Vector field, the Glossary
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.[1]
Table of Contents
104 relations: Analytic function, Angular momentum, Atmospheric pressure, Calculus, Cartesian coordinate system, Commutative property, Conservation of energy, Conservative vector field, Coordinate system, Covariance and contravariance of vectors, Curl (mathematics), Curve, Del, Derivation (differential algebra), Derivative, Diffeomorphism, Differentiable curve, Differentiable manifold, Differential calculus over commutative algebras, Differential form, Divergence, Divergence theorem, Dual space, Eisenbud–Levine–Khimshiashvili signature formula, Electric field, Equivalence class, Euclidean plane, Euclidean space, Euclidean vector, Euler characteristic, Exponential map (Lie theory), Exterior derivative, Field (physics), Field line, Field strength, Flow (mathematics), Fluid, Fluid dynamics, Force, Fraktur, Fundamental theorem of calculus, Geodesic, Gradient, Gradient descent, Gravitational field, Gravity, Hairy ball theorem, Invariant (mathematics), Iron, Lie bracket of vector fields, ... Expand index (54 more) »
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
See Vector field and Analytic function
Angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum.
See Vector field and Angular momentum
Atmospheric pressure
Atmospheric pressure, also known as air pressure or barometric pressure (after the barometer), is the pressure within the atmosphere of Earth.
See Vector field and Atmospheric pressure
Calculus
Calculus 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.
Cartesian coordinate system
In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.
See Vector field and Cartesian coordinate system
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
See Vector field and Commutative property
Conservation of energy
The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time.
See Vector field and Conservation of energy
Conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Vector field and conservative vector field are vector calculus.
See Vector field and Conservative vector field
Coordinate system
In geometry, a coordinate system is a system that 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.
See Vector field and Coordinate system
Covariance and contravariance of vectors
In physics, especially 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.
See Vector field and Covariance and contravariance of vectors
Curl (mathematics)
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 Euclidean space. Vector field and curl (mathematics) are vector calculus.
See Vector field and Curl (mathematics)
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. Vector field and Del are vector calculus.
Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator.
See Vector field and Derivation (differential algebra)
Derivative
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input.
See Vector field and Derivative
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.
See Vector field and Diffeomorphism
Differentiable curve
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.
See Vector field and Differentiable curve
Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.
See Vector field and Differentiable manifold
Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms.
See Vector field and Differential calculus over commutative algebras
Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.
See Vector field and Differential form
Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. Vector field and divergence are vector calculus.
See Vector field and Divergence
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
See Vector field and Divergence theorem
Dual space
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
See Vector field and Dual space
Eisenbud–Levine–Khimshiashvili signature formula
In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré–Hopf index of a real, analytic vector field at an algebraically isolated singularity.
See Vector field and Eisenbud–Levine–Khimshiashvili signature formula
Electric field
An electric field (sometimes called E-field) is the physical field that surrounds electrically charged particles.
See Vector field and Electric field
Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.
See Vector field and Equivalence class
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted \textbf^2 or \mathbb^2.
See Vector field and Euclidean plane
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Vector field and Euclidean space
Euclidean vector
In mathematics, 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 (or length) and direction. Vector field and Euclidean vector are vector calculus.
See Vector field and Euclidean vector
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
See Vector field and Euler characteristic
Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
See Vector field and Exponential map (Lie theory)
Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
See Vector field and Exterior derivative
Field (physics)
In science, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time.
See Vector field and Field (physics)
Field line
A field line is a graphical visual aid for visualizing vector fields. Vector field and field line are vector calculus.
See Vector field and Field line
Field strength
In physics, field strength is the magnitude of a vector-valued field (e.g., in volts per meter, V/m, for an electric field E).
See Vector field and Field strength
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid.
See Vector field and Flow (mathematics)
Fluid
In physics, a fluid is a liquid, gas, or other material that may continuously move and deform (flow) under an applied shear stress, or external force.
Fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases.
See Vector field and Fluid dynamics
Force
A force is an influence that can cause an object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces.
Fraktur
Fraktur is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand.
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions).
See Vector field and Fundamental theorem of calculus
Geodesic
In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold.
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. Vector field and gradient are vector calculus.
Gradient descent
Gradient descent is a method for unconstrained mathematical optimization.
See Vector field and Gradient descent
Gravitational field
In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself.
See Vector field and Gravitational field
Gravity
In physics, gravity is a fundamental interaction which causes mutual attraction between all things that have mass.
Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres.
See Vector field and Hairy ball theorem
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects.
See Vector field and Invariant (mathematics)
Iron
Iron is a chemical element.
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. Vector field and Lie bracket of vector fields are differential topology.
See Vector field and Lie bracket of vector fields
Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. Vector field and Lie derivative are differential topology.
See Vector field and Lie derivative
Lie group
In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
See Vector field and Lie group
Light field
A light field, or lightfield, is a vector function that describes the amount of light flowing in every direction through every point in a space.
See Vector field and Light field
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Vector field and line integral are vector calculus.
