Hurwitz's theorem (complex analysis): Difference between revisions

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{{split|list of vector quantities|discuss=talk: Vector (mathematics and physics) #A CONCEPTDAB article is needed|part=yes|date=April 2013}}
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{{TOC_right}}
 
Many special instances of the general definition of '''vector''' as an element of a [[vector space]] are listed below.
 
==Vectors==
*[[Euclidean vector]], a geometric entity endowed with magnitude and direction as well as a positive-definite [[inner product]]; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities that have both magnitude and direction, such as force, in contrast to [[Scalar (mathematics)|scalar]] quantities, which have no direction.
**[[Vector product]], or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
**[[Burgers vector]], a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
**[[Laplace–Runge–Lenz vector]], a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another
**[[Normal vector]], or surface normal, a vector that is perpendicular to a (hyper)surface at a point
* An element of a [[vector space]]
**[[Basis vector]], one of a set of vectors (a "basis") that, in linear combination, can represent every vector in a given vector space
** [[Coordinate vector]], in linear algebra, an explicit representation of an element of any abstract vector space
** [[Row vector]] or [[column vector]], a one-dimensional matrix often representing the solution of a system of linear equations
** An element of the [[real coordinate space]] {{math|'''R'''<sup>''n''</sup>}}
*[[Random vector]] or [[multivariate random variable]], in [[statistics]], a set of [[real number|real]]-valued [[random variable]]s that may be [[correlated]]
*[[Vector projection]], also known as the ''vector resolute'' or ''vector component'', a linear mapping producing a vector parallel to a second vector
*The vector part of a [[quaternion]], a mathematical entity which is one possible generalisation of a vector
*[[Null vector]], a vector whose magnitude is zero
*[[Position (vector)|Position vector]], a vector representing the position of a point in an [[affine space]] in relation to a reference point
*[[Displacement vector]], a vector that specifies the change in position of a point relative to a previous position
*[[Gradient vector]], the vector giving the magnitude and direction of maximum increase of a [[scalar field]]
*[[Poynting vector]], in physics, a vector representing the energy flux density of an [[electromagnetic field]]
*[[Wave vector]], a vector representation of the local phase evolution of a wave
*[[Tangent vector]], an element of the [[tangent space]] of a [[curve]], a [[surface]] or, more generally, a [[differential manifold]] at a given point.
*[[Gyrovector space|Gyrovector]], a hyperbolic geometry version of a vector
*[[Axial vector]], or pseudovector, a quantity that transforms like a vector under proper rotation but not generally under reflection
*[[Darboux vector]], the areal velocity vector of the Frenet frame of a space curve
*[[Four-vector]], in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space
*[[Interval vector]], in musical set theory, an array that expresses the intervallic content of a pitch-class set
*[[P-vector]], the tensor obtained by taking linear combinations of the wedge product of p tangent vectors
*[[Probability vector]], in statistics, a vector with non-negative entries that sum to one
*[[Spin vector]], or ''Spinor'', is an element of a complex vector space introduced to expand the notion of spatial vector
*[[Tuple]], an ordered list of numbers, sometimes used to represent a vector
*[[Unit vector]], a vector in a normed vector space whose length is 1
 
==Vector fields==
*[[Vector field]], a construction in vector calculus which associates a vector to every point in a subset of Euclidean space
**[[Conservative vector field]], a vector field which is the gradient of a scalar potential field
**[[Hamiltonian vector field]], a vector field defined for any energy function or Hamiltonian
**[[Killing vector field]], a vector field on a Riemannian manifold
**[[Solenoidal vector field]], a vector field with zero divergence
**[[Vector potential]], a vector field whose curl is a given vector field
*[[Vector flow]], a set of closely related concepts of the flow determined by a vector field
 
==Vector spaces==
*[[Vector space]], a mathematical structure made up of vectors, objects which may be added with another vector or multiplied by a scalar value
**[[Euclidean vector space]], an ''n''-dimensional space with notions of distance and angle that obey the Euclidean relationships
**[[Dual vector space]], a vector space consisting of all linear functionals on another, given vector space
**[[Graded vector space]], a type of vector space that includes the extra structure of gradation
**[[Normed vector space]], a vector space on which a norm is defined
**[[Ordered vector space]], a vector space equipped with a partial order
**[[Super vector space]], name for a Z<sub>2</sub>-graded vector space
**[[Symplectic vector space]], a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form
**[[Topological vector space]], a blend of topological structure with the algebraic concept of a vector space
 
==Manipulation of vectors, fields, and spaces==
*[[Vector bundle]], a topological construction which makes precise the idea of a family of vector spaces parameterized by another space
*[[Vector calculus]], a branch of mathematics concerned with differentiation and integration of vector fields
**[[Vector Analysis]], a free, online book on vector calculus first published in 1901 by Edwin Bidwell Wilson
*[[Vector decomposition]], refers to decomposing a vector of R<sup>n</sup> to several vectors, each linearly independent
*[[Vector differential]], or ''del'', is a vector differential operator represented by the nabla symbol: <math>\nabla</math>
*[[Vector Laplacian]], the vector Laplace operator, denoted by <math>\nabla^2</math> is a differential operator defined over a vector field
*[[Vector notation]], common notations used when working with vectors
*[[Vector operator]], a type of differential operator used in vector calculus
*[[Vector product]], or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
*[[Vector projection]], also known as the ''vector resolute'', a mapping of one vector onto another
*[[Vector-valued function]], a mathematical function that maps real numbers to vectors
*[[Vectorization (mathematics)]], a linear transformation which converts a matrix into a column vector
 
==Other uses in mathematics and physics==
*[[Vector autoregression]], an econometric model used to capture the evolution and the interdependencies between multiple time series
*[[Vector boson]], a boson with the spin quantum number equal to 1
*[[Vector measure]], a function defined on a family of sets and taking vector values satisfying certain properties
*[[Vector meson]], a meson with total spin 1 and odd parity
*[[Vector quantization]], a quantization technique used in signal processing
*[[Vector soliton]], a solitary wave with multiple components coupled together that maintains its shape during propagation
*[[Vector synthesis]], a type of audio synthesis
*[[Witt vector]], an infinite sequence of elements of a commutative ring
 
== See also ==
*[[Vector (disambiguation)]]
*[[Ricci calculus]]
 
==External links==
{{wiktionary|vector}}
* [http://gscim.com/phy/Physics_Basics/vectors.html Vectors in physics] Includes an embedded interactive program on vectors.
 
{{SIA}}
 
[[Category:Vectors]]

Latest revision as of 07:48, 7 April 2014

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