон:Mvar were arbitrary, this proves the formula Шаблон:EquationNote.

More generally, for any smooth functions Шаблон:Math and Шаблон:Math on Шаблон:Mvar, we define the differential Шаблон:Math-form Шаблон:Math pointwise by

<math>\alpha_p = \sum_i g_i(p) (dh_i)_p</math>

for each Шаблон:Math. Any differential Шаблон:Math-form arises this way, and by using Шаблон:EquationNote it follows that any differential Шаблон:Math-form Шаблон:Mvar on Шаблон:Mvar may be expressed in coordinates as

<math> \alpha = \sum_{i=1}^n f_i\, dx^i</math>

for some smooth functions Шаблон:Math on Шаблон:Mvar.

The second idea leading to differential forms arises from the following question: given a differential Шаблон:Math-form Шаблон:Mvar on Шаблон:Mvar, when does there exist a function Шаблон:Mvar on Шаблон:Mvar such that Шаблон:Math? The above expansion reduces this question to the search for a function Шаблон:Mvar whose partial derivatives Шаблон:Math are equal to Шаблон:Mvar given functions Шаблон:Math. For Шаблон:Math, such a function does not always exist: any smooth function Шаблон:Mvar satisfies

<math> \frac{\partial^2 f}{\partial x^i \, \partial x^j} = \frac{\partial^2 f}{\partial x^j \, \partial x^i} ,</math>

so it will be impossible to find such an Шаблон:Mvar unless

<math> \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j} = 0</math>

for all Шаблон:Mvar and Шаблон:Mvar.

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

<math> \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j = 0 ,</math>

where Шаблон:Math is defined so that:

<math> dx^i \wedge dx^j = - dx^j \wedge dx^i. </math>

This is an example of a differential Шаблон:Math-form. This Шаблон:Math-form is called the exterior derivative Шаблон:Math of Шаблон:Math. It is given by

<math> d\alpha = \sum_{j=1}^n df_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j .</math>

To summarize: Шаблон:Math is a necessary condition for the existence of a function Шаблон:Mvar with Шаблон:Math.

Differential Шаблон:Math-forms, Шаблон:Math-forms, and Шаблон:Math-forms are special cases of differential forms. For each Шаблон:Mvar, there is a space of differential Шаблон:Mvar-forms, which can be expressed in terms of the coordinates as

<math> \sum_{i_1,i_2\ldots i_k=1}^n f_{i_1i_2\ldots i_k} \, dx^{i_1} \wedge dx^{i_2} \wedge\cdots \wedge dx^{i_k}</math>

for a collection of functions Шаблон:Math. Antisymmetry, which was already present for Шаблон:Math-forms, makes it possible to restrict the sum to those sets of indices for which Шаблон:Math.

Differential forms can be multiplied together using the exterior product, and for any differential Шаблон:Mvar-form Шаблон:Mvar, there is a differential Шаблон:Math-form Шаблон:Math called the exterior derivative of Шаблон:Mvar.

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 Шаблон:Mvar. One way to do this is cover Шаблон:Mvar with coordinate charts and define a differential Шаблон:Mvar-form on Шаблон:Mvar to be a family of differential Шаблон:Mvar-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.

Intrinsic definitions

Шаблон:See also

Let Шаблон:Math be a smooth manifold. A smooth differential form of degree Шаблон:Math is a smooth section of the Шаблон:Mathth exterior power of the cotangent bundle of Шаблон:Math. The set of all differential Шаблон:Math-forms on a manifold Шаблон:Math is a vector space, often denoted Шаблон:Math.

