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Concept in linear algebra
linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular
Coordinate_vector
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces
Vector_space
Set of vectors used to define coordinates
linear program Coordinate system Change of basis – Coordinate change in linear algebra Frame of a vector space – Similar to the basis of a vector space, but
Basis_(linear_algebra)
Vector behavior under coordinate changes
particularly important for understanding how the coordinate description of a vector changes by passing from one coordinate system to another. Tensors are objects
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Geometric object that has length and direction
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude
Euclidean_vector
Mathematical operation in linear algebra
the vector on the basis. These coordinate vectors form another vector space, which is isomorphic to the original vector space. A coordinate vector is commonly
Matrix_multiplication
Broad concept generalizing scalars in mathematics and physics
of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Coordinate change in linear algebra
ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a finite
Change_of_basis
Space formed by the ''n''-tuples of real numbers
numbers, also known as coordinate vectors. Special cases are called the real line R1, the real coordinate plane R2, and the real coordinate three-dimensional
Real_coordinate_space
Assignment of a vector to each point in a subset of Euclidean space
measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically
Vector_field
Algebraic operation on coordinate vectors
sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of
Dot_product
Vector representing the position of a point with respect to a fixed origin
The coordinates of the vector r with respect to the basis vectors ei are xi. The vector of coordinates forms the coordinate vector or n-tuple (x1, x2, …
Position_(geometry)
Matrix relating a system's generalized coordinate vector and kinetic energy
derivative q ˙ {\displaystyle \mathbf {\dot {q}} } of the generalized coordinate vector q of a system and the kinetic energy T of that system, by the equation
Mass_matrix
Use of coordinates for representing vectors
Vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more
Vector_notation
Coordinate system using perpendicular axes
description of the plane was later generalized into the concept of vector spaces. Many other coordinate systems have been developed since Descartes, such as the
Cartesian_coordinate_system
Multivariate derivative (mathematics)
x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system)
Gradient
Vectors whose components are all 0 except one that is 1
basis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of
Standard_basis
Function valued in a vector space; typically a real or complex one
{k} } where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and the domain of this vector-valued function is the intersection of the
Vector-valued_function
Vector of length one
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase
Unit_vector
Mathematical operation on vector spaces
a rectangular array, the coordinate vector of x ⊗ y {\displaystyle x\otimes y} is the outer product of the coordinate vectors of x {\displaystyle x} and
Tensor_product
Computer graphics images defined by points, lines and curves
use both vector and raster graphics at times, depending on purpose. Vector graphics are based on the mathematics of analytic or coordinate geometry,
Vector_graphics
Mnemonic for 3D vectors orientations and rotations
(third coordinate vector), then the fingers curl from the positive x-axis (first coordinate vector) toward the positive y-axis (second coordinate vector).
Right-hand_rule
Topics referred to by the same term
related domains Coordinate space in mathematics Cartesian coordinate system Coordinate (vector space) Geographic coordinate system Coordinate structure in
Coordinate_(disambiguation)
Branch of physics describing the motion of objects without considering forces
particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower
Kinematics
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Mathematical description of spacetime used in relativity
| p . Applying this with f = xμ, the coordinate function itself, and X = ∂/ ∂xν , called a coordinate vector field, one obtains d x μ ( ∂ ∂ x ν )
Minkowski_spacetime
Euclidean space without distance and angles
point. These coefficients define a barycentric coordinate system for the flat through the points. Any vector space may be viewed as an affine space; this
Affine_space
Coordinate system whose directions vary in space
Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then
Curvilinear_coordinates
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
