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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 a vector field is a construction in vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
 which associates a vector to every point in a (locally) Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
.

Vector fields are often used in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
, such as the magnetic
Magnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
 or gravitational force, as it changes from point to point.

In the rigorous mathematical treatment, (tangent) vector fields are defined on manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s as section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
s of a manifold's tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 a vector field is a construction in vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
 which associates a vector to every point in a (locally) Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
.

Vector fields are often used in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
, such as the magnetic
Magnetic field

A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
 or gravitational force, as it changes from point to point.

In the rigorous mathematical treatment, (tangent) vector fields are defined on manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s as section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
s of a manifold's tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
. They are one kind of tensor field
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
 on the manifold.

Definition


Vector fields on subsets of Euclidean space

Given a subset S in Rn, a vector field is represented by a vector-valued function
Vector-valued function

A vector-valued function is a mathematical function that maps real numbers to vector s. Vector-valued functions can be defined as:* orwhere f, g and h are the coordinate functions of the Parametric equation t, and , , and are unit vectors....
 in standard Euclidean coordinates (x1, ..., xn). If there is another coordinate system y on S, then is the expression for the same vector field in the new coordinates y. In particular, a vector field is not just a collection of scalar field
Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either scalar in definition, or scalar , to every point in space....
s.

We say V is a Ck vector field if V is k times continuously
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
 differentiable. A point p in S is called stationary if the vector at that point is zero .

A vector field can be visualized as an n-dimensional space with an n-dimensional vector attached to each point. Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition

define the module
Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
 of Ck-vector fields over the ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
 of Ck-functions.

Vector fields on manifolds

Given a manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 M, a vector field on M is a continuous assignment to every point of M a tangent vector
Tangent vector

A tangent vector is a Vector that follows the direction of a curve or a surface at a given point.* Differential geometry of curves, description in the context of curves in Rn....
 to M at that point. That is, for each x in M, we have a tangent vector v(x) in TxM such that the map sending a point to the appropriate tangent vector is a continuous function from the manifold to the total space of its tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
. More precisely, a vector field is a section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
 TM. If this section is continuous/differentiable/smooth/analytic, then we call the vector field continuous/differentiable/smooth/analytic. It is important to note that these properties are invariant under the change of coordinates formula, and thus can be detected by computing the local representation in any continuous/differentiable/smooth/analytic chart.

The collection of all vector fields on M is often denoted by G(TM) or C8(M,TM) (especially when thinking of vector fields as sections); the collection of all smooth vector fields is sometimes also denoted by (a fraktur "X").

Examples

  • A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude
    Magnitude (mathematics)

    The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
    ) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
  • Velocity
    Velocity

    In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
     field of a moving fluid
    Fluid

    A fluid is defined as a substance that continually deforms under an applied shear stress. All liquids and all gases are fluids. Fluids are a subset of the Phase and include liquids, gas, Plasma physics and, to some extent, plasticity ....
    . In this case, a velocity
    Velocity

    In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
     vector is associated to each point in the fluid.
  • Streamlines, Streaklines and Pathlines
    Streamlines, streaklines and pathlines

    Fluid flow is described in general by a vector field in three or four dimensions. Pathlines, streamlines, and streaklines are field lines of different vector field descriptions of the flow....
     are 3 types of lines that can be made from vector fields. They are :
streaklines — as revealed in wind tunnel
Wind tunnel

A wind tunnel is a research tool developed to assist with studying the effects of air moving over or around solid objects.Ways that wind-speed and flow are measured in wind tunnels:...
s using smoke. streamlines (or fieldlines)— as a line depicting the instantaneous field at a given time. pathlines — showing the path that a given particle (of zero mass) would follow.
  • Magnetic field
    Magnetic field

    A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
    s. The fieldlines can be revealed using small iron
    Iron

    Iron is a chemical element with the symbol Fe and atomic number 26. Iron is a Group 8 element and period 4 element. Iron is lustrous and silvery in color....
     filings.
  • Maxwell's equations
    Maxwell's equations

    In electromagnetism, James Clerk Maxwell equations are a set of four partial differential equations that describe the properties of the electric field and magnetic field fields and relate them to their sources, charge density and current density....
     allow us to use a given set of initial conditions to deduce, for every point in Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
    , a magnitude and direction for the force
    Force

    In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
     experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field
    Electromagnetic field

    The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
    .
  • A gravitational field
    Gravitational field

    A gravitational field is a scientific model used within physics to explain how gravitation exists in the universe. In its original concept, gravity was a force between point masses....
     generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.


