In category theory, see covariant functor.
In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system.
Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it is the base which one measures against.
A contravariant vector is thus a measurement or a displacement on this space.
Thus, their relationship can be represented simply as:
Another way of defining covariant vectors is to say that "covariant vectors" are actually one-forms, that is to say, real-valued linear functions on "contravariant" vectors. These one-forms can then be said to form a dual space to the vector space they take their arguments from.
If e1, e2, e3 are contravariant basis vectors of R3 (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are: