Vector field
Overview
of a vector to each point in a subset of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane. Vector fields are often used 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 any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
, such as the magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
or gravitational force, as it changes from point to point.
The elements of differential and integral calculus extend to vector fields in a natural way.
Unanswered Questions
Encyclopedia
In vector calculus, a vector field is an assignment
of a vector to each point in a subset of Euclidean space
. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane. Vector fields are often used 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
or gravitational force, as it changes from point to point.
The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral
of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy
is exhibited as a special case of the fundamental theorem of calculus
. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence
(which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
In coordinates, a vector field on a domain in ndimensional Euclidean space can be represented as a vectorvalued function
that associates an ntuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a welldefined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets
of Euclidean space, but also make sense on other subsets such as surface
s, where they associate an arrow tangent to the surface at each point (a tangent vector
).
More generally, vector fields are defined on differentiable manifold
s, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section
of the tangent bundle
to the manifold). Vector fields are one kind of tensor field
.
in standard Cartesian coordinates (x_{1}, ..., x_{n}). If each component of V is continuous, then V is a continuous function
, and more generally V is a C^{k} vector field if each component V is k times continuously differentiable.
A vector field can be visualized as an ndimensional space with an ndimensional vector attached to each point.
Given two C^{k}vector fields V, W defined on S and a real valued C^{k}function f defined on S, the two operations scalar multiplication and vector addition
define the module
of C^{k}vector fields over the ring
of C^{k}functions.
Thus, suppose that (x_{1},...,x_{n}) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are
and suppose that (y_{1},...,y_{n}) are n functions of the x_{i} defining a different coordinate system. Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law
Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law relating the different coordinate systems.
Vector fields are thus contrasted with scalar field
s, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.
M, a vector field on M is an assignment of a tangent vector
to each point in M. More precisely, a vector field F is a mapping
from M into the tangent bundle
TM so that is the identity mapping
where p denotes the projection from TM to M. In other words, a vector field is a section
of the tangent bundle
.
If the manifold M
is smooth (respectively analytic)that is, the change of
coordinates are smooth (respectively analytic)then one can make sense
of the notion of smooth (respectively analytic) vector fields.
The collection of all smooth vector fields on a smooth manifold
M is often denoted by Γ(TM) or C^{∞}(M,TM) (especially when thinking of vector fields as section
s); the collection of all smooth vector fields is also denoted by (a fraktur "X").
s using the gradient
operator (denoted by the del
: ) 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 realvalued function (a scalar field) f on S such that
The associated flow
is called the gradient flow, and is used in the method of gradient descent
.
The path integral
along any closed curve
γ (γ(0) = γ(1)) in a gradient field is zero:
where O(n, R) 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 (and simpler) definition.
A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.
: to determine a line integral
. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at a given 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
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 γ parametrized by [0, 1] the line integral is defined as
of a vector field on Euclidean space is a function (or scalar field). In threedimensions, the divergence is defined by
with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem
.
The divergence can also be defined on a Riemannian manifold
, that is, a manifold with a Riemannian metric that measures the length of vectors.
. In threedimensions, it is defined by
The curl measures the density of the angular momentum
of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes' theorem
.
in 19th century physics, specifically in magnetism
. They were formalized by Michael Faraday
, in his concept of lines of force, who emphasized that the field itself should be an object of study, which it has become throughout physics in the form of field theory.
In addition to the magnetic field, other phenomena that were modeled as vector fields by Faraday include the electrical field and light field
.
Given a vector field V defined on S, one defines curves on S such that for each t in an interval I
By the Picard–Lindelöf theorem
, if V is Lipschitz continuous
there is a unique C^{1}curve γ_{x} for each point x in S so that
The curves γ_{x} are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (ε, +ε) 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
on S. If we drop a particle into this flow at a point p it will move along the curve γ_{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 in fluid, geodesic flow, and oneparameter subgroups and the exponential map
in Lie group
s.
of diffeomorphism
s generated by the flow along X exists for all time.
Let us convert these fields to Euclidean coordinates. The vector of length 1 in the rdirection 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 1dimensional case, as illustrated by the next example.
Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the xdirection with magnitude 1 unit of x to each point.
Now consider the coordinate ξ := 2x. If x changes one unit then ξ changes 2 units. Thus this vector field has a magnitude of 2 in units of ξ. Therefore, in the ξ coordinate the scalar field and the vector field are described by the functions
which are different.
between manifolds, , the derivative
is an induced map on tangent bundles
, . Given vector fields and , we can ask whether they are compatible under in the following sense. We say that is related to if the equation holds.
and exterior powers yields differential kforms
, and combining these yields general tensor field
s.