Gradient field


Vector fields can be constructed out of scalar field
Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either scalar in definition, or scalar , to every point in space....
s using the vector operator gradient
Gradient

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
 which gives rise to the following definition.

A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real valued function (a scalar field) f on S such that

.

The associated flow
Flow (mathematics)

In mathematics, a flow formalizes, in mathematical terms, the general idea of "a variable that depends on time" that occurs very frequently in engineering, physics and the study of ordinary differential equations....
 is called the gradient flow, and is used in the method of gradient descent
Gradient descent

Gradient descent is an optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point....
.

The path integral
Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use....
 along any closed curve
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
 ? (?(0) = ?(1)) in a gradient field is zero: .

Central field


A C8-vector field over Rn \ is called a central field if

where O(n, R) is the orthogonal group
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
. We say central fields are invariant
Invariant (mathematics)

In mathematics, an invariant is something that does not change under a set of Transformation s. The property of being an invariant is invariance....
 under orthogonal transformations
Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
 around 0.

The point 0 is called the center of the field.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Line integral


A common technique in physics is to integrate a vector field along a curve
Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curve in the Euclidean plane and in the Euclidean space by methods of differential calculus and integral calculus....
: a line integral
Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use....
. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the line integral is the work done on the particle when it travels along a certain path.

The line integral is constructed analogously to the Riemann integral
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 ....
 and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.

Given a vector field V and a curve ? parametrized by [0, 1] the line integral is defined as

Flow curves

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equation
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
s.

Given a vector field V defined on S, we can try to define curves on S such that for each t in an interval I

If V is Lipschitz continuous
Lipschitz continuity

In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for function s which is stronger than regular continuous function....
 we can find a unique C1-curve ?x for each point x in S so that

The curves ?x are called flow curves of the vector field V and partition S into equivalence class
Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
es. It is not always possible to extend the interval (-e, +e) to the whole real number line. The flow may for example reach the edge of S in a finite time.

In two or three dimensions one can visualize the vector field as giving rise to a flow
Flow

Flow may refer to:In science and technology:*Dataflow, computing term related to the flow of messages between software components*Environmental flow, the amount of water necessary in a watercourse to maintain a healthy ecosystem...
 on S. If we drop a particle into this flow at a point p it will move along the curve ?p in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p.

Typical applications are streamline
Streamline

Streamline may refer to:*Streamlines, streaklines, and pathlines, in fluid flows. See also: Drag **Streamliner, any vehicle shaped to be less resistant to air...
 in fluid, geodesic flow, and one-parameter subgroups and the exponential map
Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
 in Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
s.

Complete vector fields

A vector field is complete if its flow curves exist for all time. In particular, compactly supported vector fields on a manifold are complete. If X is a complete vector field on M, then the one-parameter group
One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G....
 of diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
s generated by the flow along X exists for all time.

Difference between scalar and vector field


The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates.

Example 1
This example is about 2-dimensional Euclidean space (R2) where we examine Euclidean (x, y) and polar (r, ?) coordinates (which are undefined at the origin). Thus x = r cos ? and y = r sin ? and also r2 = x2 + y2, cos ? = x/(x2 + y2)1/2 and sin ? = y/(x2 + y2)1/2. Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the r-direction with length 1 to each point. More precisely, they are given by the functions

Let us convert these fields to Euclidean coordinates. The vector of length 1 in the r-direction has the x coordinate cos ? and the y coordinate sin ?. Thus in Euclidean coordinates the same fields are described by the functions

We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by the next example.

Example 2
Consider the 1-dimensional Euclidean space R with its standard Euclidean coordinate x. Suppose we have a scalar field and a vector field which are both given in the x coordinate by the constant function 1,

Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the x-direction with magnitude 1 unit of x to each point.

Now consider the coordinate ? := 2x. If x changes one unit then ? changes 2 units. Thus this vector field has a magnitude of 2 in units of ?. Therefore, in the ? coordinate the scalar field and the vector field are described by the functions

which are different.

See also


External links

  • -- Mathworld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
  • -- PlanetMath
    PlanetMath

    PlanetMath is a free content, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigour, openness, pedagogy, real-time content, interlinked content, and community....
  • on vector fields
  • Java applet illustrating vectors fields
  • An interactive application to show the effects of vector fields