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" ).
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Covariant vector. I.Kh. Sabitov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Covariant_vector&oldid=14326