Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Differentiable manifold

Index Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. [1]

216 relations: Active and passive transformation, Affine connection, Affine space, Affine transformation, Albert Einstein, Algebra over a field, Algebraic geometry, Analytic function, Analytic manifold, Annals of Mathematics, Antisymmetric tensor, Arc length, Argumentum a fortiori, Atlas (topology), Banach manifold, Banach–Stone theorem, Bernhard Riemann, Biholomorphism, C*-algebra, Calculus, Carl Friedrich Gauss, Category theory, Chain rule, Christoffel symbols, Circle, Classical mechanics, Closed and exact differential forms, Compact space, Complex geometry, Complex manifold, Complex number, Connection (mathematics), Contact (mathematics), Coordinate system, Cotangent bundle, Cotangent space, Covariance, Covariance and contravariance of vectors, Cover (topology), Curl (mathematics), Curve, De Rham cohomology, Decision problem, Definite quadratic form, Derivation (differential algebra), Derivative, Diffeology, Diffeomorphism, Differentiable function, Differential form, ..., Differential geometry, Differential operator, Differential structure, Dimension, Directional derivative, Divergence, Divergence theorem, Donaldson's theorem, Dual number, Dual space, E8 manifold, Embedding, Equivalence class, Equivalence principle, Equivalence relation, Equivariant map, Euclidean space, Euclidean vector, Exact differential, Exceptional isomorphism, Exotic R4, Exotic sphere, Exterior algebra, Exterior derivative, Fiber bundle, Fibration, Finsler manifold, Fréchet manifold, Frölicher space, Function composition, Function space, Functor, G-structure on a manifold, General covariance, General linear group, General relativity, Geometrization conjecture, Germ (mathematics), Gradient, Graduate Studies in Mathematics, Green's theorem, Gregorio Ricci-Curbastro, Group (mathematics), H-cobordism, Habilitation, Hamiltonian mechanics, Hassler Whitney, Hausdorff space, Hermann Weyl, Hilbert's fifth problem, Holomorphic function, Holonomic basis, Homeomorphism, Homotopy, If and only if, Immersion (mathematics), Implicit function, Infinitesimal, Inner product space, Integral, Intersection theory, Introduction to the mathematics of general relativity, Inverse function, Inverse function theorem, James Clerk Maxwell, Jet (mathematics), Jet group, John Milnor, Kervaire manifold, Kirby–Siebenmann class, Lagrangian system, Lie algebra, Lie bracket of vector fields, Lie derivative, Lie group, Linear form, List of formulas in Riemannian geometry, Local property, Local ring, Locally finite collection, Lp space, Manifold, Maximal ideal, Möbius transformation, Metric signature, Metric tensor, Michael Freedman, Morphism of algebraic varieties, Mostow rigidity theorem, Multilinear map, Multivariable calculus, Mutatis mutandis, Nash embedding theorem, Noncommutative geometry, Nondegenerate form, Norm (mathematics), Operator theory, Orbifold, Orientability, Partition of unity, Physics, Polynomial, Presentation of a group, Principal bundle, Product rule, Projective variety, Pseudo-Riemannian manifold, Pseudogroup, Pullback, Pushforward (differential), Quaternion, Quotient group, Rank (linear algebra), Real line, Rectifiable set, Riemann curvature tensor, Riemann surface, Riemannian geometry, Riemannian manifold, Ringed space, Sard's theorem, Scheme (mathematics), Second-countable space, Set (mathematics), Sheaf (mathematics), Simon Donaldson, Simply connected space, Smooth structure, Smoothness, Sobolev space, Sophus Lie, Space (mathematics), Special linear group, Spectrum of a C*-algebra, Stokes' theorem, Submanifold, Submersion (mathematics), Support (mathematics), Surface (topology), Surgery theory, Symplectic geometry, Symplectic group, Symplectic manifold, Symplectomorphism, Synthetic differential geometry, Tangent bundle, Tangent space, Tangent vector, Taylor series, Tensor, Tensor field, Tensor product, Topological manifold, Topological space, Topos, Total derivative, Tullio Levi-Civita, University of Göttingen, Vector bundle, Vector field, Vector space, Velocity, Yang–Mills theory, 3-manifold, 4-manifold, 5-manifold. Expand index (166 more) »

Active and passive transformation

In physics and engineering, an active transformation, or alibi transformation, is a transformation which actually changes the physical position of a point, or rigid body, which can be defined even in the absence of a coordinate system; whereas a passive transformation, or alias transformation, is merely a change in the coordinate system in which the object is described (change of coordinate map, or change of basis).