The definition of a differential form may be restated as follows. At any point Шаблон:Math, a Шаблон:Math-form Шаблон:Math defines an element

<math> \beta_p \in {\textstyle\bigwedge}^k T_p^* M,</math>

where Шаблон:Math is the tangent space to Шаблон:Math at Шаблон:Math and Шаблон:Math is its dual space. This space is Шаблон:Clarify to the fiber at Шаблон:Math of the dual bundle of the Шаблон:Mathth exterior power of the tangent bundle of Шаблон:Math. That is, Шаблон:Math is also a linear functional <math display="inline">\beta_p \colon {\textstyle\bigwedge}^k T_pM \to \mathbf{R}</math>, i.e. the dual of the Шаблон:Mathth exterior power is isomorphic to the Шаблон:Mathth exterior power of the dual:

<math>{\textstyle\bigwedge}^k T^*_p M \cong \Big({\textstyle\bigwedge}^k T_p M\Big)^*</math>

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

<math>\beta_p\colon \bigoplus_{n=1}^k T_p M \to \mathbf{R}.</math>

Consequently, a differential Шаблон:Math-form may be evaluated against any Шаблон:Math-tuple of tangent vectors to the same point Шаблон:Math of Шаблон:Math. For example, a differential Шаблон:Math-form Шаблон:Math assigns to each point Шаблон:Math a linear functional Шаблон:Math on Шаблон:Math. In the presence of an inner product on Шаблон:Math (induced by a Riemannian metric on Шаблон:Math), Шаблон:Math may be represented as the inner product with a tangent vector Шаблон:Math. Differential Шаблон:Math-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

<math>\operatorname{Alt} \colon {\bigotimes}^k T^*M \to {\bigotimes}^k T^*M.</math>

For a tensor <math>\tau</math> at a point Шаблон:Math,

<math>\operatorname{Alt}(\tau_p)(x_1, \dots, x_k) = \frac{1}{k!}\sum_{\sigma \in S_k} \sgn(\sigma) \tau_p(x_{\sigma(1)}, \dots, x_{\sigma(k)}),</math>

where Шаблон:Math is the symmetric group on Шаблон:Math 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

<math>\operatorname{Alt} \colon {\textstyle\bigwedge}^k T^*M \to {\bigotimes}^k T^*M.</math>

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

Operations

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 product

The exterior product of a Шаблон:Math-form Шаблон:Math and an Шаблон:Math-form Шаблон:Math, denoted Шаблон:Math, is a (Шаблон:Math)-form. At each point Шаблон:Math of the manifold Шаблон:Math, the forms Шаблон:Math and Шаблон:Math are elements of an exterior power of the cotangent space at Шаблон:Math. 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 Шаблон:Math is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product Шаблон:Math is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is

<math>\alpha \wedge \beta = \operatorname{Alt}(\alpha \otimes \beta).</math>

If the embedding of <math>{\textstyle\bigwedge}^n T^*M</math> into <math>{\bigotimes}^n T^*M</math> is done via the map <math>n!\operatorname{Alt}</math> instead of <math>\operatorname{Alt}</math>, the exterior product is

<math>\alpha \wedge \beta = \frac{(k + \ell)!}{k!\ell!}\operatorname{Alt}(\alpha \otimes \beta).</math>

This description is useful for explicit computations. For example, if Шаблон:Math, then Шаблон:Math is the Шаблон:Math-form whose value at a point Шаблон:Math is the alternating bilinear form defined by

<math> (\alpha\wedge\beta)_p(v,w)=\alpha_p(v)\beta_p(w) - \alpha_p(w)\beta_p(v)</math>

for Шаблон:Math.

The exterior product is bilinear: If Шаблон:Math, Шаблон:Math, and Шаблон:Math are any differential forms, and if Шаблон:Math is any smooth function, then

<math>\alpha \wedge (\beta + \gamma) = \alpha \wedge \beta + \alpha \wedge \gamma,</math>
<math>\alpha \wedge (f \cdot \beta) = f \cdot (\alpha \wedge \beta).</math>

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 Шаблон:Math is a Шаблон:Math-form and Шаблон:Math is an Шаблон:Math-form, then

<math>\alpha \wedge \beta = (-1)^{k\ell} \beta \wedge \alpha .</math>

One also has the graded Leibniz rule:

<math>d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta.</math>

Riemannian manifold

On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. 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 <math>\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)</math> and the codifferential <math>\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)</math>, which has degree Шаблон:Math and is adjoint to the exterior differential Шаблон:Math.

