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Difference between revisions of "Exterior form"

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''of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371002.png" />, exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371004.png" />-form''
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$#C+1 = 14 : ~/encyclopedia/old_files/data/E037/E.0307100 Exterior form
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A homogeneous element of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371005.png" /> of the [[Exterior algebra|exterior algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371006.png" /> of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371007.png" />, i.e. an element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371008.png" />-th exterior power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e0371009.png" />. The expression  "exterior form of degree r on the space V"  usually denotes a skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710010.png" />-linear function (or a skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710011.png" /> times covariant tensor) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710012.png" />. The direct sum of the spaces of skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710013.png" />-linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710015.png" /> endowed with the [[Exterior product|exterior product]], is an algebra isomorphic to the exterior algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037100/e03710016.png" />.
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''of degree  $  r $,
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exterior  $  r $-
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form''
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A homogeneous element of degree $  r $
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of the [[Exterior algebra|exterior algebra]] $  \wedge V $
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of a vector space $  V $,  
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i.e. an element of the $  r $-
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th exterior power $  \wedge  ^ {r} V $.  
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The expression  "exterior form of degree r on the space V"  usually denotes a skew-symmetric $  r $-
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linear function (or a skew-symmetric $  r $
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times covariant tensor) on $  V $.  
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The direct sum of the spaces of skew-symmetric $  r $-
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linear functions on $  V $,
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$  r = 0, 1 \dots $
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endowed with the [[Exterior product|exterior product]], is an algebra isomorphic to the exterior algebra $  \wedge V  ^ {*} $.
  
 
Under an exterior form one also understands a [[Differential form|differential form]].
 
Under an exterior form one also understands a [[Differential form|differential form]].

Latest revision as of 19:38, 5 June 2020


of degree $ r $, exterior $ r $- form

A homogeneous element of degree $ r $ of the exterior algebra $ \wedge V $ of a vector space $ V $, i.e. an element of the $ r $- th exterior power $ \wedge ^ {r} V $. The expression "exterior form of degree r on the space V" usually denotes a skew-symmetric $ r $- linear function (or a skew-symmetric $ r $ times covariant tensor) on $ V $. The direct sum of the spaces of skew-symmetric $ r $- linear functions on $ V $, $ r = 0, 1 \dots $ endowed with the exterior product, is an algebra isomorphic to the exterior algebra $ \wedge V ^ {*} $.

Under an exterior form one also understands a differential form.

How to Cite This Entry:
Exterior form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exterior_form&oldid=46888
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article