Tangent bundle, the vector bundle of tangent spaces on a differentiable manifold. Tangent field, a section of the tangent bundle. Also called a vector field. Tangent space Torus Transversality. Two submanifolds M and N intersect transversally if at each point of intersection p their tangent spaces Tp(M) and Tp(N) generate the whole tangent s ...

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Glossary of differential geometry and topology, Glossary of differential geometry and topology - A, Glossary of differential geometry and topology - B, Glossary of differential geometry and topology - C, Glossary of differential geometry and topology - D, Glossary of differential geometry and topology - E, Glossary of differential geometry and topology - F, Glossary of differential geometry and topology - G, Glossary of differential geometry and topology - H, Glossary of differential geometry and topology - I, Glossary of differential geometry and topology - L, Glossary of differential geometry and topology - M, Glossary of differential geometry and topology - P, Glossary of differential geometry and topology - S, Glossary of differential geometry and topology - T, Glossary of differential geometry and topology - V, Glossary of differential geometry and topology - W

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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) = f(x) + g(x) (f × g)(x) = f(x) × < ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function from a X to Y, and S is any subset of X, the restriction of f to S is the function f|S from S to Y such that f|S(s) = f(s) for all s in S. The restriction f|S can also be expressed as the composition f incS,X, where incSSee also:

Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x in X, there exists some y in Y such that x is related to y. f is single-valued: for each x in X, there is at most one y in Y such that x is related to y. In some contexts, a relation that satisfies condition (1), but not necessarily (2) ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, some mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain. There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. Ordinary functions are sometimes referred to as morphisms in a concrete category. ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Diffeomorphism. Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both and its inverse are smooth functions. Doubling, given a manifold M with boundary, doubling is taking two copies of M and identifying their boundaries. As the result we get a manifold without boundary. ...

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Glossary of differential geometry and topology, Glossary of differential geometry and topology - A, Glossary of differential geometry and topology - B, Glossary of differential geometry and topology - C, Glossary of differential geometry and topology - D, Glossary of differential geometry and topology - E, Glossary of differential geometry and topology - F, Glossary of differential geometry and topology - G, Glossary of differential geometry and topology - H, Glossary of differential geometry and topology - I, Glossary of differential geometry and topology - L, Glossary of differential geometry and topology - M, Glossary of differential geometry and topology - P, Glossary of differential geometry and topology - S, Glossary of differential geometry and topology - T, Glossary of differential geometry and topology - V, Glossary of differential geometry and topology - W

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Section Submanifold. A submanifold is the image of a smooth embedding of a manifold. Submersion Surface, a two-dimensional manifold or submanifold. ...

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Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial. Principal bundle. A principal bundle is a fiber bundle P → B together with right action on P by a Lie group G that preserves the fibers of P and acts simply transitively on those fibers. Pullback ...

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Glossary of differential geometry and topology, Glossary of differential geometry and topology - A, Glossary of differential geometry and topology - B, Glossary of differential geometry and topology - C, Glossary of differential geometry and topology - D, Glossary of differential geometry and topology - E, Glossary of differential geometry and topology - F, Glossary of differential geometry and topology - G, Glossary of differential geometry and topology - H, Glossary of differential geometry and topology - I, Glossary of differential geometry and topology - L, Glossary of differential geometry and topology - M, Glossary of differential geometry and topology - P, Glossary of differential geometry and topology - S, Glossary of differential geometry and topology - T, Glossary of differential geometry and topology - V, Glossary of differential geometry and topology - W

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Fiber. In a fiber bundle, π: E → B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex. Fiber bundle Frame Frame bundle, the principal bundle of frames on a smooth manifold. Flow ...

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Glossary of differential geometry and topology, Glossary of differential geometry and topology - A, Glossary of differential geometry and topology - B, Glossary of differential geometry and topology - C, Glossary of differential geometry and topology - D, Glossary of differential geometry and topology - E, Glossary of differential geometry and topology - F, Glossary of differential geometry and topology - G, Glossary of differential geometry and topology - H, Glossary of differential geometry and topology - I, Glossary of differential geometry and topology - L, Glossary of differential geometry and topology - M, Glossary of differential geometry and topology - P, Glossary of differential geometry and topology - S, Glossary of differential geometry and topology - T, Glossary of differential geometry and topology - V, Glossary of differential geometry and topology - W

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The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the fact that |YX| = |Y||X|. See the article on cardinal numbers for more details. It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle. Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. The Möbius strip has a circle for a base B and a line segment for the fiber F. A neighborhood U of a point is an arc; in the picture, this is th ...

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Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

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List of mathematical functions - Antiderivatives of elementary functions. Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem. Exponential integral Error function: An integral important for normal random variables. Fresnel integral: related to the error function; used in optics. ...

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A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do not in general have globally-defined sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology. Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle ...

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A straightforward generalization is to allow functions depending on several arguments. For instance, g(x,y) = xy is a function which takes two inputs, x and y, and outputs their product, xy. The input (x,y) is often thought of as an ordered pair. Functions whose inputs consist of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider ...

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As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, a ...

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The image of an element x in X under f is the output f(x). The image of a subset A of X under f is the subset of Y formally defined by f[A] = {f(x) | x is in A} Usually (when subsets of the domain are not at the same time elements of the domain) one writes f(A) instead of f[A]. (An old-fashioned notation writes f'x instead of f(x ...

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X, the set of input values, is called the domain of f, sometimes denoted dom(f), and Y. The distinction between "codomain" and "range" is best understood by means of an example. Consider, once again, f(x) = x2. This function has no negative outputs, but does permit negative inputs. The domain is R, the set of all real numbers. It is often convenient to consider this as a function from R into R, and so it is possible to call R the codomain and to write f:R → R. The "range", however, is the set of all outputs, which in this ca ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called th ...

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Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (USee also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

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The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of "graph" as a picture of the function, with the ordered pairs plotted as Cartesian coordinates. Note that a binary relation on the two sets X and Y could be identified with an ordere ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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