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Vector field

In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector fields are often used in physics Physics

Physics , the most fundamental physical science [i], is concerned with the underlying principles of the ... 

 to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic Magnetic field

In physics [i], a magnetic field is that part of the electromagnetic field [i] that exists when there is ... 

 or gravitational Gravitation

In physics [i], gravitation or gravity is the tendency of objects with mass [i] to accelerate [i] ... 

 force, as it changes from point to point. In the rigorous mathematical treatment, vector fields are defined on manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

s as section Fiber bundle

In mathematics [i], in particular in topology [i], a fiber bundle is a space which locally looks like a ... 

s of the manifold's tangent bundle.

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Encyclopedia


In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.

Vector fields are often used in physics Physics

Physics , the most fundamental physical science [i], is concerned with the underlying principles of the ... 

 to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic Magnetic field

In physics [i], a magnetic field is that part of the electromagnetic field [i] that exists when there is ... 

 or gravitational Gravitation

In physics [i], gravitation or gravity is the tendency of objects with mass [i] to accelerate [i] ... 

 force, as it changes from point to point.

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

A manifold is an abstract mathematical space [i] in which every point has a neighborho ... 

s as section Fiber bundle

In mathematics [i], in particular in topology [i], a fiber bundle is a space which locally looks like a ... 

s of the manifold's tangent bundle.

Definition


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 number [i]s onto vector [i]s. ... 


in standard Euclidean coordinates . If there is another coordinate system y, then
is the expression for the same vector field in the new coordinates. In particular a vector field is not many scalar fields.

We say V is a Ck vector field if V is k times continuously differentiable Derivative

In mathematics [i], the derivative is defined as the instantaneous rate of change of a function [i] ... 

. A point p in S is called stationary if the vector at that point is zero
.

A vector field can be visualized as a n-dimensional space with a 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 of Ck-vector fields over the ring of Ck-functions.

Notes


Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space . Vector fields similarily associate a length or magnitude, as well as a direction to every point in space. For example, in the common three-space, every point in the manifold can be associated parametrically with magnitudes of x, y and z components.

The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. The first of these operations is defined in any number of dimensions . The curl however, is defined only for n=3, but it can be generalized to an arbitrary dimension using the exterior product Exterior algebra

In mathematics [i], the exterior algebra of a given vector space [i] V over a field [i] K ... 

 and exterior derivative Exterior derivative

In mathematics [i], the exterior derivative operator of differential geometry [i] extends the concept of ... 

.

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 of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure Atmospheric pressure

    Atmospheric pressure is the pressure [i] above any area in the Earth's atmosphere [i] caused by the weight [i] ... 

     map would then act as a source , and a "low" would be a sink , since air tends to move from high pressure areas to low pressure areas.
  • Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid.
  • Streamlines, Streaklines and Pathlines 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 aro... 

s using smoke.
streamlines — as a line depicting the instantaneous field at a given time.
pathlines — showing the path that a given particle would follow.
  • Magnetic field Magnetic field

    In physics [i], a magnetic field is that part of the electromagnetic field [i] that exists when there is ... 

    s. The fieldlines can be revealed using small iron Iron

    Iron is a chemical element [i] with the symbol Fe and atomic number [i] 26. ... 

     filings.
  • Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.

Gradient field


Vector fields can be constructed out of scalar fields using the vector operator gradient Gradient

A generalization of these concepts is the gradient in vector calculus [i]; and this article is mostly ab ... 

 which gives rise to the following definition.

A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on S such that
.

The path integral along any closed curve γ in a gradient field is zero:
.

Central field


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

where O is the orthogonal group. We say central fields are invariant under orthogonal transformations 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 definition.
A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Curve integral


A common technique in physics is to integrate a vector field along a curve: a line integral Line integral

In mathematics [i], a line integral is an integral [i] where the function [i] to be integrated ... 

. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path.

The curve integral is constructed analogously to the Riemann integral Riemann integral

In the branch of mathematics [i] known as real analysis [i], the Riemann integral, created by Bernhard Riemann [i]... 

 and it exists if the curve is rectifiable and the vector field is continuous.

Given a vector field V and a curve γ parametrized by [0, 1] the curve 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 [i], and particularly in analysis [i], an ordinary differential equati ... 

s.

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

If V is Lipschitz continuous we can find a unique C1-curve γx for each point x in S so that

The curves γx are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval 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 given rise to a flow on S. If we drop a particle into this flow at a point p it will move along the curve γp 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 in fluid, geodesic flow, and one-parameter subgroups and the exponential map Exponential map

There are two different notions of an exponential map in differential geometry [i], both of which genera ... 

 in Lie groups.

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 where we examine Euclidean and polar Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional [i] coordinate system [i] in which points [i] ... 

  coordinates . Thus x = r cos θ and y = r sin θ and also r2 = x2 + y2, cos θ = x/ and sin θ = y/. 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 θ and the y coordinate sin θ. 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. Also consider the coordinate ξ := 2x. Suppose we have a scalar field and a vector field which are both given in the ξ coordinates 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 ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then x changes 2 units. Then, this vector field has a magnitude of 2 in units of x. Therefore, in the x coordinate the scalar field and the vector field are described by the functions

which are different.

Example 3

In 1D, an example of a scalar field is the electric potential V, which is e.g. 20 volt at a particular point. This is a scalar, not depending on the coordinate system. An electric field at that point of 5 volt/metre in some coordinate system is −5 volt/metre in an inverse coordinate system. Since a physical quantity is not just a number, but a number times a unit, there is no change of coordinate system that gives any other than one of these two values for the electric field at the point.

See also


  • scalar field
  • tensor field
  • vector calculus
  • Lie derivative
  • differential geometry of curves
  • Time-dependent vector field
  • Vector fields in cylindrical and spherical coordinates

External links


  • -- Mathworld
  • -- PlanetMath PlanetMath

    PlanetMath is a free, collaborative, online mathematics [i] encyclopedia [i]. ... 

  • Java applet illustrating vectors fields
  • An interactive application to show the effects of vector fields