Covariance and contravariance of vectors

The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851.

Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.

In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or tuple) of numbers such as The numbers in the list depend on the choice of coordinate system.

For instance, if the vector represents position with respect to an observer (position vector), then the coordinate system may be obtained from a system of rigid rods, or reference axes, along which the components v1, v2, and v3 are measured.

For a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system.

That is to say, the components of the vectors will transform in a certain way in passing from one coordinate system to another.

meters long), the components of the measured position vector are multiplied by 100.

Similarly, if the reference axes were stretched in one direction, the components of the vector, would reduce in an exactly compensating way.

The dot product operator involving vectors is a good example of a covector.

(This can be derived by noting that we want to get the correct answer for the dot product operation when multiplying by an arbitrary vector

A scalar (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis.

The single, scalar value of mass is independent to changes in basis vectors and consequently is called invariant.

The way A relates the two pairs is depicted in the following informal diagram using an arrow.

A system of n quantities that transform oppositely to the coordinates is then a covariant vector (or covector).

More generally, in an n-dimensional Euclidean space V, if a basis is the reciprocal basis is given by (double indices are summed over), where the coefficients gij are the entries of the inverse matrix of Indeed, we then have The covariant and contravariant components of any vector are related as above by and The distinction between covariance and contravariance is particularly important for computations with tensors, which often have mixed variance.

The valence of a tensor is the number of covariant and contravariant terms, and in Einstein notation, covariant components have lower indices, while contravariant components have upper indices.

The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated index-free methods to represent tensors.

Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices.

Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a vector intermingle on going between covariant and contravariant expression.

On a manifold, a tensor field will typically have multiple, upper and lower indices, where Einstein notation is widely used.

When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one another.

Note that in general, no such relation exists in spaces not endowed with a metric tensor.

Furthermore, from a more abstract standpoint, a tensor is simply "there" and its components of either kind are only calculational artifacts whose values depend on the chosen coordinates.

Similarly, tensors of higher degree are functors with values in other representations of

However, some constructions of multilinear algebra are of "mixed" variance, which prevents them from being functors.

The first is that vectors whose components are covariant (called covectors or 1-forms) actually pull back under smooth functions, meaning that the operation assigning the space of covectors to a smooth manifold is actually a contravariant functor.

Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of (contravariant) vectors to a smooth manifold is a covariant functor.

Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather changes in the coordinate system.

Likewise, vectors with covariant components transform in the opposite way as changes in the coordinates.