Vector field structures

On a pseudo-Riemannian manifold, Шаблон:Math-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-anticommutative ("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 complex

One important property of the exterior derivative is that Шаблон:Math. This means that the exterior derivative defines a cochain complex:

<math>0\ \to\ \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M)\ \to\ \cdots \ \to\ \Omega^n(M)\ \to \ 0.</math>

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

Pullback

Шаблон:See also

Suppose that Шаблон:Math is smooth. The differential of Шаблон:Math is a smooth map Шаблон:Math between the tangent bundles of Шаблон:Math and Шаблон:Math. This map is also denoted Шаблон:Math and called the pushforward. For any point Шаблон:Math and any tangent vector Шаблон:Math, there is a well-defined pushforward vector Шаблон:Math in Шаблон:Math. However, the same is not true of a vector field. If Шаблон:Math is not injective, say because Шаблон:Math has two or more preimages, then the vector field may determine two or more distinct vectors in Шаблон:Math. If Шаблон:Math is not surjective, then there will be a point Шаблон:Math at which Шаблон:Math does not determine any tangent vector at all. Since a vector field on Шаблон:Math determines, by definition, a unique tangent vector at every point of Шаблон:Math, 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 Шаблон:Math may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential Шаблон:Math defines a linear functional on each tangent space of Шаблон:Math and therefore a differential form on Шаблон:Math. 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 Шаблон:Math be smooth, and let Шаблон:Math be a smooth Шаблон:Math-form on Шаблон:Math. Then there is a differential form Шаблон:Math on Шаблон:Math, called the pullback of Шаблон:Math, which captures the behavior of Шаблон:Math as seen relative to Шаблон:Math. To define the pullback, fix a point Шаблон:Math of Шаблон:Math and tangent vectors Шаблон:Math, ..., Шаблон:Math to Шаблон:Math at Шаблон:Math. The pullback of Шаблон:Math is defined by the formula

<math>(f^*\omega)_p(v_1, \ldots, v_k) = \omega_{f(p)}(f_*v_1, \ldots, f_*v_k).</math>

There are several more abstract ways to view this definition. If Шаблон:Math is a Шаблон:Math-form on Шаблон:Math, then it may be viewed as a section of the cotangent bundle Шаблон:Math of Шаблон:Math. Using Шаблон:I sup to denote a dual map, the dual to the differential of Шаблон:Math is Шаблон:Math. The pullback of Шаблон:Math may be defined to be the composite

<math>M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ T^*N\ \stackrel{(df)^*}{\longrightarrow}\ T^*M.</math>

This is a section of the cotangent bundle of Шаблон:Math and hence a differential Шаблон:Math-form on Шаблон:Math. In full generality, let <math display="inline">\bigwedge^k (df)^*</math> denote the Шаблон:Mathth exterior power of the dual map to the differential. Then the pullback of a Шаблон:Math-form Шаблон:Math is the composite

<math>M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ {\textstyle\bigwedge}^k T^*N\ \stackrel{{\bigwedge}^k (df)^*}{\longrightarrow}\ {\textstyle\bigwedge}^k T^*M.</math>

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

<math>{\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R},</math>

where Шаблон:Math is the trivial rank one bundle on Шаблон:Math. The composite map

<math>{\textstyle\bigwedge}^k TM\ \stackrel{{\bigwedge}^k df}{\longrightarrow}\ {\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R}</math>

defines a linear functional on each tangent space of Шаблон:Math, and therefore it factors through the trivial bundle Шаблон:Math. The vector bundle morphism <math display="inline">{\textstyle\bigwedge}^k TM \to M \times \mathbf{R}</math> defined in this way is Шаблон:Math.

Pullback respects all of the basic operations on forms. If Шаблон:Math and Шаблон:Math are forms and Шаблон:Math is a real number, then

<math>\begin{align}
            f^*(c\omega) &= c(f^*\omega), \\
      f^*(\omega + \eta) &= f^*\omega + f^*\eta, \\
 f^*(\omega \wedge \eta) &= f^*\omega \wedge f^*\eta, \\
            f^*(d\omega) &= d(f^*\omega).

