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The assertion, under certain light restrictions, that the volume integral through a volume V of the divergence of vector function (x, y, z) is equal to the surface integral of the exterior normal component of over the boundary surface S of V, or in…

A branch of mathematics concerned with the properties of space, including points, lines, curves, planes and surfaces in space, and figures bounded by them.Euclidean geometry Geometry as a high school subject is largely based on the Elements of…

The result, proved by K. Gödel in 1931, that any sufficiently advanced mathematical system must be incomplete in that there must always be a true sentence that is not provable in the system. Roughly speaking, Gödel showed how, for each such system,…

The result of the application of the distributive vector differential operator ∇, ∇ = i ∂∂x + j ∂∂y + k ∂∂z, to a differentiable scalar function S(x, y, z); thus ∇S = i ∂S∂x + j ∂S∂y + k∂S∂z. The letters i, j, k are symbols for the base vectors…

A branch of mathematics that belongs partly to combinatorial analysis and partly to topology. Its applications occur (sometimes under other names) in electrical network theory, operations research, organic chemistry, theoretical physics, and…

Procedures and techniques for visually representing on paper or a screen information pertaining to data analysis and decision making or relationships between variables by means of diagrams or charts.Structural information is usually depicted by a…

A solution of a partial differential equation for the case of a point source of unit strength within the region under examination. The Green's function is an important mathematical tool that has application in many areas of theoretical physics…

A term used variously in mathematical literature to denote either (1) the Gauss divergence theorem (shown below) or some one of several forms or immediate consequences of their theorem, or (2) the plane case of Stokes' theorem.Gauss' theoremStokes'…

Any set of elements which is equipped with an operation (called multiplication) satisfying the requirements (1), (2), (3), and (4) is called a group. Group theory is the branch of mathematics devoted to the properties of groups. Group theory has…

Any nonplanar curve all of whose tangents make the same angle with a fixed line. Other characteristic properties are that all principal normals are parallel to a plane and that the ratio of torsion to curvature is constant. If a helix has constant…

An abstract notion of great power and beauty, which has been central to the development of mathematical analysis and forms the backdrop for many applications of analysis to science and engineering. Its essence lies in the fact that the objects of…

A curve cut from a cone of revolution by a plane that intersects both nappes of the cone and does not contain the apex (Fig. 1). In analytic geometry it is shown, as shown in Fig. 2, that a hyperbola is the locus of points P in a plane, such that PF…

The hyperbolic sine and cosine of a real or complex variable z are defined by Eqs. (1). 1 Both sinh z and cosh z have a period 2πi of ez. From Dez = dezdz = ez, Eqs. (2) 2are obtained. Since by definition, Eqs. (3) hold and the series converge for…

A geometry that obeys a consistent set of axioms differing from Euclid's in implying that the angle sum of a triangle is less than 180°. In the early nineteenth century, K. F. Gauss proved the existence of such a geometry. He questioned which…

The analytic continuation of the function defined by the series in Eq. (1), 1where the shifted factorial (a)n is defined by Eq. (2). 2 3 It satisfies Eq. (3), a linear, homogeneous differential equation whose only singular points in the full complex…

The terms infinity and infinite have a variety of related meanings in mathematics. The adjective finite means “having an end,” so infinity may be used to refer to something having no end. In order to give a precise definition, the mathematical…

The application to biological systems of the theory of encoding effects on signal transmission and communications efficiency. In everyday language, information is associated with knowledge, meaning (semantics), and the influence of information on…

An equation of the form typified by Eq. (1). 1 The major problem is to decide when there is a function Φ(x) which is a solution to the equation. Equations such as (1) arise from the analysis of ordinary differential equations. For example, it is…

An integral relation between two classes of functions. For example, a relation such as Eq. (1) 1 is said to define an integral transform. More generally, the integral may be a multiple integral, and the functions f, G, φ may depend on a larger…

An operation of the infinitesimal calculus which has two aspects. The roots of one go back to antiquity, for Archimedes and other Greek mathematicians used the “method of exhaustion” to compute areas and volumes. A simple example of this is the…

A process in mathematics used to estimate an intermediate value of one (dependent) variable which is a function of a second (independent) variable when values of the dependent variable corresponding to several discrete values of the independent…

A term applied to a curve C′ that cuts at right angles all tangents of a curve C (see illustration). Each curve C has infinitely many involutes and the distance between corresponding points of any two involutes is constant. If xi = xi (s), with i =…

An integral extensively used by P. S. Laplace in the theory of probability. In simplest form it is expressed as Eq. (1). 1 It is thought of as transforming the determining function φ(t) into the generating function f(s). The variable t is real, the…

Laplace's equation in two independent variables x and y is given as Eq. (1) 1 and is of central importance in both pure mathematics and mathematical physics. A function u(x,y) having continuous first and second partial derivatives and satisfying…

The differential operator ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2, in which the symbols x, y, z denote the variables of a rectangular Cartesian coordinate system. The Laplacian is frequently denoted by the symbol ∇2 (read del square) in accordance with the fact…