New!!: Differentiable manifold and Active and passive transformation · See more »

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

New!!: Differentiable manifold and Affine connection · See more »

Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

New!!: Differentiable manifold and Affine space · See more »

Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

New!!: Differentiable manifold and Affine transformation · See more »

Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

New!!: Differentiable manifold and Albert Einstein · See more »

Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

New!!: Differentiable manifold and Algebra over a field · See more »

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

New!!: Differentiable manifold and Algebraic geometry · See more »

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

New!!: Differentiable manifold and Analytic function · See more »

Analytic manifold

In mathematics, an analytic manifold is a topological manifold with analytic transition maps.

New!!: Differentiable manifold and Analytic manifold · See more »

Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

New!!: Differentiable manifold and Annals of Mathematics · See more »

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.

New!!: Differentiable manifold and Antisymmetric tensor · See more »

Arc length

Determining the length of an irregular arc segment is also called rectification of a curve.

New!!: Differentiable manifold and Arc length · See more »

Argumentum a fortiori

Argumentum a fortiori (Latin: "from a/the stronger ") is a form of argumentation which draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in the first.

New!!: Differentiable manifold and Argumentum a fortiori · See more »

Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

New!!: Differentiable manifold and Atlas (topology) · See more »

Banach manifold

In mathematics, a Banach manifold is a manifold modeled on Banach spaces.

New!!: Differentiable manifold and Banach manifold · See more »

Banach–Stone theorem

In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

New!!: Differentiable manifold and Banach–Stone theorem · See more »

Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

New!!: Differentiable manifold and Bernhard Riemann · See more »

Biholomorphism

In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.

New!!: Differentiable manifold and Biholomorphism · See more »

C*-algebra

C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.

New!!: Differentiable manifold and C*-algebra · See more »

Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

New!!: Differentiable manifold and Calculus · See more »

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

New!!: Differentiable manifold and Carl Friedrich Gauss · See more »

Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

New!!: Differentiable manifold and Category theory · See more »

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

New!!: Differentiable manifold and Chain rule · See more »

Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

New!!: Differentiable manifold and Christoffel symbols · See more »

Circle

A circle is a simple closed shape.

New!!: Differentiable manifold and Circle · See more »

Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

New!!: Differentiable manifold and Classical mechanics · See more »

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.

New!!: Differentiable manifold and Closed and exact differential forms · See more »

Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

New!!: Differentiable manifold and Compact space · See more »

Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables.

New!!: Differentiable manifold and Complex geometry · See more »

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

New!!: Differentiable manifold and Complex manifold · See more »

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

New!!: Differentiable manifold and Complex number · See more »

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.

New!!: Differentiable manifold and Connection (mathematics) · See more »

Contact (mathematics)

In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives.

New!!: Differentiable manifold and Contact (mathematics) · See more »

Coordinate system

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

New!!: Differentiable manifold and Coordinate system · See more »

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.

New!!: Differentiable manifold and Cotangent bundle · See more »

Cotangent space

In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below).

New!!: Differentiable manifold and Cotangent space · See more »

Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

New!!: Differentiable manifold and Covariance · See more »

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.

New!!: Differentiable manifold and Covariance and contravariance of vectors · See more »

Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset.

New!!: Differentiable manifold and Cover (topology) · See more »

Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

New!!: Differentiable manifold and Curl (mathematics) · See more »

Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

New!!: Differentiable manifold and Curve · See more »

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.