are given as in finite coordinate space. The dimensionality of F∞ is countably infinite. A standard basis consists of the vectors ei which contain a 1 in
Examples_of_vector_spaces
Family of linear transformations
matrix, which rotates any 3-dimensional vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive
Lorentz_transformation
Expression that may be integrated over a region
particular, if v = ej is the jth coordinate vector then ∂v f is the partial derivative of f with respect to the jth coordinate vector, i.e., ∂f / ∂xj, where x1
Differential_form
Structure defining distance on a manifold
_{n}} where ei are the standard coordinate vectors in ℝn. When φ is applied to U, the vector v goes over to the vector tangent to M given by φ ∗ ( v )
Metric_tensor
Elements of a field, e.g. real numbers, in the context of linear algebra
algebra, every vector space has a basis. It follows that every vector space over a field K is isomorphic to the corresponding coordinate vector space where
Scalar_(mathematics)
Method to control electric motors
vectors' rotating reference-frame two-coordinate time invariant system. Such complex stator current space vector can be defined in a (d,q) coordinate
Field-oriented_control
Affine connection on the tangent bundle of a manifold
intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the
Levi-Civita_connection
Tensor in differential geometry
{\displaystyle g_{ij}} are defined by evaluating g {\displaystyle g} on coordinate vector fields, while the functions g i j {\displaystyle g^{ij}} are defined
Ricci_curvature
Shorthand notation for tensor operations
contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when
Einstein_notation
Operator in differential topology
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
Lie_bracket_of_vector_fields
System to specify locations on Earth
intersects the equatorial plane. In a geodetic coordinate system, the second point is found where the normal vector from the surface of the ellipsoid at the
Geographic_coordinate_system
Method for specifying point positions
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points
Coordinate_system
Vector operation
of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the
Outer_product
Simple machine consisting of a beam pivoted at a fixed hinge
operated by applying an input force FA at a point A located by the coordinate vector rA on the bar. The lever then exerts an output force FB at the point
Lever
Algebraic object with geometric applications
geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations
Tensor
Length in a vector space
spaces. The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. Hence,
Norm_(mathematics)
Specification of a derivative along a tangent vector of a manifold
covariant derivative of a vector field with respect to a vector field, both in a coordinate-free language and using a local coordinate system and the traditional
Covariant_derivative
Frame of reference for an orbit
} coordinate must be aligned with the eccentricity vector. Circular orbits, having no eccentricity, give no means by which to orient the coordinate system
Perifocal_coordinate_system
Horizontal angle from north or other reference cardinal direction
in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer (origin) to a point of interest
Azimuth
Coordinates comprising a distance and two angles
surveying instrument Vector fields in cylindrical and spherical coordinates – Vector field representation in 3D curvilinear coordinate systems Yaw, pitch
Spherical_coordinate_system
Direction and rate of rotation
direction (a unit vector) parallel to Earth's rotation axis ( ω ^ = Z ^ {\displaystyle {\hat {\omega }}={\hat {Z}}} , in the geocentric coordinate system). If
Angular_velocity
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
such as by rotation or stretching the coordinate system, the components of the vector also transform. The vector itself does not change, but the reference
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Vector operator in vector calculus
divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use. A vector field with zero divergence
Divergence
Similarity measure for number sequences
text mining, each word is assigned a different coordinate and a document is represented by the vector of the numbers of occurrences of each word in the
Cosine_similarity
Mathematical gradient operator in certain coordinate systems
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. This article uses the standard notation ISO 80000-2
Del in cylindrical and spherical coordinates
Del_in_cylindrical_and_spherical_coordinates
Tensor index notation for tensor-based calculations
example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector A = (A1, A2, A3) = (Ax, Ay, Az) shows a direct correspondence between
Ricci_calculus
Fundamental space of geometry
Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this
Euclidean_space
