Exterior derivative - Citizendia

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree zero, to differential forms of higher degree. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Its current form was invented by Élie Cartan. Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental

The exterior derivative d has the property that d² = 0 and is the differential (coboundary) used to define de Rham (and Alexander-Spanier) cohomology on forms. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable In Mathematics, particularly in Algebraic topology Alexander-Spanier cohomology is a Cohomology theory arising from Differential forms with Integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology of a smooth manifold. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. The theorem of de Rham shows that this map is actually an isomorphism. In this sense, the exterior derivative is the "dual" of the boundary map on singular simplices.

1 Definition
2 Examples
3 Properties
4 Invariant formula
5 The exterior derivative in calculus
6 See also
7 References

Definition

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

Given a multi-index $I=(i_1, i_2, \dots, i_k)$ with $1\le i_1, i_2, \dots, i_k \le n,$ the exterior derivative of a k-form

$\omega = f_Idx_I=f_{i_1i_2\cdots i_k}dx_{i_1}\wedge dx_{i_2}\wedge\cdots\wedge dx_{i_k}$

over Rⁿ is defined as

$d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.$

For general k-forms ω = Σ_I f_I dx_I (where the components of the multi-index I run over all the values in {1, . The Mathematical notation of multi-indices simplifies formulae used in Multivariable calculus, Partial differential equations and the theory of distributions . . , n}), the definition of the exterior derivative is extended linearly. The word linear comes from the Latin word linearis, which means created by lines. Note that whenever $i$ is one of the components of the multi-index $I,$ then $dx_i \wedge dx_I = 0$ (see wedge product).

Geometrically, the k + 1 form dω acts on each tangent space of Rⁿ in the following way: a (k + 1)-tuple of vectors (u₁,. . . ,u_{k + 1}) in the tangent space defines an oriented (k + 1)-polyhedron p. dω(u₁,. . . ,u_{k + 1}) is defined to be the integral of ω over the boundary of p, where the boundary is given the inherited orientation. Assuming the fact that every smooth manifold admits a (smooth) triangulation, this gives immediately Stokes' theorem. In advanced Geometry, in the most general meaning triangulation is a subdivision of a geometric object into simplices. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

Examples

For a 1-form $\sigma = u\, dx + v\, dy$ on R² we have, by applying the above formula to each term,

$d \sigma = \left(\frac{\partial{u}}{\partial{x}} dx \wedge dx + \frac{\partial{u}}{\partial{y}} dy \wedge dx\right) + \left(\frac{\partial{v}}{\partial{x}} dx \wedge dy + \frac{\partial{v}}{\partial{y}} dy \wedge dy\right)$

$= 0 -\frac{\partial{u}}{\partial{y}} dx \wedge dy + \frac{\partial{v}}{\partial{x}} dx \wedge dy + 0$

$= \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) dx \wedge dy.$

Properties

Exterior differentiation is by definition linear. Direct computation shows that it also has the following properties:

the wedge product rule holds (see antiderivation)

$d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge d\eta),$

and d² = 0, which follows from the equality of mixed partial derivatives. In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator In Mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking Partial derivatives of a function

It can be shown that the exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).

Differential forms in the kernel of d are said to be closed forms. 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 For instance, a 1-form is closed if on each tangent space, its integral along the boundary of the parallellogram given by any pair of tangent vectors is zero. Thus closedness is a local condition. The the image of d is said to consist of exact forms (cf. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage exact differentials). In Mathematics, a differential dQ is said to be exact, as contrasted with an Inexact differential, if the differentiable function It is immediate that exact forms are closed.

The exterior derivative is natural. If f: M → N is a smooth map and Ω^k is the contravariant smooth functor that assigns to each manifold the space of k-forms on the manifold, then the following diagram commutes

so d(f*ω) = f*dω, where f* denotes the pullback of f. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from This follows from that f*ω(·), by definition, is ω(f_*(·)), f_* being the pushforward of f. Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Thus d is a natural transformation from Ω^k to Ω^k+1. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

Invariant formula

Given a k-form ω and arbitrary smooth vector fields V₀,V₁, …, V_k we have

$d\omega(V_0,V_1,...V_k) = \sum_i(-1)^i V_i\left(\omega(V_0, \ldots, \hat V_i, \ldots,V_k)\right)$

$+\sum_{i<j}(-1)^{i+j}\omega([V_i, V_j], V_0, \ldots, \hat V_i, \ldots, \hat V_j, \ldots, V_k)$

where $[V i, V j]$ denotes Lie bracket and the hat denotes the omission of that element: $\omega(V_0, \ldots, \hat V_i, \ldots,V_k) = \omega(V_0, \ldots, V_{i-1}, V_{i+1}, \ldots, V_k).$

In particular, for 1-forms we have:

d ω(X, Y) = X (ω(Y)) - Y (ω(X)) - ω([X, Y]). In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. See Lie algebra for more on the definition of the Lie bracket and Lie derivative for the derivation In the mathematical field of Differential

The exterior derivative in 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 (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

Gradient

For a 0-form, that is, a smooth function f: Rⁿ→R, we have

$df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\, dx_i.$

This is a 1-form, a section of the cotangent bundle, that gives local linear approximation to f on each tangent space. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces

For a vector field V,

$df(V) = \langle \mbox{grad}f ,V\rangle,$

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 increase of the scalar In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

Curl

One can associate to a vector field V = (u, v, w) on R³ the 1-form

$\omega^1 _V = u dx + v dy + w dz,$

and the 2-form

$\omega^2 _V = u dy \wedge dz + v dz \wedge dx + w dx \wedge dy.$

The integral of ω¹_V over a path gives work done against -V along the path; locally, it is the dot product with V. The integral of ω²_V over a surface gives the flux of V over that surface; locally, it is the scalar triple product with V. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion.

One can check directly that

$d \omega^1 _V = \omega^2 _{curl \;V},$

where curl V denotes the curl of V. The flux of curl V over a surface is the integral of ω¹_V over the boundary of the surface.

Divergence

Similarly,

$d \omega^2 _V = \mbox{div}\; V \; dx \wedge dy \wedge dz.$

The flux of V over the boundary of a 3-polyhedron p is given by the integral of the divergence of V over p. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the

Invariant formulations of div, grad, and curl

The three operators above can be written in coordinate-free notation as follows:

$\begin{array}{rcl} \nabla f &=& \left( {\mathbf d} f \right)^\sharp \\ \nabla \times F &=& \left[ \star \left( {\mathbf d} F^\flat \right) \right]^\sharp \\ \nabla \cdot F &=& \star {\mathbf d} \left( \star F^\flat \right) \\ \end{array}$

where $\star$ is the Hodge star operator and $\flat$ and $\sharp$ are the musical isomorphisms. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Mathematics, the musical isomorphism (or canonical isomorphism is an Isomorphism between the Tangent bundle TM and the Cotangent

References

Flanders, Harley (1989). In Mathematics, and specifically Differential geometry, a connection form is a manner of organizing the data of a connection using the language of In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Mathematics, the discrete exterior calculus ( DEC) or finite element exterior calculus ( FEEC) is the extension to the method of finite Differential forms with applications to the physical sciences. New York: Dover Publications, page 20. ISBN 0-486-66169-5.
Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society, page 54. ISBN 0-8218-3702-8.
Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser, page 239. ISBN 0-8176-4134-3.
Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press, page 35. ISBN 0-521-46800-0.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents