divergence

　 　 　

WordNet

1. the act of moving away in different direction from a common point; "an angle is formed by the divergence of two straight lines" (同)divergency
2. an infinite series that has no limit (同)divergency
3. move or draw apart; "The two paths diverge here"
4. extend in a different direction; "The lines start to diverge here"; "Their interests diverged"
5. have no limits as a mathematical series
6. expel air; "Exhale when you lift the weight" (同)expire, breathe out
7. give out (breath or an odor); "The chimney exhales a thick smoke" (同)give forth, emanate
8. a conflict of peoples opinions or actions or characters (同)dissension, dissonance
9. the speech act of disagreeing or arguing or disputing
10. something that is emitted or radiated (as a gas or an odor or a light, etc.)
11. (theology) the origination of the Holy Spirit at Pentecost; "the emanation of the Holy Spirit"; "the rising of the Holy Ghost"; "the doctrine of the procession of the Holy Spirit from the Father and the Son" (同)rise, procession
12. a significant change; "the difference in her is amazing"; "his support made a real difference"
13. the quality of being unlike or dissimilar; "there are many differences between jazz and rock"

PrepTutorEJDIC

1. 分岐;それること
2. (一点から)分岐する,分かれる《+『from』+『名』》 / (定まったコースから)それる,外れる《+『from』+『名』》
3. 〈人が〉〈息・煙など〉‘を'吐き出す / 〈機関車・花などが〉〈蒸気・かおりなど〉‘を'発散する / 息(煙)を吐く / 発散する
4. 〈U〉(…間の)不一致,不調和《+between+『名』》 / 〈C〉(…の間の)意見の相違;争い,不和《+between+『名』》
5. 〈U〉発散,放射 / 〈C〉発散物
6. 『違い』,『相違点』,意見などの食い違い / 『差』,差額

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/09/08 11:51:01」(JST)

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[Wiki en表示]

For other uses, see Divergence (disambiguation).

Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient Faà di Bruno's formula
Integral Lists of integrals Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green's Stokes'
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian
Specialized Fractional Malliavin Stochastic Variations

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence has a negative value, as the region is a sink.

Definition of divergence

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. (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.)

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three-dimensional region V divided by the volume of V as V shrinks to p. Formally,

$\operatorname{div}\,\mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS$

where |V | is the volume of V, S(V) is the boundary of V, and the integral is a surface integral with n being the outward unit normal to that surface. The result, div F, is a function of p. From this definition it also becomes explicitly visible that div F can be seen as the source density of the flux of F.

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – 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.

Application in Cartesian coordinates

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial U}{\partial x} +\frac{\partial V}{\partial y} +\frac{\partial W}{\partial z }.$

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. Because applying an operator is different from multiplying the components, this is considered an abuse of notation.

The divergence of a continuously differentiable second-order tensor field is a first-order tensor field:

$\overrightarrow{\operatorname{div}}\,(\mathbf{\underline{\underline{\epsilon}}}) = \begin{bmatrix} \frac{\partial \epsilon_{xx}}{\partial x} +\frac{\partial \epsilon_{xy}}{\partial y} +\frac{\partial \epsilon_{xz}}{\partial z} \\[6pt] \frac{\partial \epsilon_{yx}}{\partial x} +\frac{\partial \epsilon_{yy}}{\partial y} +\frac{\partial \epsilon_{yz}}{\partial z} \\[6pt] \frac{\partial \epsilon_{zx}}{\partial x} +\frac{\partial \epsilon_{zy}}{\partial y} +\frac{\partial \epsilon_{zz}}{\partial z} \end{bmatrix}$

Cylindrical coordinates

For a vector expressed in cylindrical coordinates as

where e_a is the unit vector in direction a, the divergence is

$\operatorname{div}\, \mathbf F = \nabla\cdot\mathbf F = \frac1r \frac{\partial}{\partial r} (rF_r) + \frac1r \frac{\partial F_\theta}{\partial\theta} + \frac{\partial F_z}{\partial z}\, .$

Spherical coordinates

In spherical coordinates, with the angle with the z axis and the rotation around the z axis, the divergence reads

$\operatorname{div}\, \mathbf F = \nabla\cdot\mathbf F = \frac1{r^2} \frac{\partial}{\partial r}(r^2 F_r) + \frac1{r\sin\theta} \frac{\partial}{\partial \theta} (\sin\theta\, F_\theta) + \frac1{r\sin\theta} \frac{\partial F_\phi}{\partial \phi}.$

Decomposition theorem

Main article: Helmholtz decomposition

It can be shown that any stationary flux v(r) which is at least two times continuously differentiable in and vanishes sufficiently fast for |r| → ∞ can be decomposed into an irrotational part E(r) and a source-free part B(r). Moreover, these parts are explicitly determined by the respective source-densities (see above) and circulation densities (see the article Curl):

For the irrotational part one has

with

The source-free part, B, can be similarly written: one only has to replace the scalar potential Φ(r) by a vector potential A(r) and the terms −∇Φ by +∇×A, and the source-density div v by the circulation-density ∇×v.

This "decomposition theorem" is in fact a by-product 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

Main article: Vector calculus identities

The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.

$\operatorname{div}( a\mathbf{F} + b\mathbf{G} ) = a\;\operatorname{div}( \mathbf{F} ) + b\;\operatorname{div}( \mathbf{G} )$

for all vector fields F and G and all real numbers a and b.

