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What is a Vector Field?

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  • Written By: Kathy Dowling
  • Edited By: A. Joseph
  • Last Modified Date: 27 November 2016
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    Conjecture Corporation
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A vector field is a function that allocates vectors to different points in time and space. There are two types of vector fields: velocity vector fields and force fields. Vector fields are studied in vector calculus by both mathematicians and physicists.

A vector is thought of as an arrow beginning at the origin of a plane and going to a point in space. This point is basically a pair of numbers that can be plotted in Euclidean space. Vectors are studied in physics and mathematics and are used to model speed and force. When two vectors are added together, the result is a force of two single forces, applied to the same object at the same time. Many vectors constitute a vector field, and this is used to symbolize forces at all points in time and space.

The domain of a vector field is a set of points, and its range is a set of vectors. So, a vector field is essentially a function that allocates a two- or three-dimensional vector to each point in a two- or three-dimensional plane. Vector fields that are three-dimensional are usually too difficult to draw by hand and require the assistance of a computer algebra system.

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Vectors and the vector field that they constitute are applied to events that occur in everyday life. For example, they might represent wind velocities that occur during a tornado or different ocean current pattens. Velocity vector fields are indicative of speed and direction and have been used to show the speed at which air moves past airfoils. A force field is another type of vector field that correlates every point in time and space with a force vector. Such vector fields are particularly useful when modeling magnetic and gravitational forces.

Mathematicians and physicists are also able to calculate line and surface integrals of vector fields. A line integral can be thought of as a "curve" integral and is often used to find out how an object moves along a curve. Surface integrals can be used to discover the speed at which fluid moves across a surface.

A vector field might be considered conservative when the field represents a gradient of a scalar function. That is, the field represents an incline or a slope. Not all vector fields are conservative, but they do regularly emerge in physics.

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