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Difference between revisions of "Vector field"

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A term which is usually understood to mean a function of points in some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964201.png" /> whose values are vectors (cf. [[Vector|Vector]]), defined for this space in some way.
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A term which is usually understood to mean a function of points in some space $X$ whose values are vectors (cf. [[Vector|Vector]]), defined for this space in some way.
  
In the classical [[Vector calculus|vector calculus]] it is a subset of a Euclidean space that plays the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964202.png" />, while the vector field represents directed segments applied at the points of this subset. For instance, the collection of unit-length vectors tangent or normal to a smooth curve (surface) is a vector field on it.
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In the classical [[Vector calculus|vector calculus]] it is a subset of a Euclidean space that plays the part of $X$, while the vector field represents directed segments applied at the points of this subset. For instance, the collection of unit-length vectors tangent or normal to a smooth curve (surface) is a vector field on it.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964203.png" /> is an abstractly specified differentiable manifold, a vector field is understood to mean a tangent vector field, i.e. a function that associates to each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964204.png" /> an (invariantly constructed) vector tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964205.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964206.png" /> is finite-dimensional, the vector field is equivalently defined as a collection of univalent, contravariant tensors, which are depending on the points.
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If $X$ is an abstractly specified differentiable manifold, a vector field is understood to mean a tangent vector field, i.e. a function that associates to each point of $X$ an (invariantly constructed) vector tangent to $X$. If $X$ is finite-dimensional, the vector field is equivalently defined as a collection of univalent, contravariant tensors, which are depending on the points.
  
In the general case a vector field is interpreted as a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964207.png" /> with values in a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964208.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v0964209.png" /> in some way; it differs from an arbitrary vector function in that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v09642010.png" /> is defined with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v09642011.png" /> "internally"  rather than as a  "superstructure"  over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v09642012.png" />. A section of a [[Vector bundle|vector bundle]] with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096420/v09642013.png" /> is also considered to be a vector field.
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In the general case a vector field is interpreted as a function defined on $X$ with values in a [[Vector space|vector space]] $P$ associated with $X$ in some way; it differs from an arbitrary vector function in that $P$ is defined with respect to $X$ "internally"  rather than as a  "superstructure"  over $X$. A section of a [[Vector bundle|vector bundle]] with base $X$ is also considered to be a vector field.
  
  

Latest revision as of 09:45, 15 April 2014

A term which is usually understood to mean a function of points in some space $X$ whose values are vectors (cf. Vector), defined for this space in some way.

In the classical vector calculus it is a subset of a Euclidean space that plays the part of $X$, while the vector field represents directed segments applied at the points of this subset. For instance, the collection of unit-length vectors tangent or normal to a smooth curve (surface) is a vector field on it.

If $X$ is an abstractly specified differentiable manifold, a vector field is understood to mean a tangent vector field, i.e. a function that associates to each point of $X$ an (invariantly constructed) vector tangent to $X$. If $X$ is finite-dimensional, the vector field is equivalently defined as a collection of univalent, contravariant tensors, which are depending on the points.

In the general case a vector field is interpreted as a function defined on $X$ with values in a vector space $P$ associated with $X$ in some way; it differs from an arbitrary vector function in that $P$ is defined with respect to $X$ "internally" rather than as a "superstructure" over $X$. A section of a vector bundle with base $X$ is also considered to be a vector field.


Comments

Cf. also Vector field on a manifold.

How to Cite This Entry:
Vector field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field&oldid=31705
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article