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)

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1970–1975) pp. 1–5