Algebraically, vector fields can be characterized as derivations
of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras
.
of a vector to each point in a subset of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane. Vector fields are often used 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 any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
, such as the magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
or gravitational force, as it changes from point to point.
The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
is exhibited as a special case of the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
(which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
In coordinates, a vector field on a domain in ndimensional Euclidean space can be represented as a vectorvalued function
Vectorvalued function
A vectorvalued 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 infinitedimensional vectors. The input of a vectorvalued function could be a scalar or a vector...
that associates an ntuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a welldefined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets
Open set
The concept of an open set is fundamental to many areas of mathematics, especially pointset topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of Euclidean space, but also make sense on other subsets such as surface
Surface
In mathematics, specifically in topology, a surface is a twodimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary threedimensional Euclidean space R3 — for example, the surface of a ball...
s, where they associate an arrow tangent to the surface at each point (a tangent vector
Differential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....
).
More generally, vector fields are defined on differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M 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...
to the manifold). Vector fields are one kind of tensor field
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
.
Vector fields on subsets of Euclidean space
Given a subset S in R^{n}, a vector field is represented by a vectorvalued functionVectorvalued function
A vectorvalued 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 infinitedimensional vectors. The input of a vectorvalued function could be a scalar or a vector...
in standard Cartesian coordinates (x_{1}, ..., x_{n}). If each component of V is continuous, then V is a continuous function
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". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, and more generally V is a C^{k} vector field if each component V is k times continuously differentiable.
A vector field can be visualized as an ndimensional space with an ndimensional vector attached to each point.
Given two C^{k}vector fields V, W defined on S and a real valued C^{k}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, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
of C^{k}vector fields over the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of C^{k}functions.
Coordinate transformation law
In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.Thus, suppose that (x_{1},...,x_{n}) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are
and suppose that (y_{1},...,y_{n}) are n functions of the x_{i} defining a different coordinate system. Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law
Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law relating the different coordinate systems.
Vector fields are thus contrasted with scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinateindependent, meaning that any two observers using the same units will agree on the...
s, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.
Vector fields on manifolds
Given a differentiable manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
M, a vector field on M is an assignment of a tangent vector
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
to each point in M. More precisely, a vector field F is a mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
from M into the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M 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 so that is the identity mapping
where p denotes the projection from TM to M. In other words, a vector field is a section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M 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...
.
If the manifold M
is smooth (respectively analytic)that is, the change of
coordinates are smooth (respectively analytic)then one can make sense
of the notion of smooth (respectively analytic) vector fields.
The collection of all smooth vector fields on a smooth manifold
M is often denoted by Γ(TM) or C^{∞}(M,TM) (especially when thinking of vector fields as section
Section
Section may refer to:* Section * Section * Archaeological section* Histological section, a thin slice of tissue used for microscopic examination* Section, an instrumental group within an orchestra...
s); the collection of all smooth vector fields is 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 (magnitudeMagnitude (mathematics)The magnitude of an object in mathematics is its size: a property by which it can be compared as 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.  VelocityVelocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
field of a moving fluidFluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
. In this case, a velocityVelocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
vector is associated to each point in the fluid.  Streamlines, Streaklines and PathlinesStreamlines, streaklines and pathlinesFluid flow is characterized by a velocity vector field in threedimensional space, within the framework of continuum mechanics. Streamlines, streaklines and pathlines are field lines resulting from this vector field description of the flow...
are 3 types of lines that can be made from vector fields. They are :

 streaklines — as revealed in wind tunnelWind tunnelA wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...
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 fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s. The fieldlines can be revealed using small ironIronIron is a chemical element with the symbol Fe and atomic number 26. It is a metal in the first transition series. It is the most common element forming the planet Earth as a whole, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust...
filings.  Maxwell's equationsMaxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
allow us to use a given set of initial conditions to deduce, for every point in Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, a magnitude and direction for the forceForceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
experienced by a charged test particle at that point; the resulting vector field is the electromagnetic fieldElectromagnetic fieldAn electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
.  A gravitational fieldGravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. 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.
 Magnetic field
 streaklines — as revealed in wind tunnel
Gradient field
Vector fields can be constructed out of scalar fieldScalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinateindependent, meaning that any two observers using the same units will agree on the...
s using the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
operator (denoted by the del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a onedimensional domain, it denotes its standard derivative as defined in calculus...
: ) 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 realvalued function (a scalar field) f on S such that
The associated flow
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
is called the gradient flow, and is used in the method of gradient descent
Gradient descent
Gradient descent is a firstorder 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.The function to be integrated may be a scalar field or a vector field...
along any closed curve
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
γ (γ(0) = γ(1)) in a gradient field is zero:
Central field
A C^{∞}vector field over R^{n} \ {0} is called a central field ifwhere 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 × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
. We say central fields are invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under orthogonal transformations
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
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 curveDifferential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....
