covariantly

Covariantly is the adverb form of covariant, used to describe how objects or equations transform or are defined in a way that respects covariance under coordinate changes. In differential geometry and theoretical physics, a quantity is said to transform covariantly if its components change under a coordinate transformation according to the covariant transformation law, i.e., in the same way as a tensor of the corresponding type. Equations that hold for all coordinate systems are described as covariant or written in covariant form, meaning they are tensor equations not tied to any particular frame.

A key context is the distinction between covariant and contravariant components. Covariant components typically refer to lower indices and transform with the inverse Jacobian, as with one-forms; contravariant components, with upper indices, transform with the Jacobian. Tensors can have mixed index positions, and their transformation rules ensure that the tensorial character is preserved under coordinate changes. The covariant derivative is a construction that allows differentiation of tensor fields while preserving their tensor type; statements about a tensor field that remain true under any coordinate transformation are written covariantly.

Common phrases include covariantly constant (or parallel) fields, meaning their covariant derivative vanishes everywhere. In physics, expressing laws covariantly enables form-invariance of equations like those of electromagnetism and general relativity across reference frames.