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In particular, a vector field on a smooth manifold M is a choice of tangent vector at each point of M: this is a section of the tangent bundle of M. Likewise, a 1-form on M is a section of the cotangent bundle.

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## Some Related Sentences

particular and vector

**In**physics

**,**angular momentum

**,**moment

**of**momentum

**,**or rotational momentum

**is**

**a**

__vector__quantity that represents

**the**product

**of**

**a**body's rotational inertia and rotational velocity about

**a**

__particular__axis

**.**

A

__particular__particle within**the**body in**a**__particular__configuration**is**characterized by**a**position__vector__
It

**is**not**a**__vector__**field**because it depends not only**on****the**position**of****a**__particular__material**point****,**but also**on****the**local orientation**of****the**surface element as defined by its normal__vector__**.****In**modern introductory texts to functional analysis

**,**

**the**subject

**is**seen as

**the**study

**of**

__vector__spaces endowed with

**a**topology

**,**in

__particular__infinite dimensional spaces

**.**

**In**

**the**mathematical

**field**

**of**representation theory

**,**group representations describe abstract groups in terms

**of**linear transformations

**of**

__vector__spaces ; in

__particular__

**,**they can be used to represent group elements as matrices so that

**the**group operation can be represented by matrix multiplication

**.**

*

**In**__particular__**,**if V**is****a**normed__vector__space and if z**is**any element**of**V**,**then there exists**a**continuous linear map with ψ ( z ) = || z || and || ψ || ≤ 1**.**
* Neutral

__vector__( statistics ),**a**multivariate random variable**is**neutral if it exhibits**a**__particular__type**of**statistical independence seen when considering**the**Dirichlet distribution
* A " random word " may be parameterized by an integer-valued index into

**the**vocabulary**of**possible words ; or alternatively as an indicator__vector__**,**in which exactly one element**is****a**1 and**the**others are 0**,**with**the**1 indexing**a**__particular__word into**a**vocabulary**.**
If an image stored in one

__vector__file format**is**converted to another file format that supports all**the**primitive objects used in that__particular__image**,**then**the**conversion can be lossless**.**
When only

**the**magnitude and direction**of****the**__vector__matter**,**then**the**__particular__initial**point****is****of**no importance**,**and**the**__vector__**is**called**a**free__vector__**.**
The three-dimensional real

__vector__q represents**a****point**in reciprocal space**,**that**is****,**to**a**__particular__oscillation in**the**electron density as one moves in**the**direction in which q points**.**
The matrix

**M**may be regarded as**a**diagonal matrix D that has been re-expressed in coordinates**of****the**basis P**.****In**__particular__**,****the**one-to-one change**of**variable y = Pz shows that z < sup >*</ sup > Mz**is**real and positive for any complex__vector__z if and only if y < sup >*</ sup > Dy**is**real and positive for any y '; in other words**,**if D**is**positive definite**.**
Viewed in

**this**light**,****this**construction ( add transforms for**each**object**,**restrict maps to commute with these )**is**completely general**,**and does not depend**on**any__particular__properties**of**__vector__spaces**.**
The elements

**of**topological__vector__spaces are typically functions or linear operators acting**on**topological__vector__spaces**,**and**the**topology**is**often defined so as to capture**a**__particular__notion**of**convergence**of**sequences**of**functions**.****In**

__particular__

**,**

**the**power cepstrum

**is**often used as

**a**feature

__vector__for representing

**the**human voice and musical signals

**.**

The Poynting

__vector__represents**the**__particular__case**of**an energy flux__vector__for electromagnetic energy**.**
The individual variables in

**a**random__vector__are grouped together because there may be correlations among them — often they represent different properties**of**an individual statistical unit ( e**.**g**.****a**__particular__person**,**event**,**etc .).**In**linear algebra

**,**

**this**

__vector__

**is**known as

**the**kernel

**of**

**the**dimensional matrix

**,**and it spans

**the**nullspace

**of**

**the**dimensional matrix

**,**which in

**this**

__particular__case

**is**one dimensional

**.**

particular and field

The same argument proves that no subfield

**of****the**real__field__**is**algebraically closed ; in__particular__**,****the**__field__**of**rational numbers**is**not algebraically closed**.**
The

