Covariant

In category theory, see covariant functor.

In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system. 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.

In geometry, a covariant vector is a vector in a vector field in the cotangent bundle that defines the geometry of a manifold. More precisely, a covariant vector is a one-form, a real-valued linear function on the space contravariant vectors. These one-forms can then be said to form a dual space to the vector space they take their arguments from.

A tensor field in the cotangent bundle will typically have multiple indices; each index is said to be a covariant index. Note, though, that a general tensor may also have contravariant indices as well; that is, it has parts that live in the tangent bundle as well as the cotangent bundle.

Covariant and contravariant indices transform in different ways under coordinate transformations. By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.

When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one-another. Contravariant indices can be turned into covariant indices by contracting with the metric tensor. Contravariant indices can be gotten by contracting with the inverse of the metric tensor. Note that in general, no such relation exists without a metric tensor.

By a widely followed convention (including Wikipedia), covariant indices are written as lower indices, whereas contravariant indices are upper indices.

Example: covariant basis vectors in Euclidean R³

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:

<math> \mathbf{e}_1 = \frac{\mathbf{e}^2 \times \mathbf{e}^3}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3} ; \qquad \mathbf{e}_2 = \frac{\mathbf{e}^3 \times \mathbf{e}^1}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3}; \qquad \mathbf{e}_3 = \frac{\mathbf{e}^1 \times \mathbf{e}^2}{\mathbf{e}^1 \cdot \mathbf{e}^2 \times \mathbf{e}^3}

<math>

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

<math> q^1 = \mathbf{v \cdot e^1}; \qquad q^2 = \mathbf{v \cdot e^2}; \qquad q^3 = \mathbf{v \cdot e^3} <math>

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

<math> q_1 = \mathbf{v \cdot e_1}; \qquad q_2 = \mathbf{v \cdot e_2}; \qquad q_3 = \mathbf{v \cdot e_3} <math>

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

<math> \mathbf{v} = q_i \mathbf{e}^i = q_1 \mathbf{e}^1 + q_2 \mathbf{e}^2 + q_3 \mathbf{e}^3 <math>

<math> \mathbf{v} = q^i \mathbf{e}_i = q^1 \mathbf{e}_1 + q^2 \mathbf{e}_2 + q^3 \mathbf{e}_3 <math>.

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.