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Encyclopedia > Differential form

A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan. Multivariate calculus is a means of analyzing deterministic systems with multiple degrees of freedom. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... For more technical Wiki articles on tensors, see the section later in this article. ... 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 mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... 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

Gentle introduction

We initially work in an open set in mathbb{R}^n. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of mathbb{R}^n, we write it as In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... This article is about the concept of integrals in calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... 2-dimensional renderings (ie. ...

int_S f,{mathrm d}x^1 cdots {mathrm d}x^m.

Consider dx1, ...,dxn for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums. We call these and their negatives: -{mathrm d}x^1,dots,-{mathrm d}x^n basic 1-forms. In mathematics, a Riemann sum is a method for approximating the values of integrals. ... In linear algebra, a one-form on a vector space is the same as a linear functional on the space. ...


We define a "multiplication" rule wedge, the wedge product on these elements, making only the anticommutativity restraint that 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. ... A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ...

{mathrm d}x^i wedge {mathrm d}x^j = - {mathrm d}x^j wedge {mathrm d}x^i

for all i and j. Note that this implies

{mathrm d}x^i wedge {mathrm d}x^i = 0.

We define the set of all these products to be basic 2-forms, and similarly we define the set of products

{mathrm d}x^i wedge {mathrm d}x^j wedge {mathrm d}x^k

to be basic 3-forms, assuming n is at least 3. Now define a monomial k-form to be a 0-form times a basic k-form for all k, and finally define a k-form to be a sum of monomial k-forms.


We extend the wedge product to these sums by defining

(f,{mathrm d}x^I + g,{mathrm d}x^J)wedge(p,{mathrm d}x^K + q,{mathrm d}x^L) =
f cdot p,{mathrm d}x^I wedge {mathrm d}x^K + f cdot q,{mathrm d}x^I wedge {mathrm d}x^L + g cdot p,{mathrm d}x^J wedge {mathrm d}x^K + g cdot q,{mathrm d}x^J wedge {mathrm d}x^L,

etc., where dxI and friends represent basic k-forms. In other words, the product of sums is the sum of all possible products.


Now, we also want to define k-forms on smooth manifolds. To this end, suppose we have an open coordinate cover. We can define a k-form on each coordinate neighborhood; a global k-form is then a set of k-forms on the coordinate neighborhoods such that they agree on the overlaps. For a more precise definition of what that means, see manifold. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X and... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...


Properties of the wedge product

It can be proven that if f, g, and w are any differential forms, then

w wedge (f + g) = w wedge f + w wedge g.

Also, if f is a k-form and g is an l-form, then:

f wedge g = (-1)^{kl} g wedge f.

Formal definition

In differential geometry, a differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of a manifold. At any point p on a manifold, a k-form gives a multilinear map from the k-th exterior power of the tangent space at p to R. The set of all k-forms on a manifold M is a vector space commonly denoted Ωk(M). k-forms can be defined as totally antisymmetric covariant tensor fields. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... 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 differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In set theory, the adjective antisymmetric usually refers to an antisymmetric relation. ... In category theory, see covariant functor. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...


For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form. The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In linear algebra, a one-form on a vector space is the same as a linear functional on the space. ...


1-forms are a particularly useful basic concept in the coordinate-free treatment of tensors. In this context, they assign, to each point of a manifold, a linear functional on the tangent space at that point. In this setting, particularly in the physics literature, 1-forms are sometimes called "covariant vector fields", "covector fields", or "dual vector fields". In linear algebra, a one-form on a vector space is the same as a linear functional on the space. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... It has been suggested that this article or section be merged into Covariant transformation. ...


Integration of forms

Differential forms of degree k are integrated over k dimensional chains. If k = 0, this is just evaluation of functions at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc. This article is about algebraic topology. ...


Let

omega=sum a_{i_1,dots,i_k}({mathbf x}),{mathrm d}x^{i_1} wedge cdots wedge {mathrm d}x^{i_k}

be a differential form and S a set for which we wish to integrate over, where S has the parameterization

S({mathbf u})=(x^1({mathbf u}),dots,x^n({mathbf u}))

for u in the parameter domain D. Then [Rudin, 1976] defines the integral of the differential form over S as

int_S omega =int_D sum a_{i_1,dots,i_k}(S({mathbf u})) frac{partial(x^{i_1},dots,x^{i_k})}{partial(u^{1},dots,u^{k})},mathrm{d}{mathbf u}

where

frac{partial(x^{i_1},dots,x^{i_k})}{partial(u^{1},dots,u^{k})}

is the determinant of the Jacobian. For the French Revolution faction, see Jacobin. ...