\end{align}</math>

The pullback of a form can also be written in coordinates. Assume that Шаблон:Math, ..., Шаблон:Math are coordinates on Шаблон:Math, that Шаблон:Math, ..., Шаблон:Math are coordinates on Шаблон:Math, and that these coordinate systems are related by the formulas Шаблон:Math for all Шаблон:Math. Locally on Шаблон:Math, Шаблон:Math can be written as

<math>\omega = \sum_{i_1 < \cdots < i_k} \omega_{i_1\cdots i_k} \, dy^{i_1} \wedge \cdots \wedge dy^{i_k},</math>

where, for each choice of Шаблон:Math, ..., Шаблон:Math, Шаблон:Math is a real-valued function of Шаблон:Math, ..., Шаблон:Math. Using the linearity of pullback and its compatibility with exterior product, the pullback of Шаблон:Math has the formula

<math>f^*\omega = \sum_{i_1 < \cdots < i_k} (\omega_{i_1\cdots i_k}\circ f) \, df_{i_1} \wedge \cdots \wedge df_{i_k}.</math>

Each exterior derivative Шаблон:Math can be expanded in terms of Шаблон:Math, ..., Шаблон:Math. The resulting Шаблон:Math-form can be written using Jacobian matrices:

<math> f^*\omega = \sum_{i_1 < \cdots < i_k} \sum_{j_1 < \cdots < j_k} (\omega_{i_1\cdots i_k}\circ f)\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})} \, dx^{j_1} \wedge \cdots \wedge dx^{j_k}.</math>

Here, <math display="inline>\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})}</math> denotes the determinant of the matrix whose entries are <math display="inline">\frac{\partial f_{i_m}}{\partial x^{j_n}}</math>, <math>1\leq m,n\leq k</math>.

Integration

A differential Шаблон:Math-form can be integrated over an oriented Шаблон:Math-dimensional manifold. When the Шаблон:Math-form is defined on an Шаблон:Math-dimensional manifold with Шаблон:Math, then the Шаблон:Math-form can be integrated over oriented Шаблон:Math-dimensional submanifolds. If Шаблон:Math, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of Шаблон:Math 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

Let Шаблон:Math be an open subset of Шаблон:Math. Give Шаблон:Math its standard orientation and Шаблон:Math the restriction of that orientation. Every smooth Шаблон:Math-form Шаблон:Math on Шаблон:Math has the form

<math>\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n</math>

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

<math>\int_U \omega\ \stackrel{\text{def}}{=} \int_U f(x)\,dx^1 \cdots dx^n.</math>

Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, Шаблон:Math must be the negative of the integral of Шаблон:Math. 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

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

<math>\varphi \colon D \to M</math>

where Шаблон:Math. Give Шаблон:Math the orientation induced by Шаблон:Math. Then Шаблон:Harv defines the integral of Шаблон:Math over Шаблон:Math to be the integral of Шаблон:Math over Шаблон:Math. In coordinates, this has the following expression. Fix an embedding of Шаблон:Math in Шаблон:Math with coordinates Шаблон:Math. Then

<math>\omega = \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}({\mathbf x})\,dx^{i_1} \wedge \cdots \wedge dx^{i_n}.</math>

Suppose that Шаблон:Math is defined by

<math>\varphi({\mathbf u}) = (x^1({\mathbf u}),\ldots,x^I({\mathbf u})).</math>

Then the integral may be written in coordinates as

<math>\int_M \omega = \int_D \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}(\varphi({\mathbf u})) \frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\dots,u^{n})}\,du^1 \cdots du^n,</math>

where

<math>\frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\ldots,u^{n})}</math>

is the determinant of the Jacobian. The Jacobian exists because Шаблон:Math is differentiable.

In general, an Шаблон:Math-manifold cannot be parametrized by an open subset of Шаблон:Math. 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 Шаблон:Math-dimensional subsets for Шаблон:Math, and this makes it possible to define integrals of Шаблон:Math-forms. To make this precise, it is convenient to fix a standard domain Шаблон:Math in Шаблон:Math, usually a cube or a simplex. A Шаблон:Math-chain is a formal sum of smooth embeddings Шаблон:Math. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a Шаблон:Math-dimensional submanifold of Шаблон:Math. If the chain is

<math>c = \sum_{i=1}^r m_i \varphi_i,</math>

then the integral of a Шаблон:Math-form Шаблон:Math over Шаблон:Math is defined to be the sum of the integrals over the terms of Шаблон:Math:

<math>\int_c \omega = \sum_{i=1}^r m_i \int_D \varphi_i^*\omega.</math>

This approach to defining integration does not assign a direct meaning to integration over the whole manifold Шаблон:Math. 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 Шаблон:Math may be defined to be the integral over the chain determined by a triangulation.

