User Contributed Dictionary
Number
div A Roman numeral representing five hundred and four (504).
See also
 Alternate forms: DIV, CCCCCIIII, ccccciiii
 Previous: diii (five 5hundred and three, 503)
 Next: dv (five hundred and five, 505)
English
Pronunciation

 Rhymes: ɪv
Noun
div , a function, implemented in many programming languages, that returns the result of a division of two integers
 (vector calculus) short for divergence; a kind of differential operator
 A foolish person
Translations
;an operator Swedish: div
Czech
Pronunciation
Noun
divRelated terms
Swedish
Abbreviation
div div; the divergence operator
Extensive Definition
In vector
calculus, the divergence is an operator that measures the
magnitude of a vector
field's source or
sink at a given point; the
divergence of a vector field is a (signed) scalar. For a vector field
that denotes the velocity of air expanding as it
is heated, the divergence of the velocity field would have a
positive value because the air expands. If the air cools and
contracts, the divergence is negative. The divergence could be
thought of as a measure of the change in density.
A vector field that has zero divergence
everywhere is called solenoidal.
Definition
Let x, y, z be a system of Cartesian
coordinates on a 3dimensional Euclidean
space, and let i, j, k be the corresponding
basis
of unit
vectors.
The divergence of a continuously
differentiable vector field
F = Fx i + Fy j + Fz k is defined to be the scalarvalued
function:
 \operatorname\,\mathbf = \nabla\cdot\mathbf
Although expressed in terms of coordinates, the
result is invariant under orthogonal
transformations, as the physical interpretation suggests.
The common notation for the divergence
∇·F is a convenient mnemonic, where the dot denotes an
operation reminiscent of the dot product:
take the components of ∇ (see del), apply them to the components
of F, and sum the results. As a result, this is considered an
abuse of
notation.
Physical interpretation as source density
In physical terms, the divergence of a three
dimensional vector field is the extent to which the vector field
flow behaves like a source or a sink at a given point. It is a
local measure of its "outgoingness"—the extent to which
there is more exiting an infinitesimal region of space than
entering it. If the divergence is nonzero at some point then there
must be a source or sink at that position http://musr.phas.ubc.ca/~jess/hr/skept/Gradient/node4.html.
An alternate but equivalent definition, gives the
divergence as the derivative of the net flow of the
vector field across the surface of a small sphere relative to the volume of the sphere. (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.) Formally,
 ( \operatorname\,\mathbf) (p) =
where S(r) denotes the sphere of radius r about a
point p in R3, and the integral is a surface
integral taken with respect to n, the normal to that
sphere.
Instead of a sphere, any other volume \Delta V is
possible, if instead of \frac one writes \,\Delta V\,\,. From
this definition it also becomes explicitly visible that \,\, can be
seen as the source density of the flux \mathbf v(\mathbf r)
In light of the physical interpretation, a vector
field with constant zero divergence is called incompressible
– in this case, no net flow can occur across any closed
surface.
The intuition that the sum of all sources minus
the sum of all sinks should give the net flow outwards of a region
is made precise by the divergence
theorem.
Decomposition theorem
It can be shown that any stationary flux \mathbf v(\mathbf r) which is at least two times continuously differentiable in ^3 and vanishes sufficiently fast for \mathbf r\to \infty can be decomposed into an irrotational part \mathbf E(\mathbf r) and a sourcefree part \mathbf B(\mathbf r)\,. Moreover, these parts are explicitly determined by the respective sourcedensities (see above) and 'circulation densities'' (see the article Curl):For the irrotational part one has
\mathbf E=\nabla \Phi(\mathbf r)\,, with
\Phi (\mathbf r)=\int_\,^3\mathbf r'\,\frac\,.
The sourcefree part, \mathbf B, can be
simillarly written: one only has to replace the scalar potential
\Phi (\mathbf r) by a vector potential \mathbf A(\mathbf r) and the
terms \nabla \Phi by +\nabla\times\mathbf A, and finally the
sourcedensity \,\mathbf E by the circulationdensity \nabla
\times\mathbf B\,.
This "decomposition theorem" is in fact a
byproduct of the stationary case of electrodynamics. It is a
special case of the more general Helmholtz
decomposition which works in dimensions greater than three as
well.
Properties
The following properties can all be derived from
the ordinary differentiation rules of calculus. Most importantly, the
divergence is a linear
operator, i.e.
 \operatorname( a\mathbf + b\mathbf )
for all vector fields F and G and all real numbers
a and b.
There is a product rule
of the following type: if φ is a scalar valued function and
F is a vector field, then
 \operatorname(\varphi \mathbf)
or in more suggestive notation
 \nabla\cdot(\varphi \mathbf)
Another product rule for the cross
product of two vector fields F and G in three dimensions
involves the curl and reads
as follows:
 \operatorname(\mathbf\times\mathbf)
or
 \nabla\cdot(\mathbf\times\mathbf)
The Laplacian of a
scalar
field is the divergence of the field's gradient.
The divergence of the curl of any vector field
(in three dimensions) is constant and equal to zero. If a vector
field F with zero divergence is defined on a ball in R3, then there
exists some vector field G on the ball with F = curl(G). For
regions in R3 more complicated than balls, this latter statement
might be false (see Poincaré
lemma). The degree of failure of the truth of the statement,
measured by the homology
of the chain
complex
 \ \;