New!!: Differentiable manifold and De Rham cohomology · See more »

Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

New!!: Differentiable manifold and Decision problem · See more »

Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

New!!: Differentiable manifold and Definite quadratic form · See more »

Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.

New!!: Differentiable manifold and Derivation (differential algebra) · See more »

Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

New!!: Differentiable manifold and Derivative · See more »

Diffeology

In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are.

New!!: Differentiable manifold and Diffeology · See more »

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

New!!: Differentiable manifold and Diffeomorphism · See more »

Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

New!!: Differentiable manifold and Differentiable function · See more »

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.

New!!: Differentiable manifold and Differential form · See more »

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.

New!!: Differentiable manifold and Differential geometry · See more »

Differential operator

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

New!!: Differentiable manifold and Differential operator · See more »

Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

New!!: Differentiable manifold and Differential structure · See more »

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

New!!: Differentiable manifold and Dimension · See more »

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 curvilinear coordinate curves, all other coordinates being constant.

New!!: Differentiable manifold and Directional derivative · See more »

Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

New!!: Differentiable manifold and Divergence · See more »

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.

New!!: Differentiable manifold and Divergence theorem · See more »

Donaldson's theorem

In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable.

New!!: Differentiable manifold and Donaldson's theorem · See more »

Dual number

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2.

New!!: Differentiable manifold and Dual number · See more »

Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

New!!: Differentiable manifold and Dual space · See more »

E8 manifold

In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice.

New!!: Differentiable manifold and E8 manifold · See more »

Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

New!!: Differentiable manifold and Embedding · See more »

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

New!!: Differentiable manifold and Equivalence class · See more »

Equivalence principle

In the theory of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

New!!: Differentiable manifold and Equivalence principle · See more »

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

New!!: Differentiable manifold and Equivalence relation · See more »

Equivariant map

In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.

New!!: Differentiable manifold and Equivariant map · See more »

Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

New!!: Differentiable manifold and Euclidean space · See more »

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

New!!: Differentiable manifold and Euclidean vector · See more »

Exact differential

In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.

New!!: Differentiable manifold and Exact differential · See more »

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such isomorphisms.

New!!: Differentiable manifold and Exceptional isomorphism · See more »

Exotic R4

In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4.

New!!: Differentiable manifold and Exotic R4 · See more »

Exotic sphere

In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.

New!!: Differentiable manifold and Exotic sphere · See more »

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.

New!!: Differentiable manifold and Exterior algebra · See more »

Exterior derivative

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

New!!: Differentiable manifold and Exterior derivative · See more »

Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

New!!: Differentiable manifold and Fiber bundle · See more »

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle.

New!!: Differentiable manifold and Fibration · See more »

Finsler manifold

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space, allowing to define the length of any smooth curve as Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.

New!!: Differentiable manifold and Finsler manifold · See more »

Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

New!!: Differentiable manifold and Fréchet manifold · See more »

Frölicher space

In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds.

New!!: Differentiable manifold and Frölicher space · See more »

Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

New!!: Differentiable manifold and Function composition · See more »

Function space

In mathematics, a function space is a set of functions between two fixed sets.

New!!: Differentiable manifold and Function space · See more »

Functor

In mathematics, a functor is a map between categories.

New!!: Differentiable manifold and Functor · See more »

G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.

New!!: Differentiable manifold and G-structure on a manifold · See more »

General covariance

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations.

New!!: Differentiable manifold and General covariance · See more »

General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

New!!: Differentiable manifold and General linear group · See more »

General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

New!!: Differentiable manifold and General relativity · See more »

Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

New!!: Differentiable manifold and Geometrization conjecture · See more »

Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties.

New!!: Differentiable manifold and Germ (mathematics) · See more »

Gradient

In mathematics, the gradient is a multi-variable generalization of the derivative.

New!!: Differentiable manifold and Gradient · See more »

Graduate Studies in Mathematics

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).

New!!: Differentiable manifold and Graduate Studies in Mathematics · See more »

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, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

New!!: Differentiable manifold and Green's theorem · See more »

Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro (12January 1925) was an Italian mathematician born in Lugo di Romagna.

