Calculus
Textbooks
Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vector Calculus
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Vector Calculus
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 6
Created by Boundless

Vector Fields

A vector field is an assignment of a vector to each point in a subset of Euclidean space.

Learning Objective

  • Describe construction of vector fields


Key Points

    • 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 can be constructed out of scalar fields using the gradient operator.
    • Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).

Terms

  • bijective

    both injective and surjective

  • vector field

    a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space


Full Text

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 the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.

Fig 1

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 be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).

Gradient field: Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇). A vector field V defined on a 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: 

$\displaystyle{V = \nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3}, \dots ,\frac{\partial f}{\partial x_n}\right)}$

The associated flow is called the gradient flow.

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.
  • A gravitational field generated by any massive object is 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.
  • Magnetic field lines can be revealed using small iron filings.
  • In the case of the velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.
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