Integration using partitions of unity

There is another approach, expounded in Шаблон:Harv, which does directly assign a meaning to integration over Шаблон:Math, but this approach requires fixing an orientation of Шаблон:Math. The integral of an Шаблон:Math-form Шаблон:Math on an Шаблон:Math-dimensional manifold is defined by working in charts. Suppose first that Шаблон:Math is supported on a single positively oriented chart. On this chart, it may be pulled back to an Шаблон:Math-form on an open subset of Шаблон:Math. 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 Шаблон:Math is independent of the chosen chart. In the general case, use a partition of unity to write Шаблон:Math as a sum of Шаблон:Math-forms, each of which is supported in a single positively oriented chart, and define the integral of Шаблон:Math to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate Шаблон:Math-forms on oriented Шаблон:Math-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 Шаблон:Math, integrating a Шаблон:Math-form on the path is simply pulling back the form to a form Шаблон:Math on Шаблон:Math, and this integral is the integral of the function Шаблон:Math on the interval.

Integration along fibers

Шаблон:Main

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 Шаблон:Math and Шаблон:Math be two orientable manifolds of pure dimensions Шаблон:Math and Шаблон:Math, respectively. Suppose that Шаблон:Math is a surjective submersion. This implies that each fiber Шаблон:Math is Шаблон:Math-dimensional and that, around each point of Шаблон:Math, there is a chart on which Шаблон:Math looks like the projection from a product onto one of its factors. Fix Шаблон:Math and set Шаблон:Math. Suppose that

<math>\begin{align}

\omega_x &\in {\textstyle\bigwedge}^m T_x^*M, \\ \eta_y &\in {\textstyle\bigwedge}^n T_y^*N, \end{align}</math> and that Шаблон:Math does not vanish. Following Шаблон:Harv, there is a unique

<math>\sigma_x \in {\textstyle\bigwedge}^{m-n} T_x^*(f^{-1}(y))</math>

which may be thought of as the fibral part of Шаблон:Math with respect to Шаблон:Math. More precisely, define Шаблон:Math to be the inclusion. Then Шаблон:Math is defined by the property that

<math>\omega_x = (f^*\eta_y)_x \wedge \sigma'_x \in {\textstyle\bigwedge}^m T_x^*M,</math>

where

<math>\sigma'_x \in {\textstyle\bigwedge}^{m-n} T_x^*M</math>

is any Шаблон:Math-covector for which

<math>\sigma_x = j^*\sigma'_x.</math>

The form Шаблон:Math may also be notated Шаблон:Math.

Moreover, for fixed Шаблон:Math, Шаблон:Math varies smoothly with respect to Шаблон:Math. That is, suppose that

<math>\omega \colon f^{-1}(y) \to T^*M</math>

is a smooth section of the projection map; we say that Шаблон:Math is a smooth differential Шаблон:Math-form on Шаблон:Math along Шаблон:Math. Then there is a smooth differential Шаблон:Math-form Шаблон:Math on Шаблон:Math such that, at each Шаблон:Math,

<math>\sigma_x = \omega_x / \eta_y.</math>

This form is denoted Шаблон:Math. The same construction works if Шаблон:Math is an Шаблон:Math-form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber Шаблон:Math is orientable. In particular, a choice of orientation forms on Шаблон:Math and Шаблон:Math defines an orientation of every fiber of Шаблон:Math.

