Field Theory

field theory

In mathematics, the study of the structure of a set of objects (e.g., numbers) with two combining operations (e.g., addition and multiplication). Such a system, known as a field, must satisfy certain properties: associative law, commutative law, distributive law, an additive identity (“zero”), a muliplicative identity (“one”), additive inverses (see inverse function), and multiplicative inverses for nonzero elements. The sets of rational numbers, real numbers, and complex numbers are fields under ordinary addition and multiplication. The investigation of polynomial equations and their solutions led to the development of field theory.

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field theory [′fēld ‚thē·ə·rē]

(mathematics)

The study of fields and their extensions.

(physics)

A theory in which the basic quantities are fields; classically the equations governing the fields may be given; in quantum field theory the commutation rules satisfied by the field operators also are specified.

(psychology)

A psychological theory that emphasizes the importance of interactions between events in an individual's environment.

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Field Theory 

in mathematics, the study of the properties of scalar, vector, and—in the general case—tensor fields. A scalar field is a region of space or a plane with each of whose points M there is associated a number u(M), such as temperature, pressure, density, or magnetic permeability. If with each point M of a region there is associated a vector a(M)—such as the velocity of a particle in a moving fluid or the strength of a force field—we speak of a vector field. In a tensor field, a tensor—for example, the stress at a point in an elastic body or conductivity in an anisotropic body—is associated with each point in the region. Field theory makes extensive use of vector and tensor analysis.

Many concepts of the differential and integral calculus of functions of several variables are carried over into field theory. An example is the derivative. Thus, the gradient, or the derivative of a scalar field in the direction of maximum change, is of great importance for the description of scalar fields. The gradient is a vector that is invariant with respect to the choice of coordinate system. As a first approximation, changes in a vector field are characterized by two quantities: a scalar called the divergence, which characterizes the change in the field intensity or density, and a vector called the curl, or rotation, which characterizes the “rotational component” of a vector field. The operation of passing from a scalar field to its gradient and from a vector field to its divergence is often called the Hamiltonian, or del, operator. The processes of obtaining the gradient of a scalar field and the divergence and curl of a vector field are often referred to as the basic differential operations of field theory. Sometimes grouped with these operations is the Laplace operator, or the successive obtaining of the gradient and divergence. When the basic differential operations are applied to fields with certain types of symmetry, such as spherical or cylindrical symmetry, special types of curvilinear coordinates are used—for example, polar or cylindrical coordinates. As a result, calculations are simplified.

Field theory makes use of a number of integral equations and concepts that connect differentiation and integration in the study of a field or parts of a field. For example, the integral, with respect to a surface, of the scalar product of the field vector and the unit vector normal to the surface is called the flux of the vector field through the surface. The relation between the flux of a vector field and the divergence is given by Ostrogradskii’s theorem: the flux of the vector field through the surface is equal to the integral of the divergence with respect to the volume bounded by the surface. Another important concept is that of circulation, which is the line integral of a vector field around a closed contour—that is, the integral, with respect to the contour, of the scalar product of the vector field and the unit vector tangent to the contour. According to Stokes’ theorem, the circulation of a vector along a closed contour is equal to the integral of the curl with respect to any surface bounded by the given contour. On the basis of curl and divergence, a distinction is made between irrotational, solenoidal, and Laplacian fields. In irrotational fields, curl a = 0 (in Russian notation, rot a = 0). In solenoidal fields, div a = 0. In Laplacian fields the divergence and curl are both zero: Δɸ = 0.

A. B. IVANOV