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.)

  • Superposition of Fields

    • As vector fields, electric fields obey the superposition principle.
    • Possible stimuli include but are not limited to: numbers, functions, vectors, vector fields, and time-varying signals.
    • Electric fields are continuous fields of vectors, so at a given point, one can find the forces that several fields will apply to a test charge and add them to find the resultant.
    • To do this, first find the component vectors of force applied by each field in each of the orthogonal axes.
    • Then once the component vectors are found, add the components in each axis that are applied by the combined electric fields.

  • 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).

  • Electric Field Lines: Multiple Charges

    • Electric fields created by multiple charges interact as do any other type of vector field; their forces can be summed.
    • Each will have its own electric field, and the two fields will interact.
    • More field lines per unit area perpendicular to the lines means a stronger field.
    • As vector fields, electric fields exhibit properties typical of vectors and thus can be added to one another at any point of interest.
    • Thus, given charges q1, q2 ,... qn, one can find their resultant force on a test charge at a certain point using vector addition: adding the component vectors in each direction and using the inverse tangent function to solve for the angle of the resultant relative to a given axis.

  • Helical Motion

    • Helical motion results when the velocity vector is not perpendicular to the magnetic field vector.
    • In the section on circular motion we described the motion of a charged particle with the magnetic field vector aligned perpendicular to the velocity of the particle.
    • In this case, the magnetic force is also perpendicular to the velocity (and the magnetic field vector, of course) at any given moment resulting in circular motion.
    • shows how electrons not moving perpendicular to magnetic field lines follow the field lines.
    • When a charged particle moves along a magnetic field line into a region where the field becomes stronger, the particle experiences a force that reduces the component of velocity parallel to the field.

  • Linear Vector Spaces

    • We can add any two, say, force vectors and get another force vector.
    • We can also scale any such vector by a numerical quantity and still have a legitimate vector.
    • Definition of Linear Vector Space: A linear vector space over a field $F$ of scalars is a set of elements $V$ together with a function called addition from $V \times V$ into $V$ (the Cartesian product $A \times B$ of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$ ) and a function called scalar multiplication from $F \times V$ into $V$ satisfying the following conditions for all $x,y,z \in V$ and all $\alpha, \beta \in F$ :
    • The simplest example of a vector space is $\mathbf{R}^n$ , whose vectors are n-tuples of real numbers.
    • In the case of $n=1$ the vector space $V$ and the field $F$$F$ are the same.