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Encyclopedia > Exterior derivative

In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan. Mathematics is the study of quantity, structure, space and change. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

Contents


Definition

The exterior derivative of a differential form of degree k is a differential form of degree k + 1.


For a k-form ω = fI dxI over Rn, the definition is as follows:

For general k-forms ΣI fI dxI (where the multi-index I runs over all ordered subsets of {1, ..., n} of cardinality k), we just extend linearly. Note that if i = I above then (see wedge product). The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...


Properties

Exterior differentiation satisfies three important properties:

It can be shown that exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions). The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function f(x1, x2, ... , xn) of n variables. ...


The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials). In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β...


Invariant formula

Given a k-form ω and arbitrary smooth vector fields V0,V1, …, Vk we have Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

where [Vi,Vj] denotes Lie bracket and the hat denotes the omission of that element: A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...


In particular, for 1-forms we have:

dω(X,Y) = X(ω(Y)) − Y(ω(X)) − ω([X,Y]).

More generally, the Lie derivative is defined via the Lie bracket: In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as...

,

and the Lie derivative of a general differential form is closely related to the exterior derivative. The differences are primarily notational; various identities between the two are provided in the article on Lie derivatives. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as...


Connection with vector calculus

The following correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation. Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...


Gradient

For a 0-form, that is a smooth function f: RnR, we have In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

Therefore

where grad f denotes gradient of f and <•, •> is the scalar product. In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...


Curl

For a 1-form on R3, cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, Gopher, Telnet, DICT, FILE and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password authentication, file transfer resume, http proxy tunneling and...

which restricted to the three-dimensional case is Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...

Therefore, for vector field V=[u,v,w] we have where curl V denotes the curl of V, × is the vector product, and <•, •> is the scalar product. In mathematics, the cross product is a binary operation on vectors in three dimensions. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...


(what are U and W here? this assertion needs clarification - Gauge 23:37, 7 Apr 2005 (UTC))


Divergence

For a 2-form In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

For three dimensions, with we get

where V is a vector field defined by V = [p,q,r]. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...


Examples

For a 1-form on R2 we have

which is exactly the 2-form being integrated in Green's theorem. In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens Theorem was named after British scientist George Green and is a special case of the more...


See also


  Results from FactBites:
 
Cartan's Calculus: The exterior derivative (308 words)
This concept is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional exterior algebra subspace.
What is remarkable is that the operational definition of the exterior derivative works on any p-form carrying it into a p+1 form.
In the Cartan exterior calculus, the operations are the same and are represented by the same exterior derivative, but the operation operates on different algebraic subspaces.
Exterior derivative (173 words)
Exterior derivative extends the concept of the differential[?] of a function to differential forms of higher degree.
Exterior derivative of a differential form of degree k is a differential form of degree k+1.
For example, in 3 dimensional Euclidean space, exterior derivative of a 1-form corresponds to curl and exterior derivative of a 2-form corresponds to divergence.
  More results at FactBites »

 
 

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