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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,

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## Some Related Sentences

More and rigorously

__More__

__rigorously__

**,**if

**a**surface S

**is**

**a**union

**of**finitely many pieces S < sub > 1 </ sub >, …, S < sub > r </ sub > which do not overlap except

**at**their boundaries then

__More__

__rigorously__

**,**an interpretation

**of**quantum mechanics satisfies CFD if it includes in

**the**statistical population

**of**measurement results

**,**those measurements which are counterfactual

**by**virtue

**of**their being excluded

**by**

**the**quantum mechanical prohibition on simultaneous measurement

**of**certain pairs

**of**properties

**.**

__More__

__rigorously__

**,**it

**is**

**the**localization

**of**

**the**polynomial ring in

**the**multiplicative set consisting

**of**

**the**non-negative powers

**of**X

**.**

__More__

__rigorously__

**,**metonymy and synecdoche may be considered

**as**sub-species

**of**metaphor

**,**intending metaphor

**as**

**a**type

**of**conceptual substitution (

**as**Quintilian does in Institutio oratoria Book VIII ).

__More__

__rigorously__

**,**since

**the**phases are in thermodynamic equilibrium with each other

**,**

**the**chemical potentials

**of**

**the**phases must be equal

**.**

__More__

__rigorously__

**,**

**the**mathematical operation involved in classical limits

**is**

**a**group contraction

**,**approximating physical systems where

**the**relevant action

**is**much larger than Planck's constant ħ

**,**so

**the**" deformation parameter " ħ / S can be effectively taken

**to**be zero

**.**

__More__

__rigorously__

**,**any summation method that possesses these properties and which assigns

**a**finite value

**to**

**the**geometric series must assign this value

**.**

__More__serious consultation with

**a**range

**of**internationally adopted people from various professional and community-work based backgrounds needs

**to**be included before

**the**

**field**

**of**adoption study

**is**more truly representative and

__rigorously__informed

**.**

__More__

__rigorously__

**the**effective temperature corresponds

**to**

**the**temperature

**at**

**the**radius that

**is**

**defined**

**by**

**the**Rosseland optical depth

**.**

The minimal separators also form an algebraic structure: For two fixed vertices

**a**and b**of****a**given graph G**,**an (**a****,**b )- separator S can be regarded**as****a**predecessor**of**another (**a****,**b )- separator T**,**if every path from**a****to**b meets S before it meets T**.**__More____rigorously__**,****the**predecessor relation**is****defined****as**follows: Let S and T be two (**a****,**b )- separators in ' G '.__More__

__rigorously__

**,**Pickover's " biomorphs " encompass

**the**class

**of**organismic morphologies created

**by**small changes

**to**traditional convergence tests in

**the**

**field**

**of**" Julia set " theory

**.**

More and divergence

__More__technically

**,**

**the**

__divergence__represents

**the**

**volume**density

**of**

**the**outward flux

**of**

**a**

**vector**

**field**from an infinitesimal

**volume**around

**a**given

**point**

**.**

__More__precisely

**,**

**the**

__divergence__theorem states that

**the**outward flux

**of**

**a**

**vector**

**field**through

**a**closed surface

**is**equal

**to**

**the**

**volume**integral

**of**

**the**

__divergence__

**of**

**the**

**region**inside

**the**surface

**.**

__More__precisely

**,**they were only able

**to**perform

**the**calculation when

**the**theory contains one less flavor

**of**chiral matter than

**the**number

**of**colors in

**the**special unitary gauge group

**,**because in

**the**presence

**of**fewer flavors an unbroken nonabelian gauge symmetry leads

**to**an infrared

__divergence__and in

**the**case

**of**more flavors

**the**contribution in equal

**to**zero

**.**

More and vector

__More__generally

**,**functional analysis includes

**the**study

**of**Fréchet spaces and other topological

__vector__spaces not endowed with

**a**norm

**.**

__More__generally

**,**if

**a**linear operator on

**a**

__vector__space ( possibly infinite-dimensional ) has finite-dimensional range ( e

**.**g.,

**a**finite-rank operator ), then

**the**rank

**of**

**the**operator

**is**

**defined**

**as**

**the**dimension

**of**

**the**range

**.**

__More__generally yet

**,**

**the**additive groups

**of**all topological

__vector__spaces

**,**such

**as**Banach spaces or Hilbert spaces

**,**are topological groups

**.**

__More__formally

**,**

**a**multivariate random variable

**is**

**a**column

__vector__X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub >)< sup > T </ sup > ( or its transpose

**,**which

**is**

**a**row

__vector__) whose components are scalar-valued random variables on

**the**same probability space ( Ω

**,**

**,**P ), where Ω

**is**

**the**sample space

**,**

**is**

**the**sigma-algebra (

**the**collection

**of**all events ), and P

**is**

**the**probability measure (

**a**function returning every event's probability ).

__More__generally

**,**

**a**norm on

**a**

__vector__space

**is**Lipschitz continuous with respect

**to**

**the**associated metric

**,**with

**the**Lipschitz constant equal

**to**1

**.**

After

**a**basis for**the**__vector__space**is**chosen**,**this ring**is**naturally identified with**the**ring**of**n-by-n matrices with entries in K**.**__More__generally**,****the**endomorphism algebra**of****the**free module**is**naturally n-by-n matrices with entries in R**.**__More__generally

**,**every module category over

**a**ring

**is**additive

**,**and so in particular

**,**

**the**category

**of**

__vector__spaces over

**a**

**field**

**is**additive

**.**

__More__generally

**,**

__vector__fields are

**defined**on differentiable manifolds

**,**which are spaces that look like Euclidean space on small scales

**,**but may have more complicated structure on larger scales

**.**

Given

**a**differentiable manifold M**,****a**__vector__**field**on M**is**an assignment**of****a**tangent__vector__**to**each**point**in M**.**__More__precisely**,****a**__vector__**field****F****is****a**mapping from M into**the**tangent bundle TM so that**is****the**identity mapping__More__formally

**,**

**a**support

__vector__machine constructs

**a**hyperplane or set

**of**hyperplanes in

**a**high-or infinite-dimensional space

**,**which can be used for classification

**,**regression

**,**or other tasks

**.**

__More__precisely

**,**in order

**to**define

**the**covariant derivative

**of**it

**is**necessary first

**to**extend

**to**

**a**continuously differentiable

__vector__

**field**in an open set

**.**

__More__generally

**,**

**the**same construction allows one

**to**construct

**a**

__vector__

**field**for any Ehresmann connection on

**the**tangent bundle

**.**

__More__generally still

**,**

**the**general linear group

**of**

**a**

__vector__space GL (

**V**)

**is**

**the**abstract automorphism group

**,**not necessarily written

**as**matrices

**.**

__More__detailed discussion

**of**wave functions

**as**elements

**of**

__vector__spaces

**is**below

**,**following further definitions

**.**

__More__recently

**,**

**a**number

**of**loop-free distance

__vector__protocols have been developed — notable examples are EIGRP

**,**DSDV and Babel

**.**

__More__precisely

**,**

**the**vorticity

**of**

**a**

**flow**

**is**

**a**

__vector__

**field**

**,**equal

**to**

**the**curl ( rotational )

**of**its velocity

**field**v

**.**It can be expressed

**by**

**the**

__vector__analysis formula

**,**where

**is**

**the**nabla operator

**.**

More and field

__More__broadly

**,**scholars in

**the**

__field__define aesthetics

**as**" critical reflection on art

**,**culture and nature

**.**

__More__sophisticated theories are valence bond theory which includes orbital hybridization and resonance

**,**and

**the**linear combination

**of**atomic orbitals molecular orbital method which includes ligand

__field__theory

**.**

__More__precisely

**,**and technically

**,**

**a**Feynman diagram

**is**

**a**graphical representation

**of**

**a**perturbative contribution

**to**

**the**transition amplitude or correlation function

**of**

**a**quantum mechanical or statistical

__field__theory

**.**

__More__generally

**,**

**a**higher temperature and

**a**stronger magnetic

__field__lead

**to**

**a**smaller fraction

**of**

**the**electrons in

**the**superconducting band and consequently

**a**longer London penetration depth

**of**external magnetic fields and currents

**.**

__More__often than not

**,**

**a**woman serving

**as**

**a**courtesan would last in that

__field__only

**as**long

**as**she could prove herself useful

**to**her companion

**,**or companions

**.**

__More__recently

**,**this inclined geometry for

__field__stars in

**the**LMC has been confirmed via observations

**of**Cepheids

**,**core helium-burning red clump stars and

**the**tip

**of**

**the**red giant branch

**.**

__More__importantly

**,**each

**is**associated with

**a**different

__field__

**of**study ; deixis

**is**associated with linguistics

**,**while indexicality

**is**associated with philosophy

**.**

__More__precise theory shows that during

**the**action

**of**

**the**laser

__field__

**,**

**the**time-averaged effective potential

**of**H < sub > 2 </ sup >< sup >+</ sup >

**is**formed which binds

**the**electron even if very strong electromagnetic

__field__

**is**present in similarity

**to**

**the**hydrogen molecule ion and prevents ionization

**.**

__More__so than simply sparking improvements in

**the**budding

__field__

**of**home computing and gaming

**,**

**the**Z-80 also sparked

**a**revolution in electronic music

**,**

**as**

**the**first truly programmable polyphonic synthesizers (

**as**well

**as**their peripherals ) relied heavily on implementations

**of**this CPU

**.**

__More__generally

**,**

**the**general linear group

**of**degree n over any

__field__

**F**( such

**as**

**the**complex numbers ), or

**a**ring R ( such

**as**

**the**ring

**of**integers ),

**is**

**the**set

**of**n × n invertible matrices with entries from

**F**( or R ), again with matrix multiplication

**as**

**the**group operation

**.**

__More__generalized in

**the**

__field__

**of**economics

**,**cost

**is**

**a**metric that

**is**totaling up

**as**

**a**result

**of**

**a**process or

**as**

**a**differential for

**the**result

**of**

**a**decision

**.**

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