vector field

(noun)

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

Related Terms

Examples of vector field in the following topics:

  • Conservative Vector Fields

    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
    • Therefore, the curl of a conservative vector field $\mathbf{v}$ is always $0$.
    • A vector field $\mathbf{v}$, whose curl is zero, is called irrotational .
    • Such vortex-free regions are examples of irrotational vector fields.

  • Vector Fields

    • A vector field is an assignment of a vector to each point in a subset of Euclidean space.
    • In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.
    • 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:
    • A gravitational field generated by any massive object is a vector field.

  • Stokes' Theorem

    • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
    • Given a vector field, the theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface.
    • If $\mathbf{F}$ is a smooth vector field on $R^3$, then:
    • Electric field is a conservative vector field.
    • Therefore, electric field can be written as a gradient of a scalar field:

  • Surface Integrals of Vector Fields

    • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
    • Consider a vector field $\mathbf{v}$ on $S$; that is, for each $\mathbf{x}$ in $S$, $\mathbf{v}(\mathbf{x})$ is a vector.
    • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
    • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
    • Explain relationship between surface integral of vector fields and surface integral of a scalar field

  • Curl and Divergence

    • The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.
    • At every point in the field, the curl of that field is represented by a vector.
    • A vector field whose curl is zero is called irrotational.
    • The curl is a form of differentiation for vector fields.
    • (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink, and so on.)

  • The Divergence Theorem

    • The divergence theorem relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.
    • The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
    • More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
    • If $F$ is a continuously differentiable vector field defined on a neighborhood of $V$, then we have:
    • Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface

  • Line Integrals

    • The function to be integrated may be a scalar field or a vector field.
    • The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
    • In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve.
    • More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve.
    • For a vector field $\mathbf{F} : U \subseteq R^n \to R^n$, the line integral along a piecewise smooth curve $C \subset U$, in the direction of $r$, is defined as:

  • Calculus of Vector-Valued Functions

    • The input of a vector-valued function could be a scalar or a vector.
    • Vector calculus is a branch of mathematics that covers differentiation and integration of vector fields in any number of dimensions.
    • Because vector functions behave like individual vectors, you can manipulate them the same way you can a vector.
    • Vector calculus is used extensively throughout physics and engineering, mostly with regard to electromagnetic fields, gravitational fields, and fluid flow.
    • Vector functions are used in a number of differential operations, such as gradient (measures the rate and direction of change in a scalar field), curl (measures the tendency of the vector function to rotate about a point in a vector field), and divergence (measures the magnitude of a source at a given point in a vector field).

  • Fundamental Theorem for Line Integrals

    • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
    • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
    • By placing $\varphi$ as potential, $\nabla$ is a conservative field.
    • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
    • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

  • Parametric Surfaces and Surface Integrals

    • Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form.
    • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).
    • We will study surface integral of vector fields and related theorems in the following atoms.
    • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.