Covariant vector
An element of the vector space 
dual to an 
-dimensional vector space 
, that is, a linear functional (linear form) on 
. In the ordered pair 
, an element of 
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 
and 
so-called dual bases 
in 
and 
in 
, that is, bases such that 
(where 
is the Kronecker symbol); an arbitrary covariant vector 
is then expressible in the form 
(summation over 
from 1 to 
), where 
is the value of the linear form 
at the vector 
. On passing from dual bases 
and 
to dual bases 
and 
according to the formulas
 |
the coordinates 
of the contravariant vector 
change according to the contravariant law 
, while the coordinates 
of the covariant vector 
change according to the covariant law 
(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=46546