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 against which one measures.
A contravariant vector is thus a measurement or a displacement on this space.
Thus, their relationship can be represented simply as:
A note on the notation: the opposite convention (covariant as subscript, contravariant as superscript) is at least as widespread as the convention introduced here; it is used by authoritative textbooks like Landau-Lifshitz and Misner-Thorne-Wheeler.
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:
Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:
Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.
Then v can be expressed in two (reciprocal) ways, viz.
The indices of covariant coordinates, vectors, and tensors are subscripts (but see above, note on notation convention). If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts.