Difference between revisions of "Covariant vector"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
− | + | <!-- | |
+ | c0268901.png | ||
+ | $#A+1 = 32 n = 0 | ||
+ | $#C+1 = 32 : ~/encyclopedia/old_files/data/C026/C.0206890 Covariant vector | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
− | + | {{TEX|auto}} | |
+ | {{TEX|done}} | ||
− | + | An element of the vector space | |
+ | dual to an n - | ||
+ | dimensional vector space E , | ||
+ | that is, a linear functional (linear form) on E . | ||
+ | In the ordered pair ( E, E ^ {*} ) , | ||
+ | an element of E | ||
+ | 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 | + | The coordinate notation for a covariant vector is particularly simple if one chooses in E |
+ | and E ^ {*} | ||
+ | so-called dual bases e _ {1} \dots e _ {n} | ||
+ | in E | ||
+ | and e ^ {1} \dots e ^ {n} | ||
+ | in E ^ {*} , | ||
+ | that is, bases such that ( e ^ {i} e _ {j} ) = \delta _ {j} ^ {i} ( | ||
+ | where \delta _ {j} ^ {i} | ||
+ | is the [[Kronecker symbol|Kronecker symbol]]); an arbitrary covariant vector \omega \in E ^ {*} | ||
+ | is then expressible in the form \omega = f _ {i} e ^ {i} ( | ||
+ | summation over i | ||
+ | from 1 to n ), | ||
+ | where f _ {i} | ||
+ | is the value of the linear form \omega | ||
+ | at the vector e _ {i} . | ||
+ | On passing from dual bases ( e _ {i} ) | ||
+ | and ( e ^ {j} ) | ||
+ | to dual bases ( \overline{e}\; _ {i ^ \prime } ) | ||
+ | and ( \overline{e}\; {} ^ {j ^ \prime } ) | ||
+ | according to the formulas | ||
+ | |||
+ | $$ | ||
+ | \overline{e}\; _ {i ^ \prime } = \ | ||
+ | p _ {i ^ \prime } ^ {i} | ||
+ | e _ {i} ,\ \ | ||
+ | \overline{e}\; {} ^ {j ^ \prime } = \ | ||
+ | q _ {i} ^ {j ^ \prime } | ||
+ | e ^ {i} ,\ \ | ||
+ | p _ {k ^ \prime } ^ {i} | ||
+ | q _ {j} ^ {k ^ \prime } = \ | ||
+ | \delta _ {j} ^ {i} , | ||
+ | $$ | ||
+ | |||
+ | the coordinates x ^ {i} | ||
+ | of the contravariant vector $ x = x ^ {i} e _ {i} $ | ||
+ | change according to the contravariant law $ \overline{x}\; {} ^ {i ^ \prime } = q _ {i} ^ {i ^ \prime } x ^ {i} $, | ||
+ | while the coordinates f _ {i} | ||
+ | of the covariant vector \omega | ||
+ | change according to the covariant law $ \overline{f}\; _ {i ^ \prime } = p _ {i ^ \prime } ^ {i} f _ {i} $( | ||
+ | i.e. they change in the same way as the basis, whence the terminology "covariant vectorcovariant" ). | ||
====References==== | ====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> | <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> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====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> | <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> |
Latest revision as of 17:31, 5 June 2020
An element of the vector space E ^ {*}
dual to an n -
dimensional vector space E ,
that is, a linear functional (linear form) on E .
In the ordered pair ( E, E ^ {*} ) ,
an element of E
is called a 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 E and E ^ {*} so-called dual bases e _ {1} \dots e _ {n} in E and e ^ {1} \dots e ^ {n} in E ^ {*} , that is, bases such that ( e ^ {i} e _ {j} ) = \delta _ {j} ^ {i} ( where \delta _ {j} ^ {i} is the Kronecker symbol); an arbitrary covariant vector \omega \in E ^ {*} is then expressible in the form \omega = f _ {i} e ^ {i} ( summation over i from 1 to n ), where f _ {i} is the value of the linear form \omega at the vector e _ {i} . On passing from dual bases ( e _ {i} ) and ( e ^ {j} ) to dual bases ( \overline{e}\; _ {i ^ \prime } ) and ( \overline{e}\; {} ^ {j ^ \prime } ) according to the formulas
\overline{e}\; _ {i ^ \prime } = \ p _ {i ^ \prime } ^ {i} e _ {i} ,\ \ \overline{e}\; {} ^ {j ^ \prime } = \ q _ {i} ^ {j ^ \prime } e ^ {i} ,\ \ p _ {k ^ \prime } ^ {i} q _ {j} ^ {k ^ \prime } = \ \delta _ {j} ^ {i} ,
the coordinates x ^ {i} of the contravariant vector x = x ^ {i} e _ {i} change according to the contravariant law \overline{x}\; {} ^ {i ^ \prime } = q _ {i} ^ {i ^ \prime } x ^ {i} , while the coordinates f _ {i} of the covariant vector \omega change according to the covariant law \overline{f}\; _ {i ^ \prime } = p _ {i ^ \prime } ^ {i} f _ {i} ( i.e. they change in the same way as the basis, whence the terminology "covariant vectorcovariant" ).
References
[1] | P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian) |
[2] | D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian) |
[3] | J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951) |
Comments
References
[a1] | M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1970–1975) pp. 1–5 |
Covariant vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_vector&oldid=14326