Vector Fields

    By and large, a vector field is a function that assigns a vector to each point in a set of ordered pairs. Notice that it may be impossible to draw every vector in a vector field (drawing a vector at every point in space); even so, drawing a few key representatives (vectors) usually will give a reasonable impression of a vector field. Common examples of vector fields include force fields, velocity fields, gravitational fields, magnetic fields, and electric fields.     
    Vector fields can be used to quantify the amount of work done by variable force acting on a moving body. Measuring the amount of force (fluid flow, electric charge, etc.) can sometimes be achieved by computing an integral of a vector field with respect to an orientable curve or surface.

Definition (Vector Field) A vector field in is a function   that assigns a vector to each point in its domain. A vector field with domain in has the form   



where the scalar functions and are called the components of A continuous vector field is one whose components are continuous.

    To visualize a particular vector field it often helps to select a number of points in the domain of and then draw an arrow emanating from each point with the direction of and length representing the magnitude We will refer to such a representation as the graph of

Example (Vector Field) Sketch some representations for each of the following vector fields.

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Cite this as:
Vector Fields
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-fields.html