Cosines of the angles between a vector and the coordinate axes
(or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the
Direction_cosine
ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions
N-vector
Coordinate system that is defined by points instead of vectors
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle
Barycentric_coordinate_system
Space formed by the ''n''-tuples of complex numbers
n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers, also known as complex vectors. The space is
Complex_coordinate_space
Branch of mathematics
isomorphism allows representing a vector by its inverse image under this isomorphism, that is by the coordinate vector (a1, ..., am) or by the column matrix
Linear_algebra
Vector differential operator
x_{n}}\right)} where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system R 3 {\displaystyle \mathbb {R} ^{3}} with coordinates
Del
Matrix representing a Euclidean rotation
Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied
Rotation_matrix
Coordinate system in black hole physics
the timelike coordinate vector is not hypersurface orthogonal.) Note the last two fields are rotations of one-another, under the coordinate transformation
Schwarzschild_coordinates
Array of numbers describing a metric connection
R n {\displaystyle \mathbb {R} ^{n}} pulls back to a standard ("coordinate") vector basis ( ∂ 1 , ⋯ , ∂ n ) {\displaystyle (\partial _{1},\cdots ,\partial
Christoffel_symbols
Four-dimensional number system
additive inverses of 1, i, j, and k. The vector part of a quaternion can be interpreted as a coordinate vector in R 3 ; {\displaystyle \mathbb {R} ^{3};}
Quaternion
Extension of the scalar spherical harmonics for use with vector fields
the VSH are complex-valued functions expressed in the spherical coordinate basis vectors. Several conventions have been used to define the VSH. We follow
Vector_spherical_harmonics
Representation of mechanical stress at every point within a deformed 3D object
represent the vector, but the magnitude of the vector is a physical quantity (a scalar) and is independent of the Cartesian coordinate system chosen to
Cauchy_stress_tensor
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Motion of a certain space that preserves at least one point
and a unit vector for the axis, or as a Euclidean vector obtained by multiplying the angle with this unit vector, called the rotation vector (although
Rotation_(mathematics)
Directional planes
plane regions, vectors, directions, etc. A surface is horizontal if its tangent planes are everywhere perpendicular to the gravity vector at the tangent
Vertical_and_horizontal
Coordinates comprising a distance and an angle
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are the point's
Polar_coordinate_system
Mathematical function
{\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real
Function_of_a_real_variable
Central object in linear algebra; mapping vectors to vectors
When using affine transformations, the homogeneous component of a coordinate vector (normally called w) will never be altered. One can therefore safely
Transformation_matrix
Mathematical operation on vectors in 3D space
product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional
Cross_product
Graphics mode on the Super NES video game console
y_{0}} (which define the vector r 0 {\displaystyle \mathbf {r} _{0}} , the origin). Specifically, 2D screen coordinate vector r {\displaystyle \mathbf
Mode_7
Force directed to the center of rotation
analysis agrees with this one. A merit of the vector approach is that it is manifestly independent of any coordinate system. The upper panel in the image at
Centripetal_force
Process of projecting a 3D object onto a 2D plane
maps the coordinate vector p ∈ R 3 {\displaystyle p\in \mathbb {R} ^{3}} of a general point P {\displaystyle P} in space to the coordinate vector in R 2
Axonometry
Mathematical algorithm
continuously differentiable function F, a coordinate descent algorithm can be sketched as: Choose an initial parameter vector x. Until convergence is reached,
Coordinate_descent
Geometric transformation
homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py, pz, 1) would need to be multiplied
Scaling_(geometry)
One-dimensional physical quantity
changes to a vector space basis (i.e., a coordinate rotation) but may be affected by translations (as in relative speed). A change of a vector space basis
Scalar_(physics)
Cartesian vectors of position and velocity of an orbiting body in space
and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are Cartesian vectors of position ( r {\displaystyle \mathbf {r}
Orbital_state_vectors
Concept in mathematics
{x} =0,} where x {\displaystyle \mathbf {x} } is the homogeneous coordinate vector in three variables restricted so that the last variable is 1, i.e
Matrix representation of conic sections
Matrix_representation_of_conic_sections
Geometric model of the physical space