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

$\operatorname{div}(\varphi \mathbf{F}) = \operatorname{grad}(\varphi) \cdot \mathbf{F} + \varphi \;\operatorname{div}(\mathbf{F}),$

or in more suggestive notation

$\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi \;(\nabla\cdot\mathbf{F}).$

Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:

$\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl}(\mathbf{F})\cdot\mathbf{G} \;-\; \mathbf{F} \cdot \operatorname{curl}(\mathbf{G}),$

or

$\nabla\cdot(\mathbf{F}\times\mathbf{G}) = (\nabla\times\mathbf{F})\cdot\mathbf{G} - \mathbf{F}\cdot(\nabla\times\mathbf{G}).$

The Laplacian of a scalar field is the divergence of the field's gradient:

$\operatorname{div}(\nabla\varphi) = \Delta\varphi.$

The divergence of the curl of any vector field (in three dimensions) is 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 this, the 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

(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 express the divergence as a particular case of the exterior derivative, which takes a 2-form to a 3-form in R³. Define the current two form

.

It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density moving with local velocity F. Its exterior derivative is then given by

$d j = \left( \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx\wedge dy\wedge dz = (\nabla\cdot \mathbf{F}) \rho$

Thus, the divergence of the vector field F can be expressed as:

Here the superscript is one of the two musical isomorphisms, and is the Hodge dual. Note however that working with the current two form itself and the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes with a change of (curvilinear) coordinate system.

Generalizations

The divergence of a vector field can be defined in any number of dimensions. If

in a Euclidean coordinate system where and , define

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial F_1}{\partial x_1} +\frac{\partial F_2}{\partial x_2}+\cdots +\frac{\partial F_n}{\partial x_n}.$

The appropriate expression is more complicated in curvilinear coordinates.

In the case of one dimension, a "vector field" is simply a regular function, and the divergence is simply the derivative.

For any n, the divergence is a linear operator, and it satisfies the "product rule"

$\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi \;(\nabla\cdot\mathbf{F}).$

for any scalar-valued function .

The divergence can be defined on any manifold of dimension n with a volume form (or density) e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two form for a vector field on , on such a manifold a vector field X defines a n−1 form obtained by contracting X with . The divergence is then the function defined by

Standard formulas for the Lie derivative allow us to reformulate this as

This means that the divergence measures the rate of expansion of a volume element as we let it flow with the vector field.

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

where the second expression is the contraction of the vector field valued 1-form with itself and the last expression is the traditional coordinate expression used by physicists.

An equivalent expression without using connection is

where is the metric and denotes partial derivative with respect to coordinate .

Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector is given by

where is the covariant derivative.

Equivalently, some authors define the divergence of any mixed tensor by using the "musical notation #":

If T is a (p,q)-tensor (p for the contravariant vector and q for the covariant one), then we define the divergence of T to be the (p,q−1)-tensor

that is we trace the covariant derivative on the first two covariant indices.

See also

• Curl
• Del in cylindrical and spherical coordinates
• Divergence theorem
• Gradient
• Laplacian

Notes

References

1. Brewer, Jess H. (1999-04-07). "DIVERGENCE of a Vector Field". Vector Calculus. Retrieved 2007-09-28. 
2. Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0-486-41147-8. 

External links

• Hazewinkel, Michiel, ed. (2001), "Divergence", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• The idea of divergence of a vector field
• Khan Academy: Divergence video lesson

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Related Links

• divergenceの意味 - 英和辞書 - 英語辞書 - goo辞書

divergenceとは。意味や和訳。[名][C][U]1 （一点から）分かれ出ること，分岐；（意見などの）相違((between, of ...))；（…からの）逸脱((from ...)). ⇒CONVERGENCE 1a divergence in views見解の相違.2 《物理学》《気象》（大気の ...

• Divergence | Define Divergence at Dictionary.com

divergence di·ver·gence (dĭ-vûr'jəns, dī-) n. A moving or spreading apart in different directions from a common point. The degree by which things deviate or spread apart. A turning of both eyes outward from a common point or of one eye ...

• RSI_divergence - サイトマップ | NETAGE

MT4用インジケータRSI_divergenceを配布。 ... RSI_divergence RSIとダイバージェンスを表示するインジです。名前はRSI_divergenceですがコンバージェンスも表示します。

Related Pictures

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★リンクテーブル★

リンク元	「exhale」「disagreement」「diverge」「disparity」「相違」

「exhale」

　　[★]

• vt.

• (息・言葉などを)はき出す(⇔inhale)。(ガス・匂いなどを)発散させる

• vi.

• (ガス・匂いなどを)立ち上る、発散する、蒸発する(from, out of)。消散する。息を吐き出す。

関

diverge、divergence、emanate、emanation、exhalation、expiration、expiratory

　 　 　

「disagreement」

　　[★]

• n.

• 不一致、(意見の)相違、反対

関

adverse、against、contrary、difference、disagree、discordance、discrepant、disparity、divergence、inconsistency、inconsistent、object、objection、oppose、opposite、opposition、variance

　 　 　

「diverge」

　　[★]

• v.

• 分岐する、逸脱する、(数学)発散する

関

aberration、arborize、bifurcation、departure、deviate、deviation、divergence、emanate、emanation、escape、exhale、multibranch、multibranched、prolapse

　 　 　 　

「disparity」

　　[★]

• 不同性、不等、不同、相違、差異

関

difference、disagreement、discrepant、disparate、distinction、divergence、nonequivalence、nonequivalent、unequal、unequally、variation

　 　

「相違」

　　[★]

英

difference、divergence、disparity、disagreement、discrepant

関

差、反対、不一致、不等、不同、不同性、分岐、発散、開散、差異