See Vector field and Line integral
Line integral convolution
In scientific visualization, line integral convolution (LIC) is a method to visualize a vector field (such as fluid motion) at high spatial resolutions. Vector field and line integral convolution are vector calculus.
See Vector field and Line integral convolution
Line of force
A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction.
See Vector field and Line of force
Linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers).
See Vector field and Linear form
Linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.
See Vector field and Linear map
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.
See Vector field and Lipschitz continuity
Magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.
See Vector field and Magnetic field
Magnitude (mathematics)
In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind.
See Vector field and Magnitude (mathematics)
Map (mathematics)
In mathematics, a map or mapping is a function in its general sense.
See Vector field and Map (mathematics)
MathWorld
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.
See Vector field and MathWorld
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits.
See Vector field and Maxwell's equations
Meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting.
See Vector field and Meteorology
Michael Faraday
Michael Faraday (22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry.
See Vector field and Michael Faraday
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring.
See Vector field and Module (mathematics)
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra of a vector space.
See Vector field and Multivector
Number line
In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
See Vector field and Number line
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as an additive group.
See Vector field and One-parameter group
Open set
In mathematics, an open set is a generalization of an open interval in the real line.
Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
See Vector field and Orthogonal group
Orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
See Vector field and Orthogonal matrix
Parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
See Vector field and Parametric equation
Physics
Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.
Picard–Lindelöf theorem
In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution.
See Vector field and Picard–Lindelöf theorem
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia.
See Vector field and PlanetMath
Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. Vector field and Poincaré–Hopf theorem are differential topology.
See Vector field and Poincaré–Hopf theorem
Position (geometry)
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space.
See Vector field and Position (geometry)
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Vector field and Real number
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
See Vector field and Riemann integral
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.
See Vector field and Riemannian manifold
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.
See Vector field and Ring (mathematics)
Scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space.
See Vector field and Scalar field
Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi. Vector field and section (fiber bundle) are differential topology.
See Vector field and Section (fiber bundle)
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.
See Vector field and Smoothness
Space (mathematics)
In mathematics, a space is a set (sometimes known as a universe) with a definition (structure) of relationships among the elements of the set.
See Vector field and Space (mathematics)
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 (Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. Vector field and Stokes' theorem are vector calculus.
See Vector field and Stokes' theorem
Streamlines, streaklines, and pathlines
Streamlines, streaklines and pathlines are field lines in a fluid flow.
See Vector field and Streamlines, streaklines, and pathlines
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero.
See Vector field and Support (mathematics)
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold.
See Vector field and Surface (topology)
Tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Vector field and tangent bundle are differential topology.
See Vector field and Tangent bundle
Tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. Vector field and tangent space are differential topology.
See Vector field and Tangent space
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). Vector field and tensor field are differential topology.
See Vector field and Tensor field
Three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point.
See Vector field and Three-dimensional space
Time dependent vector field
In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. Vector field and time dependent vector field are vector calculus.
See Vector field and Time dependent vector field
Vector (mathematics and physics)
In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
See Vector field and Vector (mathematics and physics)
Vector calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3.
See Vector field and Vector calculus
Vector fields in cylindrical and spherical coordinates
Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. Vector field and vector fields in cylindrical and spherical coordinates are vector calculus.
See Vector field and Vector fields in cylindrical and spherical coordinates
Vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. Vector field and vector-valued function are vector calculus.
See Vector field and Vector-valued function
Velocity
Velocity is the speed in combination with the direction of motion of an object.
Volume
Volume is a measure of regions in three-dimensional space.
Wind
Wind is the natural movement of air or other gases relative to a planet's surface.
Work (physics)
In science, work is the energy transferred to or from an object via the application of force along a displacement.
See Vector field and Work (physics)
References
[1] https://en.wikipedia.org/wiki/Vector_field
Also known as Complete vector field, F-related, Gradient flow, Gradient vector field, Index of a vector field, Operations on vector fields, Tangent Bundle Section, Vector field on a manifold, Vector fields, Vector plot, Vector point function, Vector-field, Vectorfield.
, Lie derivative, Lie group, Light field, Line integral, Line integral convolution, Line of force, Linear form, Linear map, Lipschitz continuity, Magnetic field, Magnitude (mathematics), Map (mathematics), MathWorld, Maxwell's equations, Meteorology, Michael Faraday, Module (mathematics), Multivector, Number line, One-parameter group, Open set, Orthogonal group, Orthogonal matrix, Parametric equation, Physics, Picard–Lindelöf theorem, PlanetMath, Poincaré–Hopf theorem, Position (geometry), Real number, Riemann integral, Riemannian manifold, Ring (mathematics), Scalar field, Section (fiber bundle), Smoothness, Space (mathematics), Stokes' theorem, Streamlines, streaklines, and pathlines, Support (mathematics), Surface (topology), Tangent bundle, Tangent space, Tensor field, Three-dimensional space, Time dependent vector field, Vector (mathematics and physics), Vector calculus, Vector fields in cylindrical and spherical coordinates, Vector-valued function, Velocity, Volume, Wind, Work (physics).