New!!: Differentiable manifold and Gregorio Ricci-Curbastro · See more »

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

New!!: Differentiable manifold and Group (mathematics) · See more »

H-cobordism

In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps are homotopy equivalences.

New!!: Differentiable manifold and H-cobordism · See more »

Habilitation

Habilitation defines the qualification to conduct self-contained university teaching and is the key for access to a professorship in many European countries.

New!!: Differentiable manifold and Habilitation · See more »

Hamiltonian mechanics

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.

New!!: Differentiable manifold and Hamiltonian mechanics · See more »

Hassler Whitney

Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.

New!!: Differentiable manifold and Hassler Whitney · See more »

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

New!!: Differentiable manifold and Hausdorff space · See more »

Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

New!!: Differentiable manifold and Hermann Weyl · See more »

Hilbert's fifth problem

Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.

New!!: Differentiable manifold and Hilbert's fifth problem · See more »

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

New!!: Differentiable manifold and Holomorphic function · See more »

Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as where is the infinitesimal displacement vector between the point and a nearby point whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant).

New!!: Differentiable manifold and Holonomic basis · See more »

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

New!!: Differentiable manifold and Homeomorphism · See more »

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

New!!: Differentiable manifold and Homotopy · See more »

If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

New!!: Differentiable manifold and If and only if · See more »

Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

New!!: Differentiable manifold and Immersion (mathematics) · See more »

Implicit function

In mathematics, an implicit equation is a relation of the form R(x_1,\ldots, x_n).

New!!: Differentiable manifold and Implicit function · See more »

Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

New!!: Differentiable manifold and Infinitesimal · See more »

Inner product space

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

New!!: Differentiable manifold and Inner product space · See more »

Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

New!!: Differentiable manifold and Integral · See more »

Intersection theory

In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring.

New!!: Differentiable manifold and Intersection theory · See more »

Introduction to the mathematics of general relativity

The mathematics of general relativity is complex.

New!!: Differentiable manifold and Introduction to the mathematics of general relativity · See more »

Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

New!!: Differentiable manifold and Inverse function · See more »

Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.

New!!: Differentiable manifold and Inverse function theorem · See more »

James Clerk Maxwell

James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics.

New!!: Differentiable manifold and James Clerk Maxwell · See more »

Jet (mathematics)

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain.

New!!: Differentiable manifold and Jet (mathematics) · See more »

Jet group

In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point.

New!!: Differentiable manifold and Jet group · See more »

John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.

New!!: Differentiable manifold and John Milnor · See more »

Kervaire manifold

In mathematics, specifically in differential topology, a Kervaire manifold K4n+2 is a piecewise-linear manifold of dimension 4n+2 constructed by by plumbing together the tangent bundles of two 2''n''+1-spheres, and then gluing a ball to the result.

New!!: Differentiable manifold and Kervaire manifold · See more »

Kirby–Siebenmann class

In mathematics, the Kirby–Siebenmann class is an element of the fourth cohomology group which must vanish if a topological manifold M is to have a piecewise linear structure.

New!!: Differentiable manifold and Kirby–Siebenmann class · See more »

Lagrangian system

In mathematics, a Lagrangian system is a pair, consisting of a smooth fiber bundle and a Lagrangian density, which yields the Euler–Lagrange differential operator acting on sections of.

New!!: Differentiable manifold and Lagrangian system · See more »

Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

New!!: Differentiable manifold and Lie algebra · See more »

Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted.

New!!: Differentiable manifold and Lie bracket of vector fields · See more »

Lie derivative

In differential geometry, 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 defined by another vector field.

New!!: Differentiable manifold and Lie derivative · See more »

Lie group

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

New!!: Differentiable manifold and Lie group · See more »

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.

New!!: Differentiable manifold and Linear form · See more »

List of formulas in Riemannian geometry

This is a list of formulas encountered in Riemannian geometry.

New!!: Differentiable manifold and List of formulas in Riemannian geometry · See more »

Local property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.

