Covariant

Covariant is an adjective used in mathematics and physics to describe objects or quantities that transform in a way that preserves the form of physical laws under a change of coordinates. In practice, covariant behavior is defined relative to a chosen coordinate system: if coordinates are transformed, covariant objects transform to keep equations invariant. The counterpart is contravariant.

A common distinction is between covariant vectors (covectors or one-forms) and contravariant vectors. Covariant components transform with the inverse transpose of the coordinate-change matrix, while contravariant components transform with the matrix itself. Tensors with mixed indices combine both types of transformation. The metric tensor provides a bridge: it can lower an index to convert a contravariant vector into a covariant one, and raise an index to do the reverse.

Covariant differentiation is a generalization of differentiation that yields tensors that transform covariantly. Using a connection, such as the Levi-Civita connection in curved spaces, the covariant derivative ∇ respects tensor transformation rules and reduces to ordinary derivatives in Cartesian coordinates. It introduces connection coefficients (Christoffel symbols) that encode curvature and parallel transport.

In differential geometry and general relativity, covariant and contravariant components are related by the metric tensor g_ab and its inverse g^ab. The metric lowers and raises indices, making the distinction between covariant and contravariant a matter of how components transform rather than a fixed direction in space.

Note that "covariant" has a related sense in statistics through covariance, which measures how two random variables vary together; this is a separate concept from tensor covariance.