See also Stokes' theorem. Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...


Operations on forms

There are several important operations one can perform on a differential form: wedge product, exterior derivative (denoted by d), interior product, Hodge dual, codifferential and Lie derivative. One important property of the exterior derivative is that d2 = 0; see de Rham cohomology for more details. 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 mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the interior product is a degree −1 derivation on the exterior algebra of differential forms on a smooth manifold. ... In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields 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... 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. ...


The fundamental relationship between the exterior derivative and integration is given by the general Stokes' theorem, which also provides the duality between de Rham cohomology and the homology of chains. Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... 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 (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...


Differential forms 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 Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwells theory of electromagnetism. ...

textbf{F} = frac{1}{2}F_{ab}, {mathrm d}x^a wedge {mathrm d}x^b

Note that this form is a special case of the curvature form on the U(1) principal fiber bundle on which both electromagnetism and general gauge theories may be described. The current 3-form is In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... In mathematics, principal bundles are an abstract version of the notion of a frame bundle, which is a fiber bundle whose fiber over a point consists of a collection of ordered bases for a vector space (the fiber of a vector bundle over that point). ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

textbf{J} = J^a epsilon_{abcd}, {mathrm d}x^b wedge {mathrm d}x^c wedge {mathrm d}x^d

Using these definitions, Maxwell's equations can be written very compactly in geometrized units as For thermodynamic relations, see Maxwell relations. ... In physics, especially in the general theory of relativity, geometrized units or sometimes geometric units, is a physical unit system in which all physical quantities are expressed in the unit of length: meter. ...

mathrm{d}, {textbf{F}} = textbf{0}
mathrm{d}, {*textbf{F}} = textbf{J}

where * denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general. In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ...


The 2-form * mathbf{F} is also called Maxwell 2-form.


2-forms 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. In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that the exterior th power of the standard symplectic form ω, when evaluated on a simple (decomposable) -vector ζ of unit volume, is bounded above by . ... Herbert Federer, an American mathematician, is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis. ... Gromovs optimal stable 2-systolic inequality is the inequality for complex projective space, where the optimal bound is attained by the symmetric Fubini-Study metric. ... Systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner, and later developed by Mikhael Gromov and others, in its arithmetic, ergodic, and topological manifestations. ...


See also

  • Wirtinger inequality (2-forms)
  • http://www.sjsu.edu/faculty/watkins/difforms.htm

In mathematics, a complex form is a differential form on a complex manifold. ... In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential... In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that the exterior th power of the standard symplectic form ω, when evaluated on a simple (decomposable) -vector ζ of unit volume, is bounded above by . ...

References

  • David Bachman (2006). A Geometric Approach to Differential Forms. Birkhauser. ISBN 978-0-8176-4499-4. 
  • Harley Flanders (1989). Differential forms with applications to the physical sciences. Mineola, NY: Dover Publications. ISBN 0-486-66169-5. 
  • Walter Rudin (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X. 
  • Michael Spivak (1965). Calculus on Manifolds. Menlo Park, CA: W. A. Benjamin. ISBN 0-8053-9021-9. 
  • Vladimir A. Zorich (2004). Mathematical Analysis II. Springer. ISBN 3-540-40633-6. 

  Results from FactBites:
 
Differential form - Definition, explanation (878 words)
A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors.
In differential geometry, a differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of a manifold.
For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form.
CSDC : Why Differential Forms? Functional Substitution and PullBack (988 words)
Given a differential form in terms of the variables and their differentials (say x and dx) on the final state (target of the map), the differential form is well defined functionally in terms of the variables (say y and dy) on the initial state (domain of the map).
A differential form is a "scalar invariant or a density invariant" with respect to diffeomorphisms.
Construct a differential form on (meaning in terms of the variables of x, dx) the final state, and then by functional substitution and use of the Jacobian map construct the well defined functional form of the differential form on the initial state (in terms of y and dy).
  More results at FactBites »

 
 

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