The analog of Fubini's theorem is as follows. As before, Шаблон:Math and Шаблон:Math are two orientable manifolds of pure dimensions Шаблон:Math and Шаблон:Math, and Шаблон:Math is a surjective submersion. Fix orientations of Шаблон:Math and Шаблон:Math, and give each fiber of Шаблон:Math the induced orientation. Let Шаблон:Math be an Шаблон:Math-form on Шаблон:Math, and let Шаблон:Math be an Шаблон:Math-form on Шаблон:Math that is almost everywhere positive with respect to the orientation of Шаблон:Math. Then, for almost every Шаблон:Math, the form Шаблон:Math is a well-defined integrable Шаблон:Math form on Шаблон:Math. Moreover, there is an integrable Шаблон:Math-form on Шаблон:Math defined by

<math>y \mapsto \bigg(\int_{f^{-1}(y)} \omega / \eta_y\bigg)\,\eta_y.</math>

Denote this form by

<math>\bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

Then Шаблон:Harv proves the generalized Fubini formula

<math>\int_M \omega = \int_N \bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let Шаблон:Math be a compactly supported Шаблон:Math-form on Шаблон:Math. Then there is a Шаблон:Math-form Шаблон:Math on Шаблон:Math which is the result of integrating Шаблон:Math along the fibers of Шаблон:Math. The form Шаблон:Math is defined by specifying, at each Шаблон:Math, how Шаблон:Math pairs with each Шаблон:Math-vector Шаблон:Math at Шаблон:Math, and the value of that pairing is an integral over Шаблон:Math that depends only on Шаблон:Math, Шаблон:Math, and the orientations of Шаблон:Math and Шаблон:Math. More precisely, at each Шаблон:Math, there is an isomorphism

<math>{\textstyle\bigwedge}^k T_yN \to {\textstyle\bigwedge}^{n-k} T_y^*N</math>

defined by the interior product

<math>\mathbf{v} \mapsto \mathbf{v}\,\lrcorner\,\zeta_y,</math>

for any choice of volume form Шаблон:Math in the orientation of Шаблон:Math. If Шаблон:Math, then a Шаблон:Math-vector Шаблон:Math at Шаблон:Math determines an Шаблон:Math-covector at Шаблон:Math by pullback:

<math>f^*(\mathbf{v}\,\lrcorner\,\zeta_y) \in {\textstyle\bigwedge}^{n-k} T_x^*M.</math>

Each of these covectors has an exterior product against Шаблон:Math, so there is an Шаблон:Math-form Шаблон:Math on Шаблон:Math along Шаблон:Math defined by

<math>(\beta_{\mathbf{v}})_x = \left(\alpha_x \wedge f^*(\mathbf{v}\,\lrcorner\,\zeta_y)\right) \big/ \zeta_y \in {\textstyle\bigwedge}^{m-n} T_x^*M.</math>

This form depends on the orientation of Шаблон:Math but not the choice of Шаблон:Math. Then the Шаблон:Math-form Шаблон:Math is uniquely defined by the property

<math>\langle\gamma_y, \mathbf{v}\rangle = \int_{f^{-1}(y)} \beta_{\mathbf{v}},</math>

and Шаблон:Math is smooth Шаблон:Harv. This form also denoted Шаблон:Math and called the integral of Шаблон:Math along the fibers of Шаблон:Math. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the projection formula Шаблон:Harv. If Шаблон:Math is any Шаблон:Math-form on Шаблон:Math, then

<math>\alpha^\flat \wedge \lambda = (\alpha \wedge f^*\lambda)^\flat.</math>

Stokes's theorem

Шаблон:Main The fundamental relationship between the exterior derivative and integration is given by the Stokes' theorem: If Шаблон:Math is an (Шаблон:Math)-form with compact support on Шаблон:Math and Шаблон:Math denotes the boundary of Шаблон:Math with its induced orientation, then

<math>\int_M d\omega = \int_{\partial M} \omega.</math>

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If Шаблон:Math is a closed Шаблон:Math-form and Шаблон:Math and Шаблон:Math are Шаблон:Math-chains that are homologous (such that Шаблон:Math is the boundary of a Шаблон:Math-chain Шаблон:Math), then <math>\textstyle{\int_M \omega = \int_N \omega}</math>, since the difference is the integral <math>\textstyle\int_W d\omega = \int_W 0 = 0</math>.

