Vector field


In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. 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 and curl.
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 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 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.
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. Vector fields are one kind of tensor field.

Definition

Vector fields on subsets of Euclidean space

Given a subset of, a vector field is represented by a vector-valued function in standard Cartesian coordinates. If each component of is continuous, then is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space.
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 as
for some smooth functions on. The reason for this notation is that a vector field determines a linear 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 fields, defined on and a smooth function defined on, the operations of scalar multiplication and vector addition,
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 is a choice of Cartesian coordinates, in terms of which the components of the vector are
and suppose that are n functions of the xi defining a different coordinate system. Then the components 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 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, a vector field on is an assignment of a tangent vector to each point in. More precisely, a vector field is a mapping from into the tangent bundle so that is the identity mapping
where denotes the projection from to. In other words, a vector field is a section of the tangent bundle.
An alternative definition: A smooth vector field on a manifold is a linear map such that is a derivation: for all.
If the manifold is smooth or analytic—that is, the change of coordinates is smooth —then one can make sense of the notion of smooth vector fields. The collection of all smooth vector fields on a smooth manifold is often denoted by or ; the collection of all smooth vector fields is also denoted by .

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 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: the line produced by particles passing through a specific fixed point over various times
  • * pathlines: showing the path that a given particle would follow.
  • * streamlines : the path of a particle influenced by the instantaneous field.
  • Magnetic fields. The fieldlines can be revealed using small iron filings.
  • Maxwell's equations allow us to use a given set of initial and boundary 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 electric field.
  • A gravitational 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.

    Gradient field in Euclidean spaces

Vector fields can be constructed out of scalar fields using the gradient operator.
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 f on S such that
The associated flow is called the , and is used in the method of gradient descent.
The path integral along any closed curve γ = γ) in a conservative field is zero:

Central field in euclidean spaces

A -vector field over is called a central field if
where 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.

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, 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 and the vector field is continuous.
Given a vector field and a curve, parametrized by in , the line integral is defined as
To show vector field topology one can use line integral convolution.

Divergence

The divergence of a vector field on Euclidean space is a function. 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 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
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.