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# Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. [1]

111 relations: Abelian group, Anticommutativity, Antisymmetric tensor, Élie Cartan, Bilinear form, Calculus on Manifolds (book), Chain (algebraic topology), Chain complex, Circle group, Clifford algebra, Closed and exact differential forms, Cohomology, Complex differential form, Connection form, Coordinate system, Coordinate vector, Cornell University, Cotangent bundle, Covariance and contravariance of vectors, Covariant derivative, Curl (mathematics), Curvature form, De Rham cohomology, Density on a manifold, Derivative, Differentiable manifold, Differential geometry, Differential of a function, Differential operator, Directional derivative, Divergence, Divergence theorem, Duality (mathematics), Electromagnetic tensor, Electromagnetism, Equivariant differential form, Exact sequence, Exterior algebra, Exterior derivative, Fréchet derivative, Fundamental class, Fundamental theorem of calculus, Gauge theory, Geometric algebra, Geometrized unit system, Gradient theorem, Gramian matrix, Green's theorem, Gromov's inequality for complex projective space, Hausdorff measure, ... Expand index (61 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of commutativity).

## Anticommutativity

In mathematics, anticommutativity is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result.

## Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.

## Élie Cartan

Élie Joseph Cartan (9 April 1869 &ndash; 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.

## Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

## Calculus on Manifolds (book)

Michael Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965, ISBN 0-8053-9021-9) is a short text treating analysis in several variables in Euclidean spaces and on differentiable manifolds.

## Chain (algebraic topology)

In algebraic topology, a simplicial k-chain is a formal linear combination of k-simplices.

## Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology.

## Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C&times;, the multiplicative group of all nonzero complex numbers.

## Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra.

## Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα.

## Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.

## Complex differential form

In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.

## Connection form

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.

## Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.

## Coordinate vector

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis.

## Cornell University

Cornell University is an American private Ivy League and federal land-grant research university located in Ithaca, New York.

## Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

## Covariance and contravariance of vectors

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.

## Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.

## Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.

## Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle.

## De Rham cohomology

In mathematics, de Rham cohomology (after Georges de Rham) 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.

## Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable).

## Differentiable manifold

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

## Differential of a function

In calculus, the differential represents the principal part of the change in a function y.

## Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

## Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

## Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

## Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

## Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.

## Electromagnetic tensor

In electromagnetism, 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.

## Electromagnetism

Electromagnetism is a branch of physics which involves the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

## Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted by a Lie group G is a polynomial map from the Lie algebra \mathfrak.

## Exact sequence

An exact sequence is a concept in mathematics, especially in ring and module theory, homological algebra, as well as in differential geometry and group theory.

## Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

## Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

## Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.

## Fundamental class

In mathematics, the fundamental class is a homology class associated to an oriented manifold M, which corresponds to the generator of the homology group H_r(M;\mathbf)\cong\mathbf.

## Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.

## Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

## Geometric algebra

A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form.

## Geometrized unit system

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 vacuum, c, and the gravitational constant, G, are set equal to unity.

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

## Gramian matrix

In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by G_.

## Green's theorem

In mathematics, Green's 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. It is named after George Green and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

## Gromov's inequality for complex projective space

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics.

## Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in Rn or, more generally, in any metric space.

## Herbert Federer

Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician.

## Hodge dual

In mathematics, the Hodge star operator or Hodge dual is an important linear map introduced in general by W. V. D. Hodge.

## Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

## Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (Greek ὁμός (homós).

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Integral

The integral is an important concept in mathematics.

## Integration along fibers

In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where m is the dimension of the fiber, via "integration".

## Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.

## Interior product

In mathematics, the interior product or interior derivative is a degree &minus;1 antiderivation on the exterior algebra of differential forms on a smooth manifold.

## International Union of Pure and Applied Physics

The International Union of Pure and Applied Physics (IUPAP) is an international non-governmental organization whose mission is to assist in the worldwide development of physics, to foster international cooperation in physics, and to help in the application of physics toward solving problems of concern to humanity.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

## Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker, is a function of two variables, usually just positive integers.

## Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

## Lie algebra

In mathematics, a Lie algebra (not) is a vector space together with a non-associative multiplication called "Lie bracket".

## Lie derivative

In mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow of another vector field.

## Lie group

In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

## Linear function

In mathematics, the term linear function refers to two distinct, although related, notions.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Manifold

In mathematics, a manifold is a topological space that resembles Euclidean space near each point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Maurer–Cartan form

In mathematics, the Maurer&ndash;Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of.

## Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits.

## Methods of contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

## Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

## Multilinear map

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable.

## Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable: the differentiation and integration of functions involving multiple variables, rather than just one.

## One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

## Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

## Parametrization

Parametrization (or parameterization; also parameterisation, parametrisation) is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object.

## Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

## Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations &mdash; operations defined on functions by applying the operations to function values separately for each point in the domain of definition.

## Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product of a space with a group.

## Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite.

## Pullback (differential geometry)

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*.

## Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

## Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

## Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle \pi is a continuous right inverse of the function \pi.

## Skew-symmetric matrix

In mathematics, and in particular linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the condition If the entry in the and is aij, i.e. then the skew symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

## Smoothness

In mathematical analysis, smoothness has to do with how many derivatives of a function exist and are continuous.

## Stokes' theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

## Surface

In mathematics, specifically, in topology, a surface is a two-dimensional, topological manifold.

## Surface integral

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.

## Systolic geometry

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations.

## Tangent space

In mathematics, the tangent space of a manifold 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.

## Tangent vector

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.

## Tensor (intrinsic definition)

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept.

## Tensor calculus

In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study), is the study of topological spaces.

## Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of unitary matrices, with the group operation that of matrix multiplication.

## Vector field

In vector calculus, a vector field is an assignment of a vector to each point in a subset of space.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

## Vector-valued differential form

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 forms.

## Volume element

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.

## Volume form

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form (i.e., a differential form of top degree).

## Weyl algebra

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.

## Wirtinger inequality (2-forms)

In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold M, the exterior kth power of the symplectic form (Kähler form) ω, when evaluated on a simple (decomposable) (2k)-vector ζ of unit volume, is bounded above by k!.

## Yang–Mills theory

Yang–Mills theory is a gauge theory based on the SU(''N'') group, or more generally any compact, semi-simple Lie group.

## References

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