Invector calculus andphysics, avector field is an assignment of avector to each point in aspace, most commonlyEuclidean space.[1] A vector field on aplane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughoutthree dimensional space, such as thewind, or the strength and direction of someforce, such as themagnetic orgravitational force, as it changes from one point to another point.
The elements ofdifferential and integral calculus extend naturally to vector fields. When a vector field representsforce, theline integral of a vector field represents thework done by a force moving along a path, and under this interpretationconservation of energy is exhibited as a special case of thefundamental 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 thedivergence (which represents the rate of change ofvolume of a flow) andcurl (which represents the rotation of a flow).
A vector field is a special case of avector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, theposition vector of aspace curve is defined only for smaller subset of the ambient space.Likewise, ncoordinates, a vector field on a domain inn-dimensional Euclidean space can be represented as a vector-valued function that associates ann-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (covariance and contravariance of vectors) in passing from one coordinate system to the other.
Vector fields are often discussed onopen subsets of Euclidean space, but also make sense on other subsets such assurfaces, where they associate an arrow tangent to the surface at each point (atangent vector).More generally, vector fields are defined ondifferentiable manifolds, 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, asection of thetangent bundle to the manifold). Vector fields are one kind oftensor field.
Two representations of the same vector field:v(x,y) = −r. The arrows depict the field at discrete points, however, the field exists everywhere.
Given a subsetS ofRn, avector field is represented by avector-valued functionV:S →Rn in standardCartesian coordinates(x1, …,xn). If each component ofV is continuous, thenV is a continuous vector field. It is common to focus onsmooth vector fields, meaning that each component is asmooth function (differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within ann-dimensional space.[1]
One standard notation is to write for the unit vectors in the coordinate directions. In these terms, every smooth vector field on an open subset of can be written asfor some smooth functions on.[2] The reason for this notation is that a vector field determines alinear map from the space of smooth functions to itself,, given by differentiating in the direction of the vector field.
Example: The vector field describes a counterclockwise rotation around the origin in. To show that the function is rotationally invariant, compute:
Given vector fieldsV,W defined onS and a smooth functionf defined onS, the operations of scalar multiplication and vector addition,make the smooth vector fields into amodule over thering of smooth functions, where multiplication of functions is defined pointwise.
In physics, avector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. Thetransformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from acovector.
Thus, suppose that(x1, ...,xn) is a choice of Cartesian coordinates, in terms of which the components of the vectorV areand suppose that (y1,...,yn) aren functions of thexi defining a different coordinate system. Then the components of the vectorV in the new coordinates are required to satisfy the transformation law
1
Such a transformation law is calledcontravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification ofn functions in each coordinate system subject to the transformation law (1) relating the different coordinate systems.
Vector fields are thus contrasted withscalar fields, which associate a number orscalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.
An alternative definition: A smooth vector field on a manifold is a linear map such that is aderivation: for all.[3]
If the manifold is smooth oranalytic—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold is often denoted by or (especially when thinking of vector fields assections); the collection of all smooth vector fields is also denoted by (afraktur "X").
The flow field around an airplane is a vector field inR3, here visualized by bubbles that follow thestreamlines showing awingtip vortex.Vector fields are commonly used to create patterns incomputer graphics. Here: abstract composition of curves following a vector field generated withOpenSimplex noise.
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) of the arrow will be an indication of the wind speed. A "high" on the usualbarometric 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 field of a movingfluid. In this case, avelocity vector is associated to each point in the fluid.
Maxwell's equations allow us to use a given set of initial and boundary conditions to deduce, for every point inEuclidean space, a magnitude and direction for theforce experienced by a charged test particle at that point; the resulting vector field is theelectric field.
Agravitational field 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.
A vector fieldV defined on an open setS is called agradient field or aconservative field if there exists a real-valued function (a scalar field)f onS such that
The associatedflow is called thegradient flow, and is used in the method ofgradient descent.
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.
A common technique in physics is to integrate a vector field along acurve, also called determining itsline integral. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.
The line integral is constructed analogously to theRiemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.
Given a vector fieldV and a curveγ,parametrized byt in[a,b] (wherea andb arereal numbers), the line integral is defined as
Thedivergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, 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 asource or a sink for the vector flow, a result which is made precise by thedivergence theorem.
Thecurl is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with theexterior derivative. In three dimensions, it is defined by
The curl measures the density of theangular 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 byStokes' theorem.
The index of a vector field is an integer that helps describe its behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity.
Letn be the dimension of the manifold on which the vector field is defined. Take a closed surface (homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimensionn − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere Sn−1. This defines a continuous map from S to Sn−1. The index of the vector field at the point is thedegree of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself.
The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1)k around a saddle that hask contracting dimensions andn−k expanding dimensions.
The index of the vector field as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.
For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies thehairy ball theorem.
For a vector field on a compact manifold with finitely many zeroes, thePoincaré-Hopf theorem states that the vector field's index is the manifold'sEuler characteristic.
Michael Faraday, in his concept oflines of force, emphasized that the fielditself should be an object of study, which it has become throughout physics in the form offield theory.
In addition to the magnetic field, other phenomena that were modeled by Faraday include the electrical field andlight field.
In recent decades, many phenomenological formulations of irreversible dynamics and evolution equations in physics, from the mechanics of complex fluids and solids to chemical kinetics and quantum thermodynamics, have converged towards the geometric idea of "steepest entropy ascent" or "gradient flow" as a consistent universal modeling framework that guarantees compatibility with the second law of thermodynamics and extends well-known near-equilibrium results such as Onsager reciprocity to the far-nonequilibrium realm.[5]
Consider the flow of a fluid 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 defined on, one defines curves on such that for each in an interval,
The curves are calledintegral curves ortrajectories (or less commonly, flow lines) of the vector field and partition intoequivalence classes. It is not always possible to extend the interval to the wholereal number line. The flow may for example reach the edge of in a finite time.In two or three dimensions one can visualize the vector field as giving rise to aflow on. If we drop a particle into this flow at a point it will move along the curve in the flow depending on the initial point. If is a stationary point of (i.e., the vector field is equal to the zero vector at the point), then the particle will remain at.
By definition, a vector field on is calledcomplete if each of its flow curves exists for all time.[6] In particular,compactly supported vector fields on a manifold are complete. If is a complete vector field on, then theone-parameter group ofdiffeomorphisms generated by the flow along exists for all time; it is described by a smooth mappingOn a compact manifold without boundary, every smooth vector field is complete. An example of anincomplete vector field on the real line is given by. For, the differential equation, with initial condition, has as its unique solution if (and for all if). Hence for, is undefined at so cannot be defined for all values of.
The flows associated to two vector fields need notcommute with each other. Their failure to commute is described by theLie bracket of two vector fields, which is again a vector field. The Lie bracket has a simple definition in terms of the action of vector fields on smooth functions:
Given asmooth function between manifolds,, thederivative is an induced map ontangent bundles,. Given vector fields and, we say that is-related to if the equation holds.
If is-related to,, then the Lie bracket is-related to.
Algebraically, vector fields can be characterized asderivations 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 ofdifferential calculus over commutative algebras.
