In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety). A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in R2, and attach to every point of the curve the line normal to the curve at that ...

Including:

• Vector bundle - Definition and first consequences
• Vector bundle - Vector bundle morphisms
• Vector bundle - Sections and locally free sheaves
• Vector bundle - Operations on vector bundles
• Vector bundle - Variants and generalizations

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In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a indexed family of objects in C. The coproduct of the set {Xj} is an object ...

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In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. In particular, if the body rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the angular velocity and the distance of the mass to the axis. Without applying torque to the object, with respect to the reference point, the angular momentum is constant. The angular ...

Including:

• Angular momentum - Angular momentum in classical mechanics
• Angular momentum - Definition
• Angular momentum - Conservation of angular momentum
• Angular momentum - Angular momentum in relativistic mechanics
• Angular momentum - Angular momentum in quantum mechanics

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

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The Flanders' house is situated at 740 Evergreen Terrace, Springfield and is next door to 742 Evergreen Terrace, home of the Simpson family. The home phone number is 636-555-8904. The house is a two-storey detached house with a connected garage. The house itself is pink-purple in colour, with red window frames and a blue-orange canopy above the front-door. Inside the house, most if not all the rooms include religious imagery and photographs of the (late) Maude Flanders. The house is well furnished with expensive-looking furniture, which Ned claims he buys cheaply because ...

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Flanders family, Flanders family - Ned Flanders, Flanders family - Maude Flanders, Flanders family - Rod and Todd, Flanders family - Grandma Flanders, Flanders family - Other family members, Flanders family - Household, Flanders family - Other information, Flanders family - Relationship with the Simpsons, Flanders family - The Adventures of Ned Flanders, Flanders family - Everybody Hates Ned Flanders

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A real vector bundle is given by the following data: topological spaces X (the "base space") and E (the "total space") a continuous map π : E → X (the "projection") for every x in X, the structure of a real vector space on the fiber π−1({x}) satisfying the following compatibility condition: for every point in X there is an open neighborhood U, a natural number n, and a homeomorphism φ : U × See also:

Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

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Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

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Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f * T on V can be defined. That is, given a tensor and a set of vectors one then defines the pullback as . The result f * T is again a tensor, so that f *See also:

Pullback, Pullback - Pullback on tensors, Pullback - Pullback of cotangent bundles, Pullback - Pullback on tensor bundles, Pullback - Pullback of diffeomorphisms

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Clifford algebra - Relation to the exterior algebra. Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V,Q) considered as vector spaces. This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in C we define a functor Hom(A,–) : C → Set to the category of sets as follows: Hom(A,–) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A,–) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, XSee also:

Hom functor, Hom functor - Formal definition, Hom functor - Yoneda's lemma

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The class of all pointed spaces forms a category Top• with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ({•} ↓ Top) where {•} is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted {•}/Top). Objects in this category are continuous maps {•} → X. Such morphisms can be thought of as picking out a basepoint in X. Morphisms in ({•} ↓ Top) are morphisms in ...

See also:

Pointed space, Pointed space - Category of pointed spaces, Pointed space - Operations on pointed spaces

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We can denote the shift operator by where r is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system can be represented in this abstract notation by where is a function given by with the system yielding the shifted output So is an operator that advances the input vector by 1. Suppose we represent a system by an operator . This system is time-invariant if it co ...

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Time-invariant system, Time-invariant system - Simple example, Time-invariant system - Formal example, Time-invariant system - Abstract example

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The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any discrete subcategory in C. Not every family {Xi} needs to have a product, but if it does, then the product is unique in a strong sense: if πi : X → Xi and π’i : X’ → Xi are two products of the family {Xi}, then (by ...

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Product category theory, Product category theory - Definition, Product category theory - Examples, Product category theory - Discussion

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In the category of sets the pullback of f and g is the set X ×Z Y = {(x, y) ∈ X × Y | f(x) = g(y)}, together with the restrictions of the projection maps π1 and π2 to X ×Z Y . This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f o ...

See also:

Pullback category theory, Pullback category theory - Universal property, Pullback category theory - Examples

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Let I be a (possibly infinite) index set and suppose Xi is a topological space for every i in I. Set X = Π Xi, the Cartesian product of the sets Xi. For every i in I, we have a canonical projection pi : X → Xi. The product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is ...

See also:

Product topology, Product topology - Definition, Product topology - Examples, Product topology - Properties, Product topology - Relation to other topological notions

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Yamhill County ranks seventh out of Oregon's thirty-six counties in annual market value of its agricultural production. This agricultural production includes wheat, barley, horticulture, and dairy farming, with 13,201 acres (53 km²) in 1997 planted in orchards. One-third of the county is covered with commercial timber, and the economic mainstay of the western part of the county is logging and timber products. Yamhill County is also the center of Oregon's wine industry, having the largest area of any Oregon county planted in vineyards ...

See also:

Yamhill County Oregon, Yamhill County Oregon - Economy, Yamhill County Oregon - Geography, Yamhill County Oregon - Adjacent Counties, Yamhill County Oregon - Demographics, Yamhill County Oregon - History, Yamhill County Oregon - Cities and towns

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The Galilean transformation is nothing more than careful addition and subtraction of velocity vectors. Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation. The notation below describes the relationship of two coordinate systems (x′ and x) in constant relative motion (velocity u) in the x-direction. All other pa ...

See also:

Galilean transformation, Galilean transformation - History, Galilean transformation - Translation one dimension, Galilean transformation - Galilean transformations, Galilean transformation - Central extension of the Galilean group, Galilean transformation - Notes

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The most general comma category construction involves two converging functors. Typically, one of these will be a "selection" or "constant" functor: many accounts of category theory consider these special cases only, but the term is actually much more general. (A selection functor maps every object in the domain category to the same, fixed object in the codomain category, and every domain morphism to the identity morphism of that fixed object. Often, the choice of domain category is not relevant; typically, the discrete category having only one object is used.) See also:

Comma category, Comma category - Definition, Comma category - General form, Comma category - Category of objects under A, Comma category - Category of objects over A, Comma category - Other variations, Comma category - Examples of use, Comma category - Some notable categories, Comma category - Limits and universal morphisms, Comma category - Adjunctions

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Angular momentum - Definition. The traditional mathematical definition of the angular momentum of a particle about some origin is: where L is the angular momentum of the particle, r is the position of the particle expressed as a displacement vector from the origin, p is the linear momentum of the particle, and is the vector cross product. Because of the cross product, L is a pseudovector perpen ...

See also:

Angular momentum, Angular momentum - Angular momentum in classical mechanics, Angular momentum - Definition, Angular momentum - Conservation of angular momentum, Angular momentum - Angular momentum in relativistic mechanics, Angular momentum - Angular momentum in quantum mechanics

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Let C be a category with binary products and let Y and Z be objects of C. The exponential object ZY can be defined as a universal morphism from the functor –×Y to Z. (The functor –×Y from C to C maps objects X to X×Y and morphisms φ to φ×idY). Explicitly, the definition is as follows. An object ZY, together with a morphism is an exponential object if for any object X and morphism g : (X×Y) → ZSee also:

Exponential object, Exponential object - Definition, Exponential object - Examples

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