For example, if Шаблон:Math is the derivative of a potential function on the plane or Шаблон:Math, then the integral of Шаблон:Math over a path from Шаблон:Math to Шаблон:Math does not depend on the choice of path (the integral is Шаблон:Math), 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 measures

Шаблон:Details

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 Шаблон:Math-form Шаблон:Math over the interval Шаблон:Math. Assuming the usual distance (and thus measure) on the real line, this integral is either Шаблон:Math or Шаблон:Math, depending on orientation: <math>\textstyle{\int_0^1 dx = 1}</math>, while <math>\textstyle{\int_1^0 dx = - \int_0^1 dx = -1}</math>. By contrast, the integral of the measure Шаблон:Math on the interval is unambiguously Шаблон:Math (i.e. the integral of the constant function Шаблон:Math with respect to this measure is Шаблон:Math). Similarly, under a change of coordinates a differential Шаблон:Math-form changes by the Jacobian determinant Шаблон:Math, while a measure changes by the absolute value of the Jacobian determinant, Шаблон:Math, which further reflects the issue of orientation. For example, under the map Шаблон:Math on the line, the differential form Шаблон:Math pulls back to Шаблон:Math; orientation has reversed; while the Lebesgue measure, which here we denote Шаблон:Math, pulls back to Шаблон:Math; it does not change.

In the presence of the additional data of an orientation, it is possible to integrate Шаблон:Math-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, Шаблон:Math. Formally, in the presence of an orientation, one may identify Шаблон:Math-forms with densities on a manifold; densities in turn define a measure, and thus can be integrated Шаблон:Harv.

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate Шаблон:Math-forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, Шаблон:Math-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 Шаблон:Math-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 Шаблон:Math-forms over subsets for Шаблон:Math because there is no consistent way to use the ambient orientation to orient Шаблон:Math-dimensional subsets. Geometrically, a Шаблон:Math-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the Gram determinant of a set of Шаблон:Math vectors in an Шаблон:Math-dimensional space, which, unlike the determinant of Шаблон:Math vectors, is always positive, corresponding to a squared number. An orientation of a Шаблон:Math-submanifold is therefore extra data not derivable from the ambient manifold.

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

Currents

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

Applications in physics

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

<math>\textbf{F} = \frac 1 2 f_{ab}\, dx^a \wedge dx^b\,,</math>

where the Шаблон:Math are formed from the electromagnetic fields <math>\vec E</math> and <math>\vec B</math>; e.g., Шаблон:Math, Шаблон:Math, or equivalent definitions.

This form is a special case of the curvature form on the Шаблон:Math 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 Шаблон:Math, when represented in some gauge. One then has

<math>\textbf{F} = d\textbf{A}.</math>

The current Шаблон:Math-form is

<math> \textbf{J} = \frac 1 6 j^a\, \varepsilon_{abcd}\, dx^b \wedge dx^c \wedge dx^d\,,</math>

where Шаблон:Math are the four components of the current density. (Here it is a matter of convention to write Шаблон:Math instead of Шаблон:Math, i.e. to use capital letters, and to write Шаблон:Math instead of Шаблон:Math. 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 has been called <math>\vec J</math> for several decades, and by some publishers Шаблон:Math; 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

<math>\begin{align}
       d {\textbf{F}} &= \textbf{0} \\
 d {\star \textbf{F}} &= \textbf{J},

\end{align}</math> where <math>\star</math> denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.

The Шаблон:Math-form <math>{\star} \mathbf{F}</math>, which is dual to the Faraday form, is also called Maxwell 2-form.

Electromagnetism is an example of a Шаблон:Math gauge theory. Here the Lie group is Шаблон:Math, 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 Шаблон:Math in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form Шаблон:Math. The Yang–Mills field Шаблон:Math is then defined by

<math>\mathbf{F} = d\mathbf{A} + \mathbf{A}\wedge\mathbf{A}.</math>

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

Applications in geometric measure theory

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 also

Notes

Шаблон:Reflist

References

External links

Шаблон:Manifolds Шаблон:Tensors

Шаблон:Authority control

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Криптовалюты
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Разное

  1. ↑ Шаблон:Citation
  2. ↑ Шаблон:Cite book
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Differential_form&oldid=14307384