Divergence (redirected from divergences)

divergence

In mathematics, a differential operator applied to a three-dimensional vector-valued function. The result is a function that describes a rate of change. The divergence of a vector v is given by in which v₁, v₂, and v₃ are the vector components of v, typically a velocity field of fluid flow.

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divergence [də′vər·jəns]

(electronics)

The spreading of a cathode-ray stream due to repulsion of like charges (electrons).

(fluid mechanics)

The ratio of the area of any section of fluid emerging from a nozzle to the area of the throat of the nozzle.

(mathematics)

For a vector-valued function, the sum of the diagonal entries of the Jacobian matrix; it is the scalar product of the del operator and the vector.

(meteorology)

The two-dimensional horizontal divergence of the velocity field.

(nucleonics)

In a nuclear reactor, the condition wherein the number of neutrons produced increases in each succeeding generation.

(oceanography)

A horizontal flow of water, in different directions, from a common center or zone.

(physics)

The spreading apart of a beam of particles of light.

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Divergence 

in biology, the divergence of characters of organisms in the process of evolution.

The concept of divergence was proposed by C. Darwin to explain the appearance of diverse varieties of cultivated plants, breeds of domestic animals, and biological species in nature. In artificial selection divergence within each group of cultivated plants and domestic animals depends on the needs of man. Darwin used the principle of divergence to explain species formation in nature. If a species has a broad distribution and adapts to various ecological conditions, divergence takes place. Divergence is expressed in the appearance of differences in populations that were originally similar and is caused by the inevitably slightly different direction of natural selection in various parts of a species’ range. Divergence leads to the appearance of organisms that vary in structure and function; this in turn assures more complete use of environmental conditions, since, according to Darwin, the greatest “sum of life” is realized by the broadest variety of structures. Divergence is supported by the struggle for existence: usually even slightly specialized forms have selective advantages, which brings about more rapid extinction of intermediate forms and development of various types of isolation. The principle of divergence also explains the process of formation of larger (supraspecies) systematic groups and the rise of gaps between them.

A. V. IABLOKOV

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Divergence 

[of a vector field a(M) at a point (x, y, z)], the scalar quantity

div a = σP/σx + σQ/σy + σRR/σZ

where P, Q, and R are components of the vector a. The divergence is the limit of the ratio of the flux of a vector field through a closed surface surrounding the given point, to the volume delimited by it when the surface contracts toward the point. Divergence plays an important role in the applications of mathematics to physics. For example, if the vector field a(M) is considered to be a velocity field in a steady flow of an incompressible fluid, then div a at a point designates the intensity of the source (div a > 0) or of the flow (div a < 0) located at this point, or the absence of source and flow (div a = 0). The properties of divergence are:

div (a + b) = div a + div b

div (φa) = φ div a + a grad φ div rot a =0

div grad φ = Δφ

where Δ is a Laplace operator.