New!!: Differentiable manifold and Local property · See more »

Local ring

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

New!!: Differentiable manifold and Local ring · See more »

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space.

New!!: Differentiable manifold and Locally finite collection · See more »

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

New!!: Differentiable manifold and Lp space · See more »

Manifold

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

New!!: Differentiable manifold and Manifold · See more »

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.

New!!: Differentiable manifold and Maximal ideal · See more »

Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

New!!: Differentiable manifold and Möbius transformation · See more »

Metric signature

The signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.

New!!: Differentiable manifold and Metric signature · See more »

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

New!!: Differentiable manifold and Metric tensor · See more »

Michael Freedman

Michael Hartley Freedman (born 21 April 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara.

New!!: Differentiable manifold and Michael Freedman · See more »

Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.

New!!: Differentiable manifold and Morphism of algebraic varieties · See more »

Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique.

New!!: Differentiable manifold and Mostow rigidity theorem · See more »

Multilinear map

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

New!!: Differentiable manifold and Multilinear map · See more »

Multivariable calculus

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

New!!: Differentiable manifold and Multivariable calculus · See more »

Mutatis mutandis

Mutatis mutandis is a Medieval Latin phrase meaning "the necessary changes having been made" or "once the necessary changes have been made".

New!!: Differentiable manifold and Mutatis mutandis · See more »

Nash embedding theorem

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space.

New!!: Differentiable manifold and Nash embedding theorem · See more »

Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).

New!!: Differentiable manifold and Noncommutative geometry · See more »

Nondegenerate form

In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if.

New!!: Differentiable manifold and Nondegenerate form · See more »

Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

New!!: Differentiable manifold and Norm (mathematics) · See more »

Operator theory

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.

New!!: Differentiable manifold and Operator theory · See more »

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

New!!: Differentiable manifold and Orbifold · See more »

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.

New!!: Differentiable manifold and Orientability · See more »

Partition of unity

In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unit interval such that for every point, x\in X,.

New!!: Differentiable manifold and Partition of unity · See more »

Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

New!!: Differentiable manifold and Physics · See more »

Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

New!!: Differentiable manifold and Polynomial · See more »

Presentation of a group

In mathematics, one method of defining a group is by a presentation.

New!!: Differentiable manifold and Presentation of a group · See more »

Principal bundle

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

New!!: Differentiable manifold and Principal bundle · See more »

Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.

New!!: Differentiable manifold and Product rule · See more »

Projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.

New!!: Differentiable manifold and Projective variety · See more »

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, but need only be a non-degenerate bilinear form, which is a weaker condition.

New!!: Differentiable manifold and Pseudo-Riemannian manifold · See more »

Pseudogroup

In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example).

New!!: Differentiable manifold and Pseudogroup · See more »

Pullback

In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product.

New!!: Differentiable manifold and Pullback · See more »

Pushforward (differential)

Suppose that is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus.

New!!: Differentiable manifold and Pushforward (differential) · See more »

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

New!!: Differentiable manifold and Quaternion · See more »

Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

New!!: Differentiable manifold and Quotient group · See more »

Rank (linear algebra)

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.

New!!: Differentiable manifold and Rank (linear algebra) · See more »

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

New!!: Differentiable manifold and Real line · See more »

Rectifiable set

In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense.

New!!: Differentiable manifold and Rectifiable set · See more »

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.

New!!: Differentiable manifold and Riemann curvature tensor · See more »

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

New!!: Differentiable manifold and Riemann surface · See more »

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.

New!!: Differentiable manifold and Riemannian geometry · See more »

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 differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

New!!: Differentiable manifold and Riemannian manifold · See more »

Ringed space

In mathematics, a ringed space can be equivalently thought of as either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.

New!!: Differentiable manifold and Ringed space · See more »

Sard's theorem

Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0.

New!!: Differentiable manifold and Sard's theorem · See more »

Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

New!!: Differentiable manifold and Scheme (mathematics) · See more »

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.

