Results for divergence

On this page:

Dictionary:

divergence

(dĭ-vûr'jəns, dī-) pronunciation

1. The act of diverging.
2. The state of being divergent.
3. The degree by which things diverge.
Physiology. A turning of both eyes outward from a common point or of one eye when the other is fixed.
Departure from a norm; deviation.
Difference, as of opinion. See synonyms at deviation, difference.
Biology. The evolutionary tendency or process by which animals or plants that are descended from a common ancestor evolve into different forms when living under different conditions.
Mathematics. The property or manner of diverging; failure to approach a limit.
A meteorological condition characterized by the uniform expansion in volume of a mass of air over a region, usually accompanied by fair dry weather.

#SearchBtn{margin-top:14px !important;}

Library

Investment Dictionary: Divergence

A situation in which the price of an asset and an indicator, index or other related asset move in opposite directions. In technical analysis traders make transaction decisions by identifying situations of divergence, where the price of a stock and a set of relevant indicators, such as the money flow index (MFI), are moving in opposite directions.

Investopedia Says:
In technical analysis, divergence is considered either positive or negative, both of which are signals of major shifts in the direction of the price. Positive divergence occurs when the price of a security makes a new low while the indicator starts to climb upward. Negative divergence happens when the price of the security makes a new high, but the indicator fails to do the same and instead closes lower than the previous high.

Related Links:
Divergences may signal a change in market direction, but traders must also identify the speed of that change. Divergences, Momentum And Rate Of Change
Learn about this indicator that uses two exponential moving averages, and falls into the oscillator category. Moving Average Convergence Divergence - Part 1
Learn to combine this powerful tool with traditional technical tools for greater returns. Using Pivot Points In FX
Compare the results of two trades based on MACD histogram divergences. Tales From The Trenches: Trading Divergences In FX
This method, which works particularly well for currency traders, helps avoid stop-order triggers before the real reversal. Trading The MACD Divergence
Find out how to make sense of this tricky - but profitable - formation. Tales From The Trenches: Volume Confirmed Broadening Pattern
Take advantage of short-term price moves by pinpointing reversals. Candlesticks And Oscillators For Successful Swing Trades

Financial & Investment Dictionary: Divergence

In Technical Analysis, graphic plottings of prices or indicators that are moving in directions that fail to confirm a trend. See also Moving Average Convergence/Divergence (Macd).

Thesaurus: divergence

noun

The condition of being unlike or dissimilar: difference, discrepance, discrepancy, disparity, dissimilarity, dissimilitude, distinction, divarication, divergency, unlikeness. See same/different/compare.
A departing from what is prescribed: aberration, departure, deviation, divergency, diversion. See approach/retreat, correct/incorrect.
An instance of digressing: aside, deviation, digression, divagation, divergency, excursion, excursus, irrelevancy, parenthesis, tangent. See approach/retreat.
The condition of being divided, as in opinion: disunion, disunity, divergency, division, schism. See assemble/disassemble.

Antonyms: divergence

Definition: branching out; difference
Antonyms: accord, agreement, concord, convergence, harmony, sameness

Geography Dictionary: divergence

The spreading out of a vector field; mathematically, convergence is negative divergence. In meteorology, divergence is the spreading out of an air mass into paths of different directions. In the atmosphere, horizontal divergence predominates, and the word ‘horizontal’ is understood when this term is used. Divergence is linked with the vertical shrinking of the atmosphere, since, by the conservation of matter, an outflow of air must result.

It is closely related to vertical vorticity, since it is the principal agency responsible for the vorticity change experienced as a particle flows from a cyclonic to an anticyclonic circulation. Divergence in the upper air is associated with depressions at ground level. See jet stream, Rossby waves.

Britannica Concise Encyclopedia: 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.

For more information on divergence, visit Britannica.com.

Archaeology Dictionary: divergence

[Th]

The production of varied final conditions from originally similar states.

Veterinary Dictionary: divergence

A moving apart, or inclination away from a common point.

Wikipedia: divergence

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.

A vector field that has zero divergence everywhere is called solenoidal.

Definition

Let x, y, z be a system of Cartesian coordinates on a 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 = F₁ i + F₂ j + F₃ k is defined to be the scalar-valued function:

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z}.$

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, and an abuse of notation, where the dot denotes something just reminiscent of the dot product: take the components of ∇ (see del), apply them to the components of F, and sum the results.

Physical interpretation

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 [1]. 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,

$( \operatorname{div}\,\mathbf{F}) (p) = \lim_{r \rightarrow 0} \iint_{S(r)} {\mathbf{F}\cdot\mathbf{n}dS \over \frac{4}{3} \pi r^3 }$

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.

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.

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.

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

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

There is a product rule of the following type: if φ 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:

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

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

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

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

$\{\mbox{scalar fields on }U\} \;$

$\to\{\mbox{vector fields on }U\} \;$

$\to\{\mbox{scalar fields on }U\} \;$

(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 2-form to a 3-form in R³. If we define:

$\alpha=F_1\ dy\wedge dz + F_2\ dz\wedge dx + F_3\ dx\wedge dy$

its exterior derivative $d α$ is given by

$d\alpha = \left( \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx\wedge dy\wedge dz$

Generalizations

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

$\mathbf{F}=(F_1, F_2, \dots, F_n),$

define

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

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 φ.

The divergence can be defined on any manifold of dimension n with a volume form (or density) $μ$ e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two form for a vectorfield on $\mathbb{R}^3$ , on such a manifold a vectorfield X defines a n-1 form $j = i X μ$ obtained by contracting X with $μ$ . The divergence is then the function defined by

$d j = \operatorname{div}(X) \mu$

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

$\mathcal{L}_X \mu = \operatorname{div}(X) \mu$

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 $\nabla$

$\operatorname{div}(X) = \nabla\cdot X = X^a_{;a}$

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

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

External links

The idea of divergence and curl

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Best of the Web: divergence

Some good "divergence" pages on the web:

Math
mathworld.wolfram.com

Join the WikiAnswers Q&A community. Post a question or answer questions about "divergence" at WikiAnswers.

Copyrights:

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Investment Dictionary. Copyright ©2000, Investopedia.com - Owned and Operated by Investopedia Inc. All rights reserved. Read more
		Financial & Investment Dictionary. Dictionary of Finance and Investment Terms. Copyright © 2006 by Barron's Educational Series, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Antonyms. © 1999-2008 by Answers Corporation. All rights reserved. Read more
		Geography Dictionary. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Archaeology Dictionary. The Concise Oxford Dictionary of Archaeology. Copyright © 2002, 2003 by Oxford University Press. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Divergence". Read more

Search for answers directly from your browser with the FREE Answers.com Toolbar!
Click here to download now.

Get Answers your way! Check out all our free tools and products.

On this page: Print Link