 \to\ \;

 \to\ \;

 \to\ \;
(where the first map is the gradient, the second
is the curl, the third is the divergence) serves as a nice
quantification of the complicatedness of the underlying region U.
These are the beginnings and main motivations of de Rham
cohomology.
Relation with the exterior derivative
One can establish a parallel between the divergence and a particular case of the exterior derivative, when it takes a 2form to a 3form in R3. If we define: \alpha=F_1\ dy\wedge dz + F_2\ dz\wedge dx + F_3\ dx\wedge dy
 d\alpha = \left( \frac
See also Hodge
star operator.
Generalizations
The divergence of a vector field can be defined
in any number of dimensions. If
 \mathbf=(F_1, F_2, \dots, F_n),
define
 \operatorname\,\mathbf = \nabla\cdot\mathbf
For any n, the divergence is a linear operator,
and it satisfies the "product rule"
 \nabla\cdot(\varphi \mathbf)
for any scalarvalued function φ.
The divergence can be defined on any manifold of
dimension n with a volume form (or density) \mu e.g. a Riemannian
or Lorentzian
manifold. Generalising the construction of a two form for a
vectorfield on \mathbb^3, on such a manifold a vectorfield X
defines a n1 form j = i_X \mu obtained by contracting X with \mu.
The divergence is then the function defined by
 d j = \operatorname(X) \mu
Standard formulas for the Lie
derivative allow us to reformulate this as
 \mathcal_X \mu = \operatorname(X) \mu
This means that the divergence measures the rate
of expansion of a volume element as we let it flow with the
vectorfield.
On a Riemannian or Lorentzian manifold the
divergence with respect to the metric volume form can be computed
in terms of the Levi
Civita connection \nabla
 \operatorname(X) = \nabla\cdot X = X^a_
where the second expression is the contraction of
the vectorfield valued 1 form \nabla X with itself and the last
expression is the traditional coordinate expression used by
physicists.
References
 Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
External links
div in Bosnian: Divergencija
div in Bulgarian: Дивергенция (математика)
div in Catalan: Divergència
div in Czech: Divergence
div in German: Divergenz (Mathematik)
div in Spanish: Divergencia (matemática)
div in Esperanto: Diverĝenco
div in Persian: واگرایی
div in French: Divergence (mathématiques)
div in Korean: 발산 (벡터)
div in Icelandic: Sundurleitni
div in Italian: Divergenza
div in Hebrew: דיברגנץ
div in Lithuanian: Divergencija
div in Hungarian: Divergencia
(vektoranalízis)
div in Dutch: Divergentie (vectorveld)
div in Japanese: 発散
div in Polish: Dywergencja
div in Portuguese: Divergente
div in Romanian: Divergenţă
div in Russian: Дивергенция
div in Slovak: Divergencia (vektorové
pole)
div in Finnish: Divergenssi
div in Tamil: விரிதல் (திசையன் நுண்கணிதம்)
div in Vietnamese: Toán tử div
div in Turkish: Diverjans
div in Ukrainian: Дивергенція
div in Chinese: 散度