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 * e^{1}*,

*,*

**e**^{2}*are contravariant basis vectors of*

**e**^{3}**R**

*(not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are:*

^{3}

**can be obtained by the dot product of**

*v***with the contravariant basis vectors:**

*v*

**v**can be obtained from the dot product of

**v**with covariant basis vectors, viz.

**v**can be expressed in two (reciprocal) ways, viz.

- .