Covariant vector

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)

References

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