: to determine a line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at a given 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. The Riemann integral is unsuitable for many theoretical purposes...
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 γ parametrized by [0, 1] the line integral is defined as
Divergence
The divergenceDivergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of a vector field on Euclidean space is a function (or scalar field). In threedimensions, the divergence is defined by
with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
.
The divergence can also be defined on a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
, that is, a manifold with a Riemannian metric that measures the length of vectors.
Curl
The curl is an operation which takes a vector field and produces another vector field. The curl is defined only in threedimensions, but some properties of the curl can be captured in higher dimensions with the exterior derivativeExterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
. In threedimensions, it is defined by
The curl measures the density of the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
.
History
Vector fields arose originally in classical field theoryClassical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
in 19th century physics, specifically in magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
. They were formalized by Michael Faraday
Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....
, in his concept of lines of force, who emphasized that the field itself should be an object of study, which it has become throughout physics in the form of field theory.
In addition to the magnetic field, other phenomena that were modeled as vector fields by Faraday include the electrical field and light field
Light field
The light field is a function that describes the amount of light faring in every direction through every point in space. Michael Faraday was the first to propose that light should be interpreted as a field, much like the magnetic fields on which he had been working for several years...
.
Flow curves
Consider the flow of a fluid, such as a gas, through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity.Given a vector field V defined on S, one defines curves on S such that for each t in an interval I
By the Picard–Lindelöf theorem
Picard–Lindelöf theorem
In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to firstorder equations with given initial conditions.The theorem is named after Charles...
, if V is Lipschitz continuous
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
there is a unique C^{1}curve γ_{x} for each point x in S so that
The curves γ_{x} are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (ε, +ε) 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 (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
on S. If we drop a particle into this flow at a point p it will move along the curve γ_{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 in fluid, geodesic flow, and oneparameter 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 smooth 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 oneparameter groupOneparameter group
In mathematics, a oneparameter group or oneparameter 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 in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. Definition :...
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 2dimensional Euclidean space (R^{2}) where we examine Euclidean (x, y) and polar (r, θ) coordinates (which are undefined at the origin). Thus x = r cos θ and y = r sin θ and also r^{2} = x^{2} + y^{2}, cos θ = x/(x^{2} + y^{2})^{1/2} and sin θ = y/(x^{2} + y^{2})^{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 rdirection with length 1 to each point. More precisely, they are given by the functionsLet us convert these fields to Euclidean coordinates. The vector of length 1 in the rdirection 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 1dimensional case, as illustrated by the next example.
Example 2
Consider the 1dimensional 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 xdirection with magnitude 1 unit of x to each point.
Now consider the coordinate ξ := 2x. If x changes one unit then ξ changes 2 units. Thus this vector field has a magnitude of 2 in units of ξ. Therefore, in the ξ coordinate the scalar field and the vector field are described by the functions
which are different.
frelatedness
Given a smooth functionSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
between manifolds, , the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is an induced map on tangent bundles
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M 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...
, . Given vector fields and , we can ask whether they are compatible under in the following sense. We say that is related to if the equation holds.
Generalizations
Replacing vectors by pvectors (pth exterior power of vectors) yields pvector fields; taking the dual spaceDual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
and exterior powers yields differential kforms
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
, and combining these yields general tensor field
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
s.
Algebraically, vector fields can be characterized as derivations
Derivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a Kderivation is a Klinear map D: A → A that satisfies Leibniz's law: D = b + a.More...
of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras
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 also
 Eisenbud–Levine–Khimshiashvili signature formulaEisenbud–Levine–Khimshiashvili signature formulaIn mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the PoincaréHopf index of a real, analytic vector field at an algebraicially isolated singularity. It is named after David Eisenbud, Harold...
 Field lineField lineA field line is a locus that is defined by a vector field and a starting location within the field. Field lines are useful for visualizing vector fields, which are otherwise hard to depict...
 Lie derivativeLie derivativeIn mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
 Scalar fieldScalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinateindependent, meaning that any two observers using the same units will agree on the...
 Timedependent vector field
 Vector fields in cylindrical and spherical coordinatesVector fields in cylindrical and spherical coordinates* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in question. \phi is the angle between the projection of the radius vector onto the xy plane and the x axis...
External links
 Vector field  MathworldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
 Vector field  PlanetMathPlanetMathPlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, realtime content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
 3D Magnetic field viewer
 Vector Field Simulation Java applet illustrating vectors fields
 Vector fields and field lines
 Vector field simulation An interactive application to show the effects of vector fields
 Vector Fields Software 2d & 3d electromagnetic design software that can be used to visualise vector fields and field lines