Differential form - Citizendia

A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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. 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 Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental

1 Gentle introduction
2 Properties of the wedge product
3 Formal definition
4 Integration of differential forms
5 Operations on forms
6 Differential forms in physics
7 2-forms in geometric measure theory
8 See also
9 References

Gentle introduction

We initially work in an open set in $\mathbb{R}^n$ . In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in A 0-form is defined to be a smooth function f. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability When we integrate a function f over an m-dimensional subspace S of $\mathbb{R}^n$ , we write it as

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

Consider $d x 1$ , . The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it . . , $d x n$ for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. We call these and their negatives: $-{\mathrm d}x^1,\dots,-{\mathrm d}x^n$ basic 1-forms. 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

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

for all i and j. In mathematics anticommutativity refers to the property of an operation being anticommutative, i 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 $d x I$ 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be To this end, suppose we have an open coordinate cover. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

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. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since The set of all k-forms on a manifold M is a vector space commonly denoted Ω^k(M). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added k-forms can be defined as totally antisymmetric covariant tensor fields. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings 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 Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In this context, they assign, to each point of a manifold, a linear functional on the tangent space at that point. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional In this setting, particularly in the physics literature, 1-forms are sometimes called "covariant vector fields", "covector fields", or "dual vector fields".

Integration of differential forms

Differential forms of degree k are integrated over k dimensional chains. In Algebraic topology, a simplicial k - chain is a formal linear combination of k - simplices. 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.

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 differentiable k-manifold over which we wish to integrate, where S has the parameterization

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

for u in the parameter domain D. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Then [Rudin, 1976] defines the integral of the differential form over S as

$\int_S \omega =\int_D \sum a_{i_1,\dots,i_k}(S({\mathbf u})) \frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}\,du^1\ldots du^k$

where

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

is the determinant of the Jacobian. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. The Jacobian exists because S is differentiable.

See also Stokes' theorem. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

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. 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 In Mathematics, the interior product is a degree &minus1 derivation on the Exterior algebra of Differential forms on a Smooth manifold In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the One important property of the exterior derivative is that d² = 0; see de Rham cohomology for more details. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable

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. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is

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

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

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. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is In Differential geometry, the curvature form describes Curvature of a connection on a Principal bundle. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations The current 3-form is

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

Using these definitions, Maxwell's equations can be written very compactly in geometrized units as

$\mathrm{d}\, {\textbf{F}} = \textbf{0}$

$\mathrm{d}\, {*\textbf{F}} = \textbf{J}$

where $*$ denotes the Hodge star operator. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric A geometrized unit system or geometric unit system is a system of Natural units in which the base physical units are chosen so that the Speed of light In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W Similar considerations describe the geometry of gauge theories in general.

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. For other inequalities named after Wirtinger see Wirtinger's inequality. A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory. Herbert Federer, an American Mathematician, is one of the creators of Geometric measure theory, at the meeting point of Differential geometry and Mathematical The Wirtinger inequality is also a key ingredient in Gromov's inequality for complex projective space in systolic geometry. In Riemannian geometry, Gromov 's optimal stable 2- systolic inequality is the inequality \mathrm{stsys}_2{}^n \leq n! \\mathrm{vol}_{2n}(\mathbb{CP}^n For a slower-paced introduction click on Introduction to systolic geometry.

References

David Bachman (2006). In Mathematics, a complex differential form is a Differential form on a Manifold (usually a Complex manifold) which is permitted to have In Mathematics, a vector-valued differential form on a Manifold M is a Differential form on M with values in a Vector space For other inequalities named after Wirtinger see Wirtinger's inequality. 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.
Wendell H. Fleming (1965) Functions of Several Variables, Addison-Wesley. Chapter 6: Exterior algebra and differential calculus, pages 205-38. This textbook in multivariate calculus introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges. Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve
Shigeyuki Morita (2001). Geometry of Differential Forms. AMS. ISBN 0-8218-1045-6.
Walter Rudin (1976). Walter Rudin (born 1921) is an American Mathematician, currently a Professor Emeritus of Mathematics at the University of Wisconsin-Madison Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.
Michael Spivak (1965). Michael David Spivak (*1940 in Queens, New York) is a Mathematician specializing in Differential geometry, an expositor of Mathematics 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.

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