Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space formula_1. A vector field on a plane 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 throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.
The elements of differential and integral calculus extend naturally to vector fields. 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).
A vector field is a special case of a "vector-valued function", whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space.
Likewise, n coordinates, a vector field on a domain in "n"-dimensional Euclidean space formula_1 can be represented as a vector-valued function that associates an "n"-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 on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).
More generally, vector fields are defined on differentiable 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, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
Definition.
Vector fields on subsets of Euclidean space.
Given a subset "S" of R"n", a vector field is represented by a vector-valued function "V": "S" → R"n" in standard Cartesian coordinates ("x"1, …, "x""n"). If each component of "V" is continuous, then "V" is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within an "n"-dimensional space.
One standard notation is to write formula_3 for the unit vectors in the coordinate directions. In these terms, every smooth vector field formula_4 on an open subset formula_5 of formula_6 can be written as
formula_7
for some smooth functions formula_8 on formula_5. The reason for this notation is that a vector field determines a linear map from the space of smooth functions to itself, formula_10, given by differentiating in the direction of the vector field.
Example: The vector field formula_11 describes a counterclockwise rotation around the origin in formula_12. To show that the function formula_13 is rotationally invariant, compute:
formula_14
Given vector fields "V", "W" defined on "S" and a smooth function f defined on "S", the operations of scalar multiplication and vector addition,
formula_15
formula_16
make the smooth vector fields into a module over the ring of smooth functions, where multiplication of functions is defined pointwise.
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 components of the vector V are
formula_17
and suppose that ("y"1...,"y""n") are "n" functions of the "x""i" defining a different coordinate system. Then the components of the vector "V" in the new coordinates are required to satisfy the transformation law
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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 (1) relating the different coordinate systems.
Vector fields are thus contrasted with scalar fields, 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 manifold formula_18, a vector field on formula_18 is an assignment of a tangent vector to each point in formula_18. More precisely, a vector field formula_21 is a mapping from formula_18 into the tangent bundle formula_23 so that formula_24 is the identity mapping
where formula_25 denotes the projection from formula_23 to formula_18. In other words, a vector field is a section of the tangent bundle.
An alternative definition: A smooth vector field formula_28 on a manifold formula_18 is a linear map formula_30 such that formula_28 is a derivation: formula_32 for all formula_33.
If the manifold formula_18 is smooth or analytic—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 formula_18 is often denoted by formula_36 or formula_37 (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by formula_38 (a fraktur "X").
Examples.
Gradient field in Euclidean spaces.
Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).
A vector field "V" defined on an open 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
formula_39
The associated flow is called the ;templatestyles src="Template:Visible anchor/styles.css" / ;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 conservative field is zero:
formula_40
Central field in euclidean spaces.
A "C"∞-vector field over R"n" \ {0} is called a central field if
formula_41
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.
Operations on vector fields.
Line integral.
A common technique in physics is to integrate a vector field along a curve, also called determining its line 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 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 t in ["a", "b"] (where a and b are real numbers), the line integral is defined as
formula_42
To show vector field topology one can use line integral convolution.
Divergence.
The divergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by
formula_43
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.
Curl in three dimensions.
The curl 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 the exterior derivative. In three dimensions, it is defined by
formula_44
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.
Index of a vector field.
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.
Let "n 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 dimension "n" − 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 S"n"−1. This defines a continuous map from S to S"n"−1. The index of the vector field at the point is the degree 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 has "k" contracting dimensions and "n"−"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 the hairy ball theorem.
For a vector field on a compact manifold with finitely many zeroes, the Poincaré-Hopf theorem states that the vector field’s index is the manifold’s Euler characteristic.
Physical intuition.
Michael Faraday, in his concept of "lines of force," 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 by Faraday include the electrical field and light 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.
Flow curves.
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 formula_4 defined on formula_5, one defines curves formula_47 on formula_5 such that for each formula_49 in an interval formula_50,
formula_51
By the Picard–Lindelöf