to a general vector space V {\displaystyle V} , the space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically
Three-dimensional_space
On vector derivatives for rotating frames
Edmond Bour) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time
Transport_theorem
Set of coordinates where the coordinate hypersurfaces all meet at right angles
the coordinate for which the basis vector is extracted. In other words, a curve is obtained by fixing all but one coordinate; the unfixed coordinate is
Orthogonal_coordinates
Mathematical transformation that preserves distances
vector v, produces a transformed vector T(v) of the form T(v) = R v + t where RT = R−1 (i.e., R is an orthogonal transformation), and t is a vector giving
Rigid_transformation
Coordinate system
{\displaystyle \partial _{v}} , the coordinate vector field dual to the covector field d v {\displaystyle dv} , is a null vector field. Indeed, geometrically
Brinkmann_coordinates
Speed and direction of a motion
physical objects. Velocity is a vector quantity, meaning that both magnitude and direction are needed to define it (velocity vector). The scalar absolute value
Velocity
Calculus of vector-valued functions
the handedness of the coordinate system to be taken into account. Vector calculus can be defined on other 3-dimensional real vector spaces if they have
Vector_calculus
Representation of a tensor in Euclidean space
contravariance of vectors for why. The term "component" of a vector is ambiguous: it could refer to: a specific coordinate of the vector such as az (a scalar)
Cartesian_tensor
Study of geometry using a coordinate system
analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic
Analytic_geometry
Tensor field in Riemannian geometry
{\displaystyle \partial _{\mu }=\partial /\partial x^{\mu }} are the coordinate vector fields. The above expression can be written using Christoffel symbols:
Riemann_curvature_tensor
Assignment of a tensor continuously varying across a region of space
{\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.} The list of Cartesian coordinate basis vectors e k {\displaystyle \mathbf {e} _{k}} transforms as a covector,
Tensor_field
Basis used to express spherical tensors
basis and use complex numbers. A vector A in 3D Euclidean space R3 can be expressed in the familiar Cartesian coordinate system in the standard basis ex
Spherical_basis
Approach to general relativity
bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called
Tetrad_formalism
Type of derivative in differential geometry
(including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore
Lie_derivative
Geometric model of the planar projection of the physical universe
\mathbf {A} }},} the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by ∇ f = ∂ f ∂ x i + ∂ f ∂
Euclidean_plane
Mathematical methods that avoid coordinates
elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space
Coordinate-free
COORDINATE VECTOR
COORDINATE VECTOR
Boy/Male
Indian, Telugu
Coordinator; Conveyor; Become a Leader
Boy/Male
Tamil
Co-coordinator
Boy/Male
Hindu
Co-coordinator
COORDINATE VECTOR
COORDINATE VECTOR
Biblical
father
Boy/Male
Indian, Punjabi, Sikh
King of the Kings
Girl/Female
Arabic
Soft and Gentle
Girl/Female
Latin
God's gift.
Male
English
Patronymic form of English Mile, MILES means "son of Mile."
Boy/Male
Hindu
Born of water
Boy/Male
French, German
From the Little Home
Male
Swiss
, blessed.
Girl/Female
Arabic, Australian, Muslim
Beautiful
Boy/Male
Arabic
Servant of the Compassionate
COORDINATE VECTOR
COORDINATE VECTOR
COORDINATE VECTOR
COORDINATE VECTOR
COORDINATE VECTOR
v. i.
To have or indulge inordinate desire.
imp. & p. p.
of Coordinate
n.
A thing of the same rank with another thing; one two or more persons or things of equal rank, authority, or importance.
n.
The state of being coordinate; equality of rank or authority.
a.
Inordinate; irregular; vicious.
a.
Not limited to rules prescribed, or to usual bounds; irregular; excessive; immoderate; as, an inordinate love of the world.
a.
Expressing coordination.
adv.
In a coordinate manner.
n.
Joint ordinance.
p. pr. & vb. n.
of Coordinate
a.
Excessive; extravagant; inordinate.
a.
Equal in rank or order; not subordinate.
v. t.
To give a common action, movement, or condition to; to regulate and combine so as to produce harmonious action; to adjust; to harmonize; as, to coordinate muscular movements.
n.
Lines, or other elements of reference, by means of which the position of any point, as of a curve, is defined with respect to certain fixed lines, or planes, called coordinate axes and coordinate planes. See Abscissa.
v. t.
To make coordinate; to put in the same order or rank; as, to coordinate ideas in classification.
a.
Disorderly; irregular; inordinate.
a.
Not coordinate.
n.
The incense of praise; inordinate flattery.
a.
Pertaining to two coordinate species or divisions.
a.
Inordinate; disorderly.