__field__**of**chemistry uses analysis in**at**least three ways**:**to identify**the**components**of****a**__particular__chemical compound ( qualitative analysis ), to identify**the**proportions**of**components in**a**mixture ( quantitative analysis ), and to break down chemical processes and examine chemical reactions between elements**of**matter**.**
Field automorphisms are important to

**the**theory**of**__field__extensions**,**in__particular__Galois extensions**.**
The < tt > MixColumns </ tt > step can also be viewed as

**a**multiplication by**a**__particular__MDS matrix in**a**finite__field__**.**
The term argot

**is**also used to refer to**the**informal specialized vocabulary from**a**__particular____field__**of**study**,**hobby**,**job**,**sport**,**etc**.**
Ideas from it have proved influential

**,**and have been developed by others**,**in__particular__dessins d ' enfants and**a**new__field__emerging as anabelian geometry**.**
The black letter law refers to

**the**basic standard elements for**a**__particular____field__**of**law**,**which are generally known and free from doubt or dispute**.**
The International Organization for Standardization

**,**ISO**,**has**a**special technical committee for cycles**,**TC149**,**that has**the**following scope**:**" Standardization in**the**__field__**of**cycles**,**their components and accessories with__particular__reference to terminology**,**testing methods and requirements for performance and safety**,**and interchangeability**.**
Graduate programs in BME

**,**like in other scientific fields**,**are highly varied**,**and__particular__programs may emphasize certain aspects within**the**__field__**.**
However

**,**Ogden prescribed that any student should learn an additional 150-word list for everyday work in some__particular____field__**,**by adding**a**word list**of**100 words particularly useful in**a**general__field__( e**.**g., science**,**verse**,**business**,**etc**.**
Of

__particular__note**,**programs that were placed within**the**top 10 %**of**all size programs in that__field__based**on**an average**of****the**R95 and S95 rank order include Aeronautics**,**Astrophysics**,**Biochemistry and Molecular Biophysics**,**Bioengineering**,**Biology**,**Chemical Engineering**,**Chemistry**,**Electrical Engineering**,**Environmental Science and Engineering**,**Geology**,**Geophysics**,**Materials Science**,**Mechanical Engineering**,**Physics**,**Planetary Science**,**and Social Science ( Economics ).
The term

**is**synonymous with wealth ( commonly denoted as**a**person with fame and fortune ), implied with great popular appeal**,**prominence in**a**__particular____field__**,**and**is**easily recognized by**the**general public**.**
While people may gain celebrity status as

**a**result**of****a**successful career in**a**__particular____field__( primarily in**the**areas pertaining towards sports and entertainment ), in other cases**,**people become celebrities due to media attention for their extravagant lifestyle or wealth ( as in**the**case**of****a**socialite ); for their connection to**a**famous person ( as in**the**case**of****a**relative**of****a**famous person ); or even for their misdeeds ( as in**the**case**of****a**well-known criminal ).
Therefore

**,**there exists**a**contact force density or Cauchy traction__field__that represents**this**distribution in**a**__particular__configuration**of****the**body**at****a**given time**.**
The company organization

**of****each**camp had**a**dual-authority supervisory staff**:**firstly**,**Department**of**War personnel or Reserve officers ( until 1 July 1939 ),**a**" company commander " and junior officer**,**who were responsible for overall camp operation**,**logistics**,**education and training ; and secondly**,**ten to fourteen technical service civilians**,**including**a**camp " superintendent " and " foreman ," employed by either**the**Departments**of**Interior or Agriculture**,**responsible for**the**__particular____field__work**.****In**

__particular__

**,**

**this**

__field__contains all

**the**numbers named in

**the**mathematical constants article

**,**and all algebraic numbers ( and therefore all rational numbers ).

A design pattern in architecture and computer science

**is****a**formal way**of**documenting**a**solution to**a**design problem in**a**__particular____field__**of**expertise**.**
An organized collection

**of**design patterns that relate to**a**__particular____field__**is**called**a**pattern language**.**
)

**is**reserved for those who can prove**a**__particular__distinction in**the**__field__**,**usually through**a**body**of**published work or**the**submission**of****a**dissertation**.**0.144 seconds.