View source for Covariant vector

An element of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268901.png" /> dual to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268902.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268903.png" />, that is, a linear functional (linear form) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268904.png" />. In the ordered pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268905.png" />, an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268906.png" /> is called a [[Contravariant vector|contravariant vector]]. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.

The coordinate notation for a covariant vector is particularly simple if one chooses in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268908.png" /> so-called dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c0268909.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689012.png" />, that is, bases such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689013.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689014.png" /> is the [[Kronecker symbol|Kronecker symbol]]); an arbitrary covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689015.png" /> is then expressible in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689016.png" /> (summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689017.png" /> from 1 to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689018.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689019.png" /> is the value of the linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689020.png" /> at the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689021.png" />. On passing from dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689023.png" /> to dual bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689025.png" /> according to the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689026.png" /></td> </tr></table>

the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689027.png" /> of the contravariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689028.png" /> change according to the contravariant law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689029.png" />, while the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689030.png" /> of the covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689031.png" /> change according to the covariant law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026890/c02689032.png" /> (i.e. they change in the same way as the basis, whence the terminology  "covariant vectorcovariant" ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,   "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,   "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,   "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1970–1975)  pp. 1–5</TD></TR></table>