Vector Field

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 a collection of arrows with a given magnitude and direction each attached to a 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 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 (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.

Other articles related to "vector field, vector fields, vectors, vector, field, fields":

Synchronization Of Chaos - Identical Synchronization
... the simplest case of two diffusively coupled dynamics is described by where is the vector field modeling the isolated chaotic dynamics and is the coupling parameter ... Assuming that is small we can expand the vector field in series and obtain a linear differential equation - by neglecting the taylor remainder - governing the behavior of the difference where denotes ...
Lie Bracket Of Vector Fields - Definition
... Each vector field X on a smooth manifold M may be regarded as a differential operator acting on smooth functions on M ... Indeed, each vector field X becomes a derivation on the smooth functions C∞(M) when we define X(f) to be the element of C∞(M) whose value at a point p is the directional ... This Lie algebra structure can be transferred to the set of vector fields on M as follows ...
Vector Field - Generalizations
... Replacing vectors by p-vectors (pth exterior power of vectors) yields p-vector fields taking the dual space and exterior powers yields differential k-forms, and combining these yields general ... Algebraically, vector fields can be characterized as derivations of the algebra of smooth functions on the manifold, which leads to defining a vector ...
Time Dependent Vector Field
... A time dependent vector field on a manifold M is a map from an open subset on such that for every, is an element of ... For every such that the set is nonempty, is a vector field in the usual sense defined on the open set ...
Born Coordinates - Langevin Observers in The Cylindrical Chart
... From the line element we can immediately read off a frame field representing the local Lorentz frames of stationary (inertial) observers Here, is a timelike unit vector field while the others are ... Simultaneously boosting these frame fields in the direction, we obtain the desired frame field describing the physical experience of the Langevin observers, namely This frame was apparently first introduced (implicit ... Each integral curve of the timelike unit vector field appears in the cylindrical chart as a helix with constant radius (such as the red curve in the figure at right) ...

Famous quotes containing the word field:

A field of water betrays the spirit that is in the air. It is continually receiving new life and motion from above. It is intermediate in its nature between land and sky.
Henry David Thoreau (1817–1862)