New!!: Differentiable manifold and Second-countable space · See more »

Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

New!!: Differentiable manifold and Set (mathematics) · See more »

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

New!!: Differentiable manifold and Sheaf (mathematics) · See more »

Simon Donaldson

Sir Simon Kirwan Donaldson FRS (born 20 August 1957), is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.

New!!: Differentiable manifold and Simon Donaldson · See more »

Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

New!!: Differentiable manifold and Simply connected space · See more »

Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function.

New!!: Differentiable manifold and Smooth structure · See more »

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

New!!: Differentiable manifold and Smoothness · See more »

Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.

New!!: Differentiable manifold and Sobolev space · See more »

Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

New!!: Differentiable manifold and Sophus Lie · See more »

Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure.

New!!: Differentiable manifold and Space (mathematics) · See more »

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

New!!: Differentiable manifold and Special linear group · See more »

Spectrum of a C*-algebra

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.

New!!: Differentiable manifold and Spectrum of a C*-algebra · See more »

Stokes' theorem

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

New!!: Differentiable manifold and Stokes' theorem · See more »

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

New!!: Differentiable manifold and Submanifold · See more »

Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

New!!: Differentiable manifold and Submersion (mathematics) · See more »

Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

New!!: Differentiable manifold and Support (mathematics) · See more »

Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

New!!: Differentiable manifold and Surface (topology) · See more »

Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by.

New!!: Differentiable manifold and Surgery theory · See more »

Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

New!!: Differentiable manifold and Symplectic geometry · See more »

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.

New!!: Differentiable manifold and Symplectic group · See more »

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

New!!: Differentiable manifold and Symplectic manifold · See more »

Symplectomorphism

In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.

New!!: Differentiable manifold and Symplectomorphism · See more »

Synthetic differential geometry

In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory.

New!!: Differentiable manifold and Synthetic differential geometry · See more »

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

New!!: Differentiable manifold and Tangent bundle · See more »

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 that gives the displacement of the one point from the other.

New!!: Differentiable manifold and Tangent space · See more »

Tangent vector

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

New!!: Differentiable manifold and Tangent vector · See more »

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

New!!: Differentiable manifold and Taylor series · See more »

Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

New!!: Differentiable manifold and Tensor · See more »

Tensor field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

New!!: Differentiable manifold and Tensor field · See more »

Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

New!!: Differentiable manifold and Tensor product · See more »

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

New!!: Differentiable manifold and Topological manifold · See more »

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

New!!: Differentiable manifold and Topological space · See more »

Topos

In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).

New!!: Differentiable manifold and Topos · See more »

Total derivative

In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

New!!: Differentiable manifold and Total derivative · See more »

Tullio Levi-Civita

Tullio Levi-Civita, FRS (29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas.

New!!: Differentiable manifold and Tullio Levi-Civita · See more »

University of Göttingen

The University of Göttingen (Georg-August-Universität Göttingen, GAU, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany.

New!!: Differentiable manifold and University of Göttingen · See more »

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

New!!: Differentiable manifold and Vector bundle · See more »

Vector field

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

New!!: Differentiable manifold and Vector field · See more »

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.

New!!: Differentiable manifold and Vector space · See more »

Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.

New!!: Differentiable manifold and Velocity · See more »

Yang–Mills theory

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

New!!: Differentiable manifold and Yang–Mills theory · See more »

3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

New!!: Differentiable manifold and 3-manifold · See more »

4-manifold

In mathematics, a 4-manifold is a 4-dimensional topological manifold.

New!!: Differentiable manifold and 4-manifold · See more »

5-manifold

In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.

New!!: Differentiable manifold and 5-manifold · See more »

Redirects here:

Ambient manifold, Differental manifold, Differentiable manifolds, Differential manifold, Geometric structure, Manifold/rewrite/differentiable manifold, Non-smoothable manifold, Sheaf of smooth functions, Smooth manifold, Smooth manifolds.

References

[1] https://en.wikipedia.org/wiki/Differentiable_manifold

OutgoingIncoming
Hey! We are on Facebook now! »